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If you have a valid syllogism thats conclusion is true, have you gained any further truth?

I'll explain my reasoning.

The syllogism: All men are mortal. Socrates is a man Conclusion: Socrates is a mortal

This syllogism is valid and its conclusion is true it is also similar to this

If set A = mortal, contains men, And set B = men , contains socrates. Set B is contained in Set A

So from the the syllogism and the set theory you have gained no further truth. It's like saying if a set contains something, then it contains something, or if something is true, then it is true.

So with respect what's the use of syllogisms?

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Your criticism of syllogisms was made by John Stuart Mill. He put it slightly differently, but the point is similar. Mill's claim is that the syllogism, "All men are mortal; Socrates is a man; therefore, Socrates is mortal" is of no value, since we could not know that the premise, "All men are mortal" is true unless we already knew that Socrates is mortal. So we learn nothing from the syllogism. We might even say that it is question-begging.

The point might be applied to any deductively valid argument. How is a valid argument informative, if the premises seem to 'contain' the conclusion in some sense? One way to respond is to say that while deductive validity is a transitive relation, obvious deductive validity is not transitive. Each step in a long argument might be obviously valid, but the entire chain of steps might well not be, and so the argument as a whole can be informative.

This fits with what logicians do when proving an argument using natural deduction rules. They start from the premises and show how the conclusion can be reached in steps by using a small number of simple rules.

There are other ways to rebut Mill's criticism. Sometimes universals like "All men are mortal" are not mere inductive generalisations that require that we already know that Socrates is mortal. They might have a law-like nature and so we believe them to be true because they fit with our best scientific theories. Indeed this seems to be the case with "All men are mortal". That is a scientific fact, not just an observation.

Also, some universal statements are facts that we accept on authority or are just generally acknowledged. We can draw deductive inferences from them without begging the question. Other universal statements are true by fiat, i.e. they are true because some authority says they are true and that is that. This happens in legal statutes, for example.

More generally, deductive reasoning can often be informative by allowing us to combine information from multiple sources. We learn A from this source, we learn B from that source, and we put A and B together to deduce C. We didn't know C before because we didn't have all the required information. So syllogisms, and deductive reasoning generally, can allow us to gain knowledge that we didn't previously have.

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  • But, is obvious deductive validity transitive when we're only permitting categorical syllogisms? I think it might be, because they're simpler and more obvious than first order logic. I've searched a little for the computational complexity of deciding whether a conclusion follows from a set of premises via a series of categorical syllogisms, but haven't yet found anything. Do you happen to know? If the complexity of that is low enough, like O(n) or O(n log n) in the number of premises, then we might say that obvious deductive validity is transitive for that case.
    – causative
    Commented Oct 28, 2023 at 16:34
  • @causative In the case of a single categorical syllogism, I suppose the complexity must be linear because it is in effect just a simple lookup. If you identify obviousness with complexity then I agree a syllogism is obvious. In practice, it is one of those cases where everything seems obvious once you have understood it. But even simple syllogisms can be informative. The sign says: "all people whose names begin with A-D queue here"; my name begins with A-D; so I queue here. The inference is obvious but not question-begging.
    – Bumble
    Commented Oct 29, 2023 at 5:33
  • To decide a single categorical syllogism is constant time. But we aren't just asking whether a single syllogism is obvious - is a sequence of them also obvious? So the question is if we have N premises (N > 2) and we want to know whether a single conclusion follows from them by categorical syllogisms. If all statements are of the form "all A are B" then this can be done in O(N) time with a breadth-first search. Then if we also allow "no A are B" statements, I think I can see a solution that does a pair of breadth-first searches for O(N) time again. I haven't considered the other cases.
    – causative
    Commented Oct 29, 2023 at 5:53
  • @causative I wouldn't say an entire chain is obvious. The conclusion of a very long argument might be far from obvious. As to the complexity, verifying a fully specified chain would simply be linear in the length of the chain. Proving validity for a given set of premises has a much higher complexity. There is PhD thesis on the subject here: curate.nd.edu/downloads/und:8c97kp81k9k A cursory reading suggests that the problem is O(n!) and hence not polynomial. I believe the more general problem of first order validity with monadic predicates and finite structures is coNEXP-complete.
    – Bumble
    Commented Oct 29, 2023 at 6:43
  • I did find that paper when I looked earlier, but the problem he is addressing is that of finding all possible derivations to obtain the conclusion from the premises, not simply deciding whether the conclusion is obtainable from the premises. Also the paper seems to be of low quality as two paragraphs in the intro are repeated twice verbatim, and he says the problem of finding all the relevant trees "may be NP-complete" when it is obviously not even a decision problem. Perhaps you've spotted something in the paper I haven't, as I didn't read much beyond that.
    – causative
    Commented Oct 29, 2023 at 7:59
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Do you gain further truth from syllogisms

No. Syllogisms don't lead to a gain of knowledge, but to a narrowing down on the question "what subsequent assertions are permissible from a logical standpoint?". A syllogism ultimately instantiates something that beforehand was comprised in an abstraction.

A syllogism merely highlights how an assertion is consistent with [usually two] underlying premises. An inconsistency excludes that assertion from the scope of conclusions applicable, or associated to, those underlying premises.

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  • So to say: syllogisms don't extend our knowledge but our understanding? Commented Oct 28, 2023 at 21:50
  • @KristianBerry Not really. Understanding is already palpable in choosing the proper abstraction to instantiate. Commented Oct 29, 2023 at 13:08
  • I suppose it is a matter of epistemic language games, then, including various offshoots of games for words/phrases like "proper abstraction to instantiate," "understanding," "knowledge," etc. Is the deeper significance of this all relative to individual epistemic agents or is there some social value to playing the games in certain ways? Commented Oct 29, 2023 at 13:20
  • @KristianBerry Not sure I'm following your latest question, but I meant no language games. When selecting and arranging the premises to produce a syllogism, the person necessarily has a prior, thorough understanding of the meaning and scope of those premises. Otherwise the validity of that syllogism would be random. Commented Oct 29, 2023 at 13:37
  • I just meant that you are using "understanding" in one way, I was using it in another, so we are not speaking of the same thing in the same way, but the same thing in different ways and different things in the same way. Commented Oct 29, 2023 at 13:38

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