Free logic: Wikis

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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.

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 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. $\forall xA \land E!t \rightarrow \exists xA$, 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)[1], Lambert (1967), Hailperin (1957), and Mendelsohn (1989).

Interpretation

Karel Lambert wrote in 1967[2]:

"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 which 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 Quine so vigorously defended a form of logic which 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. He criticizes this approach because it puts too much ideology into a logic which is supposed to be philosophically neutral. Rather, he points out, not only does free logic provide for Quine's criterion--it even proves it! This is done by brute force, though, since he takes as axioms $\exists xFx \rightarrow (\exists x(E!Fx))$ and $Fy \rightarrow (E!y \rightarrow \exists xFx)$, which neatly formalizes Quine's dictum. So, Lambert argues, to reject his construction of free logic requires you to reject Quine's philosophy, which requires some argument and also means that whatever logic you develop is always accompanied by the stipulation that you must reject Quine to accept the logic. Likewise, if you reject Quine then you must reject free logic. This amounts to the contribution which free logic makes to ontology.

The point of free logic, though, is to have a formalism which implies no particular ontology, but which merely makes an interpretation of Quine both formally possible and simple. An advantage of this is that formalizing theories of singular existence in free logic brings out their implications for easy analysis. Lambert takes the example of the theory proposed by Wesley C. Salmon and Nahknikian[3], which is that to exist is to be self-identical. Yet, formalized by free logic, we get E!y =df y = y. Yet anything satisfies this definition, even Pegasus and round squares! This is intolerable. He then proceeds to criticize theories presented by Descartes and Henry Leonard.

References

1. ^ Jaako Hintikka (1959). Existential Presuppositions and Existential Commitments. Journal of Philosophy 56 (3):125-137.
2. ^ Free Logic and the Concept of Existence by Karel Lambert, Notre Dame Journal of Formal Logic, V.III, numbers 1 and 2, April 1967
3. ^ George Nahknikian and Wesley C. Salmon, "'Exists' as a Predicate" Philosophical Review Vol. 66: 1957 pp. 535-542
• 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.