. The central topic of logic is valid reasoning, its systematiz-ation and the study of notions relevant to it. This gives it two systematically related areas of concern, formal logic and philosophical logic (also called logical theory, though usage varies and philosophy of logic is sometimes confined to the study of logical systems and their applications).
The main task of formal logic is to axiomatize (see AXIOM SYSTEM) various subject-matters, constructing in particular the propositional and predicate CALCULI. Theorems are then proved in and about the resulting systems to bring out their properties. The important properties include CONSISTENCY, COMPLETENESS and the possession of DECISION PROCEDURES. The study of such properties is often called metalogic.
Systems complex enough to axiomatize mathematics on the basis of DECIDABLE axioms turn out to have these properties only to a severely limited extent (see GÖDEL’S THEOREMS). Also the logical PARADOXES make it harder to construct these systems. Much of modern formal logic consists in trying to avoid or minimize these limitations and difficulties. How relevant they are in the sphere of philosophical logic, or even outside it, is disputed. The main issues are the nature of truth, especially in view of the LIAR PARADOX, and the relations between formal systems and ordinary language. Some, especially adherents to linguistic PHILOSOPHY, have argued from these or other considerations that ordinary language has no exact logic (cf. also OPEN TEXTURE). The study of ordinary language from this point of view is sometimes called informal logic, though this can also refer to the study of those inferences which depend on the content rather than form of the sentences concerned. We infer ‘Smith is unmarried’ from ‘Smith is a bachelor’ because of the meaning of ‘bachelor’ not because ‘Smith is a bachelor’ has a certain form. A formal inference from ‘Smith is a bachelor’ might yield ‘Bachelors include Smith’ (see FORM); this raises some of the problems concerning the ANALYTIC.
Formal logic also studies NATURAL DEDUCTION, and formal parts of modal and deontic logic (see below).
A topic related to formal logic is set theory (see CALCULUS). This and proof theory (see METAMATHEMATICS) are normally together called mathematical logic, and lead towards the philosophy of MATHEMATICS. Modern formal logic and mathematical logic are each, or together, often called symbolic logic, to mark the more intensive use of symbols, or logistic.
Deontic logic studies logical relations between propositions containing terms like ‘obliged’, ‘commanded’, ‘permitted’, ‘forbidden’, though the term tends to be confined to the construction of formal systems using deontic terms, and the problems these systems raise. A rather similar subject is the logic of preference, which asks, for example, what sets of preferences can consistently be held together (e.g. can one prefer a to b, and b to c, but c to a?) Cf. VOTING PARADOX.
Philosophical logic examines the concepts involved in formal logic and uses its results, but is not concerned with the mechanics of the various systems; however, the boundaries between these areas of logic are not sharp. It also studies the nature of logical systems as such, and whether there can be alternative logics (see below).
As a study devoted to valid reasoning, it naturally asks about reasoning in general and how many kinds of it there are. Is all reasoning, properly speaking, deductive, or is there also inductive reasoning (see INDUCTION, philosophy of SCIENCE), and perhaps other kinds, e.g. in morals, history, aesthetics? Are there several kinds of validity? Validity is closely connected to logical necessity, and thence to necessity in general and other modal concepts such as possibility and impossibility (see MODALITIES). Modal concepts are studied by modal logic, though in practice this term, like ‘deontic logic’, tends to be confined to the study of formal systems using modal terms. General analyses of necessity, etc. belong to philosophical logic, which also asks how these modal notions are related to the A PRIORI and the ANALYTIC.
Reasoning involves passing from premises to conclusions, and so involves a relation and the things which it relates. Both of these are subjects for philosophical logic. The relation in its most general form is IMPLICATION, which raises problems about its different kinds and its relation to INFERENCE. The things it relates are SENTENCES, propositions or statements, which again raise problems about what they are.
Attempts to relate logic to epistemology may be said to lie behind the development of INTUITIONIST logic, perhaps the main alternative to two-valued classical logic (recognizing only the two TRUTHVALUES truth and falsity). But other ‘deviant logics’ have also been developed for various purposes, such as the alleged needs of quantum mechanics.
Logic also studies meaning in general, of ordinary words as well as of formal words like ‘all’, ‘and’, etc., and of both words and sentences. This broadens into the study of language in general: how it does what it does and how it relates to the world (cf. above on relating ordinary language to formal systems). How are the various roles that words fulfil, such as meaning, referring, describing, predicating, to be distinguished and related? These problems now form a subject of their own, philosophy of LANGUAGE, but they still fall broadly under logic. Two further notions important in this area are DEFINITIONS and TRUTH (see above; truth also belongs to epistemology). Logic borders on metaphysics when we ask how far various logical views commit us to asserting that certain kinds of things, like propositions, exist, and how we should analyse existential QUANTIFICATION (cf. BEING).
Formal logic effectively began with Aristotle, who systematized immediate INFERENCE and the SYLLOGISM, which remained the basis of traditional logic until about a century ago. Vigorous developments in both formal and philosophical logic were made by the Stoics, and in philosophical logic in the Middle Ages, but these were forgotten and have only recently been revived. The syllogism is now seen to form a small part of the predicate calculus.
A key feature in the history of modern logic since about the middle of the nineteenth century has been the development of a logic of relations. This comes from realizing, and taking seriously, that not all propositions consist of a subject and predicate linked by the copula, ‘is’ (‘are’). Traditional logic had unduly restricted itself by assuming that they do, and could not formalize so simple an argument as ‘Ten exceeds nine and nine exceeds eight, so ten exceeds eight’, where the main verb stands for a relation. See also TOPIC-NEUTRAL, QUANTIFIER WORDS.
*I.Copi, Introduction to Logic, Macmillan 6th edn, 1982. (Covers formal, informal, and inductive logic and has many examples, some with answers. See also W.A.Hodges, Logic, Penguin, 1977 (has exercises with answers).)
J.N.Crossley et al.,What is Mathematical Logic?, Oxford UP, 1972. (Brief. Fairly elementary.)
*A.Grayling, An Introduction to Philosophical Logic, Harvester, 1982; 2nd edn, Duckworth, 1990. (Mainly elementary with two rather harder chapters at end.)
G.E.Hughes and M.J.Cresswell, An Introduction to Modal Logic, Methuen, 1968. (Comprehensive.)
J.N.Keynes, Formal Logic, 1884, 4th (revised) edn 1906. Standard and full treatment of traditional logic.)
W. and M.Kneale, The Development of Logic, Oxford UP, 1962. (Very full treatment of historical development of the subject.)
*W.V.O.Quine, Philosophy of Logic, Prentice-Hall, 1970. (General introduction from Quine’s own point of view.)
*R.M.Sainsbury, Logical Forms: An Introduction to Philosophical Logic, Blackwell, 1991. (Claims to bridge gap between elementary formal logic and more advanced philosophical logic.)
P.F.Strawson, Introduction to Logical Theory, Methuen, 1952. (Represents ‘linguistic philosophy’ outlook, that there is no exact logic of ordinary language.)
G.H.von Wright, The Logic of Preference, Edinburgh UP, 1963. (Introduction.)
G.H.von Wright, An Essay in Deontic Logic, North-Holland, 1968. (Introduction, with bibliography.)
F.Waismann, ‘Are there alternative logics?’, Proceedings of the Aristotelian Society, 1945–6, reprinted in his How I See Philosophy, Macmillan, 1968. Cf. S.Haack, Deviant Logic, Cambridge UP, 1974, and Philosophy of Logics, Cambridge UP, 1978 (which includes glossary).
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