Free logic facts for kids

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.

Explanation

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).

See also

In Spanish: Lógica libre para niños

• Logical cube
• Logical hexagon
• Octagon of Prophecies
• Square of opposition
• Triangle of opposition
• Table of logic symbols