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Does the universal quantifier express existence? If I say "All dogs are cute", does the truth of that proposition require at least one existent dog?

My question is different from the suggested one as it is evolving in the comments section. Thanks.

Geoffrey Thomas
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user47634
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  • @Conifold I just read it. I think it does. So, do you agree with Mauro ALLEGRANZA that there is an assumption of a non-empty domain (in other words, that it implies existence)? – user47634 Jul 27 '20 at 16:47
  • @Conifold Could you give me an example of an existential proposition that is not formalised with a conditional form, so that it is false if something does not exist? – user47634 Jul 27 '20 at 16:55
  • If your question is different from the one linked by Conifold: how? – Natalie Clarius Jul 28 '20 at 15:06

2 Answers2

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"Standard" semantics for FOL assumes that there are no interpretations with empty domain.

Thus, in general, ∀xPx implies P[t/x], and thus ∃xPx.

But regarding specifically "All dogs are cute", this statement is true also when there are no dogs at all (see Vacuous truth).

The issue is related to the so-called problem of Existential import in Syllogism.

See also Free logic.

Mauro ALLEGRANZA
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  • Thanks, Mauro. Can you give me the meaning of "P[t/x]"? – user47634 Jul 27 '20 at 11:40
  • @user47634 - substitution of term t in place of x. From "Everyone is a Philosopher" to "Socrates is a Philosopher" to "There is at least one Philosopher" – Mauro ALLEGRANZA Jul 27 '20 at 11:53
  • Thanks again. So, I understand that universal propositions in the form of material conditionals will be subjected to vacuous truth. Could you give me an example of an existential proposition that is not formalised with a conditional form, so that it is false if something does not exist? – user47634 Jul 27 '20 at 16:51
  • Or this does not happen, precisely because there are no interpretations with empty domain? – user47634 Jul 27 '20 at 16:58
  • @user47634 - "an example of an existential proposition that is not formalised with a conditional form, so that it is false if something does not exist?" Example: "There is a square circle". – Mauro ALLEGRANZA Jul 28 '20 at 05:54
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I don't think so because:

1-by definition the truth is the property of being in accord with fact or reality.

2-for the predication to be meaningful, both the predicate and the subject (variable) need to be defined.

3-for the predication to be true or false the subject must exist to verify if the predicate is accord with the fact that the subject have it as a property or not.

Yassine Sifeddine
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