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My professor defines logical validity (in the English language) like so:

'An argument is logically valid if and only if there is no (uniform) interpretation (of subject-specific expressions) under which the premises are all true, and the conclusion is false.'

He contrasts subject-specific expressions (e.g., Donald Trump, Aristotle, chemical element, London), with logical expressions (i.e., if, not, if and only if, every, some). Logical expressions are not subject to re-interpretation; they keep their standard English meanings all the time.

My question is this: is the following argument valid?

P1: Santa Claus does not exist. C: Something does not exist.

Now, on the one hand, I'm inclined to say yes: if I replace 'Santa Clause' with any other noun, or I replace the property of not existing with any other property, the resulting argument is such that: if the premises are true, so too is the conclusion.

On the other hand, I'm hesitant to say yes: if I replace 'something' with, for example, 'a car', then the resulting argument seems to involve a true premise, and a false conclusion.

That said, I'd unhesitatingly say that the following argument is valid:

P1: Santa Claus does not exist. P2: Santa Claus is something. C: Something does not exist.

Could Santa Clause not be 'something'? From another angle: is 'something' a 'subject-specific expression'? I'm inclined to think it is not, but I'm not sure how to justify this thought. (My inkling is that it has something (lol) to do with the fact that 'something' is a pronoun, whilst 'a car' is a noun? Also, I'm aware that the argument in question involves a valid rule of inference in FOL. But I wonder if this is one of those cases where validity in FOL comes apart from more informal characterisations of validity in the English language (e.g., http://www.jimpryor.net/teaching/courses/intro/notes/leibniz-epist.html).)

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    Let R(x) be the predicate "x is real", then you are going from ¬R(s) to ∃x¬R(x). This is a valid case of universal generalization. But you can not replace "something" with "a car", "something" does not function like a predicate name logically, unlike "being a car". Going from ¬R(s) to ∃x(C(x)∧¬R(x)) is invalid. If Santa Clause was a car, i.e. if we also had C(s), then you could validly get there. – Conifold Oct 5 '20 at 22:00
  • In order to avoid fallacies, we have to find a way to "translate" something without using the existential quantifier, otherwise the argument will be invalid in FOL: from ** Santa=Santa** by equality axioms we have ∃x(x=Santa). FOL is consistent, and thus we cannot derive: ¬∃x(x=Santa). – Mauro ALLEGRANZA Oct 6 '20 at 14:53
  • A possible way out mat be with a predicate RoundSquare(x); in this way we may express the fact that this predicate is not instantiated: ¬∃xRoundSquare(x). If we move to Second Order Logic, we may derive ∃P¬∃xP(x), that can be a "reasonable" symbolization of "Something does not exist". – Mauro ALLEGRANZA Oct 6 '20 at 14:54
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There is a rule of first-order predicate logic (FOPL) called existential generalization that allows you to go from "Fred is a mechanic" to "something is a mechanic". But in standard logic, there are two issues with your example. Firstly, you cannot name things that do not exist, so "Fred is a mechanic" fails to be a proposition if there is no Fred. Secondly, existence is not treated as a predicate. Existing is not a property that some things have and others don't. So under these assumptions, your P1 "Santa Claus does not exist" fails because there is no thing that has the name Santa Claus, and C fails because it attempts to predicate non-existence of some thing.

One approach to fixing this is to treat names as definite descriptions in disguise. So if we treat the name Santa Claus as meaning something like "the fat man with the white hair and beard who appears at Christmas time and gives presents to children" then we can write P in predicate logic as

¬(∃x)(Fat(x)^Man(x)^Whitehair(x)^Whitebeard(x)^Appearsatxmas(x)^Givespresentstochildren(x))

That fixes P, but we cannot then get to C. Also, it is a contentious issue in the philosophy of language as to whether it is correct to treat names as definite descriptions. An important rival theory, the causal theory of names, is championed by Kripke among others.

Another approach to fixing it is to adopt one of the 'free' logics. These are logics that are similar to FOPL but have different rules for quantification. In one version, we distinguish between things that really exist and those that are fictional, and we have different rules for quantifying over each. In this logic, Santa Claus is a name, not a definite description, but it names a fictional entity. We can then read P1 as saying that Santa Claus is not a real thing, and C as saying there is at least one thing that is not real. Both would then be correct.

The problem with free logic is that it introduces a number of difficulties in specifying the semantics, compared with standard FOPL. For many logicians, the drawbacks outweigh the advantages. It is true to say, however, that standard logic imposes limitations that are not present in ordinary English usage. A canonical example is the sentence, "Sherlock Holmes is more famous than any real detective". This is perfectly understandable, and probably true, but it suffers the same problems as your example.

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    I do not think we need free logic for such elementary examples. We can simply have a domain of discourse that includes fictional entities with a reality predicate defined on it, and use the usual FOPL. – Conifold Oct 6 '20 at 0:34
  • @Jayjay45 For background, see What does Kant mean by “Existence is not a predicate”?. – J D Oct 6 '20 at 16:13
  • @Conifold It may be overkill but in my opinion it's the natural thing to do if we buy this as a compelling concern. – Noah Schweber Oct 9 '20 at 23:00
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Presumably, good logic allows us to use language to assert meaningfully, and either truly or falsely, that God does not exist.

So, yes, it is obviously true that if Santa Claus doesn't exist, then something doesn't exist, namely, whatever I am talking about when I am talking about Santa Claus, which everybody proficient in English will easily understand.

You cannot substitute "a car" to "something", because it is not true that if something is F, then a car is F. This is simply because something is not necessarily a car.

You could substitute "something" to "a car", however, since if a car is F, then something is F, which is why it is also true to say that if Santa Claus doesn't exist, then something doesn't exist.

The reason is that both a car and Santa Claus are something.

So, this follows straightforwardly from the transitivity of the implication.

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