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The various descriptions of the semantics of First Order Logic that I have seen all require that the domain is non-empty.

Why this restriction?

It means that certain sentences e.g. $\exists x \; p \lor \neg p$, which seem to me to be contingent, come out as always true.

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  • $\begingroup$ It's not a restriction out of need. That restriction is due to the fact that there's not much to discuss about the empty universe and it's troublesome to always consider the logical truth: the universe isn't empty or the universe is empty. So it's usual to assume it's not empty and ignore the empty universe. $\endgroup$
    – Git Gud
    Jul 22, 2013 at 12:31
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    $\begingroup$ It allows you to infer $\exists x \phi$ from $\forall x\phi$. More precisely, this comes from $\forall x\phi(x)\vdash \phi(a)$ and $\phi(a)\vdash\exists x\phi(x)$ and for the first o ftehse the non-empty domain is needed. $\endgroup$ Jul 22, 2013 at 12:34

2 Answers 2

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On one side, there is some irritation in having to deal with the empty structure as a special case.

  1. There are various useful identities that are only valid in a non-empty domain, for example: $$ \big ( [ (\forall x) \phi ] \to \psi \big ) \equiv (\exists x) [\phi \to \psi] $$ when $\psi$ does not mention $x$. These identities are used to put formulas in prenex normal form.

  2. Even the definition of "truth in a structure" requires the domain to be nonempty. A common way of defining truth in a structure requires a function assigning the variables in the language to objects in the domain; there is no such function when the domain is empty, and the entire definition fails in that case. By contrast, the definition of truth for an empty structure is a different, ad hoc definition.

On the other side, one of the main goals of first-order logic is to formalize mathematical objects such as groups, equivalence relations, etc. In many cases, these objects must have a nonempty domain (e.g. groups) or are uninteresting when it is empty (e.g. equivalence relations, posets). So there is little motivation to include the empty domain as a possible model, because it will either not be a model of the theories being studied, or will not be a model of mathematical interest.

It is certainly possible to study logic in which models may be empty; more generally, one studies "free logic" in which terms may not have referents. The reason that this is not done in most texts is that the authors don't see the effort as justifying the benefit.

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This is done to avoid some fairly technical problems. For example, consider the statement $\forall x \bot$. This is true in the empty structure, but from it we can (if we aren't careful) infer $\bot$, which is of course false in every structure. The simplest way to get around this is to ban the empty structure. There are other ways, however, involving things like slightly weakening the rules of inference. P.T. Johnstone does this in his "Notes on logic and set theory". He argues that it's better to make your logic slightly fiddlier than it is to make it unable to talk about perfectly good structures like (say) the empty poset.

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    $\begingroup$ Looking at Johnstone's little book for the first time in years, I now notice he gets the definition of truth in the empty structure wrong. Oh Well. $\endgroup$ Jul 22, 2013 at 12:43

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