# Linksys n validating identity

The reason why (S) is true is that, as we now know, the ether does not exist.According to classical logic, however, (S) is false, because it implies the existence of the ether.

Classical logic requires each singular term to denote an object in the domain of quantification—which is usually understood as the set of “existing” objects. Free logic is therefore useful for analyzing discourse containing singular terms that either are or might be empty.A term is empty if it either has no referent or refers to an object outside the domain.Tradition has generally taken it for granted that free logics are first-order—that is, that their quantifiers range over individuals—but Corine Besson (2009) has argued that internalist theories of natural kinds require second-order free logics, whose quantifiers range over kinds, and she finds precedent for this idea ranging as far back as Cocchiarella (1986).This article, however, focuses on first-order free logics.Section 5 samples applications to theories of description, logics of partial or non-strict functions, logics with Kripke semantics, logics of fiction and logics that are in a certain sense Meinongian. Since classical (i.e., Fregean) predicate logic requires that singular terms denote members of is, as usual, taken to be the class of existing things, free logic may be characterized as logic the referents of whose singular terms need not exist.Karel Lambert (1960) coined the term ‘free logic’ as an abbreviation for ‘logic free of existence assumptions with respect to its terms, singular and general’. Lambert was suggesting that just as classical predicate logic generalized Aristotelian logic by, , admitting predicates that are satisfied by no existing thing (‘is a Martian’, ‘is non-self-identical’, ‘travels faster than light’), so free logic generalizes classical predicate logic by admitting singular terms that denote no existing thing (‘Aphrodite’, ‘the greatest integer’, ‘the present king of France’).

Because classical logic's singular terms must denote existing things (when, as usual, ‘∃’ is read as “there exists”), classical logic is unreliable in application to statements containing singular terms whose referents either do not exist or are not known to.

Consider, for example, the true statement: using ‘the ether’ as a singular term for the light-bearing medium posited by nineteenth century physicists.

Section 1 lays out the basics of free logic, explaining how it differs from classical predicate logic and how it is related to inclusive logic, which permits empty domains or “worlds.” Section 2 shows how free logic may be represented by each of three formal methods: axiom systems, natural deduction rules and tree rules.

Varying conventions for calculating the truth values of atomic formulas containing empty singular terms yield three distinct species of free logic: negative, positive and neutral.

These are surveyed in Section 3, along with supervaluations, which were developed to augment neutral logics.

Section 4 is critical, examining three anomalies that infect most free logics. Singular terms include proper names (individual constants), definite descriptions, and such functional expressions as ‘2 + 2’.

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