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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 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).

Interpretation

Karel Lambert wrote in 1967^[1]:

"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 accomodates his famous dictum, "To be is to be the value of a variable," when the logic is supplimented with Russelian 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 $\exists xFx\rightarrow (\exists x(E!Fx))$

References

^ Free Logic and the Concept of Existence by Karel Lambert, Notre Dame Journal of Formal Logic, V.III, numbers 1 and 2, April 1967

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.

@@ Line 1: / Line 1: @@
+'''Free logic''' is a [[logic]] with no [[existential]] [[presupposition]]s. 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.
-==Speedy deletion of [[:Matt Kay]]==
-[[Image:Ambox warning_pn.svg|48px|left]] A tag has been placed on [[:Matt Kay]], requesting that it be speedily deleted from Wikipedia. This has been done under [[WP:CSD#G1|section G1 of the criteria for speedy deletion]], because the page appears to have no meaningful content or history, and the text is unsalvageably incoherent. If the page you created was a test, please use the [[Wikipedia:Sandbox|sandbox]] for any other experiments you would like to do. Feel free to leave a message on my talk page if you have any questions about this.
+:1. <math> \forall xA \rightarrow \exists xA</math>;
-If you think that this notice was placed here in error, you may contest the deletion by adding <code>{{tl|hangon}}</code> to '''the top of [[:Matt Kay|the page that has been nominated for deletion]]''' (just below the existing speedy deletion or "db" tag), coupled with adding a note on '''[[ Talk:Matt Kay|the talk page]]''' explaining your position, but be aware that once tagged for ''speedy'' deletion, if the article meets the criterion it may be deleted without delay. Please do not remove the speedy deletion tag yourself, but don't hesitate to add information to the article that would would render it more in conformance with Wikipedia's policies and guidelines. Lastly, please note that if the article does get deleted, you can contact [[:Category:Wikipedia administrators who will provide copies of deleted articles|one of these admins]] to request that a copy be emailed to you. <!-- Template:Db-nonsense-notice --> <!-- Template:Db-csd-notice-custom --> [[User:TrulyBlue|TrulyBlue]] ([[User talk:TrulyBlue|talk]]) 18:48, 10 October 2008 (UTC)
+:2. <math> \forall xA \rightarrow A(r/x)</math> (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. <math> Ar \rightarrow \exists xAx</math> (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. <math> \forall x(Fx \rightarrow Gx) \land \exists xFx \rightarrow \exists x(Fx \land Gx)</math>.
+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, &exist;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. <math> \forall xA \land E!t \rightarrow \exists xA</math>, where E! is an existence predicate (in some but not all formulations of free logic, E!t can be defined as &exist;y(y=t)).
+Similar modifications are made to other theorems with existential import (e.g. the Rule of Particularization becomes (Ar &rarr; (E!r &rarr; &exist;xAx)).
+[[Axiom|Axiomatizations]] of free-logic are given in [[Hintikka]] (1959), [[Lambert]] (1967), [[Hailperin]] (1957), and [[Mendelsohn]] (1989).
+==Interpretation==
+[[Karel Lambert]] wrote in 1967<ref><i>Free Logic and the Concept of Existence</i> by Karel Lambert, Notre Dame Journal of Formal Logic, V.III, numbers 1 and 2, April 1967</ref>:
+"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 accomodates his famous dictum, "To be is to be the value of a variable," when the logic is supplimented with Russelian 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 <math> \exists xFx \rightarrow (\exists x(E!Fx))</math>
+[[Category:Philosophical logic]]
+==References==
+{{reflist}}
+* 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.
+==See also==
+* [[Square of opposition]]
+[[cs:Volná logika]]
+[[pl:Logika wolna]]
+[[zh:自由逻辑]]
+[[es:Lógica libre]]