Existence and ontology
Because one of the basic concepts of first-order logic is that of existence, as codified by the existential quantifier “("x),” one might suppose that there is little room left for any separate philosophical problem of existence. Yet existence, in fact, does seem to pose a problem, as witnessed by the bulk of the relevant literature. Some issues are relatively easy to clarify. In the usual formulations of first-order logic, for instance, there are “existential presuppositions” present to the effect that none of the singular terms employed is without a bearer (as “Pegasus” is). It is a straightforward matter, however, to dispense with these presuppositions. Though this seems to involve the procedure, branded as inadmissible by many philosophers, of treating existence as a predicate, this can nonetheless be easily done on the formal level. Given certain assumptions, it may even be shown that this “predicate” will have to be “("x) (x = a)” (for “a exists”—literally, “There exists an x such that x is a”) or something equivalent. Furthermore, the logical peculiarities of this predicate seem to explain amply philosophers’ apparent denial of its reality.
The interest in the notion of existence is connected with the question of what entities a theory commits its holder to or what its “ontology” is. The “predicate of existence” just mentioned recalls Quine’s criterion of ontological commitment: “To be is to be a value of a bound variable”—i.e., of the x in ($x) or in ("x). According to Quine, a theory is committed to those and only those entities that in the last analysis serve as the values of its bound variables. Thus ordinary first-order theory commits one to an ontology only of individuals (particulars), whereas higher order logic commits one to the existence of sets—i.e., of collections of definite and distinct entities (or, alternatively, of properties and relations). Likewise, if bound first-order variables are assumed to range over sets (as they do in set theory), a commitment to the existence of these sets is incurred.
The doctrine that an ontology of individuals is all that is needed is known as (the modern version of) nominalism. The opposite view is known as (logical) realism. Even those philosophers who profess sympathy with nominalism find it hard, however, to maintain that mathematics could be built on a consistently nominalistic foundation.
The precise import of Quine’s criterion of ontological commitment, however, is not completely clear. Nor is it clear in what other sense one is perhaps committed by a theory to those entities that are named or otherwise referred to in it but not quantified over in it. Questions can also be raised concerning the very distinction between what in modern logic are usually called individuals (“particulars” would be a more traditional designation) and such universals as their properties and relations; and these questions can be combined with others concerning the “tie” that binds particulars and universals together in predication.
An interesting approach to these problems is the distinction made by Gottlob Frege, a pioneer of mathematical logic in the late 19th century, between individuals—he called them objects—and what he called functions (which in his view include concepts) and his doctrine of the unsaturated character of the latter, according to which a function (as it were) contains a gap, which can be filled by an object. Another approach is the “picture theory of language” of Wittgenstein’s Tractatus Logico-Philosophicus, according to which a simple sentence presents a person with an isomorphic representation (a “picture”) of reality as it would be if the sentence were true. According to this view (which was later given up by Wittgenstein), “a sentence [or proposition, Satz] is a model of reality such as we think of it as being.”
The natures of most of the so-called nonclassical logics can be understood against the background of what has here been said. Some of them are simply extensions of the “classical” first-order logic—e.g., modal logics and many versions of intensional logic. The so-called free logics are simply first-order (or modal) logics without existential presuppositions.
One of the most important nonclassical logics is intuitionistic logic, first formalized by the Dutch mathematician Arend Heyting in 1930. It has been shown that this logic can be interpreted in terms of the same kind of modal logic serving as a system of epistemic logic. In the light of its purpose to consider only the known, this isomorphism is suggestive. The avowed purpose of the intuitionist is to consider only what can actually be established constructively in logic and in mathematics—i.e., what can actually be known. Thus, he refuses to consider, for example, “Either A or not-A” as a logical truth, for it does not actually help one in knowing whether A or not-A is the case. This does not close, however, the philosophical problem about intuitionism. Special problems arise from intuitionists’ rejection (in effect) of the nonepistemic aspects of logic, as illustrated by the fact that only a part of epistemic logic is needed in this translation of intuitionistic logic into epistemic logic.
Other new logics are obtained by modifying the rules of those games that are involved in the game-theoretical interpretation of first-order logic mentioned above. The logician may reject, for instance, the assumption that he possesses perfect information, an assumption that characterizes classical first-order logic. One may also try to restrict the strategy sets of the players—to recursive strategies, for example.
Among the oldest kinds of alternative logics are many-valued logics. In them, more truth values than the usual true and false are assumed. The idea seems very natural when considered in abstraction from the actual use of logic. But a philosophically satisfactory interpretation of many-valued logics is not equally straightforward. The interest in finite-valued logics and the applicability of them are sometimes exaggerated. The idea, however, of using the elements of an arbitrary Boolean algebra—a generalized calculus of classes—as abstract truth-values has provided a powerful tool for systematic logical theory.
Logic and other disciplines
The relations of logic to mathematics, to computer technology, and to the empirical sciences are here considered.