Free logic

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Free logic is a logic with no existential presuppositions. Alternatively, it is a logic whose theorems are valid in all domains, including the empty domain.

[edit] 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. $\forall xA \rightarrow \exists xA$ ;

2. $\forall xA \rightarrow A(r/x)$ (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. $Ar \rightarrow \exists xAx$ (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. $\forall x(Fx \rightarrow Gx) \land \exists xFx \rightarrow \exists x(Fx \land Gx)$ .

Informally, if F is '=y', G is 'is Pegasus', and we sub '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. $\forall xA \land E!t \rightarrow \exists xA$ , 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).

[edit] References

Lambert, Karel, 2003. Free logic: Selected essays. Cambridge Univ. Press.
-------, 2001, "Free Logics," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
------, 1997. Free logics: Their foundations, character, and some applications thereof. Sankt Augustin: Academia.
------, ed. 1991. Philosophical applications of free logic. Oxford Univ. Press.
Morscher, Edgar, and Hieke, Alexander, 2001. New essays in free logic. Dordrecht: Kluwer.