A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain. A free logic with the latter property is an inclusive logic.

What's the difference between allowing terms that do not denote any object and allowing models that have an empty domain in free logic? Does allowing terms that do not denote any object mean you can have empty terms and allowing models that have an empty domain mean that every object in a domain can be empty terms? Doesn't the first imply the second? I don't understand the logic here.

  • See Free Logic Jun 6 at 5:55
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    If there are no objects in the domain, every term has no reference. But we can have logics with non-empty domains and terms that do not necessarily refer. Jun 6 at 5:55

Free logic being inclusive with empty domain will ensure its terms don't denote any ontic existent object, but it's not the only way. Not necessarily denoting any object in the quantificational domain is actually a crucial design feature of free logic apart from classic logic by an introduction of E!t predicate formula which is defined according to its SEP reference:

To distinguish terms that denote members of D from those that do not, free logic often employs the one-place “existence” predicate, E!. For any singular term t, E!t is true if t denotes a member of D, false otherwise.

So you may ask why we need to invent such a lax logic? Remember classical logic’s singular terms must denote existing things, so as same reference explained:

classical logic is unreliable in application to statements containing singular terms whose referents either do not exist or are not known to. Consider, for example, the true statement: (S) We detect no motion of the earth relative to the ether, using ‘the ether’ as a singular term for the light-bearing medium posited by nineteenth century physicists. The reason why (S) is true is that, as we now know, the ether does not exist. According to classical logic, however, (S) is false, because it implies the existence of the ether. Free logic allows such statements to be true despite the non-referring singular term.

In summary free logic is more ontologically inclusive than classic logic and might cover more cases during logical analysis as above example demonstrates when we're still unsure about something really exists or not.

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