# Logical Terms

# LOGICAL TERMS

The two central problems concerning "logical terms" are demarcation and interpretation. The search for a demarcation of logical terms goes back to the founders of modern logic, and within the classical tradition a partial solution, restricted to logical connectives, was established early on. The characteristic feature of logical connectives, according to this solution, is truth-functionality, and the totality of truth functions (Boolean functions from n-tuples of truth values to a truth value) determines the totality of logical connectives. In his seminal 1936 paper, "On the Concept of Logical Consequence," Alfred Tarski demonstrated the need for a more comprehensive criterion by showing that his semantic definition of logical consequence—the sentence σ is a logical consequence of the set of sentences Σ iff (if and only if) every model of Σ is a model of σ—is dependent on such a demarcation. (Thus suppose the existential quantifier is not a logical term, then its interpretation will vary from model to model, and the intuitively logically valid consequence, "Rembrandt is a painter; therefore there is at least one painter," will fail to satisfy Tarski's definition. Suppose "Rembrandt" and "is a painter" are both logical terms, then the intuitively logically invalid consequence, "Frege is a logician; therefore Rembrandt is a painter," will satisfy Tarski's definition.) Tarski, however, left the general demarcation of logical terms an open question, and it was not until the late 1950s that the first steps toward developing a systematic criterion for logical predicates and quantifiers were taken.

In his 1957 paper, "On a Generalization of Quantifiers," A. Mostowski proposed a semantic criterion for first-order logical quantifiers that generalizes Frege's analysis of the standard quantifiers as second-level cardinality predicates. Technically, Mostowski interpreted a quantifier, Q, as a function from universes (sets of objects), A, to A-quantifiers, Q_{A}, where Q_{A} is a function assigning a truth-value to each subset B of A. Thus, given a set A, the existential and universal quantifiers are defined by: for any B⊆A,∃_{A}(B) = T iff B ≠ ϕ and ∀_{A}(B) = T iff A − B = ϕ. Intuitively, a quantifier is logical if it does "not allow us to distinguish between different elements" of the underlying universe. Formally, Q is logical iff it is invariant under isomorphic structures of the type <A,B>, where B⊆A; that is, Q is a logical quantifier iff for every structure <A,B> and <A′,B′>:if<A,B>≅<A′,B′>, then Q_{A}(B) = Q_{A′}(B′). Quantifiers satisfying Mostowski's criterion are commonly called *cardinality quantifiers*, and some examples of these are "!*δ* x" ("There are exactly *δ* individuals in the universe such that …"), where *δ* is any cardinal, "Most x" ("There are more x's such that … than x's such that not …"), "There are finitely many x," "There are uncountably many x," and so forth.

In 1966, P. Lindström extended Mostowski's criterion to terms in general: A term (of type n) is logical iff it is invariant under isomorphic structures (of type n). Thus, the well-ordering predicate, W, is logical since for any A,A′, R⊆A^{2} and R′ ⊆A′^{2}: if <A,R>≅<A′,R′>, then W_{A}(R) = W_{A′}(R′). Intuitively, we can say that a term is logical iff it does not distinguish between isomorphic arguments. The terms satisfying Lindström's criterion include identity, n-place cardinality quantifiers (e.g., the 2-place "Most," as in "Most A's are B's"), relational or polyadic quantifiers like the well-ordering predicate above and "is an equivalence relation," and so forth. Among the terms not satisfying Lindström's criterion are individual constants, the first-level predicate "is red," the first-level membership relation, the second-level predicate "is a property of Napoleon," and so forth. Tarski (1966) proposed essentially the same division.

The Mostowski-Lindström-Tarski (MLT) approach to logical terms has had a considerable impact on the development of contemporary model theory. Among the central results are Lindström's characterizations of elementary logic, various completeness and incompleteness theorems for generalized (model-theoretic, abstract) logics, and so forth. (See Barwise and Feferman 1985). But whereas the mathematical yield of MLT has been prodigious, philosophers, by and large, have continued to hold on to the traditional view according to which the collection of (primitive) logical terms is restricted to truth-functional connectives, the existential and/or universal quantifier and, possibly, identity. One of the main strongholds of the traditional approach has been Willard Van Orman Quine, who (in his 1970 book) justified his approach on the grounds that (1) standard first-order logic (without identity) allows a remarkable concurrence of diverse definitions of logical consequence, and (2) standard first-order logic (with or without identity) is complete. Quine did not consider the logicality of nonstandard quantifiers such as "there are uncountably many," which allow a "complete" axiomatization. L. H. Tharp (1975), who did take into account the existence of complete first-order logics with nonstandard generalized quantifiers, nevertheless arrived at the same conclusion as Quine's.

During the 1960s and 1970s many philosophers were concerned with the interpretation rather than the identity of logical terms. Thus, Ruth Barcan Marcus (1962, 1972) and others developed a substitutional interpretation of the standard quantifiers; Michael Anthony Eardley Dummett (1973) advocated an intuitionistic interpretation of the standard logical terms based on considerations pertaining to the theory of meaning; many philosophers (e.g., van Fraassen) pursued "free" and "many-valued" interpretations of the logical connectives; Jaako Hintikka (1973, 1976) constructed a game theoretic semantics for logical terms. In a later development, G. Boolos (1984) proposed a primitive (non-set-theoretic) interpretation of "nonfirstorderizable" operators, which has the potential of overcoming ontological objections to higher-order logical operators (e.g., by Quine).

In the mid-1970s philosophers began to search for an explicit, general philosophical criterion for logical terms. The attempts vary considerably, but in all cases the criterion is motivated by an underlying notion of logical consequence. Inspired by Gerhard Gentzen's proof-theoretic work, Ian Hacking (1979) suggests that a logical constant is introduced by (operational) rules of inference that preserve the basic features of the traditional deducibility relation: the subformula property (compositionality), reflexivity, dilution (stability under additional premises and conclusions), transitivity (cut), cut elimination, and so forth. Hacking's criterion renders all and only the logical terms of the ramified theory of types genuinely logical. A. Koslow's (1992) also utilizes a Gentzen-like characterization of the deducibility relation. Abstracting from the syntactic nature of Gentzen's rules, he arrives at a "structural" characterization of the standard logical and modal constants. Both Koslow and Hacking incorporate lessons from an earlier exchange between A. N. Prior (1960, 1964) and N. Belnap (1962) concerning the possibility of importing an inconsistency into a hitherto consistent system by using arbitrary rules of inference to introduce new logical operators.

C. Peacocke (1976) approaches the task of delineating the logical terms from a semantic perspective. The basic property of logical consequence is, according to Peacocke, a priori. *α* is a logical operator iff *α* is a noncomplex n-place operator such that given knowledge of which objects (sequences of objects) satisfy an n-tuple or arguments of *α*, <*β* _{1},…,*β* _{n}>, one can know a priori which objects satisfy *α* (*β* _{1},…,*β* _{n}). Based on this criterion Peacocke counts the truth-functional connectives, the standard quantifiers, and certain temporal operators ("In the past …") as logical, while identity (taken as a primitive term), the first-order membership relation, and "necessarily" are nonlogical. Peacocke's criterion is designed for classical logic, but it is possible to produce analogous criteria for nonclassical logics (e.g., intuitionistic logic). T. McCarthy (1981) regards the basic property of logical constants as topic neutrality. He considers Peacocke's condition as necessary but not sufficient, and his own criterion conjoins Peacocke's condition with Lindström's invariance condition (MLT). The standard first-order logical vocabulary as well as various nonstandard generalized quantifiers satisfy McCarthy's criterion, but cardinality quantifiers do not (intuitively, cardinality quantifiers are not topic-neutral).

Sher (1991) considers necessity and formality as the two characteristic features of logical consequence. Treating formality as a semantic notion, Sher suggests that any formal operator incorporated into a Tarskian system according to certain rules yields consequences possessing the desired characteristics. Viewing Lindström's invariance criterion as capturing the intended notion of formal operator, Sher endorses the full-fledged MLT as delineating the scope of logical terms in classical logic.

The theory of logical terms satisfying Lindström's criterion has led, with various adjustments, to important developments in linguistic theory: a systematic account of determiners as generalized quantifiers (Barwise and Cooper, Higginbotham and May); numerous applications of "polyadic" quantifiers (van Benthem, Keenan); and an extension of Henkin's 1961 theory of standard branching quantifiers, applied to English by Hintikka (1973), to branching generalized quantifiers (Barwise and others).

** See also ** Dummett, Michael Anthony Eardley; Frege, Gottlob; Hintikka, Jaako; Logic, History of; Marcus, Ruth Barcan; Model Theory; Prior, Arthur Norman; Quine, Willard Van Orman; Tarski, Alfred.

## Bibliography

Barwise, J. "On Branching Quantifiers in English." *Journal of Philosophical Logic* 8 (1979): 47–80.

Barwise, J., and R. Cooper. "Generalized Quantifiers and Natural Language." *Linguistics and Philosophy* 4 (1981): 159–219.

Barwise, J., and S. Feferman, eds. *Model-Theoretic Logics.* New York: Springer-Verlag, 1985.

Belnap, N. "Tonk, Plonk, and Plink." *Analysis* 22 (1962): 130–134.

Boolos, G. "To Be Is to Be a Value of a Variable (or to Be Some Values of Some Variables)." *Journal of Philosophy* 81 (1984): 430–449.

Dummett, M. "The Philosophical Basis of Intuitionistic Logic." In *Truth and Other Enigmas.* Cambridge, MA: Harvard University Press, 1978.

Hacking, I. "What Is Logic?" *Journal of Philosophy* 76 (1979): 285–319.

Henkin, L. "Some Remarks on Infinitely Long Formulas." In *Infinitistic Methods.* Warsaw, 1961.

Higginbotham, J., and R. May. "Questions, Quantifiers and Crossing." *Linguistic Review* 1 (1981): 41–79.

Hintikka, J. "Quantifiers in Logic and Quantifiers in Natural Languages." In *Philosophy of Logic*, edited by S. Körner, 208–232. Oxford, 1976.

Hintikka, J. "Quantifiers vs. Quantification Theory." *Dialectica* 27 (1973): 329–358.

Keenan, E. L. "Unreducible n-ary Quantifiers in Natural Language." In *Generalized Quantifiers*, edited by P. Gärdenfors. Dordrecht: Reidel, 1987.

Koslow, A. *A Structuralist Theory of Logic.* Cambridge, U.K.: Cambridge University Press, 1992.

Lindström, P. "First Order Predicate Logic with Generalized Quantifiers." *Theoria* 32 (1966): 186–195.

Marcus, R. Barcan. "Interpreting Quantification." *Inquiry* 5 (1962): 252–259.

Marcus, R. Barcan. "Quantification and Ontology." *Noûs* 6 (1972): 240–250.

McCarthy, T. "The Idea of a Logical Constant." *Journal of Philosophy* 78 (1981): 499–523.

Mostowski, A. "On a Generalization of Quantifiers." *Fundamenta Mathematicae* 42 (1957): 12–36.

Peacocke, C. "What Is a Logical Constant?" *Journal of Philosophy* 73 (1976): 221–240.

Prior, A. N. "Conjunction and Contonktion Revisited." *Analysis* 24 (1964): 191–195.

Prior, A. N. "The Runabout Inference-Ticket." *Analysis* 21 (1960): 38–39.

Quine, W. V. O. *Philosophy of Logic.* Englewood Cliffs, NJ: Prentice-Hall, 1970.

Sher, G. *The Bounds of Logic: A Generalized Viewpoint.* Cambridge, MA: MIT Press, 1991.

Tarski, A. *Logic, Semantics, Metamathematics*, 2nd ed. Indianapolis: Hackett, 1983.

Tarski, A. "On the Concept of Logical Consequence" (1936). In *Logic, Semantics, Metamathematics*, 2nd ed., 409–420. Indianapolis: Hackett, 1983.

Tarski, A. "What Are Logical Notions?" (1966). *History and Philosophy of Logic* 7 (1986): 143–154.

Tharp, L. H. "Which Logic Is the Right Logic?" *Synthese* 31 (1975): 1–21.

van Benthem, J. "Polyadic Quantifiers." *Linguistics and Philosophy* 12 (1989): 437–464.

van Fraassen, B. C. "Singular Terms, Truth-Value Gaps, and Free Logic." *Journal of Philosophy* 63 (1966): 481–495.

*Gila Sher (1996)*

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