Free Logic

Classical predicate logic proves that something exists. The sentence $\exists x\, (x = x)$ is a theorem, derivable from the reflexivity of identity and the rule taking a formula with a term to its existential generalisation, and it is true in every model because models are required to have non-empty domains.

A logical truth that rules out the possibility of nothing existing is doing more than logic should. That is one motivation for the systems in this article. The other is that the same rule produces falsehoods from true premises whenever a term fails to denote.

The restriction is the whole of the formal change. From $\forall x\, \varphi$ one may infer $\varphi[t/x]$ only given $\exists y\, (y = t)$, usually abbreviated by an existence predicate. Everything that goes wrong in the classical treatment goes wrong at that step, and everything free logic does follows from repairing it.

The motivating cases are ordinary. Fictional and mythological names appear in true sentences: Pegasus is a winged horse, Vulcan was posited to explain Mercury's perihelion. Improper definite descriptions denote nothing: the greatest prime, the present king of France. Partial functions produce undefined terms: division by zero, the limit of a divergent sequence. Variable domain modal semantics requires terms that denote at one world and not another, which is where the previous modal article left the matter. In each case classical logic either assigns an arbitrary denotation or forbids the sentence, and neither is satisfactory.

Three variants

The systems differ over what to say about an atomic sentence containing a non-denoting term, and the three answers exhaust the options.

Positive free logic is the most permissive and the most used, since it takes at face value the truth of claims about fictional and theoretical objects. It needs a semantics that supplies something for the term to be about.

The construction is straightforward and the interpretation is contested. On a Meinongian reading the outer domain contains non-existent objects, which many find unacceptable. On a deflationary reading it is a technical device with no ontological commitment, the outer objects being surrogates rather than things, and this is how most working uses treat it.

Negative free logic avoids the outer domain by stipulating falsity, which handles identity smoothly, since $t = t$ comes out false for an empty $t$ and no object is needed. The cost is that the truths about fiction and theory are all false, and the theory must explain their apparent truth by paraphrase or by a story-operator.

Neutral free logic assigns a gap and inherits the machinery of the many-valued article, including the question of how compounds behave. It aligns with the view that a sentence with a non-denoting subject fails to express a proposition, which is Strawson's position on presupposition failure, and its supervaluational version is the standard formalisation of that position.

The empty domain

Admitting it costs one theorem, $\exists x\, (x = x)$, and gains the ability to reason about a domain one does not know to be occupied. The practical case is the one Quine noted: a theory of a hypothesised kind of particle should not have its existence as a theorem of logic, and the natural regimentation of "there are no unicorns" should not be logically false in a language containing a name for one.

The relation to definite descriptions is where free logic and Russell's theory diverge. Russell eliminated descriptions by paraphrase, so that a sentence about the present king of France is false rather than truth-valueless, and no term fails to denote because no term is a term. Free logic keeps the description as a singular term and adjusts the logic instead. The outcomes differ for negation: Russell's theory makes the negation of the false sentence true on the wide scope reading, while a neutral free logic makes both gappy, and this is the standard test case between the approaches.

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Modal logic has appeared in this tier as a formal subject with several semantics. It is also an applied one, and three of its applications have distinctive axioms, distinctive paradoxes and enough independent development to be treated together.