To theists, I say: go on being theists. You are better off being a theist than not being one. Your position is rationally defensible and the alternatives are rationally rejectable. But don't fancy that you can prove the existence of God or the opposite. In the end you must decide how you will live and what you will believe.
About "Don't fancy that you can prove the existence of God or the opposite," Owen Anderson asks:
How would we know if that claim is itself true? Isn't it is possible that one or the other can indeed be proven?
To formulate my point in the declarative rather than the exhortative mood:
P. Neither the existence nor the nonexistence of God is provable.
How do I know (P) to be true? By reflection on the nature of proof. An argument is a proof if and only if it satisfies all of the following six requirements: it is deductive; valid in point of logical form; free of such informal fallacies as petitio principii; possesses a conclusion that is relevant to the premises; has premises each of which is true; has premises each of which is known to be true.
I say that an argument is a proof if and only it is rationally compelling, or rationally coercive. But an argument needn't be rationally compelling to be a more or less 'good argument,' one that renders its conclusion more or less rationally acceptable.
Now if my definition above gives what we ought to mean by 'proof,' then it is clear that neither the existence nor the nonexistence of God can be proven. Suppose you present a theistic or anti-theistic argument that satisfies the first five requirements. I will then ask how you know that the premises are true. Suppose one of your premises is that change is the conversion of potency into act. That is a plausible thing to maintain, but how do you know that it is true? How do you know that the general-ontological framework within which the proposition acquires its very sense, namely, Aristotelian metaphysics, is tenable? After all, there are alternative ways of understanding change. That there is change is a datum, a Moorean fact, but it would be an obvious mistake to confuse this datum with some theory about it, even if the theory is true. Suppose the theory is true. This still leaves us with the question of how we know it is. Besides, the notions of potency and act, substance and accident, form and matter, and all the rest of the Aristotelian conceptuality are murky and open to question. (For example, the notion of prime matter is a necessary ingredient in an Aristotelian understanding of substantial change, but the notion of materia prima is either incoherent or else not provably coherent.)
To take a second example, suppose I give a cosmological argument the starting point of which is the seemingly innocuous proposition that there are are contingent beings, and go on to argument that this starting point together with some auxiliary premises, entails the existence of God. How do I know that existnece can be predicated of concrete individuals? Great philosophers have denied it. Frege and Russell fanmously held that existence vannot be meaningfully predicated of individuals but only of cncepts and propositional functions. I have rather less famoulsy argued that the 'GFressellina' view' is mstaken, but this is a point of controversy. Furtrhertmore, if existence cannot be meaningfully predicated of individuals, how can individuals be said to exist contingently?
The Appeal to Further Arguments
If you tell me that the premises of your favorite argument can be known to be true on the basis of further arguments that take those premises as their conclusions, then I simply iterate my critical procedure: I run the first five tests above and if your arguments pass those, then I ask how you know that their premises are true. If you appeal to still further arguments, then you embark upon a vicious infinite regress.
The Appeal to Self-Evidence
If you tell me that the premises of your argument are self-evident, then I will point out that your and my subjective self-evidence is unavailing. It is self-evident to me that capital punishment is precisely what justice demands in certain cases. I'll die in the ditch for that one, and pronounce you morally obtuse to boot for not seeing it. But there are some who are intelligent, well-meaning, and sophisticated to whom this is not self-evident. They will charge with with moral obtuseness. Examples are easily multiplied. What is needed is objective, discussion-stopping, self-evidence. But then, how, in a given case, do you know that your evidence is indeed objective? All you can go on is how things seem to you. If it seems to you that it is is objectively the case that p, that boils down to: it seems to you that, etc., in which case your self-evidence is again merely subjective.
The Appeal to Authority
You may attempt to support the premises of your argument by an appeal to authority. Now many such appeals are justified. We rightly appeal to the authority of gunsmiths, orthopaedic surgeons, actuaries and other experts all the time, and quite sensibly. But such appeals are useless when it comes to PROOF. How do you know that your putative authority really is one, and even if he is, how do you know that he is eight in the present case? How do you know he is not lying to you well he tells you you need a new sere in your semi-auto pistol?
If your argument falls afoul of petitio principii, that condemns it, and the diameter of the circle doesn't matter. A circle is a circle no matter its diameter.
Am I Setting the Bar Too High?
It seems to me I am setting it exactly where it belongs. After all we are talking about PROOF here and surely only arguments that generate knowledge count as proofs. But if an argument is to generate a known proposition, then its premises must be known, and not merely believed, or believed on good evidence, or assumed, etc.
"But aren't you assuming that knowledge entails certainty, or (if this is different) impossibility of mistake?" Yes I am assuming that. Argument here.
Can I Consistently Claim to Know that (P) is true?
Owen Anderson asked me how I know that (P) is true. I said I know it by reflection on the concept of proof. But that was too quick. Obviously I cannot consistently claim to know that (P) if knowledge entails certainty. For how do I know that my definition captures the essence of proof? How do I know that there is an essence of proof, or any essence of anything? What I want to say, of course, is that it is very reasonable to define 'proof' as I define it -- absent some better definition -- and that if one does so define it then it is clear that there are very few proofs, and, in particular, that there are no proofs of God or of the opposite.
"But then isn't it is possible that one or the other can indeed be proven?"
Yes, if one operates with a different, less rigorous, definition of 'proof.' But in philosophy we have and maintain high standards. So I say proof is PROOF (a tautological form of words that expresses a non-tautological proposition) and that we shouldn't use the word to refer to arguments that merely render their conclusions rationally acceptable.
Note also that if we retreat from the rationally compelling to the rationally acceptable, then both theism and atheism are rationally acceptable. I suspect that what Owen wants is a knock-down argument for the existence of God. But if that is what he wants, then he wants a proof in my sense of the world. If I am right, that is something very unreasonable to expect.
There is no getting around the need for a decision. In the end, after all the considerations pro et contra, you must decide what you will believe and how you will live.
Life is a venture and an adventure. You cannot live without risk. This is true not only in the material sphere, but also in the realm of ideas.
I also note a confusion that has been running through this discussion, about the meaning of ‘contradiction’. I do not mean to appeal to etymology or authority, but it’s important we agree on what we mean by it. On my understanding, a contradiction is not ‘the tallest girl in the class is 18’ and ‘the cleverest girl in the class is not 18’, even when the tallest girl is also the cleverest. Someone could easily believe both, without being irrational. The point of the Kripke puzzle is that Pierre seems to end up with an irrational belief. So it’s essential, as Kripke specifies, that he must correctly understand all the terms in both utterances, and that both utterances are logically contradictory, as in ‘Susan is 18’ and ‘Susan is not 18’.
Do we agree?
Well, let's see. The Maverick method enjoins the exposure of any inconsistent polyads that may be lurking in the vicinity. Sure enough, there is one:
An Inconsistent Triad
a. The tallest girl in the class is the cleverest girl in the class. b. The tallest girl in the class is 18. c. The cleverest girl in the class is not 18.
This trio is logically inconsistent in the sense that it is not logically possible that all three propositions be true. But if we consider only the second two limbs, there is no logical inconsistency: it is possible that (b) and (c) both be true. And so someone, Tom for example, who believes that (b) and also believes that (c) cannot be convicted of irrationality, at least not on this score. For all Tom knows -- assuming that he does not know that (a) -- they could both be true: it is epistemically possible that both be true. This is the case even if in fact (a) is true. But we can say more: it is metaphysically possible that both be true. For (a), if true, is contingently true, which implies that it is is possible that it be false.
By contrast, if Tom entertains together, in the synthetic unity of one consciousness, the propositions expressed by 'Susan is 18 years old' and 'Susan is not 18 years old,' and if Tom is rational, then he will see that the two propositions are logical contradictories of each other, and it will not be epistemically possible for him that both be true. If he nonetheless accepts both, then we have a good reason to convict him of being irrational, in this instance at least.
Given the truth of (a), (b) and (c) cannot both be true and cannot both be false. This suggests that the pair consisting of (b) and (c) is a pair of logical contradictories. But then we would have to say that the contradictoriness of the pair rests on a contingent presupposition, namely, the truth of (a). London Ed will presumably reject this. I expect he would say that the logical contradictoriness of a pair of propositions cannot rest on any contingent presupposition, or on any presupposition at all. Thus
d. Susan is 18
e. Susan is not 18
form a contradictory pair the contradictoriness of which rests on their internal logical form -- Fa, ~Fa -- and not on anything external to the propositions in question.
So what should we say? If Tom believes both (b) and (c), does he have contradictory beliefs? Or not?
The London answer is No! The belief-contents are not formally contradictory even though, given the truth of (a), the contents are such that they cannot both be true and cannot both be false. And because the belief-contents are not formally contradictory, the beliefs themselves -- where a belief involves both an occurrent or dispositional state of a person and a belief-content towards which the person takes up a propositional attitude -- are in no theoretically useful sense logically contradictory.
The Phoenix answer suggestion is that, because we are dealing with the beliefs of a concrete believer embedded in the actual world, there is sense to the notion that Tom's beliefs are contradictory in the sense that their contents are logically contradictory given the actual-world truth of (a). After all, if Susan is the tallest and cleverest girl, and the beliefs in question are irreducibly de re, then Tom believes, of Susan, that she is both 18 and not 18, even if Tom can gain epistemic access to her only via definition descriptions. That belief is de re, irreducibly, is entailed by (SUB), to which Kripke apparently subscribes:
SUB: Proper names are everywhere intersubstitutable salva veritate.
A Second Question
If, at the same time, Peter believes that Paderewski is musical and Peter believes that Paderewski is not musical, does it follow that Peter believes that (Paderewski is musical and Paderewski is not musical)? Could this conceivably be a non sequitur? Compare the following modal principle:
MP: If possibly p and possibly ~p, it does not follow that possibly (p & ~p).
For example, I am now seated, so it is possible that I now be seated; but it is also possible that I now not be seated, where all three occurrences/tokens of 'now' rigidly designate the same time. But surely it doesn't follow that it is possible that (I am now seated and I am now not seated). Is it perhaps conceivable that
BP: If it is believed by S that p and it is believed by S that ~p, it does not follow that it is believed by S that (p & ~p)?
Has anybody ever discussed this suggestion, even if only to dismiss it?
London Ed propounds a difficulty for our delectation and possible solution:
Clearly the difficulty with the intralinguistic theory is its apparent absurdity, but I am trying to turn this around. What can we say about extralinguistic reference? What actually is the extralinguistic theory? You argue that the pronoun ‘he’ inherits a reference from its antecedent, so that the pronoun does refer extralinguistically, but only per alium, not per se.
Mark 14:51 And there followed him [Jesus] a certain young man (νεανίσκος τις) , having a linen cloth (σινδόνα) cast about his naked body; and the young men laid hold on him. 14:52 And he left the linen cloth, and fled from them naked.
So the pronoun ‘he’ inherits its reference through its antecedent. But the antecedent is the noun phrase ‘a certain young man’. On your theory, does this refer extralinguistically? That’s a problem, because indefinite noun phrases traditionally do not refer, indeed that’s the whole point of them. ‘a certain young man’ translates the Latin ‘adulescens quidam’ which in turn translates the Greek ‘νεανίσκος τις’. Here ‘certain’ (Latin quidam, Greek τις) signifies that the speaker knows who he is talking about, but declines to tell the audience who this is. Many commentators have speculated that the man was Mark himself, the author of the gospel, which if true means that ‘a certain young man’ and the pronouns, could be replaced with ‘I’, salva veritate. But Mark deliberately does not tell us.
So, question 1, in what sense does the indefinite noun phrase refer, given that, on the extralinguistic theory, it has to be the primary referring phrase, from which all subsequent back-reference inherits its reference?
A. First of all, it is not clear why Ed says, ". . . indefinite noun phrases traditionally do not refer, indeed that’s the whole point of them." Following Fred Sommers, in traditional formal logic (TFL) as opposed to modern predicate logic (MPL), indefinite noun phrases do refer. (See Chapter 3, "Indefinite Reference" of The Logic of Natural Language.) Thus the subject terms in 'Some senator is a physician' and 'A physician is running for president' refer, traditionally, to some senator and to a physician. This may be logically objectionable by Fregean lights but it is surely traditional. That's one quibble. A second is that it is not clear why Ed says "that's the whole point of them."
So the whole point of a tokening of 'a certain young man' is to avoid making an extralinguistic reference? I don't understand.
B. Ed says there is a problem on my view. A lover of aporetic polyads, I shall try to massage it into one. I submit for your solution the following inconsistent pentad:
a. There are only two kinds of extralinguistic reference: via logically proper names, including demonstratives and indexicals, and via definite descriptions. b. The extralinguistic reference of a grammatical pronoun used pronominally (as opposed to quantificationally or indexically) piggy-backs on the extralinguistic reference of its antecedent. It is per alium not per se. c. 'His,' 'him,' and 'he' in the verse from Mark are pronouns used pronominally the antecedent of which is 'a certain young man.' d. 'A certain young man' in the verse from Mark is neither a logically proper name nor a definite description. e. 'A certain young man' in the verse from Mark refers extralinguistically on pain of the sentence of which it is a part being not true.
The pentad is inconsistent.
The middle three limbs strike me as datanic. So there are two possible solutions.
One is (a)-rejection. Maintain as Sommers does that indefinite descriptions can refer. This 'solution' bangs up against the critique of Peter Geach and other Fregeans.
The other is (e)-rejection. Deny that there is any extralinguistic reference at all. This, I think, is Ed's line. Makes no sense to me, though.
I wonder: could Ed be toying with the idea of using the first four limbs as premises in an argument to the conclusion that all reference is intralinguistic? I hope not.
Jean van Heijenoort was drawn to Anne-Marie Zamora like a moth to the flame. He firmly believed she wanted to kill him and yet he travelled thousands of miles to Mexico City to visit her where kill him she did by pumping three rounds from her Colt .38 Special into his head while he slept. She then turned the gun on herself. There is no little irony in the fact that van Heijenoort met his end in the same city as Lev Davidovich Bronstein, better known as Leon Trotsky. For van Heijenoort was Trotsky's secretary, body guard, and translator from 1932 to 1939.
The former 'Comrade Van' was a super-sharp logician but a romantic fool nonetheless. He is known mainly for his contribution to the history of mathematical logic. He edited From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 (Harvard University Press 1967) and translated some of the papers. The source book is a work of meticulous scholarship that has earned almost universally high praise from experts in the field.
One lesson is the folly of seeking happiness in another human being. The happiness we seek, whether we know it or not, no man or woman can provide. And then there is the mystery of self-destruction. Here is a brilliant, productive, and well-respected man. He knows that 'the flame' will destroy him, but he enters it anyway. And if you believe that this material life is the only life you will ever have, why throw it away for an unstable, pistol-packing female?
One might conclude to the uselessness of logic for life. If the heart has its reasons (Pascal) they apparently are not subject to the discipline of mathematical logic. All that logic and you still behave irrationally about the most important matters of self-interest? So what good is it? Apparently, van Heijenoort never learned to control his sexual and emotional nature. Does it make sense to be ever so scrupulous about what you allow yourself to believe, but not about what you allow yourself to love?
SOURCES (The following are extremely enjoyable books. I've read both twice.)
Anita Burdman Feferman, Politics, Logic, and Love: The Life of Jean van Heijenoort, Boston: Jones and Bartlett Publishers, 1993.
Jean van Hejenoort, With Trotsky in Exile: From Prinkipo to Coyoacan, Harvard UP, 1978.
. . . I’m confused by some of your epistemic terms. You reject [in the first article referenced below] the view that we can “rigorously prove” the existence of God, and several times say that theistic arguments are not rationally compelling, by which you mean that there are no arguments “that will force every competent philosophical practitioner to accept their conclusions on pain of being irrational if he does not.“
Okay, so far I’m tracking with you. But then you go on to say that “[t]here are all kinds of evidence” for theism (not just non-naturalism), while the atheist “fails to account for obvious facts (consciousness, self-consciousness, conscience, intentionality, purposiveness, etc.) if he assumes that all that exists is in the space-time world. I will expose and question all his assumptions. I will vigorously and rigorously drive him to dogmatism. Having had all his arguments neutralized, if not refuted, he will be left with nothing better than the dogmatic assertion of his position."
So how is the atheist not irrational on your view, assuming he is apprised of your arguments? Perhaps the positive case for theism and the negative case against naturalism don’t count as demonstrations in a mathematical sense, but I’m not sure why they’re not supposed to be compelling according to your gloss on the term.
The term 'mathematical' muddies the waters since it could lead to a side-wrangle over what mathematicians are doing when they construct proofs. Let's not muddy the waters. My claim is that we have no demonstrative knowledge of the truth of theism or of the falsity of naturalism. Demonstrative knowledge is knowledge produced by a demonstration. A demonstration in this context is an argument that satisfies all of the following conditions:
1. It is deductive 2. It is valid in point of logical form 3. It is free of such informal fallacies as petitio principii 4. It is such that all its premises are true 5. It is such that all its premises are known to be true 6. It is such that its conclusion is relevant to its premises.
To illustrate (6). The following argument satisfies all of the conditions except the last and is therefore probatively worthless:
Snow is white ergo Either Obama is president or he is not.
On my use of terms, a demonstrative argument = a probative argument = a proof = a rationally compelling argument. Now clearly there are good arguments (of different sorts) that are not demonstrative, probative, rationally compelling. One type is the strong inductive argument. By definition, no such argument satisfies (1) or (2). A second type is the argument that satisfies all the conditions except (5).
Can one prove the existence of God? That is, can one produce a proof (as above defined) of the existence of God? I don't think so. For how will you satisfy condition (5)? Suppose you give argument A for the existence of God. How do you know that the premises of A are true? By argument? Suppose A has premises P1, P2, P3. Will you give arguments for these premises? Then you need three more arguments, one for each of P1, P2, P3, each of which has its own premises. A vicious infinite regress is in the offing. Needless to say, moving in an argumentative circle is no better.
At some point you will have to invoke self-evidence. You will have to say that, e.g., it is just self-evident that every event has a cause. And you will have to mean objectively self-evident, not just subjectively self-evident. But how can you prove, to yourself or anyone else, that what is subjectively self-evident is objectively self-evident? You can't, at least not with respect to states of affairs transcending your consciousness.
I conclude that no one can prove the existence of God. But one can reasonably believe that God exists. The same holds for the nonexistence of God. No one can prove the nonexistence of God. But one can reasonably believe that there is no God.
The same goes for naturalism. I cannot prove that there is more to reality than the space-time system and its contents. But I can reasonably believe it. For I have a battery of arguments each of which satisfies conditions (1), (2), (3) and (6) and may even, as far as far as I know, satisfy (4).
"So how is the atheist not irrational on your view, assuming he is apprised of your arguments?"
He is not irrational because none of my arguments are rationally compelling in the sense I supplied, namely, they are not such as to force every competent philosophical practitioner to accept their conclusions on pain of being irrational if he does not. To illustrate, consider the following argument from Peter Kreeft (based on C. S. Lewis), an argument I consider good, but not rationally compelling. I will argue (though I will not prove!) that one who rejects this argument is not irrational.
The Argument From Desire
Premise 1: Every natural, innate desire in us corresponds to some real object that can satisfy that desire.
Premise 2: But there exists in us a desire which nothing in time, nothing on earth, no creature can satisfy.
Conclusion: Therefore there must exist something more than time, earth and creatures, which can satisfy this desire.
This something is what people call "God" and "life with God forever."
This is surely not a compelling argument. In fact, as it stands, it is not even valid. But it is easily repaired. There is need of an additional premise, one to the effect that the desire that nothing in time can satisfy is a natural desire. This supplementary premise is needed for validity, but it is not obviously true. For it might be -- it is epistemically possible that -- this desire that nothing in time can satisfy is artificially induced by one's religious upbringing or some other factor or factors.
Furthermore, is premise (1) true? Not as it stands. Suppose I am dying of thirst in the desert. Does that desire in me correspond to some real object that can satisfy it? Does the existence of my token desire entail the existence of a token satisfier? No! For it may be that there is no potable water in the vicinity, when only potable water in the immediate vicinity can satisfy my particular thirst. At most, what the natural desire for water shows is that water had to have existed at some time. It doesn't even show that water exists now. Suppose all the water on earth is suddenly rendered undrinkable. That is consistent with the continuing existence of the natural desire/need for water.
But this is not a decisive objection since repairs can be made. One could reformulate:
1* Every type of natural, innate desire in us corresponds to some real object that can satisfy some tokens of that type of desire.
But is (1*) obviously true? It could be that our spiritual desires are not artificial, like the desire to play chess, but lacking in real objects nonetheless. It could be that their objects are merely intentional. Suppose our mental life (sentience, intentionality, self-awareness, the spiritual desires for meaning, for love, for lasting happiness, for an end to ignorance and delusion and enslavement to base desires) is just an evolutionary fluke. Our spiritual desires would then be natural as opposed to artificial, but lacking in real objects.
Why do we naturally desire, water, air, sunlight? Because without them we wouldn't have come into material existence in the first place. Speaking loosely, Nature implanted these desires in us. This is what allows us to infer the reality of the object of the desire from the desire. Now if God created us and implanted in us a desire for fellowship with him, then we could reliably infer the reality of God from the desire. But we don't know whether God exists; so it may be that the natural desire for God lacks a real object.
Obviously, one cannot define 'natural desire' as a desire that has a real and not merely intentional object, and then take the non-artificiality of a desire as proof that it is natural. That would be question-begging.
My point is that (1) or (1*) is not known to be true and is therefore rationally rejectable. The argument from desire, then, is not rationally compelling.
As for premise (2), how do we know that it is true? Granting that it is true hitherto, how do we know that it will be true in the future? The utopian dream of the progressives is precisely that we can achieve here on earth final satisfaction of our deepest desires. Now I don't believe this for a second. But I don't think one can reasonably claim to know that (2) is false. What supports it is a very reasonable induction. But inductive arguments don't prove anything.
In sum, the argument from desire, suitably deployed and rigorously articulated, helps render theistic belief rationally acceptable. But it is not a rationally compelling argument.
While traipsing through the Superstition foothills Sunday morning in search of further footnotes to Plato, I happened to think of James Madison and Federalist #51 wherein we read, "If men were angels, no government would be necessary." My next thought was: "Men are not angels." But I realized it could be the formal fallacy of Denying the Antecedent were I to conclude to the truth, "Some government is necessary." (I hope you agree with me that that is a truth.)
The first premise is a counterfactual conditional, indeed, what I call a per impossibile counterfactual. To keep things simple, however, we trade the subjunctive in for the indicative. Let this be the argument under consideration:
1. If men are angels, then no government is necessary. 2. Men are not angels. ergo 3. Some government is necessary.
A prima vista, we have here an instance of the invalid argument-form, Denying the Antecedent:
If p, then q ~p ergo ~q.
But I am loath to say that the argument (as opposed to the just-depicted argument-form) is invalid. It strikes me as valid. But how could it be valid?
One could take the (1)-(3) argument to be an enthymeme where the following is the tacit premise:
1.5 If no government is necessary, then men are angels.
Add (1.5) to the premises of the original argument and the conclusion follows by modus tollendo tollens.
Might it be that 'if ___ then ___' sentences in English sometimes express biconditional propositions? Clearly, if we replace (1) with
1* Men are angels if and only if no government is necessary
the resulting argument is valid.
One might take the (1)-(3) argument as inductive. Now every inductive argument is invalid in the technical sense of 'invalid' in play here. So if there are good inductive arguments, then there are good invalid arguments. Right? If the (1)-(3) argument is inductive, then I think we should say it is a very strong inductive argument. It would then be right churlish and cyberpunkish to snort, "You're denying the antecedent!"
The question arises: are there any good examples from real argumentative life (as opposed to logic text books) of Denying the Antecedent? I mean, nobody or hardly anybody argues like this:
If Jack ran a red light, then Jack deserves a traffic citation. Jack did not run a red light. ergo Jack does not deserve a traffic citation.
I am interested in your logical or linguistic intuitions here. Consider
(*) There is someone called ‘Peter’, and Peter is a musician. There is another person called ‘Peter’, and Peter is not a musician.
Is this a contradiction? Bear in mind that the whole conjunction contains the sentences “Peter is a musician” and “Peter is not a musician”. I am corresponding with a fairly eminent philosopher who insists it is contradictory.
Whether or not (*) is a contradiction depends on its logical form. I say the logical form is as follows, where 'Fx' abbreviates 'x is called 'Peter'' and 'Mx' abbreviates 'x is a musician':
LF1. (∃x)(∃y)[Fx & Mx & Fy & ~My & ~(x =y)]
In 'canonical English':
CE. There is something x and something y such that x is called 'Peter' and x is a musician and y is called 'Peter' and y is not a musician and it is not the case that x is identical to y.
There is no contradiction. It is obviously logically possible -- and not just logically possible -- that there be two men, both named 'Peter,' one of whom is a musician and the other of whom is not.
I would guess that your correspondent takes the logical form to be
LF2. (∃x)(∃y)(Fx & Fy & ~(x = y)) & Mp & ~Mp
where 'p' is an individual constant abbreviating 'Peter.'
(LF2) is plainly a contradiction.
My analysis assumes that in the original sentence(s) the first USE (not mention) of 'Peter' is replaceable salva significatione by 'he,' and that the antecedent of 'he' is the immediately preceding expression 'Peter.' And the same for the second USE (not mention) of 'Peter.'
If I thought burden-of-proof considerations were relevant in philosophy, I'd say the burden of proving otherwise rests on your eminent interlocutor.
But I concede one could go outlandish and construe the original sentences -- which I am also assuming can be conjoined into one sentence -- as having (LF2).
So it all depends on what you take to be the logical form of the original sentence(s). And that depends on what proposition you take the original sentence(s) to be expressing. The original sentences(s) are patient of both readings.
Now Ed, why are you vexing yourself over this bagatelle when the barbarians are at the gates of London? And not just at them?
I am sometimes tempted by the following line of thought. But I am also deeply suspicious of it.
Are the 'laws of thought' 'laws of reality' as well? Since such laws are necessities of thought, the question can also be put by asking whether or not the necessities of thought are also necessities of being. It is surely not self-evident that principles that govern how we must think if we are to make sense to ourselves and to others must also apply to mind-independent reality. One cannot invoke self-evidence since such philosophers as Nagarjuna and Hegel and Nietzsche have denied (in different ways) that the laws of thought apply to the real.
Consider, for example, the Law of Identity:
Id. Necessarily, for any x, x = x.
(Id) seems harmless enough and indisputable. Everything, absolutely everything, is identical to itself, and this doesn't just happen to be the case. But what does 'x' range over? Thought-accusatives? Or reals? Or both? What I single out in an act of mind, as so singled out, cannot be thought of as self-diverse. No object of thought, qua object of thought, is self-diverse. And no object of thought, as such, is both F and not F at the same time, in the same respect, and in the same sense. So there is no question but that Identity and Non-Contradiction apply to objects of thought, and are aptly described as laws of thought. (Excluded Middle is trickier and so I leave it to one side.) What's more, these laws of thought hold for all possible finite, discursive, ectypal intellects. Thus what we have here is a transcendental principle, at least, not one grounded in the contingent empirical psychology or physiology of the type of animals we happen to be. Transcendentalism maybe, but no psychologism or physiologism!
But do Identity and Non-Contradiction apply to 'reals,' i.e., to entities whose existence is independent of their being objects of thought? Are these transcendental principles also ontological principles? Is the necessity of such principles as (Id) grounded in the transcendental structure of the finite intellect, or in being itself? Are the principles merely transcendental or are they also transcendent? (It goes without saying that I am using these 't' words in the Kantian way.)
The answer is not obvious.
Consider a pile of leaves. If I refer to something using the phrase, 'that pile of leaves,' I thereby refer to one self-identical pile; as so referred to, the pile cannot be self-diverse. But is the pile self-identical in itself (apart from my referring to it, whether in thought or in overt speech)?
In itself, in its full concrete extramental reality, the pile is not self-identical in that it is composed of many numerically different leaves, and has many different properties. In itself, the pile is both one and many. As both one and many, it is both self-identical and self-diverse. It is self-identical in that it is one pile; it is self-diverse in that this one pile is composed of many numerically different parts and has many different properties. Since the parts and properties are diverse from each other, and these parts and properties make up the pile, the pile is just as much self-diverse as it is self-identical. The pile is of course not a pure diversity; it is a diversity that constitutes one thing. So, in concrete reality, the pile of leaves is both self-identical and self-diverse.
If you insist that the pile's being self-identical excludes its being self-diverse, then you are abstracting from its having many parts and properties. So abstracting, you are no longer viewing the pile as itis in concrete mind-independent reality, but considering it as an object of thought merely. You are simply leaving out of consideration its plurality of parts and of properties. For the pile to be self-identical in a manner to exclude self-diversity, the pile would have to be simple as opposed to complex. But it is not simple in that it has many parts and many properties.
The upshot is that the pile of leaves, in concrete reality, is both one and many and therefore both self-identical and self-diverse. But this is a contradiction. Or is the contradiction merely apparent? Now the time-honored way to defuse a contradiction is by making a distinction.
One will be tempted to say that the respect in which the pile is self-identical is distinct from the respect in which it is self-diverse. The pile is self-identical in that it is one pile; the pile is self-diverse in that it has many parts and properties. No doubt.
But 'it has many parts and properties' already contains a contradiction. For what does 'it' refer to? 'It' refers to the pile which does not have parts and properties, but is its parts and properties. The pile is not something distinct from its parts and properties. The pile is a unity in and through a diversity of parts and properties. As such, the pile is both self-identical and self-diverse.
What the above reasoning suggests is that such 'laws of thought' as Identity and Non-Contradiction do not apply to extramental reality. No partite thing, such as a pile of leaves, is self-identical in a mannerto exclude self-diversity. Such things are as self-diverse as they are self-identical. So partite things are self-contradictory.
From here we can proceed in two ways.
The contradictoriness of partite entities can be taken to argue their relative unreality. For nothing that truly exists can be self-contradictory. This is the way of F. H. Bradley. One takes the laws of thought as criterial for what is ultimately real, shows that partite entities are not up to this exacting standard, and concludes that partite entities belong to Appearance.
The other way takes the lack of fit between logic and reality as reflecting poorly on logic: partite entities are taken to be fully real, and logic as a falsification. One can find this theme in Nietzsche and in Hegel.
1. The question this post raises is whether it is at all useful to speak of burden of proof (BOP) in dialectical situations in which there are no agreed-upon rules of procedure that are constitutive of the 'game' played within the dialectical situation. By a dialectical situation I mean a context in which orderly discussion occurs among two or more competent and sincere interlocutors who share the goal of arriving as best they can at the truth about some matter, or the goal of resolving some question in dispute. My main concern is with dialectical situations that are broadly philosophical. I suspect that in philosophical debates the notion of burden of proof is out of place and not usefully deployed. That is what I will now try to argue.
2. I will begin with the observation that the presumption of innocence (POI) in an Anglo-American court of law is never up for grabs in that arena. Thus the POI is not itself presumptively maintained and subject to defeat. If Jones is accused of a crime, the presumption of his innocence can of course be defeated, but that the accused must be presumed innocent until proven guilty is itself never questioned and of course never defeated. The POI is not itself a defeasible presumption. And if Rescher is right that there are no indefeasible presumptions, then the POI is not even a presumption. The POI is a rule of the 'game,' and constitutive of the 'game.' The POI in a court room situation is like a law of chess. The laws of chess, as constitutive of chess, cannot themselves be contested within a game of chess. In a particular game a dispute may arise as to whether or not a three-fold repetition of position has occurred. But that a three-fold repetition of position results in a draw is not subject to dispute. The reason there is always a definite outcome in chess (win, lose, or draw) is precisely because of the non-negotiable chess-constitutive laws. These laws, of course, are not inscribed in the nature of things, but are conventional in nature.
As I pointed out earlier, defeasible presumption (DP) and burden of proof are correlative notions. The defeasible presumption that the accused is innocent until proven guilty places the onus probandi on the prosecution. Therefore, from the fact that the POI is not itself defeasible in a court of law, it follows that neither is the BOP. Where the initating BOP lies -- the BOP that remains in force and never shifts during the proceedings -- is never subject to debate. It lies on the state in a criminal case and on the plaintiff in a civil case. If you agree to play the game, then you agree to its constitutive rules. Since these rules are constitutive of the game, they cannot be rejected on pain of ceasing to play the particular game in question.
3. But in philosophy matters are otherwise. For in philosophy everything is up for grabs, including the nature of philosophical inquiry and the rules of procedure. (This is why metaphilosophy is not 'outside of' philosophy but a branch of same.) And so where the BOP lies in a debate between, say, atheists and theists is itself a matter of debate and bitter contention. Each party seeks to put the BOP on the other, to 'bop' him if you will. The theist is inclined to say that there is a defeasible presumption in favor of the truth of theism; but of course few atheists will meekly submit to that pronunciamento. If the theist is right in his presumption, then he doesn't have to do anything except turn aside the atheist's objections: he is under no obligation to argue positively for theism any more than the accused is under an obligation to prove his innocence.
Accused to accuser: "I don't have to prove my innocence; you have to prove my guilt. I enjoy the presumption of innocence; you bear the burden of proof."
Theist to atheist: "I don't have to prove that God exists; you have to prove that God does not exist. Theism enjoys the presumption of being true; atheism bears the burden of proving that theism is not true." (This assumes that BOP and DP are legitimately deployed within broadly philosophical precincts -- which I am denying.)
Note that if the theist invokes the above presumption he needn't be committing the ad ignorantiam fallacy. He needn't be saying that theism is true because it hasn't been proved to be false. Surely the following deductive argument is invalid:
No one has ever proved that God does not exist ergo God does exist.
Just as the presumption of innocence does not entail that the accused is innocent, the presumption of truth does not entail that the proposition presumed true is true. So the mere fact that I have the presumption on my side does not amount to an argument that what I am presuming is true. If I have the presumption on my side, then my dialectical opponent bears the BOP. That's all.
4. Now we come to my tentative suggestion. There is no fact of the matter as to where the BOP lies in any dialectical context, legal, philosophical or any other: it is a matter of decision and agreement upon what has been conventionally decided. In chess, for example, the rules had to be decided and the players have to agree to accept them. No one thinks that these rules are inscribed in rerum natura. The same goes for BOP and DP. It had to be decided that in court room discourse and dialectic the accused enjoys the DP and the accuser(s) the BOP.
In philosophical discourse, however, there are no procedural rules regarding DP and BOP that we will all agree on.
For example, according to Douglas N. Walton, ". . . the basic rule of burden of proof in reasonable dialogue is: He who asserts must prove." (Informal Logic, p. 59) That is clearly false. If I assert that that you left the door open, there is no need for me to prove my assertion. A proof is an argument having premises and conclusion. Surely there is no need to argue for matters evident to sense perception. In fact, it would be unreasonable to do so. Or suppose I assert the Law of Noncontradiction. There is no way I can (non-circularly) prove it. So I cannot be under any epistemic obligation to prove it. 'Ought' implies 'can.'
And how would this work in a dispute between theist and atheist? I assert that God exists and you assert that God does not exist. We both assert. So we both bear the BOP, and we both enjoy DP? But then BOP and DP have no application in this area.
I have heard it said that the BOP lies on the one who makes a positive (affirmative) assertion. But surely both theist and atheist make positive assertions about reality. 'Reality is such that God exists.' 'Reality is such that God does not exist.' Both propositions are logically affirmative.
Suppose our atheist denies God by saying 'God is an unconscious anthropomorphic projection.' Logically, that is an affirmative proposition. Will you conclude that the BOP is on the atheist?
Some say that presumptions are essentially conservative: there is a presumption in favor of the existing and the established and against the novel, the far-out, and what runs contrary to prevailing opinion. "If it ain't broke, don't fix it." Suppose I give the following speech:
There is a presumption in favor of every existing institution, long-standing way of doing things, and well-entrenched and widespread way of belief. Now the consensus gentium is that God exists. And so I lay it down that there is a defeasible presumption in favor of theism and that the burden of proof lies squarely on the shoulders of the atheist. Theism is doxastically innocent until proven guilty. The theist need only rebut the atheist's objections; he needn't make a positive case for his side.
Not only would the atheist not accept this declaration, he would be justified in not accepting it, for reasons that are perhaps obvious. For my declaration is as much up for grabs as anything else in philosophy. And of course if I make an ad baculum move then I remove myself from philosophy's precincts altogether. In philosophy the appeal is to reason, never to the stick.
The situation in philosophy could be likened to the situation in a court of law in which the contending parties are the ones who decide on the rules of procedure, including BOP and DP rules. Such a trial could not be brought to a conclusion. That's the way it is in philosophy. Every procedural rule and methodological maxim is further fodder for philosophical Forschung. (Sorry, couldn't resist the alliteration.)
My tentative conclusion is as follows. In philosophy no good purpose is served by claims that the BOP lies on one side or the other of a dispute, or that there is a DP in favor of this thesis but not in favor of that one. For there is no fact of the matter as to where the BOP lies. BOP considerations are usefully deployed only in dialectical situations in which there is an antecedent conventional agreement on the rules of procedure, rules that constitute the dialectical 'game' in question, and that are agreed upon by the players of the game and never contested by them while playing it.
It occurred to me this morning that there is a connection between the two.
Suppose a person asserts that abortion is morally wrong. Insofar forth, a bare assertion which is likely to elicit the bare counter-assertion, 'Abortion is not morally wrong.' What can be gratuitously asserted may be gratuitously denied without breach of logical propriety, a maxim long enshrined in the Latin tag Quod gratis asseritur, gratis negatur. So one reasonably demands arguments from those who make assertions. Arguments are supposed to move us beyond mere assertions and counter-assertions. Here is one:
Infanticide is morally wrong There is no morally relevant difference between abortion and infanticide Ergo Abortion is morally wrong.
Someone who forwards this argument in a concrete dialectical situation in which he is attempting to persuade himself or another asserts the premises and in so doing provides reasons for accepting the conclusion. This goes some distance toward removing the gratuitousness of the conclusion. THe conclusion is supported by reasons that are independent of the conclusion. But suppose he gave this argument:
Abortion is the deliberate and immoral termination of an innocent pre-natal human life Ergo Abortion is morally wrong.
The second argument is a clear example of petitio principii, begging the question. While the premise entails the conclusion, it does not support it with a reason independent of the conclusion. The argument 'moves in a circle' presupposing the very thing it needs to prove.
So the second 'argument' merely appears to be an argument: it us really just an assertion in the guise of an argument, and a gratuitous assertion at that. But what is gratuitously asserted can be gratuitously denied.
So there we have the connection between Quod gratis asseritur, gratis negatur and Petitio principii.
Recognizing your praise for Critical Rationalism and Morris Raphael Cohen, I believe his page (and also the Karl Popper page) in my PDF Logic Gallery will interest you.
Of course, I hope the book's entire theme/content will also interest you.
Your comments will surely interest ME.
In these dark days of the Age of Feeling, when thinking appears obsolete and civilization is under massive threat from Islamism and its 'liberal' and leftist enablers, it seems fitting that I should repost with additions my old tribute to Morris Raphael Cohen. So here it is:
Tribute to Morris R. Cohen: Rational Thought as the Great Liberator
Morris Raphael Cohen (1880-1947) was an American philosopher of naturalist bent who taught at the City College of New York from 1912 to 1938. He was reputed to have been an outstanding teacher. I admire him more for his rationalism than for his naturalism. In the early 1990s, I met an ancient lady at a party who had been a student of Cohen's at CCNY in the 1930s. She enthusiastically related how Cohen had converted her to logical positivism, and how she had announced to her mother, "I am a logical positivist!" much to her mother's incomprehension.
We best honor a thinker by critically re-enacting his thoughts. Herewith, a passage from Cohen's A Preface to Logic, Dover, 1944, pp. 186-187:
...the exercise of thought along logical lines is the great liberation, or, at any rate, the basis of all civilization. We are all creatures of circumstance; we are all born in certain social groups and we acquire the beliefs as well as the customs of that group. Those ideas to which we are accustomed seem to us self-evident when [while?] our first reaction against those who do not share our beliefs is to regard them as inferiors or perverts. The only way to overcome this initial dogmatism which is the basis of all fanaticism is by formulating our position in logical form so that we can see that we have taken certain things for granted, and that someone may from a purely logical point of view start with the denial of what we have asserted. Of course, this does not apply to the principles of logic themselves, but it does apply to all material propositions. Every material proposition has an intelligible alternative if our proposition can be accurately expressed.
These are timely words. Dogmatism is the basis of all fanaticism. Dogmatism can be combatted by the setting forth of one's beliefs as conclusions of (valid) arguments so that the premises needed to support the beliefs become evident. By this method one comes to see what one is assuming. One can also show by this method that arguments 'run forward' can just as logically be 'run in reverse,' or, as we say in the trade, 'One man's modus ponens is another man's modus tollens.' These logical exercises are not merely academic. They bear practical fruit when they chasten the dogmatism to which humans are naturally prone.
In Cohen's day, the threats to civilization were Fascism, National Socialism, and Communism. Today the main threat is Islamo-totalitarianism, with a secondary threat emanating from the totalitarian Left. Then as now, logic has a small but important role to play in the defeat of these threats. The fanaticism of the Islamic world is due in no small measure to the paucity there of rational heads like Cohen.
But I do have one quibble with Cohen. He tells us that "Every material proposition has an intelligible alternative..." (Ibid.) This is not quite right. A material proposition is one that is non-logical, i.e., one that is not logically true if true. But surely there are material propositions that have no intelligible alternative. No color is a sound is not a logical truth since its truth is not grounded in its logical form. No F is a G has both true and false substitution-instances. No color is a sound is therefore a material truth. But its negation Some color is a sound is not intelligible if 'intelligible' means possibly true. If, on the other hand, 'intelligible' characterizes any form of words that is understandable, i.e., is not gibberish, then logical truths such as Every cat is a cat have intelligible alternatives: Some cat is not a cat, though self-contradictory, is understandable. If it were not, it could not be understood to be self-contradictory. By contrast, Atla kozomil eshduk is not understandable at all, and so cannot be classified as true, false, logically true, etc.
So if 'intelligible' means (broadly logically or metaphysically) possibly true, then it is false that "Every material proposition has an intelligible alternative . . . ."
If you accept truthmakers, and two further principles, then you can maintain that a deductive argument is valid just in case the truthmakers of its premises suffice to make true its conclusion. Or as David Armstrong puts it in Sketch of a Systematic Metaphysics (Oxford UP, 2010), p. 66,
In a valid argument the truthmaker for the conclusion is contained in the truthmaker for the premises. The conclusion needs no extra truthmakers.
For this account of validity to work, two further principles are needed, Truthmaker Maximalism and the Entailment Principle.Truthmaker Maximalism is the thesis that every truth has a truthmaker. Although I find the basic truthmaker intuition well-nigh irresistible, I have difficulty with the notion that every truth has a truthmaker. Thus I question Truthmaker Maximalism. (The hyperlinked entry sports a fine photo of Peter L.)
Armstrong, on the other hand, thinks that "Maximalism flows from the idea of correspondence and I am not willing to give up on the idea that correspondence with reality is necessary for any truth." (63) Well, every cygnet is a swan. Must there be something extramental and extralinguistic to make this analytic truth true? And let's not forget that Armstrong has no truck with so-called abstract objects. His brand of naturalism excludes them. So he can't say that there are the quasi-Platonic properties being a cygnet and being a swan with the first entailing the second, and that this entailment relation is the truthmaker of 'Every cygnet is a swan.'
The Entailment Principle runs as follows:
Suppose that a true proposition p entails a proposition q. By truthmaker Maximalism p has a truthmaker. According to the Entailment Principle, it follows that this truthmaker for p is also a truthmaker for q. [. . .] Note that this must be an entailment. If all that is true is that p --> q, the so-called material conditional, then this result does not follow.
I would accept a restricted Entailment Prinicple that does not presuppose Maximalism. To wit, if a proposition p has a truthmaker T, and p entails a proposition q, then T is also a truthmaker for q. For example, if Achilles' running is the truthmaker of 'Achilles is running,' then, given that the proposition expressed by this sentence entails the proposition expressed by 'Achilles is on his feet,' Achilles' running is also the truthmaker of the proposition expressed by 'Achilles is on his feet.'
. . . my old copy of Alan Hamilton, Logic for Mathematicians, CUP 1978, uses 'statement variables' in his account of the 'statement calculus', as here. The justification for 'variable' is surely that statements have values, namely truth and falsehood. The truth value of a compound statement is calculated from the truth values of its component simple statements by composition of the truth functions corresponding to the logical connectives. This is analogous to the evaluation of an arithmetic expression by composition of arithmetic functions applied to the values of arithmetic variables.
I detect a possible conflation of two senses of 'value.' There is 'value' in the sense of truth value, and there is 'value' in the sense of the value of a variable.
If I am not mistaken, talk of truth values in the strict sense of this phrase enters the history of logic first with Gottlob Frege (1848-1925). Truth and Falsity for him are not properties of propositions, but values of propositional functions. Thus the propositional function denoted by 'x is wise' has True for its value with Socrates as argument, and False for its value with Nero as argument. Please note the ambiguity of 'argument.' We are now engaging in MathSpeak. The analogy with mathematics is obvious. The squaring function has 4 for its value with 2 or -2 as arguments. Propositional functions map their arguments onto the two truth values.
But we also speak in a different sense of the value of a variable. The bound variables in
(x)(x is a man --> x is mortal)
range over real items. These items are the values of the bound variables but they are not truth values. Therefore, one should not confuse 'value' in the sense of truth value with 'value' in the sense of value of a variable. When Quine famously stated that "To be is to be the value of a [bound] variable" he was not referring to truth values.
Brightly says that "The justification for 'variable' is surely that statements have values, namely truth and falsehood." I think that is a mistake that trades on the confusion just exposed. Agreed, statements have truth values. But it doesn't follow that that placeholders for statements are variables.
I was pleased to see that Hamilton observes the distinction I drew several times between an abbreviation and a placeholder. He uses 'label' for 'abbreviation,' but no matter. But I distinguish a placeholder from a variable while Hamilton doesn't.
To appreciate the distinction, first note that with respect to variables we ought to make a three-way distinction among the variable, say 'x,' the value, say Socrates, and the substituend, say 'Socrates.' Now consider the argument:
Tom is tall or Tom is fat Tom is not tall ------- Tom is fat
This argument has the form of the Disjunctive Syllogism:
P v Q ~P ------- Q.
Obviously, 'P' and 'Q' are not abbreviations (labels); if they were then the second display would not display an argument form. It would be an abbreviated argument. But it doesn't follow that 'P' and 'Q' are variables. For if they were variables, then they would have both substituends andf values. But while they have substituends, e.g., the sentences 'Tom is tall' and 'Tom is fat,' they don't have values. Why not? Because we are not quantifying over propositions (or statements if you prefer). There are no quantifiers in the form diagram. (This is not to say that one cannot quantify over propositions.)
'Tom' is tall' is one of many possible substituends for 'P.' But 'Tom is tall' is not the value of 'P.' For we are not quantifying over sentences. We are not quantifying over propositions either. So *Tom is tall* is also not a value of 'P.'
My thesis is that placeholders in the propositional calculus are arbitrary propositional constants. Since they are constants, they are not variables. It is a subtle distinction, I'll grant you that, but it seems necessary if we are to think precisely about these matters. But then one man's necessary distinction is another man's hair-splitting.
You also argue that London must wrongly decide that 'if roses are red then roses are red' (RR) is a contingency, because we say it can be seen as having the form 'P-->Q' and in general statements of this form are contingencies. Indeed they are. But we don't so decide. We say this is a special case in which P and Q stand for the same simple sentence, 'roses are red', not different ones. P and Q are therefore either both true or both false and either way the truth function for --> returns true. Hence this special case is tautologous. We disagree that the move from RR to 'P-->Q' must be seen as an abstraction. We retain the information that P and Q stand for specific substatements within RR, which may themselves have internal structure. 'Form' is a device for making such structure explicit.
So you are saying that 'P --> Q' has a special case that is tautologous. But that makes no sense to me if RR has both forms. A sentence (understood to have one definite meaning) is tautologous if its logical form is tautologous, and if RR has the form 'P--> Q' then it it is not tautologous as an instance of that form. So you seem committed to saying that RR is both tautologous and not tautologous.
Isn't that obvious? If one and same sentence (understood to have one definite meaning) has two logical forms, one tautologous and the other non-tautologous, then one and the same sentence is both tautologous and non-tautologous -- which is a contradiction.
One solution, as I have suggested several times already, is to say that, while 'P --> P' is a special case of 'P -->Q,' namely the case in which P = Q, the two forms are not both forms of 'If roses are red, then roses are red.' Only one of them is, the first one. The second is a form of the first form, not a form of the English sentence.
Putting the problem as an aporetic hexad:
1. 'P -->P' is a special case of 'P --> Q' 2. If a proposition s instantiates form F, and F is a special case of form G, then s instantiates G. 3. 'P --> P' is a tautologous form. 4. 'P --> Q' is a non-tautologous form. 5. No one proposition instantiates both a tautologous and a non-tautologous form. 6. 'If roses are red, then roses are red' instantiates the form 'P --> P.'
The hexad is inconsistent. Phoenix and London agree on (1), (3), (4), and (6). The Phoenician solution is to reject (2). The Londonian solution is reject (5).
But the Phoenicians have an argument for (5):
7. The logical form of a proposition is not an accidental feature of it but determines the very identity of the proposition. Ergo 8. If s instantiates form F, then necessarily, s instantiates F. ergo 5. No one proposition instantiates both a tautologous and a non-tautologous form.
London Ed refers us to Understanding Arguments: an Introduction to Informal Logic, Robert Fogelin and Walter Sinnott-Armstrong, and provides this quotation:
Perhaps a bit more surprisingly, our definitions allow 'roses are red and roses are red' to be a substitution instance of 'p & q'. This example makes sense if you compare it to variables in mathematics. Using only positive integers, how many solutions are there to the equation 'x + y = 4'? There are three: 3+1, 1+3, and 2+2. The fact that '2+2' is a solution to 'x + y = 4' shows that '2' can be substituted for both 'x' and 'y' in the same solution. That's just like allowing 'roses are red' to be substituted for both 'p' and 'q', so that 'roses are red and roses are red' is a substitution instance of 'p & q' in propositional logic.
In general, then, we get a substitution instance of a propositional form by uniformly replacing the same variable with the same proposition throughout, but different variables do not have to be replaced with different propositions. The rule is this:
Different variables may be replaced with the same proposition [Ed: Let's call this the London rule], but different propositions may not be replaced with the same variable.
Suppose I am given the task of determining whether the conditional English sentence 'If roses are red, then roses are red' is a tautology, a contradiction, or a contingency. How do I proceed?
Step One is translation, or encoding. Let upper case letters serve as placeholders for propositions. Let '-->' denote the truth-functional connective known in the trade as the material or Philonian conditional. I write 'P --> P.'
Step Two is evaluation. Suppose for reductio that the truth value of 'P -->P' is false. Then, by the definition of the Philonian conditional, we know that the antecedent must be true, and the consequent false. But antecedent and consequent are the same proposition. Therefore, the same proposition is both true and false. This is a contradiction. Therefore, the assumption that conditional is false is itself false. Therefore the conditional is a tautology.
Now that obviously is the right answer since you don't need logic to know that 'If roses are red, then roses are red' is a tautology. (Assuming you know the definition of 'tautology.') But if if Fogelin & Co. are right, and the 'P -->Q' encoding is permitted, then we get the wrong answer, namely, that the English conditional is a contingency.
I am assuming that if 'P-->Q' is a logical form of 'If roses are red, then roses are red,' then 'P -->Q' is a legitimate translation of 'If roses are red, then roses are red.' As Heraclitus said, the way up and the way down are the same. The assumption seems correct.
If I am right, then there must be something wrong with the mathematical analogy. Now there is no doubt that Fogelin and his side kick are right when it comes to mathematics. And I allow that what they say is true about variables in general. Suppose I want to translate into first-order predicate logic with identity the sentence, 'There is exactly one wise man.' I would write, '[(Ex)Wx & (y)(Wy --> x = y)].' Suppose Siddartha is the unique wise man. Then Siddartha is both the value of 'x' and the value of 'y.'
So different variables can have the same value. And they can have the same substituend. In the example, Siddartha is the value and 'Siddartha' is the substituend. But is a placeholder the same as a variable? I don't think so. Here is a little argument:
No variable is a constant Every placeholder is an arbitrary constant Every arbitrary constant is a constant ------- No placeholder is a variable.
A placeholder is neither an abbreviation, nor a variable. It is an arbitrary constant. Thus the logical form of 'Al is fat' is Fa, not Fx. Fa is a proposition, not a propositional function. 'F' is a predicate constant. 'a' is an individual constant. We cannot symbolize 'Al is fat' as Fx. For Fx is not a proposition but a propositional function. If 'a' were not an arbitrary constant, then Fa would not depict the logical form of 'Al is fat,' a form it shares with other atomic sentences.
Here is another argument:
Every variable is either free or bound by a quantifier No placeholder is either free or bound by a quantifier ------- No placeholder is a variable.
Here is a third argument:
Every variable has a domain over which it ranges No placeholder has a domain over which it ranges ------- No placeholder is a variable.
A fourth argument:
There is no quantification over propositions in the propositional calculus ------- There are no propositional variables in the propositional calculus If there are no propositional variables in the propositional calculus, then the placeholders in the propositional calculus cannot be variables ------- The placeholders in the proposition calculus cannot be variables.
Punchline: because placeholders are not variables, the fact that the different variables can have the same value and the same substituend does not show that different placeholders can have the same substituend. 'If roses are red, then roses are red' does not have the logical form 'P -->Q' and the latter form does not have as a substitutution-instance 'If roses are red, then roses are red.'
As I have said many times already, one cannot abstract away from the fact that the same proposition is both antecedent and consequent.
What one could say, perhaps, is that 'P --> P' has the higher order form 'P --> Q.' But this latter form is not a form of the English sentence but a form of the form of the English sentence.
Ed can appeal to authority all he wants, but that is an unphilosophical move, indeed an informal fallacy. He needs to show where I am going wrong.
"The most conspicuous purpose of logic, in its applications to science and everyday discourse, is the justification and criticism of inference." (Emphasis added, Willard Van Orman Quine, Methods of Logic, 2nd revised ed., Holt, Rinehart & Winston, 1959, p. 33.
Perhaps the dispute in the earlier thread could be resolved if we all could agree on the following.
1. The most specific logical form of a deductive argument A is the form relevant for assessing whether the reasoning embodied in A is valid or invalid.
2. Every deductive argument has exactly one most specific form.
3. Symmetry Thesis: if the most specific form of A is valid, then A is valid; if the most specific form of A is invalid, then A is invalid.
In case 'most specific logical form' needs explanation, consider the difference between the following valid form from the predicate calculus and the following invalid form from the propositional calculus:
Fa Ga ------- (Ex)(Fx & Gx)
p q ------- r.
The former is the most specific logical form of 'Al is fat, Al is gay, ergo, something is both fat and gay.' The latter, if a form of the argument at all, is less specific: it abstracts from the internal subpropositional logical structure of the constituent propositions.
Now three examples in illustration of (1)-(3).
Example One. Call the following argument 'Charley':
Tom is tall ------- Tom is tall.
Although the above display, which is a written expression of the argument and not the argument itself, shows two tokens of the sentence type 'Tom is tall,' the argument consists of exactly one proposition. Anyone who executes the reasoning displayed infers the proposition *Tom is tall* from itself. (I am using asterisks to mention propositions. So '*Tom is tall*' is an abbreviation of 'the proposition expressed by a tokening of the sentence type "Tom is tall".')
It is perfectly clear that the reasoning embodied by Charley is valid and that its form is 'P ergo P.' The reasoning is not from P to some proposition that may or may not be identical to P. Therefore the concrete episode of reasoning does not have the form 'P ergo Q.'
But let us irenically concede that if one wished, for whatever reason, to abstract not only from the content of the argument but also from the plain fact that the argument involves exactly one proposition, one could view the form 'P ergo P' as a special case of 'P ergo Q.' And I will also concede, to keep peace between Phoenix and London, that the argument instantiates the second invalid form, even though I don't believe that this is the case.
Either way, the Symmetry Thesis stands and the Asymmetry Thesis falls. For as G. Rodrigues in the earlier thread pointed out, 'P ergo P' is the most specific form of Charley.
Example Two. Call the following argument 'Kitty Kat.'
If cats like cream, then cats like cream Cats like cream ------- Cats like cream.
Please note that there is no equivocation in this example: 'Cats like cream' has the same sense in all four of its occurrences.
Kitty Kat's most specific form is 'P --> P, P, ergo P.' This form is valid. So Kitty Kat is valid, notwithstanding the fact, if it is a fact, that Kitty Kat also instantiates the formal fallacy, Affirming the Consequent: P --> Q, Q, ergo P. By (1) above, the fact, if it is a fact, that Kitty Kat instantiates Affirming the Consequent is irrelevant to the assessment of the validity/invalidty of the reasoning embodied in Kitty Kat.
Example Three. Call the following example 'Massey':
If God created something , then God created everything. God created everything. ------- God created something.
This argument fits the pattern of the formal fallacy, Affirming the Consequent:
If p then q q ------- p.
But the argument also has a valid form:
Every x is such that Cgx ------- Some x is such that Cgx.
Please note that if an argument is valid, adding a premise can't make it invalid; this principle is what allows us to disregard the first line.
(Example adapted from Gerald J. Massey, "The Fallacy behind Fallacies," Midwest Studies in Philosophy VI (1981), pp. 489-500)
The most specific form of Massey is the predicate logic form above displayed. Since it is valid, Massey is valid.
For the 'Londonistas,' Ed and David, partners in logical investigations. We are unlikely ever to agree, but clarification of differences is an attainable and worthwhile goal, here, and in every arena of controversy. Have at it, boys.
1. Suppose someone reasons as follows. 'Some Englishmen are Londoners; therefore, some Londoners are Englishmen.' To reason is one thing, to reason correctly another. So one can ask: Is this specimen of reasoning correct or incorrect? This is the sort of question with which logic deals. Logic is the study of inference and argument from a normative point of view. It seeks to articulate the criteria of correct and incorrect reasoning. It is analogous to ethics which seeks to articulate the criteria of correct and incorrect action.
2. We all take for granted that some reasoning is correct and some incorrect, and we are all more or less naturally good at reasoning correctly. Almost everyone grasps immediately that if Tom is an Englishman and some Englishmen are Londoners, it does not follow that Tom is a Londoner. What distinguishes the logician is his reflective stance. He reflects upon reasoning in general and tries to extract and systematize the principles of correct reasoning. 'Extract' is an apt metaphor. The logician develops a theory from his pre-theoretical understanding of argumentative correctness. As every teacher of logic comes to learn, one must already be logical to profit from the study of logic just as one must already be ethical to profit from the study of ethics. It is a matter of making explicit and raising to the full light of awareness what must already be implicitly present if the e-duc-ation, the drawing out into the explicit is to occur. This is why courses in logic and ethics are useless for many and positively harmful for some. But they do make some of us more logical and more ethical.
3. Correctness in deductive logic is called validity, and incorrectness invalidity. Since one can argue correctly from false premises and incorrectly from true premises, we distinguish validity from truth. Consider the following argument:
Some Englishmen are Londoners ------- Some Londoners are Englishmen.
We say of neither the premise nor the conclusion that it is either valid or invalid: we say that it is either true or false. And we do not say of the argument that it is true or false, but that it is either valid or invalid. We also speak of inferences as either valid or invalid.
4. What makes a valid argument valid? It can't be that it has true premises and a true conclusion. For there are invalid arguments that satisfy this condition. Some say that what makes a valid argument valid is the impossibility of the premises' being true and the conclusion false. Theirs is a modal explanation of validity. Equivalently,
D1. Argument A is valid =df necessarily, if A's premises are all true, then A's conclusion is true.
This necessity is plainly the necessity of the consequence (necessitas consequentiae), not the necessity of the consequent (necessitas consequentiis): in the majority of cases the premises and conclusion are all contingent propositions.
The modal explanation of validity in (D1) is fine as far as it goes, but it leads to the question: what is the ground of the necessity? If validity is explained by the RHS of (D1), what explains the necessity? What explains the necessitas consequentiae of the conditional on the RHS of (D1)?
Enter logical form.
The validity of a given valid argument evidently resides in something distinct from the given argument. What is this distinct something? It is the logical form of the argument, the argument form. The form F of an argument A is distinct from A because F is a universal (a repeatable) while A is a particular (an unrepeatable). Thus the form
All S are M All M are P ------- All S are P
is a one-in-many, a repeatable. It is repeated in every argument of that form. It is the form of indefinitely many syllogisms, although it is not itself a syllogism, any more than 'All S are M' is a proposition. A proposition is either true or false, but 'All S are M' is neither true nor false. To appreciate this, bear in mind that 'S' and 'M' are not abbreviations but placeholders. If the letters above were abbreviations, then the array above would be an (abbreviated) argument, not an argument form. An argument form is not an argument but a form of indefinitely many arguments.
Now validity is a property of argument forms primarily, and secondarily of arguments having valid forms. What makes a valid argument valid is the validity of its form:
D2. Argument A is valid =df A is an instance of a valid argument form.
D3. Argument form F is valid =df no instance of F has true premises and a false conclusion.
Validity is truth-preserving: a valid argument form will never take you from true premises to a false conclusion. (Exercise for the reader: show that invalidity is not falsehood preserving.) In sum, an argument is valid in virtue of having a valid form, and a form is valid if no argument of that form has true premises and a false concusion. The logical form of a valid argument is what makes it impossible for the premises to be true and the conclusion false.
5. If a valid argument is one with a valid form, one will be tempted to to say that an invalid argument is one with an invalid form. Call this the Symmetry Thesis:
ST. If an argument is an instance of a valid form, then it is valid, and if it is an instance of an invalid form, then it is invalid.
But there are examples that appear to break the symmetry, e.g.:
If God created something , then God created everything. God created everything. ------- God created something.
This argument fits the pattern of the formal fallacy, Affirming the Consequent:
If p then q q ------- p.
But the argument also has a valid form:
Every x is such that Cgx ------- Some x is such that Cgx.
(Example adapted from Gerald J. Massey, "The Fallacy behind Fallacies," Midwest Studies in Philosophy VI (1981), pp. 489-500)
So which is it? Is the argument valid or invalid? It can't be both and it can't be neither. One option is to abandon the Symmetry Thesis and maintain that having a valid form is sufficient for an argument to be valid, but that having an invalid form is not sufficient for it to be invalid. One would then be adopting the following Asymmetry Thesis:
AT. Having a valid form suffices for an argument to be valid, but having an invalid form does not suffice for an argument to be invalid.
Another option is to hold to the Symmetry Thesis and maintain that the Massey argument is really two arguments, not one. But before exploring this option, let us consider the unintuitive consequences of holding that one and the same argument can have two different forms, one valid, the other invalid.
6. Consider any valid syllogism. A syllogism, by definition, consists of exactly three different propositions: a major premise, a minor premise, and a conclusion. So every valid syllogism has the invalid form: p, q, ergo r. Generalizing, we can say that any argument whose validity hinges upon the internal subpropositional logical structure of its constituent propositions will instantiate an invalid form from the propositional calculus (PC). For example, any argument of the valid form, Some S are P; ergo, Some P are S, is an instance of the invalid PC form, p, ergo q.
To think of a valid syllogism as having the invalid form p, q, ergo r is to abstract away from the internal subpropositional logical structure that the syllogism's validity pivots on. But if this abstraction is permitted, one may permit oneself to abstract away from the requirement that the same terms in an argument be replaced by the same placeholders. One might then maintain that
All men are mortal Socrates is a man ------- Socrates is mortal
has the invalid logical form
All Fs are Gs a is an H ------- a is a G
But why stop there? By the same 'reasoning,' the Socrates syllogism has the invalid form:
All Fs are Gs a is an H ------- b is an I.
But if one abstracts away from the requirement that the same term or sentence be replaced by the same placeholder, then we get the result that the obviously valid
Tom is tall ------- Tom is tall
has the valid form p ergo p and the invalid form p ergo q. Here we are abstracting away from the fact that a proposition entails itself and ascending to the higher level of abstraction at which a proposition entails a proposition. After all, it is surely true that in our example a proposition entails a proposition.
I submit, however, that our example's having an invalid form is an intolerable result. Something has gone wrong. Surely the last argument has no invalid form. Surely one cannot lay bare the form of an argument, in an serious sense of 'argument,' if one abandons the requirement that the same term or sentence be replaced by the same placeholder. To do that is to engage in vicious abstraction. It is vicious because an argument in any serious sense of the term is not just a sequence of isolated propositions, but a sequence of propositions together with the idea that one of them is supposed to follow from the others. An argument in any serious sense of the term is a sequence of propositions that has the property of being putatively such that one of them, the conclusion, follows from the others, the premises. But no sequence of propositions can have this property if the argument's form allows for different terms/propositions to have different placeholders.
7. So I suggest that we abandon the Asymmetry Thesis and adopt the Symmetry Thesis according to which no valid argument has any invalid forms. Let me now try to motivate this proposal.
An argument form is an abstraction from an argument. But it is also true that an argument is an abstraction from a concrete episode of reasoning by a definite person at a definite time. Clearly, the same argument can be enacted by the same person at different times, and by the same or different persons at different times. I can 'run through' the argument that the null set is unique any number of times, and so can you. An argument in this sense is not a concrete episode of arguing (reasoning) but a sequence of propositions. A proposition, of course, is not the same as a sentence used to express it.
Now I grant that an argument taken in abstraction from an episode of reasoning (and as the content of that reasoning) can instantiate two or more argument forms. But I deny that a concrete episode of reasoning by a definite person at a definite time can instantiate two or more argument forms. So my claim is that while an argument in abstracto can have two or more forms, an argument in concreto, i.e. a concrete episode of reasoning cannot have more than one form. If this form is valid the argument in concreto is valid. If invalid, the argument in concreto is invalid. To illustrate:
Suppose I know that no Democrat supports capital punishment. Then I learn that Jones is a Democrat. Putting together these two pieces of information, I infer that Jones does not support capital punishment. By 'the concrete episode of reasoning,' I mean the reasoning process together with its content. One first thinks of the first proposition, then the second, then one infers the third, and all of this in the unity of one consciousness. The content is the argument considered in abstraction from any particular diachronic mental enactment by a particular person at a particular time. The reasoning process as a datable temporally extended mental process is also an abstraction from the concrete episode of reasoning which must include both, the reasoning and its content.
Now the concrete episode of reasoning embodies a pattern. In the example, I reason in accordance with this pattern:
(x) (Fx --> ~Gx) Fa ------- ~Ga
Which is also representable as follows:
No Fs are Gs a is an F ------- a is not a G.
The pattern or logical form of my concrete episode of reasoning is assuredly not: p, q, ergo r. This is consistent with saying that the argument in abstracto instantiates the invalid form p, q, ergo r in addition to the valid form above.
The point I am making is this. If we take an argument in abstraction from the concrete episode of reasoning in which it is embodied, then we may find that it instantiates more than one form. There is no denying that every valid syllogism, considered by itself and apart from the mental life of an agent who thinks it through, instantiates the invalid form p, q, ergo r. But no one who reasons syllogistically reasons in accordance with that invalid form. Syllogistic reasoning, whether correct or incorrect, is reasoning that is sensitive to the internal subpropositional logical structure of the syllogism's constituent propositions. The invalid form is not a form of the argument in concreto.
One must distinguish among the following:
The temporally extended event of Jones' reasoning. This is a particular mental process.
The content of this reasoning process, the argument in abstracto as sequence of propositions.
The concrete episode of reasoning (i.e. the argument in concreto) which involves both the reasoning and its content.
The verbal expression in written or spoken sentences of the argument.
The form or forms of the argument in abstracto.
The verbal expression of a form or forms in a form diagram(s).
The form of the argument in concreto.
My point, again, is that we can uphold the Symmetry Thesis if we make a distinction between arguments in the concrete and arguments in the abstract. But this is a distinction we need in any case. The Symmetry Thesis holds for arguments in the concrete. But these are the arguments that matter because these are the ones people actually give.
Applying this to the Massey example above, we can say that while the abstract argument expressed by the following display has two forms, one invalid, the other valid:
If God created something , then God created everything. God created everything. ------- God created something
there is no one concrete argument, no one concrete episode of reasoning, that the display expresses. One who reasons in a way that is attentive to the internal subpropositional structure of the constituent propositions reasons correctly. But one who ignores this internal structure reasons incorrectly.
In this way we can uphold the Symmetry Thesis and avoid the absurdities to which the Asymmetry Thesis leads.
I read and excerpted the chapter. I am not mistaken. Also, what he says seems correct to me.
He claims that logic is not formal, insofar as it is concerned with the 'laws of thought'. He says "Thought is a psychical phenomenon, and psychical phenomena have no extension. What is meant by the form of an object that has no extension?" I can't fault this.
I take it that the argument is this:
1. Only spatially extended objects have forms. 2. Neither acts of thinking, nor such objects of thought as propositions, are spatially extended. Therefore 3. If logic studies either acts of thinking or objects of thought, then logic is not a formal study, a study of forms.
If this is the argument, I am not impressed. Premise (1) is false. L.'s notion of form is unduly restrictive. There are forms other than shapes. Consider a chord and an arpeggio consisting of the same notes. The 'matter' is the same, the 'form' is different. In a chord the notes sound at the same time; in an arpeggio at different times. The arrangement of the notes is different. Arrangement and structure are forms. Examples are easily multiplied.
Nor, he says, is it the object of logic to investigate how we are thinking or how we ought to think. "The first task belongs to psychology, the second to a practical art of a similar kind to mnemonics". And then he says "Logic has no more to do with thinking than mathematics has". Isn't that correct?
We can agree that logic is not a branch of psychology: it is not an empirical study and its laws are not empirical generalizations. LNC, for example, is not an empirical generalization. But a case can be made for logic's being normative. It does not describe how we do think, but it does prescribe how we ought to think if we are to arrive at truth. If so, then logic does have a practical side and issues hypothetical imperatives, e.g., "If you want truth, avoid contradictions!"
In a similar vein he notes the formalism of Aristotelian logic. The whole Aristotelian theory of the syllogism is built up on the four expressions 'every' (A), 'no' (E), 'some' (I) and 'not every' (O). "It is obvious that such a theory has nothing more in common with our thinking than, for instance, the theory of the relations of greater and less in the field of numbers". Brilliant.
Why do you call it "brilliant"? Husserl and Frege said similar things. It's old hat, isn't it? Psychologism died with the 19th century at least in the mainstream. Given propositions p, q, logic is concerned with such questions as: Does p entail q? Are they consistent? Are they inconsistent? We could say that logic studies certain relations between and among propositions, which are the possible contents of judgings, but are not themselves judgings or entertainings or supposings or anything else that is mental or psychological.
Again, on the need for logic and science to focus on the expression of thought rather than 'thought', he says "Modern formal logic strives to attain the greatest possible exactness. This aim can be reached only by means of a precise language built up of stable, visually perceptible signs. Such a language is indispensable for any science. Our own thoughts not formed in words are for ourselves almost inapprehensible and the thoughts of other people, when not bearing an external shape [my emphasis] could be accessible only to a clairvoyant. Every scientific truth in order to be perceived and verified, must be put into an external form [my emphasis] intelligible to everybody."
I can't fault any of this. What do you think?
Sorry, but I am not impressed. It is fundamentally wrongheaded. First of all this is a howling non sequitur:
1. Logic does not study mental processes; Therefore 2. Logic studies visually perceptive signs.
Surely it is a False Alternative to suppose that logic must either study mental processes or else physical squiggles and such. There is an easy way between the horns: logic studies propositions, which are neither mental nor physical.
In my last post I can gave two powerful arguments why a perceptible string of marks is not identical to the proposition those marks are used to express.
L. speaks of an external form intelligible to everybody. But what is intelligible (understandable) is not the physical marks, but the proposition they express. We both can see this string:
Yash yetmis ish bitmish
but only I know what it means. (Assuming you don't know any Turkish.) Therefore, the meaning (the proposition), is not identical to the physical string.
There is also an equivocation on 'thought' to beware of, as between thinking and object of thought. As you well know, in his seminal essay Der Gedanke Frege was not referring to anything psychological.
I will grant L. this much, however. Until one has expressed a thought, it is not fully clear what that thought is. But I insist that the thought -- the proposition -- must not be confused with its expression.
The real problem here is that you wrongly think that one is multiplying entities beyond necessity if one makes the sorts of elementary distinctions that I am making.
We can't say that an argument is invalid because it instantiates an invalid form. The argument Socrates is a man; all men are mortal; ergo Socrates is mortal instantiates the invalid form a is F; all Hs are G; ergo a is G, but modulo equivocation, it is truth-preserving. Instantiation of form is just pattern-matching, and the argument does match the pattern of the invalid form.
I reject this of course.The sample argument is an example of correct reasoning. But anyone who argues in accordance with the schema argues incorrectly. Why? Because the schema is not truth-preserving. Therefore the sample argument does not instantiate the invalid form.
I don't think Brightly understands 'truth-preserving.' This is a predicate of argument forms, primarily, and the same goes for 'valid' and 'invalid.' Here are some definitions:
D1. An argument form is truth-preserving =df no argument of that form has true premises and a false conclusion.
D2. An argument form F is valid =df F is truth-preserving.
D3. A particular argument A is valid =df A instantiates a valid form. (This allows for the few cases in which an argument has two forms, one valid and one invalid.)
D4. A particular argument A is invalid =df there is no valid form that it instantiates.
Now what is it for an argument to instantiate an argument form? To answer this question we need to know what an argument is. Since deductive arguments alone are under consideration, I define:
D5. A deductive argument is a sequence of propositions together with the claim that one of them, the conclusion, follows from the others, the premises, taken together.
If the claim holds, the argument is valid; if not, invalid.
Now the main point for present purposes is that an argument is composed of propositions. A proposition is not a complex physical object such as a string of marks on paper. Thus what you literally SEE when you see this:
7 + 5 = 12
is not a proposition, but a spatiotemporal particular, a physical item subject to change: it can be deleted. But the proposition it expresses cannot be deleted by deleting what you just literally SAW. That suffices to show that the proposition expressed by what you saw is not identical to what you saw. Whatever propositions are (and there are different theories), they are not physical items.
What's more, you did not SEE (with your eyes) the proposition, or that it is true, but you UNDERSTOOD the proposition and that it is true. (A proposition and its being true are not the same even if the proposition is true.) So this is a second reason why a proposition is not identical to its physical expression.
Now what holds for propositions also holds for arguments: you cannot delete an argument by deleting physical marks, and you cannot understand an argument merely by seeing a sequence of strings of physical marks.
An argument is not a pattern of physical marks. So there is no question of matching this physical pattern with some other physical pattern. Instantiation of logical form is not just pattern-matching.
If a sentence contains a sign like 'bank' susceptible of two or more readings, then no one definite proposition is expressed by the sentence. Until that ambiguity is resolved one does not have a definite proposition, and without definite propositions no definite argument. But once one has a definite argument then one can assess its validity. If it instantiates a valid form, then it is valid; if it instantiates an invalid form, then it is invalid.
It is as simple as that. But one has to avoid the nominalist mistake of thinking that arguments are just collections of physical items.
It is a well-known and puzzling fact that proper names are ambiguous. According to the US telephone directory, Frodo Baggins is a real person (who lives in Ohio). But according to LOTR, Frodo Baggins is a hobbit. Not a problem. The name ‘Frodo Baggins’ as used in LOTR, clearly has a different meaning from when used to talk about the person in Ohio. So the argument below is invalid:
Frodo Baggins is a hobbit Frodo Baggins is not a hobbit Some hobbit is not a hobbit.
This is because both premisses could be true, but the conclusion could not be true. So your claim that the validity of arguments using fictional names has ‘nothing to do with any semantic property’ is incorrect.
Well, ex contradictione quodlibet. Since anything follows from a contradiction, the conclusion of the above syllogism follows from the premises. So the above argument is valid in that it instantiates a valid argument-form, namely:
p ~p --- q
Obviously, there is no argument of the above form that has true premises and a false conclusion. So every argument of that form is valid or truth-preserving.
You invoke a Moorean fact. But we have to be very clear as to the identity of this fact.
It is a Moorean fact that proper names, taken in abstraction from the circumstances of their thoughtful use, are not, well, proper. They are common, or ambiguous as you say. It is no surprise that some dude in Ohio rejoices under the name 'Frodo Baggins.'
But so taken, a name has no semantic properties: it doesn't mean anything. It is just a physical phenomenon, whether marks on paper or a sequence of sounds, etc. Pronounce the sounds corresponding to 'bill,' 'john, 'dick.' Is 'dick' a name or a common noun, and for what? How many dicks in this room? How many detectives? How many penises? How many disagreeable males, 'pricks'? How many men named 'Dick'? Consider the multiple ambiguity of 'There are more dicks than johns in the room but the same number of bills.'
A name that has meaning (whether or not it refers to anything) is always a name used by a mind (not a voice synthesizing machine) in definite circumstances. For example, if the context is a discussion of LOTR, then my use and yours of 'Frodo' has meaning: it means a character in that work, despite the fact that in reality there is no individual named. And as long as we stay in that context, the name has the same meaning.
And the same holds in the context of argument. In your argument above 'Frodo Baggins' has the same meaning in both premises.
You can't have it both ways: you can't maintain that 'Frodo Baggins' is a meaningless string that could mean anything in any occurrence (a fictional character, a real man, his dog, a rock group, a town, etc.) AND that it figures as a term in an argument.
To sum up. Whether a deductive argument is valid or not depends on its logcal form. If there is a valid form it instantiates, then it is valid. The validity of the form is inherited by the argument having that form. But form abstracts from semantic content. So the specific meaning of a name is irrelevant to the evaluation of the validity of an argument in which the name figures. But of course it is always assumed that names are used in the same sense in all of their occurrences in an argument. So only in this very abstract sense is meaning relevant to the assessment of validity.
Cicero was a Roman Tully was a philosopher ----- Some Roman was a philosopher.
Quite simply, there is no middle term. The example is an instance of the dreaded quaternio terminorum. But of course we learned at Uncle Willard's knee that Cicero = Tully. Add that fact as a premise and the above argument becomes valid. As a general rule, any invalid argument can be rendered valid by adding one or more premises.
So sameness of reference is not sufficient for sameness of name. 'Cicero' and 'Tully' have the same reference, but they are different names. They are both token- and type-different. Since they are different names, that fact must be accommodated in the form diagram, which looks like this:
Fa Gb --- (Ex)(Fx & Gx).
This form is clearly invalid. The most one can squeeze out of these premises using Existential Generalization is '(Ex)Fx & (Ex)Gx.'
It is worth pointing out that the use of the different signs 'a' and 'b' does not entail that a is not identical to b; it leaves open both the possibility that a = b and the possibility that ~(a = b). It is because of the second of these possibilities that the argument-form is invalid.
Commenter Edward Ockham in a comment on the old blog wanted to know why, given that we had to add a premise to make the Cicero argument valid, we don't have to add a premise to make the Alexander argument valid. That argument, from the days when men were men and went around 'seizing' women, proceeds thusly:
Alexander seized Helen Alexander did not seize Helen ----- Someone seized and did not seize Helen.
Ockham wants to know why we don't have to add an identity premise to secure the validity of this argument. But what premise would he have us add? It can't be 'Alexander is Alexander' for that is necessarily true and therefore true whether or not both occurrences of 'Alexander' in the original argument are coreferential. Presumably, Ockham wants us to supply '"Alexander" is coreferential in both of its occurrences.' But this goes without saying. There in no need to affirm this in a separate premise since it is implied by the fact that 'Alexander' in both occurrences is a token of the same word-type. We needn't say what is plainly shown. (He said with a sidelong glance in old Ludwig's direction.)
Ockham is bothered by the possibility of equivocation. Well, either there is an equivocation on 'Alexander' or there isn't. If there is an equivocation, then the argument instantiates an invalid form, and Ockham's contention collapses. If there is no equivocation, then the argument instantiates a valid form but it is not the case that both premises are true; so again Ockham's contention collapses. Either way, his contention collapses.
Either we capture the reference [of a name] in the form, and my objection collapses. Or you concede that the form covers only the visible or audible outward form of the word. In which case, my specious Alexander argument really does have the right form, and we have to add on the condition about reference, and my point stands.
I grasp something like the first horn. If 'a' occurs two or more times in a form diagram, then no argument of that form has an equivocation on a term whose place is held by 'a.' This is to say that the form diagram enforces coreferentiality on any terms whose place is held by 'a' in the form schema. Otherwise, the argument would not be of the form in question.
Ockham wants to have it both ways at once. He wants his argument A to be of valid form F without F enforcing coreferentiality on the occurrences of 'a' in A. This is just impossible. If there is an equivocation on 'a,' then A does not instantiate F. But if A does instantiate F, then there cannot be any equivocation of 'a.' Why? Because the form does not permit it. The form enforces coreferentiality.
Now look back at the Cicero argument. It is invalid because its form (depicted above) is invalid and the argument has no valid form. But I don't say that the invalid form enforces lack of coreferentiality on the singular terms whose place is held in the diagram by 'a' and 'b.' I say instead that the invalid form permits coreferentiality of these terms. Thus there is an asymmetry between the Alexander and Cicero cases.
I demanded an argument valid in point of logical form all of whose premises are purely factual but whose conclusion is categorically (as opposed to hypothetically or conditionally) normative. Recall that a factual proposition is one which, whether true or false, purports to record a fact, and that a purely factual proposition is a factual proposition containing no admixture of normativity.
My demand is easily, if trivially, satisfied.
Ex contradictione quodlibet. From a contradiction anything, any proposition, follows. This is rigorously provable within the precincts of the PC (the propositional calculus). As follows:
1. p & ~p 2. p (from 1 by Simplification) 3. p v q (from 2 by Addition) 4. ~p & p (from 1 by Commutation) 5. ~p (from 4 by Simplification) 6. q (from 3, 5 by Disjunctive Syllogism)
Now plug in 'Obama is a liar' for p and 'One ought to be kind to all sentient beings' for q. The result is:
Obama is a liar Obama is not a liar Ergo One ought to be kind to all sentient beings.
My demands have been satisifed. The above is an argument valid in point of logical form whose premises are all purely factual and whose conclusion is categorically normative.
I thank Tully Borland for pushing the discussion in this fascinating direction.
Affirming the Consequent is an invalid argument form. Ergo One ought not (it is obligatory that one not) give arguments having that form.
Modus Ponens is valid Ergo One may (it is permissible to) give arguments having that form.
Correct deductive reasoning is in every instance truth-preserving. Ergo One ought to reason correctly as far as possible.
An argument form is valid just in case no (actual or possible) argument of that form has true premises and a false conclusion. An argument form is invalid just in case some (actual or possible) argument of that form has true premises and a false conclusion. Deductive reasoning is correct just in case it proceeds in accordance with a valid argument form. 'Just in case' is but a stylistic variant of 'if and only if.'
Now given these explanations of key terms, it seems that validity, invalidity, and correctness are purely factual, and thus purely non-normative, properties of arguments/reasonings. If so, how the devil do we get to the conclusions of the three arguments above?
View One: We don't. A, B, and C are each illicit is-ought slides.
View Two: Each of the above arguments is valid. Each of the key terms in the premises is normatively loaded from the proverbial 'git-go,' in addition to bearing a descriptive load.. Therefore, there is no illict slide. The move is from the normative to the normative. Validity, invalidity, and correctness can be defined only in terms of truth and falsity which are normative notions.
View Three: We have no compelling reason to prefer one of the foregoing views to the other. Each can be argued for and each can be argued against. Thus spoke the Aporetician.
Consider the argument: Bill is a brother ----- Bill is a sibling.
Is this little argument valid or invalid? It depends on what we mean by 'valid.' Intuitively, the argument is valid in the following generic sense:
D1. An argument is (generically) valid iff it is impossible that its premise(s) be true and its conclusion false.
(D1) may be glossed by saying that there are no possible circumstances in which the premises are true and the conclusion false. Equivalently, in every possible circumstance in which the premises are true, the conclusion is true. In short, validity is immunity to counterexample.
(D1), though correct as far as it goes, leaves unspecified the source or ground of a valid argument's validity. This is the philosophically interesting question. What makes a valid argument valid? What is the ground of the impossibility of the premises' being true and the conclusion being false? One answer is that the source of validity is narrowly logical or purely syntactic: the validity of a valid argument derives from its subsumability under logical laws or (what comes to the same thing) its instantiation of valid argument-forms.
Now it is obvious that the validity of the above argument does not derive from its logical form. The logical form is
Fa ----- Ga
where 'a' is an arbitrary individual constant and 'F' an arbitrary predicate constant. The above argument-form is invalid since it is easy to interpret the place-holders so as to make the premise true and the conclusion false: let 'a' stand for Al, 'F' for fat and 'G' for gay.
We now introduce a second, specific sense of 'valid,' one that alludes to the source of validity:
D2. An argument is syntactically valid iff it is narrowly-logically impossible that there be an argument of that form having true premises and a false conclusion.
According to (D2), a valid argument inherits its validity from the validity of its form, or logical syntax. So on (D2) it is primarily argument-forms that are valid or invalid; arguments are valid or invalid only in virtue of their instantiation of valid or invalid argument-forms. (D2) is thus a specification of the generic (D1).
But there is a second specification of (D1) according to which validity/invalidity has its source in the constituent propositions of the arguments themselves and so depends on their extra-syntactic content:
D3. An argument is extra-syntactically valid iff (i) it is impossible that its premises be true and its conclusion false; and (ii) this impossibility is grounded neither in any contingent matter of fact nor in logic proper, but in some necessary connection between the senses or the referents of the extra-logical terms of the argument.
A specification of (D3) is
D4. An argument is semantically valid iff (i) if it is impossible that its premises be true and its conclusion false; and (ii) this impossibility is grounded in the senses of the extra-logical terms of the argument.
Thus to explain the semantic validity of the opening argument we can say that the sense of 'brother' includes the sense of 'sibling.' There is a necessary connection between the two senses, one that does not rest on any contingent matter of fact and is also not mediated by any law of logic. Note that logic allows (does not rule out) a brother who is not a sibling. Logic would rule out a non-sibling brother only if 'x is F & x is not G' had only false substitution-instances -- which is not the case. To put it another way, a brother that is not a sibling is a narrowly-logical possibility. But it is not a broadly-logical possibility due to the necesssary connection of the two senses.
So it looks as if analytic entailments like Bill is a brother, ergo, Bill is a sibling show that subsumability under logical laws is not necessary for (generically) valid inference. Sufficient, but not necessary. Analytic entailments appear to be counterexamples to the thesis that inferences in natural language can be validated only by subsumption under logical laws.
One might wonder what philosophers typically have in mind when they speak of validity. I would say that most philosophers today have in mind (D1) as specified by (D2). Only a minority have in mind (D3) and its specification (D4). I could easily be wrong about that. Is there a sociologist of philosophers in the house?
Consider the Quineans and all who reject the analytic/synthetic distinction. They of course will have no truck with analytic entailments and talk of semantic validity. Carnapians, on the other hand, will uphold the analytic/synthetic distinction but validate all entailments in the standard (derivational) way by importing all analytic truths as meaning postulates into the widened category of L-truths.
Along broadly Carnapian lines one could argue that the above argument is an enthymeme which when spelled out is
Every brother is a sibling Bill is a brother ----- Bill is a sibling.
Since this expanded argument is syntactically valid, the original argument -- construed as an enthymeme -- is also syntactically valid. When I say that it is syntactically valid I just mean that the conclusion can be derived from the premises using the resources of standard logic, i.e. the Frege-inspired predicate calculus one finds in logic textbooks such as I. Copi's Symbolic Logic. In the aboveexample, one uses two inference rules, Universal Instantiation and Modus Ponens, to derive the conclusion.
If this is right, then the source of the argument's validity is not in a necessary connection between the senses of the 'brother' and 'sibling' but in logical laws.
Here is a little puzzle I call the Stromboli Puzzle. An earlier post on this topic was defective. So I return to the topic. The puzzle brings out some of the issues surrounding existence. Consider the following argument.
Stromboli exists. Stromboli is an island volcano. Ergo An island volcano exists.
This is a sound argument: the premises are true and the reasoning is correct. It looks to be an instance of Existential Generalization. How can it fail to be valid? But how can it be valid given the equivocation on 'exists'? 'Exists' in the conclusion is a second-level predicate while 'exists' in the initial premise is a first-level predicate. Although Equivocation is standardly classified as an informal fallacy, it induces a formal fallacy. An equivocation on a term in a syllogism induces the dreaded quaternio terminorum, which is a formal fallacy. Thus the above argument appears invalid because it falls afoul of the Four Term Fallacy.
Objection 1. "The argument is valid without the first premise, and as you yourself have pointed out, a valid argument cannot be made invalid by adding a premise. So the argument is valid. What's your problem?"
Reply 1. The argument without the first premise is not valid. For if the singular term in the argument has no existing referent, then the argument is a non sequitur. If 'Stromboli' has no referent at all, or has only a nonexisting Meinongian referent, then Existential Generalization could not be performed, given, as Quine says, that "Existence is what existential quantification expresses."
Objection 2: "The first premise is redundant because we presuppose that the domain of quantification is a domain of existents."
Reply 2: Well, then, if that is what you presuppose, then you can state your presupposition by writing, 'Stromboli exists.' Either the argument without the first premise is an enthymeme or it is invalid. If it is an enthymeme, then we need the first premise to make it valid. If it is invalid, then it is invalid.
Therefore, we are stuck with the problem of explaining how the original argument is valid, which it surely is.
My answer is that the original argument is an enthymeme an unstated premise of which links the first- and second-level uses of 'exist(s)' and thus presupposes the admissibility of the first-level uses. Thus we get:
A first-level concept F exists (is instantiated) iff it is instantiated by an individual that exists in the first-level way. Stromboli is an individual that exists in the first-level way. Stromboli is an island volcano. Ergo The concept island volcano exists (is instantiated). Ergo And island volcano exists.
Now what does this rigmarole show? It shows that Frege and Russell were wrong. It shows that unless we admit as logically kosher first-level uses of 'exist(s)' and cognates, a simple and obviously valid argument like the the one with which we started cannot be made sense of.
'Exists(s)' is an admissible predicate of individuals, and existence belongs to individuals: it cannot be reduced to, or eliminated in favor of, instantiation. This has important consequences for metaphysics.
Nicholas Rescher cites this example from Buridan. The proposition is false, but not self-refuting. If every proposition is affirmative, then of course *Every proposition is affirmative* is affirmative. The self-reference seems innocuous, a case of self-instantiation. But *Every proposition is affirmative* has as a logical consequence *No proposition is negative.* This follows by Obversion, assuming that a proposition is negative if and only if it is not affirmative.
Paradoxically, however, the negative proposition, unlike its obverse, is self-refuting. For if no proposition is negative then *No proposition is negative* is not negative. So if it is, it isn't. Plainly it is. Ergo, it isn't.
Rescher leaves the matter here, and I'm not sure I have anything useful to add.
It is strange, though, that here we have two logically equivalent propositions one of which is self-refuting and the other of which is not. The second is necessarily false. If true, then false; if false, then false; ergo, necessarily false. But then the first must also be necessarily false. After all, they are logically equivalent: each entails the other across all logically possible worlds.
What is curious, though, is that the ground of the logical necessity seems different in the two cases. In the second case, the necessity is grounded in logical self-contradiction. In the first case, there does not appear to be any self-contradiction.
It is impossible that every proposition be affirmative. And it is impossible that no proposition be negative. But whereas the impossibility of the second is the impossibility of self-referential inconsistency, the impossibility of the first is not. (That is the 'of' of apposition.)
Can I make an aporetic polyad out of this? Why not?
1. Logically equivalent logically impossible propositions have the same ground of their logical impossibility.
2. The ground of the logical impossibility of *Every proposition is affirmative* is not in self-reference.
3. The ground of the logical impossibility of *No proposition is negative* is in self-reference.
The limbs of this antilogism are individually plausible but collectively inconsistent.
Nicholas Rescher, Paradoxes: Their Roots, Range, and Resolution, Open Court, 2001, pp. 21-22.
G. E. Hughes, John Buridan on Self-Reference, Cambidge UP, 1982, p. 34. Cited by Rescher.
I had the pleasure of meeting London Ed, not in London, but in Prague, in person, a few days ago. Ed, a.k.a. 'Ockham,' and I have been arguing over existence for years. So far he has said nothing to budge me from my position. Perhaps some day he will. The following entry, from the old Powerblogs site, whose archive is no more, was originally posted 25 May 2008. Here it is again slightly redacted.
I am racking my brains over the question why commenter 'Ockham' cannot appreciate that standard quantificational accounts of existence presuppose rather than account for singular existence. It seems so obvious to me! Since I want to put off as long as possible the evil day when I will have to call him existence-blind, I will do my level best to try to understand what he might mean.
Consider the following renditions of a general and a singular existence statement respectively, where 'E' is the 'existential' or, not to beg any questions, the particular quantifier:
1. Cats exist =df (Ex)(x is a cat)
2. Max (the cat) exists =df (Ex)(x = Max).
Objectually as opposed to substitutionally interpreted, what the right-hand sides of (1) and (2) say in plain English is that something is a cat and that something is (identical to) Max, respectively. Let D be the domain of quantification. Now the right-hand side (RHS) of (1) is true iff at least one member of D is a cat. And the RHS of (2) is true iff exactly one member of D = Max. Now is it not perfectly obvious that the members of D must exist if (1) and (2) are to be true? To me that is obvious since if the members of D were Meinongian nonexistent items, then (1) and (2) would be false. (Bear in mind that there is no logical bar to quantifying over Meinongian objects, whatever metaphysical bar there might be. Meinongians, and there are quite a few of them, do it all the time with gusto.)
Therefore, 'Something is a cat' is a truth-preserving translation of 'Cats exist' only if 'Something is a cat' is elliptical for 'Something that exists is a cat.' And similarly for 'Something is Max.' But here is where 'Ockham' balks. He sees no difference between 'something' and 'something that exists' where I do see a difference.
I am sorely tempted to call anyone who cannot understand this difference 'existence-blind' and cast him into the outer darkness, that place of fletus et stridor dentium, along with qualia-deniers, eliminative materialists, deniers of modal distinctions, and the rest of the terminally benighted. But I will resist this temptation for the moment.
And were I to label 'Ockham' existence-blind he might return the 'compliment' by saying that I am hallucinating, or suffering from double-vision. "You've drunk so much Thomist Kool-Aid that you see a distinction where there isn't one!" But then we get a stand-off in which we sling epithets at each other. Not good for those of us who would like to believe in the power and universality of reason. It should be possible for one of us to convince the other, or failing that, to prove that the issue is rationally undecidable.
The issue that divides us may be put as follows. (Of course, it may be that we have yet to locate the exact bone of contention, and in our dance around each other we have succeeded only in 'dislocating' it.)
BV: Because the items in the domain of quantification exist, there has to be more to existence than can be captured by the so-called 'existential' quantifier. Existence is not a merely logical topic. Pace Quine, it is not the case that "Existence is what existential quantification expresses." Existence is a 'thick' topic: there is room for a metaphysics of existence. One can legitimately ask: What is it for a concrete contingent individual to exist? and one can expect something better than the blatantly circular, 'To exist is to be identical to something.' To beat on this drum one more time, this is a circular explanation because D is a domain all of whose members exist. One moves in a circle of embarrassingly short diameter if one maintains that to exist is to be identical to something that exists. Note that I wrote circular explanation, not circular definition. Note also that I am assuming that there is such a thing as philosophical explanation, which is not obvious, and is denied by some.
O: Pace BV, the items in the domain of quantification admit of no existence/nonexistence contrast. Therefore, 'Something is a cat' is indistinguishable from 'Something that exists is a cat.' There is no difference at all between 'something' and 'something that exists,' and 'something' is all we need. Now 'something' is capturable without remainder using the resources of standard first-order predicate logic with identity. 'Exist(s)' drops out completely. There is no (singular) existence and there are no (singular) existents. There are just items, and one cannot distinguish an item from its existence.
Now if that is what O means, then I understand him, but only on the assumption that for individuals
3. Existence = itemhood.
For if to exist = to be an item, if existence reduces to itemhood, then there cannot be an existence/nonexistence contrast at the level of items. It is a logical truth that every item is an item, and therefore an item that is not an item would be a contradiction: 'x is an item' has no significant denial. Therefore, on the assumption that existence = itemhood, there is no difference between 'Some item is a cat' and 'Some item that exists is a cat.' And if there is no such difference, then existence is fully capturable by the quantifier apparatus.
But now there is a steep price to pay. For now we are quantifying over items and not over existents, and sentences come out true that ought not come out true. 'Dragons exist,' for example, which is false, becomes 'Some item is a dragon' which is true. To block this result, O would have to recur to a first-level understanding of existence as contrasting with nonexistence. He would have to say that every item exists, that there are no nonexisting items. But then he can no longer maintain that 'something' and 'something that exists' are indistinguishable.
In defiance of Ed's teacher, C. J. F. Williams, I deny that the philosophy of existence must give way to the philosophy of someness. (Cf. the latter's What is Existence? Oxford, 1981, p. 215) The metaphysics of existence cannot be supplanted by the logic of 'exist(s).' Existence is not a merely logical topic.
Here is an obituary of Williams written by Richard Swinburne.
It is interesting that 'nothing' has two opposites. One is 'something.' Call it the logical opposite. The other is 'being.' Call it the ontological opposite. Logically, 'nothing' and 'something' are interdefinable:
D1. Nothing is F =df It is not the case that something is F
D2. Something is F =df it is not the case that nothing is F.
These definitions give us no reason to think of one term as more basic than the other. Logically, they are on a par. Logically, they are polar opposites. Anything you can say with the one you can say with the other, and vice versa.
Ontologically, however, being and nothing are not on a par. They are not polar opposites. Being is primary, and nothing is derivative. (Note the ambiguity of 'Nothing is derivative' as between 'It is not the case that something is derivative' and 'Nothingness is derivative.' The second is meant.)
Suppose we try to define the existential 'is' in terms of the misnamed 'existential' quantifier. (The proper moniker is 'particular quantifier.') We try this:
y is =df for some x, y = x.
In plain English, for y to be or exist is for y to be identical to something. For Quine to be or exist is for Quine to be identical to something. This thing, however, must exist. Thus
Quine exists =df Quine is identical to something that exists
Pegasus does not exist =df nothing that exists is such that Pegasus is identical to it.
The conclusion is obvious: one cannot explicate the existential 'is' in terms of the particular quantifier without circularity, without presupposing that things exist.
I have now supplied enough clues for the reader to advance to the insight that the ontological opposite of 'nothing,' is primary.
Mere logicians won't get this since existence is "odious to the logician" as George Santayana observes. (Scepticism and Animal Faith, Dover, 1955, p. 48, orig. publ. 1923.)
James N. Anderson and Greg Welty have published a paper entitled The Lord of Non-Contradiction: An Argument for God from Logic. Having worked out similar arguments in unpublished manuscripts, I am very sympathetic to the project of arguing from the existence of necessary truths to the necessary existence of divine mind.
Here is a quick sketch of the Anderson-Welty argument as I construe it:
1. There are laws of logic, e.g., the law of non-contradiction.
2. The laws of logic are truths.
3. The laws of logic are necessary truths.
4. A truth is a true proposition, where propositions are the primary truth-bearers or primary vehicles of the truth values.
5. Propositions exist. Argument: there are truths (from 1, 2); a truth is a true proposition (3); if an item has a property such as the property of being true, then it exists. Ergo, propositions exist.
6. Necessarily true propositions necessarily exist. For if a proposition has the property of being true in every possible world, then it exists in every possible world. Remark: in play here are 'Fregean' as opposed to 'Russellian' propositions. See here for an explanation of the distinction as I see it. If the proposition expressed by 'Socrates is Socrates' is Russellian, then it has Socrates himself, warts and all, as a constituent. But then, though the proposition is in some sense necessarily true, being a truth of logic, it is surely not necessarily existent.
7. Propositions are not physical entities. This is because no physical entity such as a string of marks on paper could be a primary truth-bearer. A string of marks, if true, is true only derivatively or secondarily, only insofar as as it expresses a proposition.
8. Propositions are intrinsically intentional. (This is explained in the post which is the warm-up to the present one.)
9. The laws of logic are necessarily existent, nonphysical, intrinsically intentional entities.
10. Thoughts are intrinsically intentional.
The argument now takes a very interesting turn. If propositions are intrinsically intentional, and thoughts are as well, might it be that propositions are thoughts?
The following invalid syllogism must be avoided: "Every proposition is intrinsically intentional; every thought is intrinsically intentional; ergo, every proposition is a thought." This argument is an instance of the fallacy of undistributed middle, and of course the authors argue in no such way. They instead raise the question whether it is parsimonious to admit into our ontology two distinct categories of intrinsically intentional item, one mental, the other non-mental. Their claim is that the principle of parsimony "demands" that propositions be constued as mental items, as thoughts. Therefore
11. Propositions are thoughts.
12. Some propositions (the law of logic among them) are necessarily existent thoughts. (From 8, 9, 10, 11)
13. Necessarily, thoughts are thoughts of a thinker.
14. The laws of logic are the thoughts of a necessarily existent thinker, and "this all men call God." (Aquinas)
A Stab at Critique
Line (11) is the crucial sub-conclusion. The whole argument hinges on it. Changing the metaphor, here is where I insert my critical blade, and take my stab. I count three views.
A. There are propositions and there are thoughts and both are intrinsically intentional.
B. Propositions reduce to thoughts.
C. Thoughts reduce to propositions.
Now do considerations of parsimony speak against (A)? We are enjoined not to multiply entities (or rather types of entity) praeter necessitatem. That is, we ought not posit more types of entity than we need for explanatory purposes. This is not the same as saying that we ought to prefer ontologies with fewer categories. Suppose we are comparing an n category ontology with an n + 1 category ontology. Parsimony does not instruct us to take the n category ontology. It instructs us to take the n category ontology only if it is explanatorily adequate, only if it explains all the relevant data but without the additional posit. Well, do we need propositions in addition to thoughts for explanatory purposes? It is plausible to say yes because there are (infinitely) many propositions that no one has ever thought of or about. Arithmetic alone supplies plenty of examples. Of course, if God exists, then there are no unthought propositions. But the existence of God is precisely what is at issue. So we cannot assume it. But if we don't assume it, then we have a pretty good reason to distinguish propositions and thoughts as two different sorts of intrinsically intentional entity given that we already have reason to posit thoughts and propositions.
So my first critical point is that the principle of parsimony is too frail a reed with which to support the reduction of propositions to thoughts. Parsimony needs to be beefed-up with other considerations, e.g., an argument to show why an abstract object could not be intrinsically intentional.
My second critical point is this. Why not countenance (C), the reduction of thoughts to propositions? It could be like this. There are all the (Fregean) propostions there might have been, hanging out in Frege's Third Reich (Popper's world 3). The thought that 7 + 5 = 12 is not a state of an individul thinker; there are no individual thinkers, no selves, no egos. The thought is just the Fregean proposition's temporary and contingent exemplification of the monadic property, Pre-Personal Awareness or Bewusst-sein. Now I don't have time to develop this suggestion which has elements of Natorp and Butchvarov, and in any case it is not my view.
All I am saying is that (C) needs excluding. Otherwise we don't have a good reason to plump for (B).
My conclusion? The Anderson-Welty argument, though fascinating and competently articulated, is not rationally compelling. Rationally acceptable, but not rationally compelling. Acceptable, because the premises are plausible and the reasoning is correct. Not compelling, because one could resist it without quitting the precincts of reasonableness.
To theists, I say: go on being theists. You are better off being a theist than not being one. Your position is rationally defensible and the alternatives are rationally rejectable. But don't fancy that you can prove the existence of God or the opposite. In the end you must decide how you will live and what you will believe.
Addendum (26 February): Steven comments, "I have my doubts about "crap" meaning "anything." I think it means "nothing", but appears in acceptable double-negative propositions which, because of widespread colloquial usage. The evidence I bring forth is the following. "You've done shit to help us" means "You've done nothing to help us," not "You've done anything to help us."
BV: I see the point and it is plausible. But this is also heard: 'You haven't done shit to help us.' I take that as evidence that 'shit' can be used to mean 'anything.' Steven would read the example as a double-negative construction in which 'shit' means 'nothing.' I see no way to decide between my reading and his.
Either way, it is curious that there are quantificational uses of 'shit,' 'crap,' etc!
In the chapter "Atheism as a Purification" in Gravity and Grace (Routledge 1995, tr. Emma Craufurd from the French, first pub. in 1947), the first entry reads as follows:
A case of contradictories which are true. God exists: God does not exist. Where is the problem? I am quite sure that there is a God in the sense that I am quite sure that my love is not illusory. I am quite sure that there is not a God in the sense that I am quite sure nothing real can be anything like what I am able to conceive when I pronounce this word. But that which I cannot conceive is not an illusion. (103)
What are we to make of writing like this? Contradictories cannot both be true and they cannot both be false. By their surface structure, God exists and God does not exist are contradictories. So, obviously, they cannot both be true if taken at face value.
Faced with an apparent contradiction, the time-tested method for relieving the tension is by making a distinction, thereby showing that the apparent contradiction is merely apparent. Suppose we distinguish, as we must in any case, between the concept God and God. Obviously, God is not a concept. This is true even if God does not exist. Interestingly, the truth that God is not a concept is itself a conceptual truth, one that we can know to be true by mere analysis of the concept God. For what we mean by 'God' is precisely a being that does not, like a concept, depend on the possibility or actuality of our mental operations, a being that exists in sublime independence of finite mind.
Now consider these translations:
God does not exist: Nothing in reality falls under the concept God.
God exists: There is an inconceivable reality, God, and it is the target of non-illusory love.
These translations seem to dispose of the contradiction. One is not saying of one and the same thing, God, that he both exists and does not exist; one is saying of a concept that it is not instantiated and of a non-concept that it is inconceivable. That is not a contradiction, or at least not an explicit contradiction. Weil's thesis is that there is a divine reality, but it is inconceivable by us. She is saying that access to the divine reality is possible through love, but not via the discursive intellect. There is an inconceivable reality.
Analogy: just as there are nonsensible realities, there are inconceivable realities. Just as there are realities beyond the reach of the outer senses (however extended via microscopes, etc.), there is a reality beyond the reach of the discursive intellect. Why not?
An objection readily suggests itself:
If you say that God is inconceivable, then you are conceiving God as inconceivable. If you say that nothing can be said about him, then you say something about him, namely, that nothing can be said about him. If you say that there exists an inconceivable reality, then that is different from saying that there does not exist such a reality; hence you are conceiving the inconceivable reality as included in what there is. If you say that God is real, then you are conceiving him as real as opposed to illusory. Long story short, you are contradicting yourself when you claim that there is an inconceivable reality or that God is an inconceivable reality, or that God is utterly beyond all of our concepts, or that no predications of him are true, or that he exists but has no attributes, or that he is real but inconceivable.
The gist of the objection is that my translation defense of Weil is itself contradictory: I defuse the initial contradiction but only by embracing others.
Should we concede defeat and conclude that Weil's position is incoherent and to be rejected because it is incoherent?
Not so fast. The objection is made on the discursive plane and presupposes the non-negotiable and ultimate validity of discursive reason. The objection is valid only if discursive reason is 'valid' as the ultimate approach to reality. So there is a sense in which the objection begs the question, the question of the ultimate validity of the discursive intellect. Weil's intention, however, is to break through the discursive plane. It is therefore no surprise that 'There is an inconceivable reality' is self-contradictory. It is -- but that is no objection to it unless one presupposes the ultimate validity of discursive reason and the Law of Non-Contradiction.
Mystic and logician seem to be at loggerheads.
Mystic: "There is a transdiscursive, inconceivable reality."
Logician: "To claim as much is to embroil yourself in various contradictions."
Mystic: "Yes, but so what?"
Logician: "So what?! That which is or entails a contradiction cannot exist! Absolutely everything is subject to LNC."
Mystic: "You're begging the question against me. You are simply denying what I am asserting, namely, that there is something that is not subject to LNC. Besides, how do you know that LNC is a law of all reality and not merely a law of your discursive thinking? What makes your thinking legislative as to the real and the unreal?"
Logician: "But doesn't it bother you that the very assertions you make, and must make if you are verbally to communicate your view, entail logical contradictions?"
Mystic: "No. That bothers you because you assume the ultimate and non-negotiable validity of the discursive intellect. It doesn't both me because, while I respect the discursive intellect when confined to its proper sphere, I do not imperialistically proclaim it to be legislative for the whole of reality. You go beyond logic proper when you make the metaphysical claim that all of reality is subject to LNC. How are you going to justify that metaphysical leap in a non-circular way?"
Logician: "It looks like we are at an impasse."
Mystic: "Indeed we are. To proceed further you must stop thinking and see!"
How then interpret the Weilian sayings? What Weil is saying is logically nonsense, but important nonsense. It is nonsense in the way that a Zen koan is nonsense. One does not solve a koan by making distinctions, distinctions that presuppose the validity of the Faculty of Distinctions, the discursive intellect; one solves a koan by "breaking through to the other side." Mystical experience is the solution to a koan. Visio intellectualis, not more ratiocination.
A telling phrase from GG 210: "The void which we grasp with the pincers of contradiction . . . ."
But of course my writing and thinking is an operating upon the discursive plane. Mystical philosophy is not mysticism. It is, at best, the discursive propadeutic thereto. One question is whether one can maintain logical coherence by the canons of the discursive plane while introducing the possibility of its transcendence.
Or looking at it the other way round: can the committed and dogmatic discursivist secure his position without simply assuming, groundlessly, its ultimate and non-negotiable validity -- in which event he has not secured it? And if he has not secured it, why is it binding upon us -- by his own lights?
My tendency has long been to use 'reification' and 'hypostatization' interchangeably. But a remark by E. J. Lowe has caused me to see the error of my ways. He writes, "Reification is not the same as hypostatisation, but is merely the acknowledgement of some putative entity's real existence." ("Essence and Ontology," in Novak et al. eds, Metaphysics: Aristotelian, Analytic, Scholastic, Ontos Verlag, 2012, p. 95) I agree with the first half of Lowe's sentence, but not the second.
Lowe's is a good distinction and I take it on board. I will explain it in my own way. Something can be real without being a substance, without being an entity logically capable of independent existence. An accident, for example, is real but is not a substance. 'Real' from L. res, rei. Same goes for the form of a hylomorphic compound. A statue is a substance but its form, though real, is not. The smile on a face and the bulge in a carpet are both real but incapable of independent existence. So reification is not the same as hypostatization. To consider or treat x as real is not thereby to consider or treat x as a substance.
Lowe seems to ignore that 'reification' and 'hypostatization' name logico-philosophical fallacies, where a fallacy is a typical mistake in reasoning, one that occurs often enough and is seductive enough to be given a label. On this point I diverge from him. For me, reification is the illict imputation of ontological status to something that does not have such status. For example, to treat 'nothing' as a name for something is to reify nothing. If I say that nothing is in the drawer I am not naming something that is in the drawer. Nothing is precisely no thing. As I see it, reification is not acknowledgment of real existence, but an illict imputation of real existence to something that lacks it. I do not reify the bulge in a carpet when I acknowledge its reality.
Or consider the internal relation being the same color as. If two balls are (the same shade of) red, then they stand in this relation to each other. But this relation is an "ontological free lunch" not "an addition to being" to borrow some phaseology from David Armstrong. Internal relations have no ontological status. They reduce to their monadic foundations. The putatively relational fact Rab reduces to the conjunction of two monadic facts: Fa & Fb. To bring it about that two balls are the same color as each other it suffices that I paint them both red (or blue, etc.) I needn't do anything else. If this is right, then to treat internal relations as real is to commit the fallacy of reification. Presumably someone who reifies internal relations will not be tempted to hypostatize them.
To treat external relations as real, however, is not to reify them. On my use of terms, one cannot reify what is already real, any more than one can politicize what is already political. To bring it about that two red balls are two feet from each other, it does not suffice that I create two red balls: I must place them two feet from each other. The relation of being two feet from is therefore real, though presumably not a substance.
To hypostatize is is to treat as a substance what is not a substance. So the relation I just mentioned would be hypostatized were one to consider it as an entity capable of existing even if it didn't relate anything. Liberals who blame society for crime are often guilty of the fallacy of hypostatization. Society, though real, is not a substance, let alone an agent to which blame can be imputed.
If I am right then this is mistaken:
First, I have given good reasons for distinguishing the two terms. Second, the mistake of treating what is abstract as material is not the same as reification or hypostatization. For example, if someone were to regard the null set as a material thing, he would be making a mistake, but he would not be reifying or hypostatizing the the null set unless there were no null set.
Or consider the proposition expressed by 'Snow is white' and 'Schnee ist weiss.' This proposition is an abstact object. If one were to regardit as a material thing one would be making a mistake, but one would not be reifying it because it is already real. Nor would one be hypostatizing it since (arguably) it exists independently.
Chapter III of Etienne Gilson's Being and Some Philosophers is highly relevant to my ongoing discussion of common natures. Gilson appears to endorse the classic argument for the doctrine of common natures in the following passage (for the larger context see here):
Out of itself, animal is neither universal nor singular. Indeed, if, out of itself, it were universal, so that animality were universal qua animality, there could be no singular animal, but each and every animal would be a universal. If, on the contrary, animal were singular qua animal, there could be no more than a single animal, namely, the very singular to which animality belongs, and no other singular could be an animal. (77)
This passage contains two subarguments. We will have more than enough on our plates if we consider just the first. The first subargument, telescoped in the second sentence above, can be put as follows:
1. If animal has the property of being universal, then every animal would be a universal. But:
2. It is not the case that every animal is a universal. Therefore:
3. It is not the case that animal has the property of being universal.
This argument is valid in point of logical form, but are its premises true? Well, (2) is obviously true, but why should anyone think that (1) is true? It is surely not obvious that the properties of a nature must also be properties of the individuals of that nature.
There are two ways a nature N could have a property P. N could have P by including P within its quidditative content, or N could have P by instantiating P. There is having by inclusion and having by instantiation.
For example, 'Man is rational' on a charitable reading states that rationality is included within the content of the nature humanity. This implies that everything that falls under man falls under rational. Charitably interpreted, the sentence does not state that the nature humanity or the species man is rational. For no nature, as such, is capable of reasoning. It is the specimens of the species who are rational, not the species.
This shows that we must distinguish between inclusion and instantiation. Man includes rational; man does not instantiate rational.
Compare 'Man is rational' with 'Socrates is rational.' They are both true, but only if 'is' is taken to express different relatons in the two sentences. In the first it expresses inclusion; in the second, instantiation. The nature man does not instantiate rationality; it includes it. Socrates does not include rationality; he instantiates it.
The reason I balk at premise (1) is because it seems quite obviously to trade on a confusion of the two senses of 'is' lately distinguished. It confuses inclusion with instantiation. (1) encapuslates a non sequitur. It does not follow from a nature's being universal that everything having that nature is a universal. That every animal would be a universal would follow from humanity's being universal only if universality were included in humanity. But it is not: humanity instantiates universality. In Frege's jargon, universality is an Eigenschaft of humanity, not a Merkmal of it.
Since the first subargument fails, there is no need to examine the second. For if the first subargment fails, then the whole Avicennian-Thomist argument fails.
Intuitively, if something is identical to Venus, it follows that something is identical to something. In the notation of MPL, the following is a correct application of the inference rule, Existential Generalization (EG):
1. (∃x)(x = Venus) 2. (∃y)(∃x)(x = y) 1, EG
(1) is contingently true: true, but possibly false. (2), however, is necessarily true. Ought we find this puzzling? That is one question. Now consider the negative existential, 'Vulcan does not exist.'
3. ~(∃x)( x = Vulcan) 4. (∃y)~(∃x)(x = y) 3, EG
(3) is contingently true while (4) is a logical contradiction, hence necessarily false. The inference is obviously invalid, having taken us from truth to falsehood. What went wrong?
Diagnosis A: "You can't existentially generalize on a vacuous term, and 'Vulcan' is a vacuous term."
The problem with this diagnosis is that whether a term is vacuous or not is an extralogical (extrasyntactic) question. Let 'a' be an arbitrary constant, and thus neither a place-holder nor a variable. Now if we substitute 'a' for 'Vulcan' we get:
3* ~(∃x)( x = a) 4. (∃y)~(∃x)(x = y) 3*, EG
The problem with this inference is with the conclusion: we don't know whether 'a' is vacuous or not. So I suggest
Diagnosis B: Singular existentials cannot be translated using the identity sign as in (1) and (3). This fact, pace van Inwagen, forces us to beat a retreat to the second-level analysis. We have to analyze 'Venus exists' in terms of
where 'V' is a predicate constant standing for the haecceity property, Venusity. Accordingly, what (5) says is that Venusity is instantiated. Similarly, 'Vulcan does not exist' has to be interpreted as saying that Vulcanity is not instantiated. Thus
where 'W' is a predicate constant denoting Vulcanity.
It is worth noting that we can existentially generalize (6) without reaching the absurdity of (4) by shifting to second-order logic and quantifying over properties:
That says that some property is such that it is not instantiated. There is nothing self-contradictory about (7).
But of course beating a retreat to the second-level analysis brings back the old problem of haecceities. Not to mention the circularity problem.
The thin theory is 'cooked' no matter how you twist and turn.
That puts me in mind of the old idea of John Stuart Mill and others that the laws of logic are empirical generalizations from what we do and do not perceive. Thus we never perceive rain and its absence in the same place at the same time. The temptation is to construe such logic laws as the Law of Non-Contradiction -- ~(p & ~p) -- as generalizations from psychological facts like these. If this is right, then logical laws lack the a priori character and epistemic ‘dignity’ that some of us are wont to see in them. They rest on psychological facts that might have been otherwise and that are known a posteriori.
London Ed might want consider this reductio ad absurdum:
1. The laws of logic are empirical generalizations. (Assumption for reductio) 2. Empirical generalizations, if true, are merely contingently true. (By definition of ‘empirical generalization’: empirical generalizations record what happens to be the case, but might not have been the case.) Therefore, 3. The laws of logic, if true, are merely contingently true. (From 1 and 2) 4. If proposition p is contingently true, then it is possible that p be false. (Def. of ‘contingently true.’)Therefore, 5. The laws of logic, if true, are possibly false. (From 3 and 4)Therefore, 6. LNC is possibly false: there are logically possible worlds in which ‘p&~p’ is true. (From 5 and the fact that LNC is a law of logic.) 7. But (6) is absurd (self-contradictory): it amounts to saying that it is logically possible that the very criterion of logical possibility, namely LNC, be false. Corollary: if laws of logic were empirical generalizations, we would be incapable of defining ‘empirical generalization’: this definition requires the notion of what is the case but (logically) might not have been the case.
In my earlier posts on this topic here and here I did not analyze an example. I make good that deficit now.
Suppose a person asserts that abortion is morally wrong. Insofar forth, a bare assertion which is likely to elicit the bare counter-assertion, 'Abortion is not morally wrong.' What can be gratuitously asserted may be gratuitously denied without breach of logical propriety, a maxim long enshrined in the Latin tag Quod gratis asseritur, gratis negatur. So one reasonably demands arguments from those who make assertions. Here is one:
Infanticide is morally wrong There is no morally relevant difference between abortion and infanticide Ergo Abortion is morally wrong.
Someone who forwards this argument in a concrete dialectical situation in which he is attempting to persuade himself or another asserts the premises and in so doing provides reasons for accepting the conclusion. This goes some distance toward removing the gratuitousness of the conclusion. But what about the premises? If they are mere assertions, then the conclusion, though proximately non-gratuitous (because supported by reasons), is not ultimately non-gratuitous (because no support has been provided for the premises).
Of course, it is better to give the above argument than merely to assert its conclusion. The point of the original post, however, is that one has not escaped from the realm of assertion by giving an argument. And this for the simple reason that (a) arguments have premises, and (b) arguments that do dialectical work must have one or more asserted premises, the assertions being made by the person forwarding the argument with the intention of rationally persuading himself or another of something.
Our old friend Lukas Novak proposes a counterexample to (b): the reductio ad absurdum (RAA)argument. If I understand him, what Novak is proposing is that some such arguments can be used to rationally justify the assertion of the conclusion without any of the premises being asserted by the producer of the argument. Suppose argument A with conclusion C has premises P1, P2, P3. Suppose further that the premise set entails a contradiction. We may then validly conclude and indeed assert that either P1 is not true or P2 is not true or P3 is not true. We may in other words make a disjunctive assertion, an assertion the content of which is a disjunctive proposition. And this without having asserted P1 or P2 or P3. What we have, then, is an argument with an asserted conclusion but no asserted promises.
I think Professor Novak is technically correct except that the sort of RAA argument he describes is not very interesting. Suppose the asserted conclusion is this: Either the null set is not empty, or the null set is not a set, or the Axiom of Extensionality does not hold, or the null set is not unique. Who would want to assert that disjunctive monstrosity? An interesting RAA argument with this subject matter would establish the uniqueness of the null set on the basis of several asserted premises and one unasserted premise, namely, The null set is not unique, the premise assumed for reductio.
So I stick to my guns: 'real life' arguments that do dialectical work must have one or more asserted premises. Novak's comment did, however, give me the insight that not every premise of a 'real life' dialectically efficacious argument must be asserted.
Now back to the abortion argument. My point, again, is that providing even a sound argument for a conclusion -- and I would say that the above argument is sound, i.e., valid in point of logical form and having true premises -- does not free one from the need to make assertions. For example, one has to assert that infanticide is morally wrong. But if no ground or grounds can be given for this assertion, then the assertion is gratuitous. To remove the gratuitousness one can give a further argument: The killing of innocent human beings is morally wrong; (human) infants are innocent human beings; ergo, etc. The first premise in this second argument is again an assertion, and so on.
Eventually we come to assertions that cannot be argued. That is not to say that these assertions lack support. They are perhaps grounded in objective self-evidence.
Note that I am not endorsing what is sometimes called the Münchhausen trilemma, also and perhaps better known as Agrippa's Trilemma, according to which a putative justification either
a. Begets an infinite regress, or b. Moves in a circle, or c. Ends in dogmatism, e.g., in an appeal to self-evidence that can only be subjective, or in an appeal to authority.
All I am maintaining -- and to some this may sound trivial -- is that every real-life argument that does dialectical work must have one or more asserted premises. And so while argument is in general superior to bare assertion, argument does not free us of the need to make assertions. I insist on this so that we do not make the mistake of overvaluing argumentation.
To put it aphoristically, the mind's discursivity needs for its nourishment intuitive inputs that must be affirmed but cannot be discursively justified.
It is one thing to abbreviate an argument, another to depict its logical form. Let us consider the following argument composed in what might be called 'canonical English':
1. If God created some contingent beings, then he created all contingent beings. 2. God created all contingent beings. ----- 3. God created some contingent beings.
The above is an argument, not an argument-form. The following abbreviation of the argument is also an argument, not an argument-form:
1. P --> Q 2. Q --- 3. P
Both are arguments; it is just that the second is an abbreviation of the first in which sentences are replaced with upper-case letters and the logical words with symbols from the propositional calculus. But it is easy to confuse the second argument with the following argument-form:
1. p --> q 2. q --- 3. p
An argument-form is a one-over-many: many arguments can have the same form. And the same goes for its constituent propositional forms: each is a one-over-many. 'p --> q' is the form of indefinitely many conditional statements. But an argument, whether spelled out or abbreviated, is a particular, and as such uninstantiable. One cannot substitute different statements for the upper-case 'P' and 'Q' above.
Some of you will call this hair-splitting. But I prefer to think of it as a distinction essential to clear thinking in logic. For suppose you confuse the second two schemata. Then you might think that the original argument, the one in 'canonical English,' is an instance of the formal fallacy of Affirming the Consequent. But the second schema, though it is an instance of the third, is also an instance of a valid argument-form:
(x)(Cgx) --- (Ex)(Cgx).
In sum, the confusion of abbreviations with place-holders aids and abets the mistake of thinking that an argument that instantiates an invalid form is invalid. Validity and invalidity are asymmetrical: if an argument instantiates a valid form, then it is valid; but if it instantiates an invalid form, then it may or may not be invalid.
(1) An assertion is a mere assertion unless argued. (2) Mere assertions are gratuitous. (3) The premises of arguments are assertions. (4) One cannot argue for every premise of every argument.
This is an accurate summary except for (3). I did not say that the premises of arguments are assertions since I allow that the premises of an argument may be unasserted propositions. The constituent propositions of arguments considered in abstracto, as they are considered in formal logic, as opposed to arguments used in concrete dialectical situations to convince oneself or someone else of something, are typically unasserted.
Since the conclusion of an argument cannot be any stronger (or less gratuitous) than its premises, doesn't it follow from these claims that the conclusion of every argument is gratuitous?
Well, if the conclusion follows from the premises, then it has the support of those premises, and is insofar forth less gratuitous than they are. Your point is better put by saying that, if the premises are gratuitious, then the conclusion canot be ultimately non-gratuitous, but only proximately non-gratuitous.
You distinguish between 'making' arguments and 'entertaining' arguments, but that doesn't offer a way out here because the kind of argument required in (1) and (3) is a 'made' argument rather than an 'entertained' argument.
Isn't the answer here to reject (1) and to grant that some assertions (e.g., the assertion that your cats are on the desk) can be neither mere assertions nor argued assertions? We need a category like 'justified' assertions: no justified assertion is a mere assertion and not every justified assertion is an argued assertion.
Professor Anderson has put his finger on a real problem with the post, and I accept his criticism. I began the post with the sentence, "Mere assertions remain gratuitous until supported by arguments." But that is not quite right. I should have written: "Mere assertions remain gratuitous until supported, either by argument, or in some other way." Thus my assertion that two black cats are lounging on my writing table is not a mere assertion although it is and must be unargued; it is an assertion justified by sense perception.
Expressed more clearly, the main point of the post was that ultimate justification via argument alone cannot be had. Sooner or late one must have recourse to propositions unsupportable by argument. Argument does not free us of the need to make assertions. (I am assuming that there is no such thing as infinitely regressive support or circular support. Not perfectly obvious, I grant: but very plausible.)
Mere assertions remain gratuitous until supported by arguments. Quod gratis asseritur, gratis negatur. That which is gratuitously assertible is gratuitously deniable. Thus one is right to demand arguments from those who make assertions. It is worth pointing out, however, that the difference between making an assertion and giving an argument is not absolute. Since no argument can prove its own premises, they must remain mere assertions from within the context of the argument. No doubt they too can be supported by further arguments, but eventually one comes to ultimate premises that can only be asserted, not argued.
Argument cannot free us of assertion since every argument has premises and they must be asserted if one is making an argument as opposed to merely entertaining one. One who makes an argument is not merely asserting its conclusion; he is asserting its conclusion on the basis of premises that function as reasons for the assertion; and yet the premises themselves are merely asserted. There is no escaping the need to make assertions.
If you refuse to accept ultimate premisses, then you are bound for a vicious infinite regress or a vicious circle, between which there is nothing to choose. (The viciousness of a logical circle is not mitigated by increasing its 'diameter.') This shows the limited value of argument and discursive rationality. One cannot avoid the immediate taking of something for true. For example, I immediately take it to be true, on the basis of sense perception, that a couple of black cats are lounging on my desk:
This is a query which I hope you can answer. Is there such a distinction as 'logical contingency' vs 'metaphysical contingency', and 'logical necessity' vs 'metaphysical necessity'? And if there is, can you explain it? Thank you.
A short answer first. Yes, there are these distinctions. They amount to a distinction between logical modality and metaphysical modality. The first is also called called narrowly logical modality while the second is also called broadly logical modality. Both contrast with nomological modality.
Now a long answer. The following nine paragraphs unpack the notion of broadly logical or metaphysical modality and contrast it with narrowly logical modality.
1. There are objects and states of affairs and propositions that can be known a priori to be impossible because they violate the Law of Non-Contradiction (LNC). Thus a plane figure that is both round and not round at the same time, in the same respect, and in the same sense of 'round,' is impossible, absolutely impossible, simply in virtue of its violation of LNC. I will say that such an object is narrowly logically (NL) impossible. Hereafter, to save keystrokes, I will not mention the 'same time, same respect, same sense' qualification which will be understood to be in force.
2. But what about a plane figure that is both round and square? Is it NL-impossible? No. For by logic alone one cannot know it to be impossible. One needs a supplementary premise, the necessary truth grounded in the meanings of 'round' and 'square' that nothing that is round is square. We say, therefore, that the round square is broadly logically (BL) impossible. It is not excluded from the realm of the possible by logic alone, which is purely formal, but by logic plus a 'material' truth, namely the necessary truth just mentioned.
3. If there are BL-impossible states of affairs such as There being a round square, then there are BL-necessary states of affairs such as There being no round square. Impossibility and necessity are interdefinable: a state of affairs is necessary iff its negation is impossible. It doesn't matter whether the modality is NL, BL, or nomological (physical). It is clear, then, that there are BL-impossible and BL-necessary states of affairs.
4. We can now introduce the term 'BL-noncontingent' to cover the BL-impossible and the BL-necessary.
5. What is not noncontingent is contingent. (Surprise!) The contingent is that which is possible but not necessary. Thus a contingent proposition is one that is possibly true but not necessarily true, and a contingent state of affairs is one that possibly obtains but does not necessarily obtain. We can also say that a contingent proposition is one that is possibly true and such that its negation is possibly true. The BL-contingent is therefore that which is BL-possible and such that its negation is BL-possible.
6. Whatever is NL or BL or nomologically impossible, is impossible period. If an object, state of affairs, or proposition is excluded from the realm of possible being, possible obtaining, or possible truth by logic alone, logic plus necessary semantic truths, or the (BL-contingent) laws of nature, then that object, state of affairs or proposition is impossible, period, or impossible simpliciter.
7. Now comes something interesting and important. The NL or BL or nomologically possible may or may not be possible, period. For example, it is NL-possible that there be a round square, but not possible, period. It is BL-possible that some man run a 2-minute mile but not possible, period. And it is nomologically possible that I run a 4-minute mile, but not possible period. (I.e., the (BL-contingent) laws of anatomy and physiology do not bar me from running a 4-minute mile; it is peculiarities not referred to by these laws that bar me. Alas, alack, there is no law of nature that names BV.)
8. What #7 implies is that NL, BL, and nomological possibility are not species or kinds of possibility. If they were kinds of possibility then every item that came under one of these heads would be possible simpliciter, which we have just seen is not the case. A linguistic way of putting the point is by saying that 'NL,' 'BL,' and 'nomological' are alienans as opposed to specifying adjectives: they shift or 'alienate' ('other') the sense of the noun they modify. From the fact that x is NL or BL or nomologically possible, it does not follow that x is possible. This contrasts with impossibility. From the fact that x is NL or BL or nomologically impossible, it does follow that x is impossible. Accordingly, 'NL,' 'BL,' and 'nomological' do not shift or alienate the sense of 'impossible.'
9. To appreciate the foregoing, you must not confuse senses and kinds. 'Sense' is a semantic term; 'kind' is ontological. From the fact that 'possible' has several senses, it does not follow that there are several species or kinds of possibility. For x to be possible it must satisfy NL, BL, and nomological constraints; but this is not to say that these terms refer to species or kinds of possibility.
Fred Sommers' "Intellectual Autobiography" begins as follows:
I did an undergraduate major in mathematics at Yeshiva College and went on to graduate studies in philosophy at Columbia University in the 1950s. There I found that classical philosophical problems were studied as intellectual history and not as problems to be solved. That was disappointing but did not strike me as unreasonable; it seemed to me that tackling something like "the problem of free will" or "the problem of knowledge" could take up one's whole life and yield little of permanent value. I duly did a dissertation on Whitehead's process philosophy and was offered a teaching position at Columbia College. Thereafter I was free to do philosophical research of my own choosing. My instinct was to avoid the seductive, deep problems and to focus on finite projects that looked amenable to solution. (The Old New Logic: Essays on the Philosophy of Fred Sommers, ed. Oderberg, MIT Press, 2005, p. 1)
Sommers says something similar in the preface to his The Logic of Natural Language (Oxford, 1982), p. xii:
My interest in Ryle's 'category mistakes' turned me away from the study of Whitehead's metaphysical writings (on which I had written a doctoral thesis at Columbia University) to the study of problems that could be arranged for possible solution.
What interests me in these two passages is the reason that Sommers gives for turning away from the big 'existential' questions of philosophy (God, freedom, immortality, and the like) to the problems of logical theory. I cannot see that it is a good reason. (And he does seem to be giving a reason and not merely recording a turn in his career.)
The reason is that the problems of logic, but not those of metaphysics, can be "arranged for possible solution." Although I sympathize with Sommers' sentiment, he must surely have noticed that his attempt to rehabilitate pre-Fregean logical theory issues in results that are controversial, and indeed just as controversial as the claims of metaphysicians. Or do all his colleagues in logic agree with him?
The problems that Sommers tackles in his magisterial The Logic of Natural Language are no more amenable to solution than the "deep, seductive" ones that could lead a philosopher astray for a lifetime. The best evidence of this is that Sommers has not convinced his MPL (modern predicate logic) colleagues. At the very most, Sommers has shown that TFL (traditional formal logic) is a defensible rival system.
If by 'pulling in our horns' and confining ourselves to problems of language and logic we were able to attain sure and incontrovertible results, then there might well be justification for setting metaphysics aside and working on problems amenable to solution. But if it turns out that logical, linguistic, phenomenological, epistemological and all other such preliminary inquiries arrive at results that are also widely and vigorously contested, then the advantage of 'pulling in our horns' is lost and we may as well concentrate on the questions that really matter, which are most assuredly not questions of logic and language — fascinating as these may be.
Given that the "deep, seductive" problems and those of logical theory are in the same boat as regards solubility, Sommer's' reason for devoting himself to logic over the big questions is not a good one. The fact that philosophy of logic is often more rigorous than 'big question' philosophy is not to the point. The distinction between the rigorous and the unrigorous cuts perpendicular to that between the soluble and the insoluble. And in any case, any philosophical problem can be tackled as rigorously as you please.
Sommers' is a rich and fascinating book. But, at the end of the day, how important is it to prove that the inference embedded in 'Some girl is loved by every boy so every boy loves a girl' really is capturable, pace the dogmatic partisans of modern predicate logic, by a refurbished traditional term logic? (See pp. 144-145) As one draws one's last breath, which is more salutary: to be worried about a silly b agatelle such as the one just mentioned, or to be contemplating God and the soul?
And shouldn't we philosophers who are still a ways from our last breaths devote our main energies to such questions as God and the soul over the trifles of logic?
It would be nice if we could set philosophy on the "sure path of science" (Kant) by abandoning metaphysics and focusing on logic (or phenomenology or whatever one considers foundational). But so far, this narrowing of focus and 'pulling in of one's horns' has availed nothing. Philosophical investigation has simply become more technical, labyrinthine, and specialized. All philosophical problems are in the same boat with respect to solubility. A definitive answer to 'Are there atomic propositions?' (LNL, ch. 1) is no more in the offing than a definitive answer to 'Does God exist?' or 'Is the will libertarianly free?'
Ask yourself: what would be more worth knowing if it could be known?
The unduly modest David Brightly has begun a weblog entitled tillyandlola, "scribblings of no consequence." In a recent post he criticizes my analysis of the invalidity of the argument: Man is a species; Socrates is a man; ergo, Socrates is a species. I claimed that the argument equivocates on 'is.' In the major premise, 'is' expresses a relation of conceptual inclusion: the concept man includes the subconcept species. In the minor premise, however, the 'is' is the 'is' of predication: Socrates falls under man, he doesn't fall within it.
I am afraid that my analysis is faulty, however, and for the reasons that David gives. There is of course a difference between the 'is' of inclusion and the 'is' of predication. 'Man is an animal' expresses the inclusion of the concept animal within the concept man. 'Socrates is a man,' however, does something different: it expresses the fact that Socrates falls under the concept man.
But as David notes, it is not clear that species is included within the concept man. If we climb the tree of Porphyry we will ascend from man to mammal to animal; but nowhere in our ascent will we hit upon species.
Philosophers always refer to their arguments as 'arguments' and never as 'proofs'. This is because there is nothing in the entire, nearly three thousand year history of philosophy that would count as a proof of anything. Nothing.
This obiter dictum illustrates how, by exaggerating and saying something that is strictly false, one can still manage to convey a truth. The truth is that there is very little in the history of philosophy that could count as a proof of anything. But of course some philosophers do refer to their arguments as proofs. Think of those Thomists who speak of proofs of the existence of God. And though no Thomist accepts the ontological 'proof,' there are philosophers who refer to the ontological argument as a proof. The Germans also regularly speak of der ontologische Gottesbeweis rather than of das ontologische Argument. For example, Frege in a famous passage from the Philosophy of Arithmetic writes, Weil Existenz Eigenschaft des Begriffes ist, erreicht der ontologische Beweis von der Existenz Gottes sein Ziel nicht. (sec. 53)
These quibbles aside, an argument is not the same as a proof. 'Prove' is a verb of success. The same goes for 'disprove' and 'refute.' But 'argue' is not. I may argue that p without establishing that p. But if I prove that p, then I establish that p. Indeed, I establish it as true.
Why has almost nothing ever been proven in the history of philosophy?
It is because for an argument to count as a proof in philosophy -- I leave aside mathematics which may not be so exacting -- certain exceedingly demanding conditions must be met. First, a proof must be deductive: no inductive argument proves its conclusion. Second, a proof must be valid: it must be a deductive argument such that its corresponding conditional is a narrowly-logical truth, where an argument's corresponding conditional is a conditional proposition the protasis of which is the conjunction of the argument's premises, and the apodosis of which is the argument's conclusion.
Third, although a valid argument needn't have true premises, a proof must have all true premises. In other words, a proof must be a sound argument. Fourth, a proof cannot commit any infomal fallacy such as petitio principii. An argument from p to p is deductive, valid, and sound. But it is obviously no proof of anything.
Fifth, a proof must have premises that are not only true, but known to be true by the producers and the consumers of the argument. This is because a proof is not an argument considered in abstracto but a method for generating knoweldge for some cognizer. For example, if I do not know that I am thinking,then I cannot use that premise in a proof that I exist.
Sixth, a proof in philosophy must have premises all of which are known to be true in a sense of 'know' that entails absolute impossibilty of mistake. Why set the bar so high? Well, if you say that you have proven the nonexistence of God, say, or that the self is but a bundle of perceptions, or that freedom of the will is an illuison, or whatever, and one of your premises is such that I can easily conceive its being false, then you haven't proven anything. You haven't rationally compelled me to accept your conclusion. You may have given a 'good' argument in the sense of a 'reasonable' argument where that is one which satisfies my first four conditions; but you haven't given me a compelling argument, an argument which is such that, were I to reject it I would brand myself as irrational. (Of course the only compulsion here at issue is rational compulsion, not ad baculum (ab baculum?) compulsion.)
Given my exposition of the notion of proof in philosophy, I think it is clear that very little has ever been proven in philosophy. I am pretty sure that London Ed, as cantankerous and contrary as he is known to be, will agree. But he goes further: he says that nothing has ever been proven in philosophy.
But hasn't the sophomoric relativist been refuted? He maintains that it is absolutely true that every truth is relative. Clearly, the sophomoric relativist contradicts himself and refutes himself. One might object to this example by claiming that no philosopher has ever been a sophomoric relativist. But even if that is so, it is a possible philosophical position and one that is provably mistaken. Or so say I.
Or consider a sophist like Daniel Dennet who maintains (in effect) that consciousness is an illusion. That is easily refuted and I have done the job more than once in these pages. But it is such a stupid thesis that it is barely worth refuting. Its negation -- that consciousness is not an illusion -- is hardly a substantive thesis. A substantive thesis would be: Consciousness is not dependent for its existence on any material things or processes.
There is also the stupidity of that fellow Krauss who thinks that nothing is something. Refuting this nonsense hardly earns one a place in the pantheon of philosophers.
Nevertheless, I am in basic agreement with London Ed: Nothing of any real substance has ever been proven in philosophy. No one has ever proven that God exists, that God does not exist, that existence is a second-level property, that there is a self, that there is no self, that the will is free, that the will is not free, and so on.
Or perhaps you think you have a proof of some substantive thesis? Then I'd like to hear it. But it must be a proof in my exacting sense.
I dedicate this post to London Ed, who likes sophisms and scholastic arcana.
Consider these two syllogistic arguments:
A1. Man is an animal; Socrates is a man; ergo, Socrates is an animal. A2. Man is a species; Socrates is a man; ergo, Socrates is a species.
The first argument is valid. On one way of accounting for its validity, we make two assumptions. First, we assume that each of the argument's constituent sentences is a predication. Second, we assume the principle of the Transitivity of Predication: if x is predicable of y, and y is predicable of z, then x is predicable of z. This principle has an Aristotelian pedigree. At Categories 3b5, we read, "For all that is predicated of the predicate will be predicated also of the subject." So if animal is predicable of man, and man of Socrates, then animal of Socrates.
Something goes wrong, however, in the second argument. The question is: what exactly? Let's first of all see if we can diagnose the fallacy while adhering to our two assumptions. Thus we assume that each occurrence of 'is' in (A2) is an 'is' of predication, and that predication is transitive. One suggestion -- and I take this to be the line of some Thomists -- is that (A2) equivocates on 'man.' In the major, 'man' means 'man-in-the-mind,' 'man as existing with esse intentionale.' In the minor, 'man' means 'man-in-reality,' 'man as existing with esse naturale.' We thus diagnose the invalidity of (A2) by saying that it falls afoul of quaternio terminorum, the four-term fallacy. On this diagnosis, Transitivity of Predication is upheld: it is just that in this case the principle does not apply since there are four terms.
But of course there is also the modern Fregean way on which we abandon both of our assumptions and locate the equivocation in (A2) elsewhere. On a Fregean diagnosis, there is an equivocation on 'is' in (A2) as between the 'is' of inclusion and the 'is' of predication. In the major premise, 'is' expresses, not predication, but inclusion: the thought is that the concept man includes within its conceptual content the subconcept species. In the minor and in the conclusion, however, the 'is' expresses predication: the thought is that Socrates falls under the concepts man and species. Accordingly, (A2) is invalid because of an equivocation on 'is,' not because of an equivocation on 'man.'
The Fregean point is that the concept man falls WITHIN but not UNDER the concept animal, while the object Socrates falls UNDER but not WITHIN the concepts man and animal. Man does not fall under animal because no concept is an animal. Animal is a mark (Merkmal) not a property (Eigenschaft) of man. In general, the marks of a concept are not its properties. But concepts do have properties. The property of being instantiated, for example, is a property of the concept man. But it is not a mark of it. If it were a mark, then man by its very nature would be instantiated and it would be a conceptual truth that there are human beings, which is false.
Since on the Fregean scheme the properties of concepts needn't be properties of the items that fall under the concepts, Transitivity of Predication fails. Thus, the property of being instantiated is predicable of the concept philosopher, and the concept philosopher is predicable of Socrates; but the property of being instantiated is not predicable of Socrates.
Here is a passage from Chapter 3 of Thomas Aquinas, On Being and Essence(tr. Robert T. Miller, emphasis added):
The nature, however, or the essence thus understood can be considered in two ways. First, we can consider it according to its proper notion, and this is to consider it absolutely. In this way, nothing is true of the essence except what pertains to it absolutely: thus everything else that may be attributed to it will be attributed falsely. For example, to man, in that which he is a man, pertains animal and rational and the other things that fall in his definition; white or black or whatever else of this kind that is not in the notion of humanity does not pertain to man in that which he is a man. Hence, if it is asked whether this nature, considered in this way, can be said to be one or many, we should concede neither alternative, for both are beyond the concept of humanity, and either may befall the conception of man. If plurality were in the concept of this nature, it could never be one, but nevertheless it is one as it exists in Socrates. Similarly, if unity were in the notion of this nature, then it would be one and the same in Socrates and Plato, and it could not be made many in the many individuals. Second, we can also consider the existence the essence has in this thing or in that: in this way something can be predicated of the essence accidentally by reason of what the essence is in, as when we say that man is white because Socrates is white, although this does not pertain to man in that which he is a man.
What intrigues me about this passage is the following argument that it contains:
1. A nature can be considered absolutely (in the abstract) or according to the being it has in this or that individual. 2. If a nature is considered absolutely, then it is not one. For if oneness were included in the nature of humanity, e.g., then humanity could not exist in many human beings. 3. If a nature is considered absolutely, then it is not many. For if manyness were included in the nature of humanity, e.g., then humanity could not exist in one man, say, Socrates. Therefore 4. If a nature is considered absolutely, then it is neither one nor many, neither singular nor plural.
I find this argument intriguing because I find it extremely hard to evaluate, and because I find the conclusion to be highly counterintuitive. It seems to me obvious that a nature or essence such as humanity is one, not many, and therefore not neither one nor many!
The following is clear. There are many instances of humanity, many human beings. Therefore, there can be many such instances. It follows that there is nothing in the nature of humanity to preclude there being many such instances. But there is also nothing in the nature of humanity to require that there be many instances of humanity, or even one instance. We can express this by saying that the nature humanity neither requires nor precludes its being instantiated. This nature, considered absolutely, logically allows multiple instantiation, single instantiation, and no instantiation. It logically allows that there be many men, just one man, or no men.
But surely it does not follow that the nature humanity is neither one nor many. What Aquinas is doing above is confusing what Frege calls a mark (Merkmal) of a concept with a property (Eigenschaft) of a concept. The marks of a concept are the subconcepts which are included within it. Thus man has animal and rational as marks. But these are not properties of the concept man since no concept is an animal or is rational. Being instantiated is an example of a property of man, a property that cannot be a mark of man. In general, the marks of a concept are not properties thereof, and vice versa. Exercise for the reader: find a counterexample, a concept which is such that one of its marks is also a property of it.
Aquinas has an insight which can be expressed in Fregean jargon as follows. Being singly instantiated -- one in reality -- and being multiply instantiated -- many in reality -- are not marks (Merkmale) of the nature humanity. But because he (along with everyone else prior to 1884) confuses marks with properties (Eigenschaften), he concludes that the nature itself cannot be either one or many.
To put it another way, Aquinas confuses the 'is' of predication ('Socrates is a man') with the 'is' of subordination ('Man is an animal'). Man is predicable of Socrates, but animal is not predicable of man, pace Aristotle, Categories 3b5: no concept or nature is an animal. Socrates falls underman; Animalfalls withinman. Animal is superordinate to man while man is subordinate to animal.
For these reasons I do not find the argument from De Ente et Essentia compelling. But perhaps there is a good Thomist response.
Something that in its very nature is political cannot be politicized. (Example here.) Similarly, that which in its mode of being is substantial cannot be hypostatized or reified. Hypostatization is illicit reification.