Free logic is a logic free of existential presuppositions. Alternatively, it is a logic whose theorems are valid in all, including the empty, domain. Jump to: navigation, search Logic (from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). ...

In classical logic there are theorems with existential import. Consider the following classically valid theorems. Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ...

1. ;

2. (where r does not occur free for x in Ax and A(r/x) is the result of substituting r for all free occurrences of x in Ax);

3. (where r does not occur free for x in Ax).

From (1) in the unrestricted predicate calculus, if A is 'is a unicorn' it can be inferred that a unicorn exists. A simple solution is to restrict the inference (1) to First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ...

1a. .

However, in identity theory (i.e. the predicate calculus with equality) a substitution instance of (1) is

1b. .

If F is '=y', G is 'is Pegasus', and we sub 'Pegasus' for y, then (1a) implies from 'everything identical with Pegasus is Pegasus' that something is identical with Pegasus. The problem comes from substituting nondesignating constants for variables. Classically, ∃x(x=y) is deducible from the open equality axiom y=y by particularization (i.e. (3) above). Hence in (1b) a valid inference is from ∀x(Fx → Gx) to ∃x(Fx ∧ Gx).

In free logic, (1a) is replaced with

1c. , where E! is an existence predicate defined as ∃y(y=x).

Similar modifications are made to other theorems with existential import (e.g. the Rule of Particularization becomes (Ar → (E!r → ∃xAx)).

Axiomatizations of free-logic are given in Hintikka (1959), Lambert (1967), Hailperin (1957), and Mendelsohn (1989).

Sources

K. Lambert, "Existential Import Revisited", Notre Dame Journal of Formal Logic, October 1963, p.288-292