the arena of philosophy devoted to examining the scope and nature of logic. Aristotle considered logic an organon, or foundation, of knowledge. Certainly, inference is the source of much human knowledge. Logic judges inferences good or bad and tries to justify those that are good. One need not agree with Aristotle, therefore, to see logic as essential to epistemology. Philosophers such as Wittgenstein, additionally, have held that the structure of language reflects the structure of the world. Because inferences have elements that are themselves linguistic or are at least expressible in language, logic reveals general features of the structure of language. This makes it essential to linguistics, and, on a Wittgensteinian view, to metaphysics. Moreover, many philosophical battles have been fought with logical weaponry. For all these reasons, philosophers have tried to understand what logic is, what justifies it, and what it tells us about reason, language, and the world.
The nature of logic. Logic might be defined as the science of inference; inference, in turn, as the drawing of a conclusion from premises. A simple argument is a sequence, one element of which, the conclusion, the others are thought to support. A complex argument is a series of simple arguments. Logic, then, is primarily concerned with arguments. Already, however, several questions arise. (1) Who thinks that the premises support the conclusion? The speaker? The audience? Any competent speaker of the language? (2) What are the elements of arguments? Thoughts? Propositions? Philosophers following Quine have found these answers unappealing for lack of clear identity criteria. Sentences are more concrete and more sharply individuated. But should we consider sentence tokens or sentence types? Context often affects interpretation, so it appears that we must consider tokens or types-in-context. Moreover, many sentences, even with contextual information supplied, are ambiguous. Is a sequence with an ambiguous sentence one argument (which may be good on some readings and bad on others) or several? For reasons that will become clear, the elements of arguments should be the primary bearers of truth and falsehood in one’s general theory of language. (3) Finally, and perhaps most importantly, what does ‘support’ mean? Logic evaluates inferences by distinguishing good from bad arguments. This raises issues about the status of logic, for many of its pronouncements are explicitly normative. The philosophy of logic thus includes problems of the nature and justification of norms akin to those arising in metaethics. The solutions, moreover, may vary with the logical system at hand. Some logicians attempt to characterize reasoning in natural language; others try to systematize reasoning in mathematics or other sciences. Still others try to devise an ideal system of reasoning that does not fully correspond to any of these. Logicians concerned with inference in natural, mathematical, or scientific languages tend to justify their norms by describing inferential practices in that language as actually used by those competent in it. These descriptions justify norms partly because the practices they describe include evaluations of inferences as well as inferences themselves. The scope of logic. Logical systems meant to account for natural language inference raise issues of the scope of logic. How does logic differ from semantics, the science of meaning in general? Logicians have often treated only inferences turning on certain commonly used words, such as ‘not’, ‘if’, ‘and’, ‘or’, ‘all’, and ‘some’, taking them, or items in a symbolic language that correspond to them, as logical constants. They have neglected inferences that do not turn on them, such as My brother is married. Therefore, I have a sister-in-law. Increasingly, however, semanticists have used ‘logic’ more broadly, speaking of the logic of belief, perception, abstraction, or even kinship. Such uses seem to treat logic and semantics as coextensive. Philosophers who have sought to maintain a distinction between the semantics and logic of natural language have tried to develop non-arbitrary criteria of logical constancy.
An argument is valid provided the truth of its premises guarantees the truth of its conclusion. This definition relies on the notion of truth, which raises philosophical puzzles of its own. Furthermore, it is natural to ask what kind of connection must hold between the premises and conclusion. One answer specifies that an argument is valid provided replacing its simple constituents with items of similar categories while leaving logical constants intact could never produce true premises and a false conclusion. On this view, validity is a matter of form: an argument is valid if it instantiates a valid form. Logic thus becomes the theory of logical form. On another view, an argument is valid if its conclusion is true in every possible world or model in which its premises are true. This conception need not rely on the notion of a logical constant and so is compatible with the view that logic and semantics are coextensive.
Many issues in the philosophy of logic arise from the plethora of systems logicians have devised. Some of these are deviant logics, i.e., logics that differ from classical or standard logic while seeming to treat the same subject matter. Intuitionistic logic, for example, which interprets the connectives and quantifiers non-classically, rejecting the law of excluded middle and the interdefinability of the quantifiers, has been supported with both semantic and ontological arguments. Brouwer, Heyting, and others have defended it as the proper logic of the infinite; Dummett has defended it as the correct logic of natural language. Free logic allows non-denoting referring expressions but interprets the quantifiers as ranging only over existing objects. Many-valued logics use at least three truthvalues, rejecting the classical assumption of bivalence – that every formula is either true or false.
Many logical systems attempt to extend classical logic to incorporate tense, modality, abstraction, higher-order quantification, propositional quantification, complement constructions, or the truth predicate. These projects raise important philosophical questions. Modal and tense logics. Tense is a pervasive feature of natural language, and has become important to computer scientists interested in concurrent programs. Modalities of several sorts – alethic (possibility, necessity) and deontic (obligation, permission), for example – appear in natural language in various grammatical guises. Provability, treated as a modality, allows for revealing formalizations of metamathematics. Logicians have usually treated modalities and tenses as sentential operators. C. I. Lewis and Langford pioneered such approaches for alethic modalities; von Wright, for deontic modalities; and Prior, for tense. In each area, many competing systems developed; by the late 1970s, there were over two hundred axiom systems in the literature for propositional alethic modal logic alone. How might competing systems be evaluated? Kripke’s semantics for modal logic has proved very helpful. Kripke semantics in effect treats modal operators as quantifiers over possible worlds. Necessarily A, e.g., is true at a world if and only if A is true in all worlds accessible from that world. Kripke showed that certain popular axiom systems result from imposing simple conditions on the accessibility relation. His work spawned a field, known as correspondence theory, devoted to studying the relations between modal axioms and conditions on models. It has helped philosophers and logicians to understand the issues at stake in choosing a modal logic and has raised the question of whether there is one true modal logic. Modal idioms may be ambiguous or indeterminate with respect to some properties of the accessibility relation. Possible worlds raise additional ontological and epistemological questions. Modalities and tenses seem to be linked in natural language, but attempts to bring tense and modal logic together remain young. The sensitivity of tense to intra- and extralinguistic context has cast doubt on the project of using operators to represent tenses. Kamp, e.g., has represented tense and aspect in terms of event structure, building on earlier work by Reichenbach. Truth. Tarski’s theory of truth shows that it is possible to define truth recursively for certain languages. Languages that can refer to their own sentences, however, permit no such definition given Tarski’s assumptions – for they allow the formulation of the liar and similar paradoxes. Tarski concluded that, in giving the semantics for such a language, we must ascend to a more powerful metalanguage. Kripke and others, however, have shown that it is possible for a language permitting self-reference to contain its own truth predicate by surrendering bivalence or taking the truth predicate indexically.
Higher-order logic. First-order predicate logic allows quantification only over individuals. Higher-order logics also permit quantification over predicate positions. Natural language seems to permit such quantification: ‘Mary has every quality that John admires’. Mathematics, moreover, may be expressed elegantly in higher-order logic. Peano arithmetic and Zermelo-Fraenkel set theory, e.g., require infinite axiom sets in firstorder logic but are finitely axiomatizable – and categorical, determining their models up to isomorphism – in second-order logic.
Because they quantify over properties and relations, higher-order logics seem committed to Platonism. Mathematics reduces to higher-order logic; Quine concludes that the latter is not logic. Its most natural semantics seems to presuppose a prior understanding of properties and relations. Also, on this semantics, it differs greatly from first-order logic. Like set theory, it is incomplete; it is not compact. This raises questions about the boundaries of logic. Must logic be axiomatizable? Must it be possible, i.e., to develop a logical system powerful enough to prove every valid argument valid? Could there be valid arguments with infinitely many premises, any finite fragment of which would be invalid?
With an operator for forming abstract terms from predicates, higher-order logics easily allow the formulation of paradoxes. Russell and Whitehead for this reason adopted type theory, which, like Tarski’s theory of truth, uses an infinite hierarchy and corresponding syntactic restrictions to avoid paradox. Type-free theories avoid both the restrictions and the paradoxes, as with truth,