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I was thinking about this and I believe that some of the popular equivalences, like this one: $\neg \forall x P(x) \equiv \exists x \neg P(x)$ won't hold up anymore. Is this correct? But equivalences like that are very useful, what else would we lose if we admit empty Domain of Discourse? If we were to admit an empty set as the UD what else would change?

zlaaemi
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    Every "for all" statement would be true and every "there exists" statement would be false. (In other words, there wouldn't be any point in having quantifiers at all.) In particular, the equivalence you mention does in fact hold up. – Greg Martin Apr 20 '23 at 21:45
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    Allowing the universe to be empty means that the universe is possibly empty (though it might be nonempty); @Greg's observation does apply for a certainly-empty universe. – ryang Apr 21 '23 at 04:06

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Most authors require the domain to be nonempty, but not all. For example, Hodges’ A Shorter Model Theory allows empty domains. As you might guess, some results become simpler and others more complicated.

As an easy example, Hodges defines the substructure $\langle Y\rangle$ generated by a subset $Y$ of the domain of a model. If $Y=\varnothing$ and the language has no constants, then $\langle Y\rangle=\varnothing$. One has to add a clause that $\langle Y\rangle$ is undefined in this situation, with the usual convention.

As a more elaborate example, Hodges has a criterion (Theorem 7.4.1) for a theory $T$ to have quantifier elimination. The theorem can be rescued, but becomes a touch messier.

Mendelsohn's Introduction to Mathematical Logic (5th ed.) has a section "Quantification Theory Allowing Empty Domains". He says, "an interpretation with an empty domain has little or no importance in applications of logic", but obviously Hodges would disagree. Even with this negative judgment, Mendelson is able without much trouble to come up with semantics and an axiom system. He also includes these references:

(1) Hailperin, T. (1944) A set of axioms for logic, JSL, 9, 1–19. (1953) Quantification theory and empty individual domains, JSL, 18, 197–200.

(2) Mostowski, A. On the rules of proof in the pure functional calculus of the first order. JSL, 16, 107–111.

(3) Quine, W.V. (1954) Quantification and the empty domain, JSL, 19, 177–179 (reprinted in Quine, Selected Logical Papers, Random House).

Finally, see here for a possible role for an empty domain in a particular categorical approach to first-order logic.

Michael Weiss
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