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Friday, February 13, 2009

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An argument against premise two

God need not justify the existence of unjustified evil, he only must justify allowing for it to exist. The argument is as follows:

Argument one.

1. All events that can be judged good or evil must be the actions of an agent. (Agentless events occuring are niether good nor evil)
2. We have free will.(via Peter Van Inwagens consequence argument)
3. Our actions are a result of our choices.
4. God is not culpable for our choices and is therfore not culpable for our actions or any evil resulting from them.

Argument two
1. God must only justify what he is culpable for.
2. God is culpable for allowing humans the possiblity of doing all possible evil. (by this I mean evil of which they are capable, I do not mean that humans can do every possible evil act, surely doing some evil acts must preclude other evil acts, IE if I incinerate your car immediatly upon seeing, that precludes me from stealing it to run people over in it)
3. God must justify allowing the possibility of all possible evil actions.


Anticipating that inevitable reply that God is unjustified in allowing the possibililty of all possible evil, here is an argument proving God is justified in allowing the possibility of all possible evil actions.

Argument three.

1. All evil is the result of the choices of Agents(argument one premise one).
2. Evil is only possible if agents make choices.
3. Agents make choices.
4. Choices are only possible given free will (I am following VanInwagens Consequence argument here and so compatibilism is out).
5. The possibility of Love requires free will to exist.
6. If the possibility of love is justified, free will is justified.
7. The possibility of Love is justified.*
8. God is justified in giving us free will.


Argument four.

1. If free will is justified then allowing the possibility of all possible actions is justified because it is entailed in the concept of free will.
2. If allowing the possibility of all possible actions is justified then allowing the possibility of all possible evil actions is justified.
3. If allowing the possibility of all possible evil actions is justified, allowing all possible evil actions is justified.
4. Allowing all possible evil actions is justified.

I thus conclude the atheistic modal argument is a failure.

*In response to those who will inevitabley respond that the possibility of love is not justified if it entails the possibility of evil, I say this: Love must entail the possiblity of evil. As such to argue against love being justified on the grounds that it entails evil is to argue the possibility of love in unjustified. I consider this conclusion to be a reductio ad absurdum

Bill,

You say: "So what we need to say is that a valid deductive argument is one whose form is such that no actual or possible argument of that form has true premises and a false conclusion. This shows that validity is a modal concept."

I'm not convinced (yet). Consider this valid argument form:

1. All Xs are Ys
2. All Ys are Zs
----------------
3. All Xs are Zs.

Can't we express the validity of this by quantification over the classes X, Y, and Z? For all X, Y, Z, 3 is never false when 1 and 2 are true. Where is the modality in this?

I'm reluctant to accept that my notion of proof rests on modal concepts. This seems to be putting the cart before the horse.

David,

You wrote:

"For all X, Y, Z, 3 is never false when 1 and 2 are true."

Shouldn't that be "For all possible X, Y, Z..."? If so, it becomes clear where the modality comes in. Wouldn't your use of "never" also carry modal force? You intend something like "when 1 and 2 are true, 3 is true no matter what could happen".

Matt.

The argument is plainly valid. But it isn't sound. Premise (2) is false. God might be a necessary being, but not be omniscient or omnipotent, for instance, so might not know about or be able to prevent unjustified evil. To shore it up, you might assume a traditional (say, Anselmian) conception of God. But even that won't work, given free will defenses. Assuming that some version of the FWD succeeds, as I think it does (it had better succeed!), it is possible that there exist moral evil E that is such that, had E not occured, the world would have been better. So, the evil is unjustified. But, the argument goes, God is justified in not preventing it. In short, the world would not be better if God prevented it, but it would be better if the agent did!

David,

"Can't we express the validity of this by quantification over the classes X, Y, and Z? For all X, Y, Z, 3 is never false when 1 and 2 are true. Where is the modality in this?"

To say that an argument is valid is not merely to say that if the premises are true then the conclusion "is never false". If I understand you correctly, you use the term 'never' in a temporal sense, something along the lines of: for every time t, P is not false at t.

However, the logical notion of validity requires something stronger. It requires the notion that when the premises are true, the conclusion *could not be false*.

Would a weaker notion suffice, one which substitutes the temporal notion *is never false* for the modal *could not be false*?
I do not think so. For that would allow for cases where the conclusion is a sentence that as a matter of fact is always true in the temporal sense, but it does not deductively follow from the premises. Suppose all of the sentences (1)-(3) are in fact true.

1. All lawyers are rich.
2. All rich people are happy.
Therefore,
3. All ravens are black

(3) happens to be true. Suppose ravens remain black colored for the duration of their existence. Then (3) is "never false". But clearly the argument I have just given is not valid; the conclusion has nothing to do with the premises. Therefore, the truth-value of the conclusion does not depend in any way on the truth-value of the premises. But, now, how can we express this in a formal way so that we do not need to examine the specific content of the sentences involved in each case of a valid argument. It is worth remembering in this context something Bill noted in his original post. The notion of validity is intended to apply to sets of premises and conclusions that have not, are not, and never will be examined.

The intuition we wish to express here is that in all cases of valid arguments the truth of the premises rules out any possible circumstance in which the conclusion is false. O/w any collection of sentences that happens to be never false, in the temporal sense, can be arranged to form a valid argument.

You note: "I'm reluctant to accept that my notion of proof rests on modal concepts. This seems to be putting the cart before the horse."

Despite what I have noted above, the reluctance you express here is not without its defenders in philosophy of logic. The subject under consideration involves the notion of "logical consequence", a subject that has been extensively debated recently. There are roughly two approaches to the nature of "logical consequence" in the literature on the subject: (a) the *proof theoretical* approach which explicates logical consequence in terms of there being a proof of the conclusion from the premises; (b) the *model theoretic* approach focuses on explicating logical consequence in terms of truth in alternative models for the interpretation of the premises and conclusion. The model theoretic approach explicitly relies upon the modal element Bill emphasized in the notion of logical consequence. The proof theoretic approach attempts to eschew any modal notion and rely exclusively on the notion of proof. Whether this program can work is of course the reason the notion of "logical consequence" is still debated.

So I suspect that your resistance to involve modal notions in your notion of proof shows that you favor the proof theoretic approach. So instead of the temporal element you proposed, perhaps you ought to explore the proof theoretic approach and see if it is to your liking.

peter



Hello Matt,

I agree that we do use 'possible' and 'never' (or 'nowhere') in this way but I think these words are redundant and do not carry any modal implications. The kind of thing I have in mind is a finite domain of, say, 3 objects and thus 2**3=8 classes (predicates). We can inspect all the 8**3=512 distinct assignments of classes to the variables X, Y, Z and see if P3 is false when P1 and P2 are true in each case. It's pure symbol shuffling and we could write a computer program to do this. I don't see how 'what could happen' comes into it.

Mike,

Good point. (2)is false for the reasons you give. Mike Young says something similar at the top of the thread.

David, Matt, Peter,

Thanks for the comments. I hope we all agree that, strictly speaking, 'never' is a purely temporal word. It means 'at no time.' Something that is impossible not only never happens but is also such that it could not ever happen. Similarly, 'always' used strictly is a purely temporal word. It means 'at every time.' Something that always exists exists at every time -- which is consistent with the thing's being modally contingent.

Peter,

Your explanation of the difference between proof-theoretic and model-theoretic approaches to logical consequence looks to be the nub of the difference between me and you and Matt, on the one hand, and David B. on the other.

David,

I am wondering if you may not be making a tacit appeal to modal notions. Given exactly three individuals a, b, c, there are a total of 8 possible combinations: a, b, c, -- you could call those singleton combinations -- ab, bc, ac, abc, and the null combination.

Suppose a, b, and c are tiles, not touching each other, laid out on a flat surface. These are given. They actually exist. But the various further combinations don't actually exist. For example a-touching-b doesn't exist, and a-touching-b-touching-c doesn't exist. Yet they are possible.

What you may be doing is considering all the possible combos as actual or really existent. But then I think you are making surreptitious use of modal notions.

I realize I am not being very clear. Perhaps Peter understands what I am driving at and can put it more clearly.

Hello Peter,

Many thanks for your last comment which crossed with mine to Matt. My use of 'never' was careless slipping into math-speak, I think, as I didn't intend any temporal sense. 'Not' would have done.

But wouldn't my quantification method detect the invalidity of your ravens argument? It has the form

P1. All Ls are Rs
P2. All Rs are Hs
---------------
P3. All Vs are Bs

Let X be an arbitrary class. The assignment L=R=H=X, V=X and B=~X makes P1 and P2 true but P3 false.

I am probably temperamentally inclined towards the proof theoretical approach but my comments here are surely in the model theoretic vein I think. I guess I can vaguely see a structural analogy with possible worlds semantics here:

p is necessary <--> p isn't false in any possible world

parallels

A is valid <--> A isn't countered by any possible assignment of meanings.

but surely modality introduces more than just *logical* necessity? Could you perhaps say a little more on this? (or offer us a reference!)

Bill said: “Let's apply the result of the preceding section to the question whether the modal distinction between the necessary and the contingent applies outside the sphere of human volition. The followers of Ayn Rand maintain that it does not, that it applies only within the sphere of the man-made, the sphere of that which can be affected by human will and choice. Whether an argument is valid or invalid, however, is independent of human volition. “

As a long-time student of Objectivism, I think part of the problem between the debaters is that Objectivism does not rely exclusively on deductive logic (modal logic), but realizes that there is such a thing as inductive logic. In fact, Miss Rand defines the term “logic” as “the non-contradictory identification [of the facts of reality as given by observation]”. So that if someone were to say, “All pigs are tigers, ” Objectivists would not consider this to be a logical statement because it contradicts the facts of reality. So, logic, in Objectivism, is reliant on observation; and the observation and the non-contradictory identification of the facts must precede having the premise in the first place.

As to whether or not logic relies on volition, Objectivists claim that it does because a man has no a prior mechanism to keep his mind in proper alliance with reality. In other words, logic – the method of non-contradictory identification of the facts of reality as given by observation – must be performed by a man of his own free will, since it is not automatic.

And volition is verified via introspection, a type of observation – that is, we observe that we have options (such as to post or not to post) and that we can decide to post or not to post. So Objectivists would consider the following syllogism to be valid in the modal sense, but it is not logical in the Objectivist sense:

All of reality is deterministic
Man is within reality
Therefore man is deterministic

This is not a logical statement, according to Objectivism, because the first premise is false, all of reality is not deterministic because man does have free will. In other words, the first premise contradicts the facts of reality, since we observe that we have free will.

Thomas:
>As a long-time student of Objectivism, I think part of the problem between the debaters is that Objectivism does not rely exclusively on deductive logic (modal logic), but realizes that there is such a thing as inductive logic.

Thanks Thomas, for briefly outlining what you consider makes Objectivist logic different - indeed, on another thread Travis is claiming it is "radically different."

Yet I am struggling to see what is "Objectivist" about this example:

p1: All of reality is deterministic
p2: Man is within reality
c: Therefore man is deterministic

If these premises are true, the conclusion is true. If these premises are false, the conclusion is false. There seems to be no original Objectivist logic in this whatsoever.

The only thing that seems to be happening is that you are saying that this syllogism is "logically valid" whilst simultaneously saying that it is "not a logical statement" according to Objectivism. So again we have a failure to communicate, in that Objectivists have decided to call what would typically be called a logical syllogism (or by "statement" do you mean the conclusion?) not logical.

Well, what would Objectivists call "logical" then? Your example indicates that in order for a syllogism to be Objecto-logical, the premises would have to be inductively proven true. Yet of course, what has become known as Hume's problem of induction shows that induction is deductively invalid; and furthermore, Rand admits herself she has no solution to this clash. (see ITOE p304)

Hence I consider this supposed "Ob-logic" to be rhetorical rather than real.

David,

"Let X be an arbitrary class. The assignment L=R=H=X, V=X and B=~X makes P1 and P2 true but P3 false."

This is not going to work because validity in a model theoretic sense is defined in terms of truth in all *possible* models; where you can think of a model as an interpretation (assignment of extensions) to all non-logical constants. In the example above you consider only one (partial) assignment (or model). So while you can assign truth-values relative to that model, you cannot define validity that way.

Notice that the model-theoretic definition of validity is not going to work if you drop "possible" and just define it in terms of "all", because then one might ask: All models that....what? That one entertains? Writes down? These won't work because one might fail to consider a model in which the premises are true (according to a given interpretation), while the conclusion is false. Under such circumstances, an argument might end up being valid in light of the narrower class of models entertained or written down, but not valid when it is evaluated with respect to all *possible* models.

I think you should explore the proof-theoretic option more seriously.

peter

Bill Wrote:

"Let's apply the result of the preceding section to the question whether the modal distinction between the necessary and the contingent applies outside the sphere of human volition. The followers of Ayn Rand maintain that it does not, that it applies only within the sphere of the man-made, the sphere of that which can be affected by human will and choice. Whether an argument is valid or invalid, however, is independent of human volition."

But arguments are CONSTRUCTED. If there had been no humans, how many arguments would exist in the natural world? None. Arguments do not exist unless made by humans. The contingency of arguments comes from the fact that humans can choose to make them or not.

Bob,

"Arguments do not exist unless made by humans. The contingency of arguments comes from the fact that humans can choose to make them or not."

Question: How many positive natural numbers human being actually counted?
Answer: Some finite number.
Question: How many positive natural numbers are there?
Answer: Infinitely many

So the *act* of counting numbers does not bring them into existence; for if it did, then there would have been only finitely many numbers, since humans can count only finitely many numbers.

1) You confuse between the notion of an *argument* and the *act* of giving or producing an argument. Arguments exist whether or not human beings exist. The act (if produced by a human)of making an argument, on the other hand, depends upon the existence of human beings just like any other activity of humans.

2) The *act* of producing this or that argument by this or that person is of course contingent: for the person could have refrained from the act of making this particular argument. However, it does not follow from the fact that the *act* of making a given argument is contingent that the argument itself is contingent; that this act somehow brought the argument into existence. This does not follow just like the act of proving that number two exists does not bring the number in question into existence.

"But arguments are CONSTRUCTED. If there had been no humans, how many arguments would exist in the natural world?"

3) Let us ask a different question first:
How many arguments are there in the natural world?
If by 'natural' you mean within space and time, the answer is: None!
Why? Because arguments are not concrete entities that can be caused to move here and there or explode or get wet. They are abstract just like numbers.
How many instances are there of people giving an argument?
I have no idea: most likely a large finite number of them.

4) So whether humans exist or not, there are no arguments in the natural world: there are only *acts* of making arguments (provided creatures who can do so exist). Of course, there are infinitely many arguments in the world, whether human beings exist or not; but, of course, the world contains many more things than the *natural* world including numbers, arguments, propositions, and perhaps much more.

peter


I am sure that all of you have, at some stage in your lives, been required to solve Pythagoras's theorem, and we all left school convinced that we really had proved that a(2)=b(2) + c(2).

What we weren't told at school (at least I never was) is that the validity of the majority of mathematical theorems only hold true in the Platonic world of mathematics where perfect triangles can exist; but not in the real, physical world that you and I inhabit.

It seems a trite thing to say here, but whenever I drew a circle or a triangle or any geometrical object, and then drew tangents or other lines that would dissect the main shapes, it always struck me that account was never taken of the finite width of these lines, and which would consequently render much of the proofs invalid, especially where you might want to use the area of one shape to be the sum of the areas of other shapes, but which was unfortunately compromised by the volume occupied by the lines.

Of course, in our minds, we ignored this, since to include it in our calculations might mess up the whole trigonometry curriculum.

However, the die had been cast. Because no one bothered to tell us that the beauty, power and accuracy of mathematics only actually lie inside the realm of the perfect world of Platonic mathematics, we would have in effect spent our entire adult lives living a curious kind of lie.

In his brilliant book, 'The Road To Reality', Professor Roger Penrose lays bare the intellectual landscape with his disclosure that life is a heady mixture of three worlds - Platonic, Real World and Mentality. This is the mentality of sentient, conscious beings - us - and it is our brains that can take images out of the Platonic world and superimpose them on the real world to give a semblance of meaning, i.e while Pythagoras may not exist in reality, he lives meaningfully enough in Plato's world to allow us to make use of it in the real world.

Our imagination, and our capacity for logical deductive reasoning allow us to define an entity known as God, and, like Pythagoras's theorem, we can carry the concept or the image in our heads, and if we attempt to define phenomena such as beauty, harmony, mystery, the coming into existence of space and time from a zero point when 15 billions years ago none existed, then the term God would appear to be a useful label.

Pythagoras's theorem does not exist in the real world; I could, of course, get a computer to draw an exceptionally accurate image of what we term a right angled triangle, but then, as with Google Earth, if I drill down to the atoms that make up the triangle, all semblance of a triangle vanishes, and then I might ask myself - what connection if any exists between 1 atom that makes up one side, with another atom makes up another side, and which from the atom's point of view, is a galaxy away?

We human beings construct worlds that basically lie inside our brains, and nowhere else. The world living in my head is different to the world inside your heads, but at the end of the day, we all retreat to our own little universes and the world of our dreams.

Arguments about belief or otherwise in God are arguments between men and women, and have precious little to do with the real world. A much better way of spending one's time, in my view, would be for every one to listen to the last 3 movements of Mahler's 2nd symphony (the resurrection) and then for each of us to describe how the music affects them, and whether or not Mahler has succeeded in revealing the true meaning (or purpose) of life, and why we are here at all.

For me, the real meaning of life is contained in the word 'evolution' which, if you read it backwards, says 'No! I t'u love'.

For me, the driving force behind evolution is to find better ways to love each other.

Maverick philosopher begins this post with the following challenge;

1) If God is possible, then God is a necessary being.
2) If God is a necessary being, then unjustified evil is impossible.
3) Unjustified evil is possible.
Therefore, God is not possible

The argument as to whether or not God exists is an emotional one, not an intellectual one, which is why people are never convinced one way or the other by rational argument.

Philosophers aside, most people are happy to remain firmly entrenched in whatever it is they happen to believe or disbelieve, and no amount of rational argument will enable them to change their view on this.

In fact, one of Man's greatest dilemmas must be - why do some people believe, and others not believe. The technology that allows mobile phones to exist is complicated, and most people would admit that they don't have a clue how they work, but this does not stop them from knowing that mobile phones came about as a result of years of research and development by Man alone, and God's role in its invention was zilch.

For all we know, the potential for sentient, conscious beings capable of analytical and rational thought may well have been the primary driving force behind the Big Bang if you wish to take a theistic point of view - but somewhere along the line, a dislocation occurred - a perturbation in an otherwise orderly pattern of evolution.

Suppose, just suppose, that Man did indeed evolve from four legged creatures. What precisely does this mean? From the physiological point of view, the only difference between Man and the rest of the mammalian animal kingdom is that our spines enter the skull directly from below - meaning that Man can stand comfortably on 2 legs, comfortably being the operative word.

What did this mean for Man? How about all you philosophers out there describing what impact this made on Man?

As far as I can ascertain, it would have taken 2 million years for the spine to shift a centimetre or so in the forward direction so that he and she could stand comfortably on 2 legs - something that took place between 8 and 6 million years ago.

What might have been the driving force behind this?

My hypothesis is that Mankind suddenly found himself confronted with one huge dilemma, and that it is this dilemma which is responsible for all of Man's current ills.

I don't wish to say any more at the moment because I am much more interested to know what any one else thinks on the subject.

Peter wrote:

"Question: How many positive natural numbers human being actually counted?
Answer: Some finite number.
Question: How many positive natural numbers are there?
Answer: Infinitely many

So the *act* of counting numbers does not bring them into existence; for if it did, then there would have been only finitely many numbers, since humans can count only finitely many numbers."


Are you saying that numbers have an existence that is independent of humans? If so, then where in the natural world are they? Are you claiming numbers exist in some sort of Platonic "other world?"


Peter: "1) You confuse between the notion of an *argument* and the *act* of giving or producing an argument. Arguments exist whether or not human beings exist."


In that case, WHERE do they exist? Trees and rocks can exist with no human beings. But where exactly are these arguments?


Peter: "2) The *act* of producing this or that argument by this or that person is of course contingent: for the person could have refrained from the act of making this particular argument. However, it does not follow from the fact that the *act* of making a given argument is contingent that the argument itself is contingent; that this act somehow brought the argument into existence. This does not follow just like the act of proving that number two exists does not bring the number in question into existence."


Did arguments and numbers exist before the solar system came into being? Did they exist before the Big Bang? Where were they then?


Peter wrote: "3) Let us ask a different question first:
How many arguments are there in the natural world?
If by 'natural' you mean within space and time, the answer is: None!
Why? Because arguments are not concrete entities that can be caused to move here and there or explode or get wet. They are abstract just like numbers."


Yes. Agreed. Exactly the point. Abstractions exist in minds (working brains) and were created by those minds in order to understand the external world. If there are no working brains, there are no abstractions.


Peter wrote: "4) So whether humans exist or not, there are no arguments in the natural world: there are only *acts* of making arguments (provided creatures who can do so exist)."


Again, agreed.


Peter wrote: "Of course, there are infinitely many arguments in the world, whether human beings exist or not; but, of course, the world contains many more things than the *natural* world including numbers, arguments, propositions, and perhaps much more."


Disagree. Where exactly in the world do those numbers, arguments, propositions reside? Inside working brains, do they not? If not, then where else are they?

Bob isn’t the disconnect from necessity/contingency arguments the fact that contingent stuff can offer no epistemological story for the necessary?

peterlupu and mark: I believe that Mark did specify a finite domain, in which case all models can be constructed without modal concepts by offering a procedure for constructing the models. Then "for all models" means "for all models that can be constructed with this procedure". You could do the same with any class of models that can be recursively enumerated, not only finite models.

I'll bet that model theories of logic that are restricted to recursively enumerable models are formally equivalent to a proof theories (in the sense that there is a one-to-one correspondence. Certainly there is a correspondence in one direction using Herbrand models), and that they are formally less comprehensive than model theories that do not restrict the possible models.

However, there is a fundamental problem with proof theories that I don't think they can possibly overcome: the question of what value the proof mechanism is. Suppose you give me a calculus that is the basis for your proof-theoretic account of logic and tell me that a valid argument is defined as an argument that can be derived from this calculus. My question is, "so why do I care about that concept of validity?" Why not take some arbitrary rule and change the rule and call that new system the standard for validity? What's the difference? The difference, of course, is that one calculus generates arguments that are actually valid while the other does not. Anyone who tries to come up with a calculus to "define" validity is implicitly acknowledging that there is already a property of arguments that he is trying to mimic using formal methods. It is that original property --the one he is trying to mimic-- that should be called "validity".

Kathy wrote:

"Bob isn’t the disconnect from necessity/contingency arguments the fact that contingent stuff can offer no epistemological story for the necessary?'

Sorry Kathy, but I'm not sure what you mean here. Could you please spell it out in greater detail? Thanks.

Bob,

You ignored this part of what I said:

Note that it would be irrelevant to point out that people make arguments by uttering declarative sentences and arranging these sentences with words like 'since' and 'therefore.' No doubt people freely do things like utter and write down sentences. Sentences and arrays of sentences are man-made. But it doesn't follow that the validity/invalidity of the arguments expressed in these man-made sentences is itself man-made.

Of course there is a sense in which arguments are constructed: sentences on a blackboard are constructed, and the mental acts of a thinker are in a sense constructed by that thinker. But as
Peter points out, you fail to distinguish between arguments thremselves and their expression. Suppose a mathematician wonders whether there is a valid argument from one proposition to
another, from the Axiom of Choice, say, to Zorn's Lemma. Either there is or there isn't. So he sets to work to find it. If he finds it, then he finds something that does not depend on his making
or will. A fortiori, the validity of the argument does not depend on anything he does.

To ask WHERE arguments exist is a pseudoquestion based on the false assumption that arguments are in space.

Peter and Bob,

Even if arguments depended for their existence entirely on the free choices of human beings, the validity/invalidity of an argument cannot so depend. Whether or not an argument is valid does not depend on someone's decision. Validity is not a man-made fact. But it is a modal fact as argued above. So the modal distinction cannot derive from human free agency.

Bill wrote:

"Let's apply the result of the preceding section to the question whether the modal distinction between the necessary and the contingent applies outside the sphere of human volition. The followers of Ayn Rand maintain that it does not, that it applies only within the sphere of the man-made, the sphere of that which can be affected by human will and choice. Whether an argument is valid or invalid, however, is independent of human volition."

I responded:

But arguments are CONSTRUCTED. If there had been no humans, how many arguments would exist in the natural world? None. Arguments do not exist unless made by humans. The contingency of arguments comes from the fact that humans can choose to make them or not.

Bill responded:

"Of course there is a sense in which arguments are constructed: sentences on a blackboard are constructed, and the mental acts of a thinker are in a sense constructed by that thinker. But as
Peter points out, you fail to distinguish between arguments thremselves and their expression. Suppose a mathematician wonders whether there is a valid argument from one proposition to
another, from the Axiom of Choice, say, to Zorn's Lemma. Either there is or there isn't. So he sets to work to find it. If he finds it, then he finds something that does not depend on his making
or will. A fortiori, the validity of the argument does not depend on anything he does."

Good point. However, all you are saying is that the mathematician has NO CHOICE as to whether the argument is valid or not. Therefore this isn't this merely another case of something in the non-human part of nature being necessitated, which is what Rand claimed?

Bill wrote:

"To ask WHERE arguments exist is a pseudoquestion based on the false assumption that arguments are in space."

If they don't exist in space, then in what sense do they exist? If a member of a debate team doesn't prepare and has nothing to say, is it a pseudoquestion to ask them "Where are your arguments?"

Hope you had a good half-marathon. Let me know if you ever run the New York Marathon. I'll cheer you on from the sidelines when you reach 1st Avenue.

Bill

I just thought of something else. From what I know about Rand, her claim that human actions are made by choice and non-human events had to be, she always used examples from the PHYSICAL world. The volcano HAD to erupt at that time. The lightning HAD to strike that tree. Arguments are not part of the physical world, are they? For one thing, arguments don't exist in space. Therefore your point about arguments is not a refutation of her position.

Hello Peter,

It seems to me that if we are claiming that "All lawyers are rich; all rich people are happy; ergo all lawyers are happy" is valid by virtue of its logical form, then our claim is that "All Xs are Ys; All Ys are Zs; ergo All Xs are Zs" is truth-preserving whatever predicates we substitute for X, Y, and Z. It's a claim about all possible substitutions of the predicates in the actual world rather than about evaluating "All lawyers are rich; all rich people are happy; ergo all lawyers are happy" over all possible worlds. I find this much simpler to understand than the modal claim; it seems nicely to capture the idea of logical form in contrast with the meanings of non-logical terms. You haven't given us a counter-example that shows that it cannot offer an adequate account of validity, so I don't feel at all coerced into the modal view.

Here is a quote from Beall and Restall's SEP article Logical Consequence, section 3:

Just as one can ask after the philosophical import of proof systems, so too one can (and philosophers often do) ask about the philosophical import of the model-centered approach. How, for example, are we to understand the “nature” of models? How are we to understand variation of truth-values across models? John Etchemendy (1990) discusses the philosophical ramifications of taking such variation to be “re-interpretation” of (non-logical) vocabulary versus taking it to reflect variation of “possible worlds”. On one account, models simply model different possible worlds (and so, logical consequence defined by those models is a model of necessary truth preservation). On the other, models provide different interpretations of the non-logical vocabulary of our language (and so, logical consequence is not necessary truth preservation, but rather, truth preservation on the basis of the meanings of the logical vocabulary.)

I guess I'm just taking the latter account.

David,

I am not sure I see the difference between evaluating your example in terms of "all possible substitutions of the predicates in the actual world" vs. evaluating it "over all possible worlds."

A model is a "possible" assignment of extensions to the non-logical vocabulary. Each model, in the sense of such an interpretation, corresponds to a "possible world" in the modal-talk. An argument is evaluated with respect to all possible models: hence, with respect to all possible worlds (in modal talk). It is true that the set of all objects is taken from the actual world, but each model arranges them into different collections (sets). In one model a predicate will be assigned one set, in another another set, and so on.
So the distinction you seem to draw does not mark a difference with respect to the relationship between the model theoretic approach and modality.

Now, David Goodman in a remark dated Feb. 15th. makes the following point:
"I'll bet that model theories of logic that are restricted to recursively enumerable models are formally equivalent to a proof theories (in the sense that there is a one-to-one correspondence.)"

I will not be surprised if he is right about the technical point. But David Goodman also raises the philosophically deeper question of whether and to what extent should we be impressed by such equivalences. After all, David G. argues, we evaluate the adequacy of an arbitrary proof theoretic apparatus to the extent that it produces results that conform to an antecedent notion of validity. So one cannot devise a system of "proofs" that say assigns the property of 'has a proof in this system' based lets say on coin tosses (heads up has a proof; heads down does not) and maintain that to the extent that this system yields results that deviate from our prior notion of validity, then so much the worst for our prior notion. No one is going to buy this response. Therefore, unlike the crazy proof system above, even a proof system that produces the "correct results" is so precisely because it yields results that coincide with our prior notion of validity.

Now, there are of course responses to the line of argument suggested by David G., but it is certainly one of the more important points and it deserves to be further explored.

peter

David and Peter,

Profitable discussion. I'm learning something. I am not very well-versed in metalogic, but it still seems to me that David is helping himself to unreduced modal notions, as Peter suggests. David writes, >>It's a claim about all possible substitutions of the predicates in the actual world rather than about evaluating "All lawyers are rich; all rich people are happy; ergo all lawyers are happy" over all possible worlds.<<

First off, note that David uses the phrase 'actual world.' Surely actuality is a modal concept. The actual is that which is neither impossible nor merely possible. How could one understand 'actual' without understanding 'possible'? Then David speaks of "possible substitutions of predicates in the actual world." This too is a modal notion. Note that the possible substitutions outrun the actual substitutions. Note further that there are more possible predicates than actual predicates. So shouldn't David have spoken of all possible substitutions of all POSSIBLE predicates? Only then will the notion of validity be properly nailed down.

I fail to see how validity could fail to be an irreducibly modal notion. Of course, there is the sort of reduction of modality that David Lewis proposes, but I cannot take that seriously and I don't think Peter can either. Lewis quantifies over a plurality of possibles worlds all equally real.

Hi,

I have a question about your definition of a valid argument. You say "Let A be an argument with premises P1 and P2 and conclusion C. A is valid iff its corresponding conditional is necessarily true. Thus A is valid iff necessarily, if P1 & P2, then C." But consider this argument:

A
P1. The declaration of independence was signed in 1776.
P2. Pigs cannot fly.
C. 2+2=4

Given that 2+2=4 is a necessary truth, and that a conditional is true iff its consequent is true as long as its antecedent is as well, "necessarily: if P1 & P2 then C" is true. So A is a valid argument. But it doesn't look valid to me.

Bill, as usual these types of arguments founder on what type of "possible" you are talking about. Consider this dialogue about the modal ontological argument.

T: Surely, you think it's possible that there is a God, surely. I mean, maybe he exists, doncha think. I mean, you aren't one of those dogmatic atheists, are you.

A: Yes, of course, I admit that it's possible that God exists.

T: Gotcha! Gotcha! Gotcha! Since the existence of God is either necessarily true or necessarily false, the S5 axiom says that if a necessary truth claim is possibly true, it must be necessarily true. Since you have admitted that God possibly exists, you must therefore conclude that God necessarily exists!!!

A: Uh, could we maybe restrict the accessibility relation or something?

Bill,

But couldn't David define "actual" non-modally as the domain of unrestricted quantification? Then as long as he is not a Lewsian, that will come out alright. I share your suspicions about David's use of possible predicate substitutions though.

Matt.

Hi Victor,

I don't think you stated the characteristic S5 axiom properly. It is: Poss p --> Nec Poss p. What you say is: Poss Nec p --> Nec p.
In any case, what you say is true as well.

Even if we assume the S5 system of modal logic, the theist still has the problem of giving a good reason for accepting the premise 'God is possible' which is the crucial premise of modal ontological arguments. Either God exists in all worlds or in none. That is Anselm's great insight. So if God exists in one world, then he exists in all, including the actual world. But how prove that God exists in just one world? That's the main difficulty as I see it. It is not enough to say that the existence of God is conceivable.

Matt,

Well, suppose David B takes his quantifiers 'wide open' as Lewis used to say -- may peace be upon him! -- still they can range over only what actually exists and not over any merely possible items. I can't be sure, but I suspect that David may be making a Lewis type move: he thinks of all the merely possible predicates and merely possible substitutions as actual stuff out there -- if you catch my drift.

Angus,

Good comment. You are bringing up the well known paradox of strict implication. But as I said in the original post, 'valid' is a terminus technicus. The argument you cite, though probatively worthless, is technically valid: it is impossible for an argument of that form to have true premises and a false conclusion. Every argument with a necessary conclusion is valid and it doesn't matter what the premises are.

Bear in mind two things: validity is only one requisite of a good deductive argument among several. And the definition of 'valid' given is not an attempt to capture the full meaning of any ordinary language use of 'valid.'

Bill et al,

Bill insists that "Validity is a modal concept". I take this to mean that one needs to understand modal language, and in particular, the 'possible worlds' apparatus, in order to understand the model theoretic notion of a valid argument. I'm not sure about this.

1. In model theory interpretations are usually taken to be mappings into the world of mathematical objects such as sets and numbers. What we are usually looking for is a proof that some argument form is truth-preserving. We hope to be able to show that, for any model, if it satisfies the premises of the argument form it also satisfies the conclusion. For example, the validity of our example syllogistic form is readily understood, as it was by Boole, from the characteristic sets of the predicates and the transitivity of set inclusion.

2. I'm as guilty as anyone of slipping modal terms like 'possible' into my comments, but I'm not sure that they add anything to the meaning. Consider

a. All possible triangles have an angle sum of 180 degrees.
b. The possible outcomes of tossing 3 coins are HHH, HHT,...,TTT.
c. The possible sequences of length 3 over the letters H and T are HHH, HHT,...
d. There are 8 possible functions from the set {1,2,3} to the set {H,T}
e. It's impossible to find a rational number whose square is 2.

Pace Peter, I claim that the word 'possible' can be dropped from each of these, or the assertion reworded, without loss of meaning. The ones which present most difficulty are (b) and (e) which are couched in terms of (future? indeterminate?) human activity. Suggested rewordings:

b* The outcome of tossing 3 coins will be one of HHH, etc
e* No rational number squares to 2.

Bill suggests that I may be slipping in modal concepts by presupposing the existence of mathematical objects and Peter has suggested that there's an equivalence between our two views. Without selling the pass, I think there's something right in that. But can anyone make the connections clearer?

"1) If God is possible, then God is a necessary being. "

Then again, "1) If Socrates is possible, then Socrates is a necessary being. "

David,

I am not saying one needs the possible worlds apparatus, but I am saying that one must grant the meaningfulness of modal terms, and that validity cannot be understood except in modal terms. Suppose I flip a coin at time t and it comes up heads. Could it have come up tails at t instead of heads? If yes, then you are admitting unrealized possibilities.

No. Read Anselm.

Hi Bill,

Yes, the coin might have come up tails. I accept the modality of the physical world, through quantum indeterminacy or deterministic chaos. What I'm not convinced of is how this applies to arguments, and in particular to mathematical proof. What I suspect may be happening is that our notion of modality in the physical relates to a sense that the physical is bound by rules but that the rules don't always fix the outcome exactly. There is also a sense that the mathematical is bound by rules and that they too don't always fix the outcome (eg, two sides and one angle don't always uniquely determine a triangle). We transfer the language of modality we use for the physical into the mathematical. The categories are quite distinct but there are analogies between them.

DavidB,

One quick question? Do you accept modality regarding mathematical truths? i.e., that mathematical truths are necessary, if true. I do not mean here that modality is involved in the validity of arguments or the meta-properties of proofs? I mean just that if a certain mathematical proposition is true, then it is necessarily true?

peter

Bill wrote:

"I am not saying one needs the possible worlds apparatus, but I am saying that one must grant the meaningfulness of modal terms, and that validity cannot be understood except in modal terms. Suppose I flip a coin at time t and it comes up heads. Could it have come up tails at t instead of heads? If yes, then you are admitting unrealized possibilities."

If a person flips the coin, the answer is yes. If a machine flips the coin, the answer is no. The results of a coin flipped by a machine only appear random to us. If one had a Laplace demon and could calculate ALL of the forces working on that coin, the result would not be random.

I think the logic is fine but the theology is problematic.
1. God cannot be, even conceptually, "necessary" as that places a (human) limitation on God, who is limitless.
Maimonides remarked, "He has no body and those who conceive of bodies [i.e. humans] cannot conceive of Him." Let alone ascribe to Him qualities such as being a necessity (or otherwise).
3. According to several traditions (Jewish, Christian) God has decided to hide His face, as it were, i.e. limit His involvement in history, it being the only way (logically too) of granting humans freedom. Human-generated unjustified evil, facilitated by God's (partial?) withdrawal, is therefore possible.

I'm an atheist, by the way,

Absalom

Absalom,

There is indeed a tradition of negative theology according to which nothing can be predicated of God. But if you are an atheist, then presumably of the predicate pair 'exists'/ 'does not exist' you ascribe the latter to God. One wonders if that is consistent with the claim that God is beyond predication.

Bob,

Once again you are confusing causal necessity with broadly-logical necessity.

Bill wrote:

"Bob, Once again you are confusing causal necessity with broadly-logical necessity."

Bill, what I'm confused about is the use of "possibility." You sid earlier:

""I am not saying one needs the possible worlds apparatus, but I am saying that one must grant the meaningfulness of modal terms, and that validity cannot be understood except in modal terms. Suppose I flip a coin at time t and it comes up heads. Could it have come up tails at t instead of heads? If yes, then you are admitting unrealized possibilities."

What does it mean that something is "logically possible" even though it is "causally impossible?" It is logically possible right now that Mt. Everest no longer exists because it was destroyed last year by a meteor impact. It is also logically possible that this impact happened two years ago, or three, or four, or five. We could also use seconds as the time interval and get millions of "logically possible events." We can also imagine bacteria from outer space (or a mutant strain from Earth, for that matter) that destroy Mt. Everest by eating and digesting all of its rocks. The "unrealized possibilities" are near infinite. But what effect does this have on the way things really are? And how does that refute the determinist claim that actual physical events "had to be?"

Hello Peter,
Do I accept modality regarding mathematical truths? Yes, in the following sense. Before a conjecture has been proven or disproven mathematicians will say that it's possibly true. This is epistemic. Once a proof has been found there is indeed a feeling that the new theorem is necessarily true. The proof reveals how 'it could not have been otherwise'. (Interestingly, some proofs are more convincing in this regard than others). However, if we restrict ourselves to work within the axiomatic method, and this covers virtually all of formal mathematics these days, I think, then the truth we have uncovered takes the form: If some object satisfies certain conditions, then it satisfies some conclusion. That is, it takes the form of an argument. Hence the sense that 'it is necessarily so' hinges on the validity of the proof. In other words the modality derives from the validity and not the other way round.

DavidB,

"Once a proof has been found there is indeed a feeling that the new theorem is necessarily true. The proof reveals how 'it could not have been otherwise'."

Well, of course, the question is whether the mathematical proposition itself is necessary, not whether we feel a strong compulsion to accept it.
Secondly, however, if the proof *reveals* that the mathematical proposition is necessary, then the property of necessity had to be already possessed by the proposition even before the proof revealed it.

You say: "Hence the sense that 'it is necessarily so' hinges on the validity of the proof. In other words the modality derives from the validity and not the other way round."

Modality in the non-epistemic sense cannot derive from the proof because then it would be a temporal matter: i.e., prior to the proof the proposition fails to have the modal property of necessity, whereas after the proof it features the modal property of necessity. But whatever you think of modality (to be distinguished from epistemic certainty) it cannot be time sensitive in this way. Just like 'truth', if a proposition is true, then it is always true, which is why it is somewhat problematical to identify truth with 'verified' or even 'verifiable' (temporally indexed propositions must be treated in a special way)

"However, if we restrict ourselves to work within the axiomatic method, and this covers virtually all of formal mathematics these days, I think, then the truth we have uncovered takes the form: if some object satisfies certain conditions, then it satisfies some conclusion. That is, it takes the form of an argument."

I do not see what compels us to view this in the manner you suggest. Sure, we can think of propositions in an axiomatic system as taking the form of a conditional along the lines you suggest: if an object satisfies the axioms, then it satisfies certain conditions. We can now ask about these conditionals whether, if they are true, they are necessarily true? Of course we can then try to prove the truth of this conditional by taking the antecedent as a premise and proving the consequence.
But my question is whether you think that if the mathematical proposition is true, it must be true; where the *must* is not merely an epistemic notion (such as a feeling of compulsion to accept it), but rather a property of the proposition itself.

peter

Bob Marks,

"if a person flips the coin, the answer is yes. If a machine flips the coin, the answer is no. The results of a coin flipped by a machine only appear random to us. If one had a Laplace demon and could calculate ALL of the forces working on that coin, the result would not be random."

This makes no sense whatsoever.

Suppose we have a Laplace demon which "could calculate ALL of the forces working on that" COIN; then if it can calculate all the forces on the coin-tossed-by-machine so that it is fully determined how the coin will land, then what prevents your demon from calculating all the forces working on the coin-tossed-by-person so that it is fully determined how the coin will land? The same forces presumably operate on the *coin* whether the coin is tossed by a machine or a person.

The machine perhaps cannot calculate the *intention* for which the coin is tossed by the person; but the intention has nothing to do with the question of whether the coin's behavior once tossed is determined or random.

Peter,

You say: "Modality in the non-epistemic sense cannot derive from the proof because then it would be a temporal matter:" Our *discovering* the proof might be a temporal matter, but proofs, like propositions themselves and their modality, might be eternal. Our *awareness* of the necessity comes only with possession of the proof.

I'm not claiming we are compelled to my view. Bill claims we are compelled to his view and I'm raising doubts (I hope) about this by putting forward what I think is a viable alternative.

Perhaps I should come clean. I have real difficulty understanding the notion of a necessary general proposition. The best handle I know on this is the 'possible worlds' scenario and that seems fraught with problems. Bill insists on placing understanding the validity of an argument into the context of understanding necessity in a general proposition. I put myself forward as a counterexample to this claim. My intuition is that it introduces far more machinery than is needed for the job in hand. For example, to understand the validity of arguments we need to understand only the necessity of conditionals, not props in full generality. And I think we understand the necessity of conditionals by understanding proof, which I claim is a lot simpler. I'm not saying that the problem can't be placed in the more general context, just that we don't have to go this far.

So, yes, I will accept that a true mathematical proposition (a conditional) indeed must be true.

Peter wrote:

"Suppose we have a Laplace demon which "could calculate ALL of the forces working on that" COIN; then if it can calculate all the forces on the coin-tossed-by-machine so that it is fully determined how the coin will land, then what prevents your demon from calculating all the forces working on the coin-tossed-by-person so that it is fully determined how the coin will land? The same forces presumably operate on the *coin* whether the coin is tossed by a machine or a person.

The machine perhaps cannot calculate the *intention* for which the coin is tossed by the person; but the intention has nothing to do with the question of whether the coin's behavior once tossed is determined or random."

But the way the person tosses the coin DOES affect the way the person tosses the coin. Denial of this would mean intentionality is epiphenomenal. The coin-tossing machine is pre-programmed. The person is not. The Laplace demon can calculate the outcome of the machine's toss BEFORE the machine tosses the coin. However, it cannot calculate the outcome for the person's toss until they actually toss it. I do agree though that once they toss it and it is out of they hands (literally) the Laplace demon can then calculate the outcome. But not before, like it ccan do with the machine.

Bob Marks - So your position must be that the laplacian demon could predict precisely what forces will be impinging on the coin at the precise moment it will be tossed, if it's tossed by a machine, but could not make a similar prediction for a human tossed coin. On what grounds do you exclude humans from the realm of such laplacian predictability?

Bob K asked Bob M:
>On what grounds do you exclude humans from the realm of such laplacian predictability?

Objectivists usually assert that man has volition (free will in non-Objectivist speak). The evidence they use in support of this assertion is that they introspect their own consciousness, and observe that they have volition. This, they claim, is sufficient "scientific" or "empirical" evidence to prove their case.

Bob Koepp wrote:

"Bob Marks - So your position must be that the laplacian demon could predict precisely what forces will be impinging on the coin at the precise moment it will be tossed, if it's tossed by a machine, but could not make a similar prediction for a human tossed coin. On what grounds do you exclude humans from the realm of such laplacian predictability?"

But that's NOT my position. The Laplace demon can predict the forces impinging on the coin BEFORE the machine tosses it because the machine is also run by deterministic laws. In the case of the human, the demon cannot make a determination before the human tosses the coin since the human has free will and could always choose otherwise. If a human is tossing the coin, the demon has to wait until it is literally out of the human's hands.

Bob M, Peter, Bob K,

I am far from being well versed in these matters so please ignore the ignorant...

It seems to me that in order to calculate any outcome you have to know everything that is going to happen (and thus also everything that has happened).

The demon can calculate the forces on the coin before the machine tosses it at time t, just because he already knows what forces are acting on the coin at time t.

What if the machine (it is mechanical and much can go wrong) does not toss the coin properly? What if an unforeseen force alters the coin (or its path) somehow after it is flipped?

Given that the demon knows everything that is going to happen, it will also know how the coin will land; it cannot be a temporal being. If the demon is at a loss in determining the outcome because some 'volition' might change it anytime, then he, like us, is stuck in time, unsure of the future.

The only thing physical determinism tells us is that the laws of nature are necessary in every possible world we can concieve. In any possible world, if X interacts with Y in some way I, the outcome Z will necessarily be same in all those worlds. The interaction is far from necessary though, because there is time, a medium that allows the events to be altered until the time t when the event actually comes to pass.

In order for physical determinism to be meaningful, the universe will have to be a both temporally and spatially finite and closed.

Bill,

Apologies for banging on about this. I think I see more clearly where we part company. I have gone back to my logic textbook (Hamilton, 'Logic for Mathematicians', CUP, 1988) Where you say A is valid iff

Necessarily, if P1 & P2, then C

Hamilton would say A is valid iff

P1 & P2 --> C

is *logically valid*, where logically valid is defined as true in every interpretation (of the underlying first order language). 'Truth in an interpretation' also has a technical meaning that's needed to clarify how quantification and the other logical operators are to be understood. An interpretation is a mathematical object, typically a set, with sufficient structure to model the constants, predicates, and functions of the first order language, together with a mapping from the language symbols to the math object which models their meanings. Roughly, sentences in the language are 'true in the interpretation' if they map to assertions about the object that are true. This gives us enough machinery to be able to prove that, say, (∀x(P(x)->Q(x)) & ∀x(Q(x)->R(x))) --> ∀x(P(x)->R(x)) is logically valid, in this technical sense.

Now I guess the philosophical issue here is the extent to which all this mathematical apparatus adequately captures the relevant concepts of logic. If you agree it does but then say that I am helping myself to modal concepts somewhere in the notion of 'true in every interpretation', that I should be talking about every *possible* interpretation, perhaps, then I will ask what is special about the application of maths to model logic? You must ask mathematicians to litter their speech with 'possible' in all areas of their subject. And I hope I have shown by the examples in an earlier comment that they find no need to do this. Although it often seems to help an informal understanding of a topic to slip in a few 'possibles' now and then, they are not required.

Could I make one last point? Let me quote again what you say under 'Validity as a Modal Concept':

"Some will be tempted to say that a valid deductive argument is one the logical form of which is such that no argument of that form has true premises and a false conclusion. But that leaves out something essential. There could be an argument A whose form F is such that no argument instantiating F has ever been given or ever will be given in which the premises are true and the conclusion false. But it doesn't follow that F is a valid argument form. So what we need to say is that a valid deductive argument is one whose form is such that no actual or possible argument of that form has true premises and a false conclusion. "

Would you agree that this has rather an empirical flavour to it? To demonstrate validity of form we must find all *actual* arguments of that form, spoken, written, or thought, and check their premises and conclusions against the state of the *actual* world. Furthermore, we have to repeat this procedure in every possible world. Only if we never find a false conclusion when premises are all true can we conclude that the argument has a valid form. Now I'd have thought that this is not how we conceptualise validity of form. We identify valid forms by intuiting something from their structure, or by applying the methods of mathematical logic (eg, truth tables). Of course, if we could complete your procedure then we'd find no discrepancy between your extensional approach and my intensional one.

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