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Category# Free logic

Summary

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.
In classical logic there are theorems that clearly presuppose that there is something in the domain of discourse. Consider the following classically valid theorems.
1.
2.
3.
A valid scheme in the theory of equality which exhibits the same feature is
4.
Informally, if F is '=y', G is 'is Pegasus', and we substitute 'Pegasus' for y, then (4) appears to allow us to infer from 'everything identical with Pegasus is Pegasus' that something is identical with Pegasus. The problem comes from substituting nondesignating constants for variables: in fact, we cannot do this in standard formulations of first-order logic, since there are no nondesignating constants. Classically, ∃x(x=y) is deducible from the open equality axiom y=y by particularization (i.e. (3) above).
In free logic, (1) is replaced with
1b. , where E! is an existence predicate (in some but not all formulations of free logic, E!t can be defined as ∃y(y=t))
Similar modifications are made to other theorems with existential import (e.g. existential generalization becomes .
Axiomatizations of free-logic are given by Theodore Hailperin (1957), Jaakko Hintikka (1959), Karel Lambert (1967), and Richard L. Mendelsohn (1989).
Karel Lambert wrote in 1967: "In fact, one may regard free logic... literally as a theory about singular existence, in the sense that it lays down certain minimum conditions for that concept." The question that concerned the rest of his paper was then a description of the theory, and to inquire whether it gives a necessary and sufficient condition for existence statements.
Lambert notes the irony in that Willard Van Orman Quine so vigorously defended a form of logic that only accommodates his famous dictum, "To be is to be the value of a variable," when the logic is supplemented with Russellian assumptions of description theory.

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