"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.
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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.
A
Affirming the Consequent is an invalid argument form. Ergo One ought not (it is obligatory that one not) give arguments having that form.
B
Modus Ponens is valid Ergo One may (it is permissible to) give arguments having that form.
C
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.
REFERENCES
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
and
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.)
Therefore
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.
Therefore
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.
Therefore
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.
'He doesn't know shit, so he doesn't know shit from shinola.' In its first occurrence, 'shit' functions as a logical quantifier; in its second, as a non-logical word, a mass term.
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
5. (∃x)(Vx)
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
6. ~(∃x)(Wx)
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:
7. (∃P)~(∃x)Px.
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.
UPDATE: London Ed does an excellent job of misunderstanding the following post. Bad comments incline me to keep my ComBox closed. But his is open.
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.
A reader asks, "What is meant by 'closure' or 'closed under'? I've heard the terms used in epistemic contexts, but I've not been able to completely understand them."
Let's start with some mathematical examples. The natural numbers are closed under the operation of addition. This means that the result of adding any two natural numbers is a natural number. What is a natural number? On one understanding of the term, the naturals are the positive integers, the counting numbers, the members of the set {1, 2, 3, 4, 5, . . .}. On a second understanding, the naturals are the positive integers and zero: {0, 1, 2, 3, 4 . . .}. Either way, it is easy to see that adding any two elements of either set yields an element of the same set. It is also easy to see that the naturals are also closed under multiplication. But they are not closed under subtraction. If you subtract 9 from 7, the result (-2) is not an element of the set of natural numbers.
Now consider the squaring operation. The square of any real number is a real number. So the reals are closed under the operation of squaring. But the reals are not closed under the square root operation. The square root of -4 cannot be 2 since 2 squared is 4; it cannot be -2 either since -2 squared is 4. The square root of -2 is the complex number 2i where the imaginary number i is the square root of -1. The square roots of negative numbers are complex; hence, the reals are not closed under the square root operation.
Generalizing, we can say that a set S is closed under a binary operation O just in case, for any elements x and y in S, xOy is an element of S. In Group Theory, a set S together with an operation O constitutes a group only if S is closed under O.
Now for some philosophical examples. Meinongian objects (M-objects) are not closed under entailment. The M-object, the yellow brick road, although yellow is not colored even though in reality nothing can be yellow without being colored. M-objects are incomplete objects. They have all and only the properties specified in their descriptions. So we say that the properties of M-objects are not closed under property-entailment. Property P entails property Q iff necessarily, if anything x has P, then x has Q.
What goes for M-objects goes for intentional objects. (On my reading of Meinong, an M-object is not the same as an intentional object: there are M-objects that are not the accusatives of any actual intending.) Suppose I am gazing out my window at the purple majesty of Superstition Mountain. The intentional object of my perception has the property of being purple, but not the properties of being colored or being extended even though in reality nothing can be purple without being both colored and extended. Phenomenologically, what is before my mind is an instance of purple, but not an instance of colored item. What I see I see as purple but not as colored.
Now consider: If S knows that p, and S knows that p entails q, then S also knows that q. If you acquiesce in the bolded thesis, then you acquiesce in the closure of 'knows' under known entailment. For what you are then committing yourself to is the proposition that a proposition q entailed by a proposition p you know -- assuming you know that p entails q -- is a member of the set of propositions you know.
Both sentences are true; both are meaningful; and the second follows from the first. How do we translate the argument into the notation of standard first-order predicate logic with identity? Taking a cue from Quine we may formulate (1) as
1*. For some x, x = Stromboli. In English:
1**. Stromboli is identical with something.
But how do we render (2)? Surely not as 'For some x, x exists' since there is no first-level predicate of existence in standard logic. And surely no ordinary predicate will do. Not horse, mammal, animal, living thing, material thing, or any other predicate reachable by climbing the tree of Porphyry. Existence is not a summum genus. (Aristotle, Met. 998b22, AnPr. 92b14) What is left but self-identity? Cf. Frege's dialog with Puenjer.
So we try,
2*. For some x, x = x. In plain English:
2**. Something is self-identical.
So our original argument becomes:
1**. Stromboli is identical with something. Ergo 2**. Something is self-identical.
But what (2**) says is not what (2) says. The result is a murky travesty of the original luminous argument.
What I am getting at is that standard logic cannot state its own presuppositions. It presupposes that everything exists (that there are no nonexistent objects) and that something exists. But it lacks the expressive resources to state these presuppositions. The attempt to state them results either in nonsense -- e.g. 'for some x, x' -- or a proposition other than the one that needs expressing.
It is true that something exists, and I am certain that it is true: it follows immediately from the fact that I exist. But it cannot be said in standard predicate logic.
What should we conclude? That standard logic is defective in its treatment of existence or that there are things that can be SHOWN but not SAID? In April 1914. G.E. Moore travelled to Norway and paid a visit to Wittgenstein where the latter dictated some notes to him. Here is one:
In order that you should have a language which can express or say everything that can be said, this language must have certain properties; and when this is the case, that it has them can no longer be said in that language or any language. (Notebooks 1914-1916, p. 107)
Applied to the present example: A language that can SAY that e.g. island volcanos exist by saying that some islands are volcanos or that Stromboli exists by saying that Stromboli is identical to something must have certain properties. One of these is that the domain of quantification contains only existents and no Meinongian nonexistents. But THAT the language has this property cannot be said in it or in any language. Hence it cannot be said in the language of standard logic that the domain of quantification is a domain of existents or that something exists or that everything exists or that it is not the case that something does not exist.
Well then, so much the worse for the language of standard logic! That's one response. But can some other logic do better? Or should we say, with the early Wittgenstein, that there is indeed the Inexpressible, the Unsayable, the Unspeakable, the Mystical? And that it shows itself?
Es gibt allerdings Unaussprechliches. Dies zeigt sich, es ist das Mystische. (Tractatus Logico-Philosphicus 6.522)
Here is another puzzle London Ed may enjoy. Is the following argument valid or invalid:
An island volcano exists. Stromboli is an island volcano. Ergo Stromboli exists.
The argument appears valid, does it not? But it can't be valid if it falls afoul of the dreaded quaternio terminorum, or 'four-term fallacy.' And it looks like it does. On the standard Frege-Russell analysis, 'exists' in the major is a second-level predicate: it predicates of the concept island volcano the property of being instantiated, of having one or more instances. 'Exists' in the conclusion, however, cannot possibly be taken as a second-level predicate: it cannot possibly be taken to predicate instantiation of Stromboli. "Exists' in the conclusion is a first-level predicate. Since 'exists' is used in two different senses, the argument is invalid. And yet it certainly appears valid. How solve this?
(Addendum, Sunday morning: this is not a good example for reasons mentioned in the ComBox. But my second example does the trick.)
The same problem arise with this argument:
Stromboli exists. Stromboli is an island volcano. Ergo An island volcano exists.
This 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'? Please don't say the the first premise is redundant. If Stromboli did not exist, if it were a Meinongian nonexistent object, then Existential Generalization could not be performed, given, as Quine says, that "Existence is what existential quantification expresses."
Ed of Beyond Necessityreports that he has translated some chapters on induction from Ockham's Summa Logicae. He goes on:
Ockham says that induction "is a progression from singulars to the universal", which is pretty much the modern understanding of the term.
That is not wrong, but it is not quite right either. On a well-informed modern understanding induction need not involve "a progression from singulars to the universal."
Suppose that every F I have encountered thus far is a G, and that I conclude that the next F I will encounter will also be a G. This is clearly an inductive inference, but it is one that moves from a universal statement to a statement about an individual. The inference is from Every F thus far encountered is a G to The next F I will encounter will be a G. So it is simply not the case that every inductive inference proceeds from singular cases to a universal conclusion. Some such inferences do, but not all. This is a common misunderstanding.
London Ed of Beyond Necessitydoes a good job patiently explaining the 'morning star' - 'evening star' example to one of his uncomprehending readers. But I don't think Ed gets it exactly right. I quibble with the following:
Summarising: (1) The sentence “the morning star is the evening star” has informational content. (2) The sentence “the morning star is the morning star” does not have informational content. (3) Therefore, the term “the morning star” does not have the same informational content as “the evening star”.
One quibble is this. Granted, the two sentences differ in cognitive value, Erkenntniswert. (See "On Sense and Reference" first paragraph.) The one sentence expresses a truth of logic, and thus a truth knowable a priori. The other sentence expresses a factual truth of astronomy, one knowable only a posteriori. But note that Frege says that they differ in cognitive value, not that the one has it while the other doesn't. Ed says that the one has it while the other doesn't -- assuming Ed is using 'informational content' to translate Erkenntniswert. There is some annoying slippage here.
More importantly, I don't see how cognitive value/informational content can be had by such subsentential items as 'morning star' and 'evening star.' Thus I question the validity of the inference from (1) & (2) to (3). Neither term gives us any information. So it cannot be that they differ in the information they give. Nor can they be contrasted in point of giving or not giving information. Information is conveyable only by sentences or propositions.
I say this: neither of the names Morgenstern (Phosphorus) or Abendstern (Hesperus) have cognitive value or informational content. (The same holds, I think, if they are not proper names but definite descriptions.) Only indicative sentences (Saetze) and the propositions (Gedanken) they express have such value or content. As I see it, for Frege, names have sense (Sinn) and reference (Bedeutung), and they may conjure up subjective ideas (Vorstellungen) in the minds of their users. But no name has cognitive value. Sentences and propositions, however, have sense, reference, and cognitive value. Interestingly, concept-words (Begriffswoerter) or predicates also have sense and reference, but no cognitive value.
I also think Ed misrepresents the Compositionality Principle. Frege is committed to compositionality of sense (Sinn), not compositionality of informational content/cognitive value. So adding the C. P. to his premise set will not validate the above inference.
Although the fallacy of composition is standardly classified as an informal fallacy, I see it is a formal fallacy, one rooted in logical form. Let W be any sort of whole (whether set, mereological sum, aggregate, etc.) Suppose each of the proper parts (if any) of W has some property P (or, for the nominalistically inclined, satisfies some predicate F). Does it follow that W has P or satisfies F? No it doesn't. To think otherwise is to commit the fallacy of composition: it is to argue in accordance with the following invalid schema: 1. Each member of W is F Therefore 2. W is F.
To show that an argument form is invalid, it suffices to present an argment of that form having true premises and a false conclusion. (This is because valid inference is truth-preserving: it cannot take one from true premises to a false conclusion. But it doesn't follow that invalid inference is falsehood-preserving: there are valid arguments with false premises and a true conclusion. Exercise for the reader: give examples.) Here is a counterexample that shows the invalidity of the above pattern: Each word in a given sentence is meaningful; ergo, the sentence is meaningful. (Let the sentence be 'Quadruplicity drinks procrastination.') Since the premise is true and the conclusion false, the argument pattern is invalid. So every argument of that form is invalid, even in the case in which the premises and conclusion are both true.
Why then is Composition standardly grouped with the informal fallacies? Petitio principii is a clear example of an informal fallacy. If I argue p, therefore p, I move in a circle of embarrassingly short diameter. But the inference is valid. (Bear in mind that 'valid' is a terminus technicus.) And if p is true, the argument is sound. Nevertheless, any argument of this form is probatively worthless: it it does not prove, but presupposes, its conclusion. Since this defect is not formal, we call it informal!
So there are clear examples of informal fallacies. But what about Equivocation? It is usually classed with the informal fallacies. Consider the syllogistic form Barbara (AAA-1):
All M are P All S are M All S are P.
Suppose there is an equivocation on the middle term 'M.' Although this is an informal defect (in that it has not to do with logical syntax, but with semantics) it translates into a formal defect, the dreaded quaternio terminorum or four-term fallacy, which is of course a formal fallacy: no syllogism with more than three terms is valid. (A syllogism by definition is a deductive argument having exactly two premises and exactly three terms.)
It can be shown that every equivocation on a key term in an argument induces a formal defect. So the standard classification of Equivocation as an informal fallacy cannot be taken too seriously. By contrast, Petitio Principii is seriously informal in its probative defectiveness.
I say that Composition is like Equivocation: it is a formal fallacy in informal disguise. (And the same goes for Division, which is roughly Composition in reverse.) So I disagree with the author of a logic book who writes:
. . . the fallacy of composition is indeed an informal fallacy. It cannot be discovered by a mere inspection of the form of an argument , that is, by the mere observation that an attribute is being transferred from parts onto the whole. . . . The critic must be certain that, given the situation, the transference of this particular attribute is not allowed. . . .
So the fallacy of composition is not always a fallacy, but only when it is a fallacy? That is the silliness that the author seems to be espousing. He is saying in effect the following: if you transfer an attribute from parts to whole, that is fallacious except in those cases in which it is not fallacious, i.e. those cases in which the transfer can legitimately be made.
But then what is the point of isolating a typical error in reasoning called Composition? What is the point of this label? Why not just say: there are many different part-whole relationships, and it is only be close acquaintance with the actual subject-matter that one can tell whether the attribute transfer is legitimate?
Logic is formal: it abstracts from subject-matter. So mistakes in logic are also formal. A mistake that is typical (recurrent) and sufficiently seductive to warrant a label is called a fallacy. To say or imply that the fallaciousness of a fallacy depends on the particular subject-matter of the argument is to abandon logic and embrace confusion.
Example. Every brick in this pile weighs more than five lbs; ergo, the pile weighs more than five lbs. This is an example of the fallacy of composition despite the fact that it is nomologically impossible that the pile not weigh more than five lbs.
Another example. Every being in the universe is contingent; ergo, the universe is contingent. This too is the fallacy of composition. And this despite the fact that it is metaphysically impossible that a universe all of whose members are contingent be necessary.
'Neologism' is not a new word, but an old word. Hence, 'neologism' is not a neologism. 'Paleologism' is not a word at all; or at least it is not listed in the Oxford English Dictionary. But it ought to be a word, so I hereby introduce it. Who is going to stop me? Having read it and understood it, you have willy-nilly validated its introduction and are complicit with me.
Now that we have 'paleologism' on the table, and an unvast conspiracy going, we are in a position to see that 'neologism' is a paleologism, while 'paleologism' is a neologism. Since the neologism/paleologism classification is both exclusive (every word is either one or the other )and exhaustive (no word is neither), it follows that 'neologism' is not a neologism, and 'paleologism' is not a paleologism.
Such words are called heterological: they are not instances of the properties they express. 'Useless' and 'monosyllabic' are other examples of heterological expressions in that 'useless' is not useless and 'monosyllabic' is not monosyllabic. A term that is not heterological is called autological. Examples include 'short' and 'polysyllabic.' 'Short' is short and 'polysyllabic' is polysyllabic. Autological terms are instances of the properties they express.
Now ask yourself this question: Is 'heterological' heterological? Given that the heterological/autological classification is exhaustive, 'heterological' must be either heterological or else autological. Now if the former, then 'heterological' is not an instance of the property it expresses, namely, the property of not being an instance of the property it expresses. But this implies that 'heterological' is autological. On the other hand, if 'heterological' is autological, then it is an instance of the property it expresses, namely the property of not being an instance of the property it expresses. But this implies that 'heterological' is heterological.
Therefore, 'heterological' is heterological if and only if it is not. This contradiction is known in the trade as Grelling's Paradox. It is named after Kurt Grelling, who presented it in 1908.
Bill Clinton may have brought the matter to national attention, but philosophers have long appreciated that much can ride on what the meaning of 'is' is.
Edward of London has a very good post in which he raises the question whether the standard analytic distinction between the 'is' of identity and the 'is' of predication is but fallout from an antecedent decision to adhere to an absolute distinction between names and predicates. If the distinction is absolute, as Frege and his epigoni maintain, then names cannot occur in predicate position, and a distinction between the two uses of 'is' is the consequence. But what if no such absolute distinction is made? Could one then dispense with the standard analytic distinction? Or are there reasons independent of Frege's function-argument analysis of propositions for upholding the distinction between the two uses of 'is'?
To illustrate the putative distinction, consider
1. George Orwell is Eric Blair
and
2. George Orwell is famous.
Both sentences feature a token of 'is.' Now ask yourself: is 'is' functioning in the same way in both sentences? The standard analytic line is that 'is' functions differently in the two sentences. In (1) it expresses identity; in (2) it expresses predication. Identity, among other features, is symmetrical; predication is not. That suffices to distinguish the two uses of 'is.' 'Famous' is predicable of Orwell, but Orwell is not predicable of 'famous.' But if Blair is Orwell, then Orwell is Blair.
Now it is clear, I think, that if one begins with the absolute name-predicate distinction, then the other distinction is also required. For if 'Eric Blair' in (1) cannot be construed as a predicate, then surely the 'is' in (1) does not express predication. The question I am raising, however, is whether the distinction between the two uses of 'is' arises ONLY IF one distinguishes absolutely and categorially between names and predicates.
Fred Sommers seems to think so. Referencing the example 'The morning star is Venus,' Sommers writes, "Clearly it is only after one has adopted the syntax that prohibits the predication of proper names that one is forced to read 'a is b' dyadically and to see in it a sign of identity." (The Logic of Natural Language, Oxford 1982, p. 121, emphasis added) The contemporary reader will of course wonder how else 'a is b' could be read if it is not read as expressing a dyadic relation between a and b. How the devil could the 'is' in 'a is b' be read as a copula?
This is what throws me about the scholastic stuff peddled by Ed and others. In 'Orwell is famous' they seem to be wanting to say that 'Orwell' and 'famous' refer to the same thing. But what could that mean?
First of all, 'Orwell' and 'famous' do not have the same extension: there are many famous people, but only one Orwell. But even if Orwell were the only famous person, Orwell would not be identical to the only famous person. Necessarily, Orwell is Orwell; but it is not the case that, necessarily, Orwell is the only famous person, even if it is true that Orwell is the only famous person, which he isn't.
If you tell me that only 'Orwell' has a referent, but not 'famous,' then I will reply that that is nominalism for the crazy house. Do you really want to say or imply that Orwell is famous because in English we apply the predicate 'famous' to him? That's ass-backwards or bass-ackwards, one. We correctly apply 'famous' to him because he is, in reality, famous. (That his fame is a social fact doesn't make it language-dependent.) Do you really want to say or imply that, were we speaking German, Orwell would not be famous but beruehmt? 'Famous' is a word of English while beruehmt is its German equivalent. The property, however, belongs to neither language. If you say there are no properties, only predicates, then that smacks of the loony bin.
Suppose 'Orwell' refers to the concrete individual Orwell, and 'famous' refers to the property, being-famous. Then you get for your trouble a different set of difficulties. I don't deny them! But these difficulties do not show that the scholastic view is in the clear.
This pattern repeats itself throughout philosophy. I believe I have shown that materialism about the mind faces insuperable objections, and that only those in the grip of naturalist ideology could fail to feel their force. But it won't do any good to say that substance dualism also faces insuperable objections. For it could be that both are false/incoherent. In fact, it could be that every theory proposed (and proposable by us) in solution of every philosophical problem is false/incoherent.
Do you remember the prediction, made in 1999, that the DOW would reach 36,000 in a few years? Since that didn't happen, I am inclined to say that Glassman and Hasset's prediction was wrong and was wrong at the time the prediction was made. I take that to mean that the content of their prediction was false at the time the prediction was made. Subsequent events merely made it evident that the content of the prediction was false; said events did not first bring it about that the content of the prediction have a truth-value.
And so I am not inclined to say that the content of their irrationally exuberant prediction was neither true nor false at the time of the prediction. It had a truth-value at the time of the prediction; it was simply not evident at that time what that truth-value was. By 'the content of the prediction' I mean the proposition expressed by 'The DOW will reach 36,000 in a few years.'
I am also inclined to say that the contents of some predictions are true at the time the predictions are made, and thus true in advance of the events predicted. I am not inclined to say that these predictions were neither true nor false at the time they were made. Suppose I predict some event E and E comes to pass. You might say to me, "You were right to predict the occurrence of E." You would not say to me, "Although the content of your prediction was neither true nor false at the time of your prediction, said content has now acquired the truth-value, true."
It is worth noting that the expression 'come true' is ambiguous. It could mean 'come to be known to be true' or it could mean 'come to have the truth-value, true.' I am inclined to read it the first way. Accordingly, when a prediction 'comes true,' what that means is that the prediction which all along was true, and thus true in advance of the contingent event predicted, is now known to be true.
So far, then, I am inclined to say that the Law of Excluded Middle applies to future-tensed sentences. If we assume Bivalence (that there are exactly two truth-values), then the Law of Excluded Middle (LEM)can be formulated as follows. For any proposition p, either p is true or p is false. Now consider a future-tensed sentence that refers to some event that is neither impossible nor necessary. An example is the DOW sentence above or 'Tom will get tenure in 2014.' Someone who assertively utters a sentence such as this makes a prediction. What I am currently puzzling over is whether any predictions, at the time that they are made, have a truth-value, i.e., (assuming Bivalence), are either true or false.
Why should I be puzzling over this? Well, despite the strong linguistic inclinations recorded above, there is something strange in regarding a contingent proposition about a future event as either true or false in advance of the event's occurrence or nonoccurrence. How could a contingent proposition be true before the event occurs that alone could make it true?
Our problem can be set forth as an antilogism or aporetic triad:
1. U-LEM: LEM applies unrestrictedly to all declarative sentences, whatever their tense. 2. Presentism: Only what exists at present exists. 3. Truth-Maker Principle: Every contingent truth has a truth-maker.
Each limb of the triad is plausible. But they can't all be true. The conjunction of any two entails the negation of the third. Corresponding to our (inconsistent) antilogism there are three (valid) syllogisms each of which is an argument to the negation of one of the limbs from the other two limbs.
If there is no compelling reason to adopt one ofthese syllogisms over the other two, then I would say that the problem is a genuine aporia, an insoluble problem.
People don't like to admit that there are insolubilia. That may merely reflect their dogmatism and overpowering need for doxastic security. Man is a proud critter loathe to confess the infirmity of reason.
According to Fred Sommers (The Logic of Natural Language, p. 166), ". . . one way of saying what an atomic sentence is is to say that it is the kind of sentence that contains only categorematic expressions." Earlier in the same book, Sommers says this:
In Frege, the distinction between subjects and predicates is not due to any difference of syncategorematic elements since the basic subject-predicate propositions are devoid of such elements. In Frege, the difference between subject and predicate is a primitive difference between two kinds of categorematic expressions. (p. 17)
Examples of categorematic (non-logical) expressions are 'Socrates' and 'mammal.' Examples of syncategorematic (logical) expressions are 'not,' 'every,' and 'and.' As 'syn' suggests, the latter expressions are not semantic stand-alones, but have their meaning only together with categorematic expressions. Sommers puts it this way: "Categorematic expressions apply to things and states of affairs; syncategorematic expressions do not." (164)
At first I found it perfectly obvious that atomic sentences have only categorematic elements, but now I have doubts. Consider the atomic sentence 'Al is fat.' It is symbolized thusly: Fa. 'F' is a predicate expression the reference (Bedeutung) of which is a Fregean concept (Begriff) while 'a' is a subject-expression or name the reference of which is a Fregean object (Gegenstand). Both expressions are categorematic or 'non-logical.' Neither is syncategorematic. And there are supposed to be no syncategorematic elements in the sentence: there is just 'F' and 'a.'
But wait a minute! What about the immediate juxtaposition of 'F' and 'a' in that order? That juxtaposition is not nothing. It conveys something. It conveys that the referent of 'a' falls under the referent of 'F'. It conveys that the object a instantiates the concept F. I suggest that the juxtaposition of the two signs is a syncategorematic element. If this is right, then it is false that atomic sentence lack all syncategorematic elements.
Of course, there is no special sign for the immediate juxtaposition of 'F' and 'a' in 'Fa.' So I grant that there is no syncategorematic element if such an element must have its own separate and isolable sign. But there is no need for a separate sign; the immediate juxtaposition does the trick. The syncategorematic element is precisely the juxtaposition.
Please note that if there were no syncategorematic element in 'Fa' there would not be any sentence at all. A sentence is not a list. The sentence 'Fa' is not the list 'F, a.' A (declarative) sentence expresses a thought (Gedanke) which is its sense (Sinn). And its has a reference (Bedeutung), namely a truth value (Wahrheitswert). No list of words (or of anything else) expresses a thought or has a truth value. So a sentence is not a list of its constituent words. A sentence depends on its constituent words, but it is more than them. It is their unity.
So I say there must be a syncategorematic element in 'Fa' if it is to be a sentence. There is need of a copulative element to tie together subject and predicate. It follows that, pace Sommers, it is false that atomic sentences are devoid of syntagorematic elements.
Note what I am NOT saying. I am not saying that the copulative element in a sentence must be a separate sign such as 'is.' There is no need for the copulative 'is.' In standard English we say 'The sea is blue' not 'The sea blue.' But in Turkish one can say Deniz mavi and it is correct and intelligible. My point is not that we need the copulative 'is' as a separate sign but that we need a copulative element which, though it does not refer to anything, yet ties together subject and predicate. There must be some feature of the atomic sentence that functions as the copulative element, if not immediate juxtaposition then something else such as a font difference or color difference.
At his point I will be reminded that Frege's concepts (Begriffe) are unsaturated (ungesaettigt). They are 'gappy' or incomplete unlike objects. The incompleteness of concepts is reflected in the incompleteness of predicate expressions. Thus '. . . is fat' has a gap in it, a gap fit to accept a name such as 'Al' which has no gap. We can thus say that for Frege the copula is imported into the predicate. It might be thought that the gappiness of concepts and predicate expressions obviates the need for a copulative element in the sentence and in the corresponding Thought (Gedanke) or proposition.
But this would be a mistake. For even if predicate expressions and concepts are unsaturated, there is still a difference between a list and a sentence. The unsaturatedness of a concept merely means that it combines with an object without the need of a tertium quid. (If there were a third thing, then Bradley's regress would be up and running.) But to express that a concept is in fact instantiated by an object requires more than a listing of a concept-word (Begriffswort) and a name. There is need of a syncategorical element in the sentence.
So I conclude that if there are any atomic sentences, then they cannot contain only categorematic expressions.
I should issue a partial retraction. I wrote earlier,"The TFL representation of singular sentences as quantified sentences does not capture their logical form, and this is an inadequacy of TFL, and a point in favor of MPL." ('TFL' is short for 'traditional formal logic'; 'MPL' for 'modern predicate logic with identity.' )
The animadversions of Edward the Nominalist have made me see that my assertion is by no means obvious, and may in the end be just a dogma of analytic philosophy which has prevailed because endlessly repeated and rarely questioned. Consider again this obviously valid argument:
1. Pittacus is a good man 2. Pittacus is a wise man ----- 3. Some wise man is a good man.
The traditional syllogistic renders the argument as follows:
Every Pittacus is a wise man Some Pittacus is a good man ----- Some wise man is a good man.
This has the form:
Every P is a W Some P is a G ----- Some W is a G.
This form is easily shown to be valid by the application of the syllogistic rules.
In my earlier post I then repeated a stock objection which I got from Peter Geach:
But is it logically acceptable to attach a quantifier to a singular term? How could a proper name have a sign of logical quantity prefixed to it? 'Pittacus' denotes or names exactly one individual. 'Every Pittacus' denotes the very same individual. So we should expect 'Every Pittacus is wise' and 'Pittacus is wise' to exhibit the same logical behavior. But they behave differently under negation.
The negation of 'Pittacus is wise' is 'Pittacus is not wise.' So, given that 'Pittacus' and 'every Pittacus' denote the same individual, we should expect that the negation of 'Every Pittacus is wise' will be 'Every Pittacus is not wise.' But that is not the negation (contradictory) of 'Every Pittacus is wise'; it is its contrary. So 'Pittacus is wise' and 'Every Pittacus is wise' behave differently under negation, which shows that their logical form is different.
My objection, in nuce, was that 'Pittacus is wise' and 'Pittacus is not wise' are contradictories, not contraries, while 'Every Pittacus is wise' and 'Every Pittacus is not wise' ('No Pittacus is wise') are contraries. Therefore, TFL does not capture or render perspicuous the logical form of 'Pittacus is wise.'
To this, Edward plausibly objected:
As I have argued here before, ‘Pittacus is wise’ and ‘Pittacus is not wise’ are in fact contraries. For the first implies that someone (Pittacus) is wise. The second implies that someone (Pittacus again) is not wise. Both imply the existence of Pittacus (or at least – to silence impudent quibblers - that someone is Pittacus). Thus they are contraries. Both are false when no one is Pittacus.
I now concede that this is a very good point. A little later Edward writes,
The thing is, you really have a problem otherwise. If 'Socrates is wise' and 'Socrates is not wise' are contradictories, and if 'Socrates is not wise' implies 'someone (Socrates) is not wise', as standard MPC holds, you are committed to the thesis that the sentence is not meaningful when Socrates ceases to exist (or if he never existed because Plato made him up). Which (on my definition) is Direct Reference.
So you have this horrible choice: Direct reference or Traditional Logic.
But must we choose? Consider 'Vulcan is uninhabited.' Why can't I, without jettisoning any of the characteristic tenets of MPL, just say that this sentence, though it appears singular is really general because 'Vulcan' is not a logically proper name but a definite description in disguise? Accordingly, what the sentence says is that a certain concept -- the concept planet between Mercury and the Sun -- has as a Fregean mark (Merkmal) the concept uninhabited.
Now consider the pair 'Socrates is dead' - 'Socrates is not dead.' Are these contraries or contradictories? If contraries, then they can both be false. Arguably, they are both false since Socrates does not exist, given that presentism is true. Since both are false, both are meaningful. But then 'Socrates ' has meaning despite its not referring to anything. So 'Socrates' has something like a Fregean sense. But what on earth could this be, given that 'Socrates' unlike 'Vulcan' names an individual that existed, and so has a nonqualitative thisnsess incommunicable to any other individual?
If, on the other hand, the meaning of 'Socrates' is its referent, then, given that presentism is true and Socrates does not exist, there is no referent in which case both sentences are meaningless.
So once again we are in deep aporetic trouble. The proper name of a past individual cannot have a reference-determining sense. This is because any such sense would have to be a Plantingian haecceity-property, and I have already shown that these cannot exist. But if we say that 'Socrates' does not have a reference-determining sense but refers directly in such a way as to require Socrates to exist if 'Socrates' is to have meaning, then, given presentism, 'Socrates' and the sentence of which it is a part is meaningless.
(By popular demand, I repost the following old Powerblogs entry.)
"The mark of a mark is a mark of the thing itself." I found this piece of scholasticism in C. S. Peirce. (Justus Buchler, ed., Philosophical Writings of Peirce, p. 133) It is an example of what Peirce calls a 'leading principle.'
Let's say you have an enthymeme:
Enoch was a man ----- Enoch died.
Invalid as it stands, this argument can be made valid by adding a premise. (Any invalid argument can be made valid by adding a premise.) Add 'All men die' and the argument comes out valid. Peirce writes:
The leading principle of this is nota notae est nota rei ipsius. Stating this as a premiss, we have the argument,
Nota notae est nota rei ipsius Mortality is a mark of humanity, which is a mark of Enoch ----- Mortality is a mark of Enoch.
But is it true that the mark of a mark is a mark of the thing itself? There is no doubt that mortality is a mark of humanity in the following sense: The concept humanity includes within its conceptual content the superordinate concept mortal, which implies that, necessarily, if anything is human, then it is mortal. But mortality is not a mark, but a property, of Enoch. I am alluding to Frege's distinction between a Merkmal and an Eigenschaft. Frege explains this distinction in various places, one being The Foundations of Arithmetic, sec. 53. But rather than quote Frege, I'll explain the distinction in my own way using a totally original example.
Consider the concept bachelor. This is a first-order or first-level concept in that the items that fall under it are not concepts but objects. The marks of a first-order concept are properties of the objects that fall under the concept. Now the marks of bachelor are unmarried, male, adult, and not a member of a religious order. These marks are themselves concepts, concepts one can extract from bachelor by analysis. Given that Tom falls under bachelor, he has these marks as properties. Thus unmarried, etc. are not marks of Tom, but properties of Tom, while unmarried, etc. are not properties of bachelor but marks of bachelor.
To appreciate the Merkmal (mark)-Eigenschaft (property) distinction, note that the relation between a concept and its marks is entirely different from the relation between a concept and its instances. A first-order concept includes its marks without instantiating them, while an object instantiates its properties without including them.
This is a very plausible line to take. It makes no sense to say of a concept that it is married or unmarried, so unmarried cannot be a property of the concept bachelor. Concepts don't get married or remain single. But it does make sense to say that a concept includes certain other concepts, its marks. On the other hand, it makes no sense to say of Tom that he includes certain concepts since he could do such a thing only if he were a concept, which he isn't. But it does make sense to say of Tom that he has such properties as being a bachelor, being unmarried, being an adult, etc.
Reverting to Peirce's example, mortality is a mark of humanity, but not a mark of Enoch. It is a property of Enoch. For this reason the scholastic formula is false. Nota notae NON est nota rei ipsius. The mark of a mark is not a mark of the thing itself but a property of the thing itself.
No doubt commenter Edward the Nominalist will want to wrangle with me over this slight to his scholastic lore, and I hope he does, since his objections will aid and abet our descent into the labyrinth of this fascinating cluster of problems. But for now, two quick applications.
One is to the ontological argument, or rather to the ontological argument aus lauter Begriffen as Kant describes it, the ontological argument "from mere concepts." So we start with the concept of God and analyze it. God is omniscient, etc. But 'surely' existence is also contained in the concept of God. For a God who did not exist would lack a perfection, a great-making property; such a God would not be id quo maius cogitari non posse. He would not be that than which no greater can be conceived. To conceive God, then, is to conceive an existing God, whence it follows that God exists! For if you are conceiving a nonexistent God, then you are not conceiving God.
Frege refutes this version of the OA -- not the only or best version I hasten to add -- in one sentence: Weil Existenz Eigenschaft des Begriffes ist, erreicht der ontologische Beweis von der Existenz Gottes sein Ziel nicht. (Grundlagen der Arithmetik, sec. 53) "Because existence is a property of concepts, the ontological argument for the existence of God fails to attain its goal." What Frege is saying is that the OA "from mere concepts" rests on the mistake of thinking of existence as a mark of concepts as opposed to a property of concepts. No concept for Frege is such that existence is included within it. Existence is rather a property of concepts, the property of having an instance.
The other application of my rejection of the scholastic formula above is to the logical question of the correct interpretation of singular propositions. The scholastics treat singulars as if they are generals as I explained fully in previous posts. But if Frege is right, this is a grave logical error since it rides roughshod over the mark/property distinction. To drag this all into the full light of day will take many more posts.
1. Pittacus is a good man 2. Pittacus is a wise man ----- 3. Some wise man is a good man.
That this argument is valid I take to be a datum, a given, a non-negotiable point. The question is whether traditional formal logic (TFL) is equipped to account for the validity of this argument. As I have already shown, it is quite easy to explain the validity of arguments like the above in modern predicate logic (MPL). In MPL, the logical form of the above argument is
Gp Wp --- (Ex)(Wx & Gx).
In order to evaluate the argument within TFL, it must be put into syllogistic form, otherwise the rules of the syllogism cannot be applied to it. Thus,
Every Pittacus is a wise man Some Pittacus is a good man ----- Some wise man is a good man.
This has the form:
Every P is a W Some P is a G ----- Some W is a G.
It is easy to prove that this form is valid by using a Venn diagram (not to be confused with an Euler diagram), or by applying the syllogistic rules. You will notice that I have rigged the argument so that those who deny that universal propositions have existential import will be satisfied that it is valid. Note also that the Venn diagram test would not work if the argument were given the following form:
Every P is a W Every P is a G ----- Some W is a G.
You can verify for yourself that if you diagram the premises you will not thereby have diagrammed the conclusion.
But is it logically acceptable to attach a quantifier to a singular term? How could a proper name have a sign of logical quantity prefixed to it? 'Pittacus' denotes or names exactly one individual. 'Every Pittacus' denotes the very same individual. So we should expect 'Every Pittacus is wise' and 'Pittacus is wise' to exhibit the same logical behavior. But they behave differently under negation.
The negation of 'Pittacus is wise' is 'Pittacus is not wise.' So, given that 'Pittacus' and 'every Pittacus' denote the same individual, we should expect that the negation of 'Every Pittacus is wise' will be 'Every Pittacus is not wise.' But that is not the negation (contradictory) of 'Every Pittacus is wise'; it is its contrary. So 'Pittacus is wise' and 'Every Pittacus is wise' behave differently under negation, which shows that their logical form is different. My argument can be put as follows:
a. Genuinely singular sentences have contradictories but not contraries. b. Sentences like 'Every Pittacus is wise' have both contradictories and contraries. Therefore c. Sentences like 'Every Pittacus is wise' are not genuinely singular. d. 'Pittacus is wise' is genuinely singular. Therefore e. The TFL representation of singular sentences as quantified sentences does not capture their logical form, and this is an inadequacy of TFL, and a point in favor of MPL.
In Modern Predicate Logic (MPL), logical quantity comes in three 'flavors,' universal, particular, and singular. Thus 'All bloggers are self-absorbed' and 'No bloggers are self-absorbed' are universal; 'Some bloggers are self-absorbed' and 'Some bloggers are not self-absorbed' are particular; 'Bernie is self-absorbed' and 'Bernie is not self-absorbed' are singular. Traditional Formal Logic (TFL), however, does not admit a separate category of singular propositions.
So, just to draw out commenter Edward the Nominalist and Co., how would a defender of TFL account for the validity of the following obviously valid argument:
1. Mars is red 2. Mars is a planet ----- 3. Some planet is red.
A supporter of MPL could construct a derivation as follows:
4. Mars is a planet & Mars is red. (From 1, 2 by Conjunction) 5. There is an x such that: x is a planet & x is red. (From 4 by Existential Generalization) 3. Some planet is red. (From 5 by translation back into ordinary language)
No sweat for the MPL boys, but how do you TFL-ers do it? (Of course I am aware that it can be done. The point of this post is mainly didactic.)
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