Free Logic - Explanation

Explanation

In classical logic there are theorems which clearly presuppose that there is something in the domain of discourse. Consider the following classically valid theorems.

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).

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. 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).