A free logic is a logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 with fewer existential

Existential clause

Existential clauses are clauses that indicate only an existence. In English, they are formed with the dummy subject construction with "there", e.g. "There are boys in the yard". Many languages do not require a dummy subject, e.g. Finnish, where the sentence Pihalla on poikia is literally "On the...

 presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models

Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....

 that have an empty domain

Empty domain

In first-order logic the empty domain is the empty set having no members. In traditional and classical logic domains are restrictedly non-empty in order that certain theorems be valid. Interpretations with an empty domain are shown to be a trivial case by a convention originating at least in 1927...

. A free logic with the latter property is an inclusive logic.

Explanation

In classical logic

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

 there are theorems which clearly presuppose that there is something in the domain of discourse

Domain of discourse

In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...

. Consider the following classically valid theorems.

1. ;

2. (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. (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. .

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

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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. the Rule of Particularization becomes (Ar → (E!r → ∃xAx)).

Axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

atizations of free-logic are given in Hintikka

Jaakko Hintikka

Kaarlo Jaakko Juhani Hintikka is a Finnish philosopher and logician.Hintikka was born in Vantaa. After teaching for a number of years at Florida State University, Stanford, University of Helsinki, and the Academy of Finland, he is currently Professor of Philosophy at Boston University...

 (1959), Lambert

Karel Lambert

Karel Lambert is a philosopher and logician at the University of California, Irvine and the University of Salzburg. He has written extensively on the subject of free logic, a term which he coined.-Lambert's Law:...

 (1967), Hailperin (1957), and Mendelsohn (1989).

Interpretation

Karel Lambert

Karel Lambert

Karel Lambert is a philosopher and logician at the University of California, Irvine and the University of Salzburg. He has written extensively on the subject of free logic, a term which he coined.-Lambert's Law:...

 wrote in 1967:

"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 Willard Van Orman Quine

Willard Van Orman Quine

Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...

 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

Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

 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 and , 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

Wesley C. Salmon

Wesley C. Salmon was a metaphysician and contemporary philosopher of science concerned primarily with the topics of causation and explanation....

 and George Nahknikian, which is that to exist is to be self-identical.