# Free logic

A **free logic** is a

**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^{[clarification needed]} 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))
^{[1]}^{[2]}^{[3]}^{[4]}

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),^{[5]} Jaakko Hintikka (1959),^{[6]} Karel Lambert (1967),^{[7]} and Richard L. Mendelsohn (1989).^{[8]}

## Interpretation

Karel Lambert wrote in 1967:^{[7]} "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 that 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

The point of free logic, though, is to have a formalism that implies no particular ontology, but that 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 George Nahknikian,^{[9]} which is that to exist is to be self-identical.

## See also

- Logical cube
- Logical hexagon
- Square of opposition
- Triangle of opposition
- Table of logic symbols

## Notes

**^**Reicher, Maria (1 January 2016). Zalta, Edward N. (ed.).*Nonexistent Objects – The Stanford Encyclopedia of Philosophy*. Metaphysics Research Lab, Stanford University – via Stanford Encyclopedia of Philosophy.- ISBN 9780300024043.
**^**Zalta, Edward N. (1983).*Abstract Objects. An Introduction to Axiomatic Metaphysics*. Dordrecht: Reidel.**^**Jacquette, Dale (1996).*Meinongian Logic. The Semantics of Existence and Nonexistence*. Perspectives in Analytical Philosophy 11. Berlin–New York: de Gruyter.- S2CID 34062434.
**.**- ^ .
**.****.**

**
**## References

- .
- ———, 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.

## External links

- Nolt, John. "Free logic". In Zalta, Edward N. (ed.).
*Stanford Encyclopedia of Philosophy*.