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.

  • Not very clear... if we assume the empty domain with an existence predicate that is considered a "usual" predicate, then the interpretation of E will be obviously the empty set. Jan 7 at 8:20
  • Having said that, if we consider traditional syllogism without the assumption that every term (universal) must be instantiated, if we assume that at least one predicate is not-emptythis means that the domain itself is not-empty. Jan 7 at 8:40
  • 1
    Not sure I understand the question, but it looks like you are talking about translating syllogisms into predicate logic, in which case the only predicates that would have to be non-empty are those that are part of the language of syllogisms. E would be part of the metalanguage of predicate logic. Jan 7 at 14:59
  • Per Lambert free logic is more ontologically neutral than its more stringent cousin aka classic logic and makes Quine's famous dictum to be is to be a value of a bound variable possible, thus it would not imply any singular existence based upon its own semantics or any model therein without further assumptions... Jan 8 at 1:37
  • @DavidGudeman I edited the question; hopefully it's clearer. I'm not trying to translate syllogisms into predicate logic per se. I'm trying to understand one possible choice for giving them a semantics with sets of entities. It feels like we're faced with a choice of either a) accepting that we have at least one real and one non-real entity in all of our models or b) positing new inference rules specific to existence. Since neither (a) nor (b) appear in what I've read so far, I'm wondering if I'm missing something about what a free-logic-style semantics for term logic is supposed to look like. Jan 8 at 3:02

1 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.

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .