Does the free-logic-style semantics for traditional syllogisms imply that something exists?

Question

I'm looking at Option 3 in Bumble's answer here which references this article on existential import in term logic in the Logic Museum website.

Does the free-logic-style semantics for term logic presuppose the existence of at least one thing? Is there a way to avoid this problem without introducing new inference rules specific to existence?

There are a few features of term logic that are hard to salvage with a direct translation into first-order logic, such as the syllogism below.

AaB    All A are B
AaC    All A are C
---- -------------
BiC   Some B are C

First, let me define two different semantics for term logic: the first-order semantics and the free semantics.

The first-order semantics consists of a non-empty domain of discourse D, and a possibly-empty subset of D associated with each category A, B, C and so on. The existence predicate E is just the universal set.

The free-logic-style semantics consists of a non-empty domain of discourse D, as well as a necessarily-nonempty subset of D associated with each category A, B, C, and so on, and their infinite negations !A, !B, !C, and so on. Existence E is treated as an ordinary category/predicate, subject to the same rules as all the others. Free-logic-style semantics is not a special case of free logic. Free logic doesn't require that every predicate is inhabited in the full domain, which is needed to get AaB, AaC |- BiC to go through.

The first-order semantics deliberately makes bad choices when used to give an account of term logic; it does so in order to illustrate the difference between term logic and modern logic.

I'm assuming that option 3 means picking a semantics like the free-logic-style semantics defined above. D is analogous to the full domain (i.e. the union of the inner domain and outer domain in dual-domain treatments of free logic).

The free-logic-style semantics has the benefit of validating inferences like AaB, AaC |- BiC and of being parsimonious by not treating existence differently from any other predicate. It has the drawback of forcing us to commit to at least one real thing and at least one non-real thing in our semantics.

However, the article on existential import in the Logic Museum mentions neither presupposing the existence of at least one thing nor the presence of rules of inference specific to existence.

This makes me wonder whether I've missed something about what option 3 is intended to mean.

Answer 1

In a non-inclusive free logic with a single domain, (∃x)E!x is a theorem, i.e. it is a given that something exists. The existence predicate E! cannot be used like a predicate in a many-sorted logic. (∃x)(¬E!x) does not hold. In fact (∀x)E!x, i.e. "everything exists" is an axiom of free logic, at least in its traditional variants, so (∃x)(¬E!x) is ruled out.

If you want to apply free logic semantics to syllogistic logic, along the lines of Option 3 - no existential import, you are better off using a dual-domain approach. You have an inner domain of things that exist, to which the quantifiers ∀ and ∃ apply, so that (∀x)E!x holds, and an outer domain of possibilia or things that subsist, with separate quantifiers Π and Σ for universal and existential quantification. This allows you to write, (Σx)(¬E!x), i.e. some things do not exist, without contradicting (∀x)E!x.

This approach allows you to formalise "All A are B; all A are C; therefore, some B are C" using the Π and Σ quantifiers, and the conclusion may be true, even if no A's exist, i.e. when the extension of A is empty in the inner domain but non-empty in the outer.