Wednesday, May 27, 2015

Classification-Based Validity

Perhaps the most commonly used account of validity is modal; it is the idea that an argument is valid when, its premises being true, the conclusion cannot be false. This account of validity makes the following argument valid:

Socrates is human;
therefore Socrates is an animal.

However, it would often be said that this is not valid in terms of its form; it is not formally valid but materially valid. Since anything formally valid that does not equivocate would certainly have also to be materially valid, the question arises as to what the principle is identifying an argument as formally valid. The usual principle suggested is some variant of Buridan's idea that it has to hold for all terms, keeping the form common to all of them the same. Thus people would usually say that the above argument is not formally valid because it would not be invariant under a consistent but arbitrary substitution of terms:

New Orleans is a city;
therefore New Orleans is a river.

But I'm not sure we should let this pass so easily. For one thing, 'keeping the form the same' has to apply to the actual logical principles used in drawing the conclusion, and it seems clear enough that the New Orleans argument does not use the same logical principles to draw its conclusion that the Socrates argument does. The conclusion in the Socrates argument 'works' because it simply moves the predicate from specific to general; that's what pretty much anyone making the argument would be doing. But this is not involved at all in the New Orleans argument.

One could perhaps argue that this is somehow not part of the argument's form, but it's difficult to see how it wouldn't be. It certainly would be if I did something like this:

Socrates is an animal that is rational;
therefore Socrates is an animal.

But there's not any obvious way in which this is different from the other. Logical terms are not words but meanings, so if 'human' includes 'animal' as part of its definition, the movement from 'human' to 'animal' should be as formal as the movement from 'animal that is rational' to 'animal', if they both proceed on the same principles, which they appear to do.

One could perhaps argue that the original is an enthymeme:

Socrates is human;
(everything human is an animal);
therefore Socrates is an animal.

This is certainly formally valid. But it's unclear that the implicit premise actually adds anything that is not already in the first premise, for exactly the reason just noted.

If one took each logical term to be a 'slot', so to speak, in a classification system, then an argument's form relates these 'slots' to each other; but then one would expect species-to-genus inference to be as formal as subalternation, universal instantiation, or inference involving the dictum de omni et nullo.

It has been noted by others that, while we can fairly easily handle the notion of logical form in particular logical systems, we don't have any general account of logical form. This is a related issue, I think, since there appears to be no useful account, applicable to the full range of arguments people would want to consider formally valid, that also obviously rules out the possibility that the classification of the terms can be part of the formal structure of an argument.

11 comments:

  1. John West11:50 PM

    Hello Brandon,


    Socrates is human;
    therefore Socrates is an animal.

    However, it would often be said that this is not valid in terms of its form; it is not formally valid but materially valid.


    Could you explain what you mean by "material validity"? I'm assuming material validity means something like "The argument follows in the case of that premise and conclusion", but I want to be sure. Thank you in advance.

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  2. John West12:39 AM

    Sorry. "... something like 'the conclusion follows from the premise in the case of that premise and that conclusion.'"

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  3. branemrys8:33 AM

    Hi, John,

    The only difference between material and formal validity -- which are old, old terms still in occasional use by logicians -- is that with formal validity the conclusion follows 'in virtue of the logical form'. Both material and formal validity are exactly the same otherwise.Thus, if you had a modal account, it would be that 'if the premises are true, the conclusion must be true'.

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  4. John West11:13 AM

    Thanks.

    Okay. And as a real essentialist you would say that in material validity the conclusion follows in virtue of not merely the internal content or meaning of a term, but in virtue of the essence of the thing being discussed?

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  5. branemrys12:21 PM

    The post is not getting into anything like that; the argument would work just as well in either case, since on either view terms can be related by being more specific and more general, which is all that's actually required. Material validity, like formal validity, is a term that is intended only to identify how an argument works; it tells us nothing about what grounds the argument, if anything.

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  6. John West12:58 PM

    Okay. Thanks Brandon. I'm really not familiar with the terminology of material validity (also somewhat sleep deprived), but it sounds like it's like intension. A few comments, hopefully not too tangental to the argument.

    Perhaps the most commonly used account of validity is modal; it is the idea that an argument is valid when, its premises being true, the conclusion cannot be false.

    I think modality is nowhere near as well behaved as the extensional concept of validity (Herodotus is a man and has a beard is a valid form (Fa and Ga, therefore Fa). The inference holds good for all reinterpretations of F and G.) So, I want to get away from the modal concept of validity.

    Instead of modality, another option is to understand validity as determined by the theorems of (say) first-order logic, which can be done without appeal to modal notions. For example, ∀x(if Fx then Fx) is logically valid since every choice of a predicate F makes the scope true for every value of x.

    It has been noted by others that, while we can fairly easily handle the notion of logical form in particular logical systems, we don't have any general account of logical form. This is a related issue, I think, since there appears to be no useful account, applicable to the full range of arguments people would want to consider formally valid, that also obviously rules out the possibility that the classification of the terms can be part of the formal structure of an argument.

    I may misunderstand. But maybe a distinction can be drawn between theorems that are valid in reality and theorems that are valid in a system, fictionally, but not in reality. If that's so, I'm not sure it's that big a problem that “we can fairly easily handle the notion of logical form in particular logical systems, [but] don't have any general account of logical form,” because surely, not every logical system lines up with the way reality is. Only some of them would be valid in reality and so as long as arguments are valid in those systems, that seems fine.

    Maybe this would make parts of logic into branches of metaphysics, but I don't see what's wrong with that. There doesn't seem to me anything wrong with saying, for example, that some of the most basic truths of logic and some of the most basic truths of reality are one, like non-contradiction.

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  7. branemrys3:20 PM

    There is no pre-modal concept of validity; validity itself is a modal notion, because it has to cover all possibilities. (You yourself slip it back in with the notion of 'all reinterpretations'. All actual reinterpretations that just happen to be made by Joe on Tuesday? No, all possible reinterpretations.) But what you are calling the extensional concept of validity doesn't affect the argument one way or another; it starts with the concept of logical form, and it is the concept of logical form, not modality, that is at issue in the post.

    (x) (Fx->Fx) is not an argument, so it is not valid or invalid unless you are equivocating between validity of statements and validity of arguments. But that aside, your example runs into the obvious problem that it cannot be a general account of validity, because it becomes completely useless once we move outside of predicate calculus. It would actually make it impossible, for instance, to determine whether natural language arguments are valid, because natural language arguments are not predicate calculus arguments. (And there is certainly no one-to-one correspondence between natural language arguments and predicate calculus arguments; it is extremely easy to come up with examples of natural language arguments that can be given invalid or valid predicate calculus translations depending entirely on whether you think the original natural language argument is valid or not.) Your account would not be of any use for any of the examples of arguments I actually use in the post, for instance, because none of them are in the predicate calculus.

    I don't understand what you mean by 'valid in reality' and 'valid in a system, fictionally', or why introducing metaphysics here would be useful.

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  8. John West3:52 PM

    There is no pre-modal concept of validity; validity itself is a modal notion, because it has to cover all possibilities. (You yourself slip it back in with the notion of 'all reinterpretations'. All actual reinterpretations that just happen to be made by Joe on Tuesday? No, all possible reinterpretations.)


    Hm. I'll have to get back to you on this, but I did run such a question by a professor who specializes in the field via email to make sure I wasn't talking pointlessly before writing my comment to you. He replied: “Validity is an amodal, or premodal concept. Modality is still being sorted out, whereas validity is well understood.” as if it were as obvious as the sun in the sky. Obviously, there's an inconsistency between these two replies that I need to look at.

    But what you are calling the extensional concept of validity doesn't affect the argument one way or another; it starts with the concept of logical form, and it is the concept of logical form, not modality, that is at issue in the post.

    Well, precisely, which is why I didn't present it as an objection to your argument, but rather as part of comments on statements in it. I was talking on the back of the first pargraph you've contested.

    (x)(Fx->Fx) is not an argument, so it is not valid or invalid

    I never claimed it as an argument. Theorems can be valid or invalid as well.

    unless you are equivocating between validity of statements and validity of arguments.

    The point was that you can deduce directly from such axioms and theorems.

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  9. branemrys4:23 PM

    It's possible that the professor in question had some particular notion of "amodal" or "premodal" in mind. But I don't know what he would mean. Historically, validity has most commonly been explicitly defined in modal terms (e.g., necessary truth-preservation); it's still commonly introduced to undergraduates in modal terms (and usually in ways treating the modal account as our intuitive notion of validity); it's still often discussed in such terms in philosophy of logic work on validity; and while one can certainly formulate an account of validity for a system without explicit appeal to modal notions, this is distinct from saying it is itself an amodal or premodal concept. Also, I don't know how one would completely sever the two given that validity in modal logic is always put in modal terms.

    Theorems can be valid or invalid as well.

    Yes, I know, but this sort of validity is not in view here, and I don't think that there is any sort of universal connection between this kind of validity and the validity of arguments, so I would regard switching from one to the other without specific justification to be problematic. Validity of theorems is a useless concept, for instance, when dealing with natural language arguments.

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  10. John West4:39 PM

    It's possible that the professor in question had some particular notion of "amodal" or "premodal" in mind. But I don't know what he would mean. Historically, validity has most commonly been explicitly defined in modal terms (e.g., necessary truth-preservation); it's still commonly introduced to undergraduates in modal terms (and usually in ways treating the modal account as our intuitive notion of validity); it's still often discussed in such terms in philosophy of logic work on validity; and while one can certainly formulate an account of validity for a system without explicit appeal to modal notions, this is distinct from saying it is itself an amodal or premodal concept. Putting it in model-theoretic terms for a specific logical system, for instance, could probably make it possible to avoid specific appeal to modal notions, but it wouldn't tell us anything about whether validity were itself modal or not. Also, I don't know how one would completely sever the two given that validity in modal logic is always put in modal terms.

    Yeah, I once ran an argument to this effect. That any person making an argument for modal anti-realism tacitly makes use of the very notions they deny, and was given a reply along the lines of my "theorems of first-order logic" paragraph. I'll report back after I look into this further.

    Yes, I know, but this sort of validity is not in view here, and I don't think that there is any sort of universal connection between this kind of validity and the validity of arguments, so I would regard switching from one to the other without specific justification to be problematic. Validity of theorems is a useless concept, for instance, when dealing with natural language arguments.



    Okay. I confess, my initial reaction to your earlier paragraph about natural language arguments was a semi-flippant: "Well, if that's all it is, of course natural language arguments are imprecise! Why should I be bothered by that. That's why we... "


    But on further reflection I can see the value in this, so if it's not too much trouble, could you point me to where I can read more about natural language arguments? Thanks.

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  11. branemrys5:11 PM

    I don't know of any text looking at natural language arguments as distinct from other arguments -- there are, as you say, reasons why people tend to formalize, although most people would, I think, take a an account of validity to be something of a failure if "All dogs are canines; therefore all dogs are canines" couldn't be evaluated at all as a valid or invalid argument on the sole grounds that it is in English. A considerable amount of early work in analytic philosophy, of the Russell type, was an attempt to work through how the formal notions could apply to natural language arguments. But you might also find some interesting things in cases of people developing logical systems that try to do justice to natural language arguments -- Fred Sommers, for instance.

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