Page 279 - Encyclopedia of Philosophy of Language

systems has grown up, with the hegemony of'classical' or 'standard' systems challenged by a variety of 'non- classical' or 'nonstandard' logics.
1. Scope and Background
'Classical logic' refers, not, as 'classical' might suggest, to Aristotelian syllogistic logic, but to a class of sys- tems of modern deductive logic: two-valued prepo- sitional calculus and first-order predicate calculus with identity. There are many different formulations of classical logic; the term refers, not to a particular set of axioms and/or rules of inference expressed in terms of a particular primitive vocabulary and rep- resented in a particular notation, but to a class of equivalent systems: those which, differences of notation, primitive vocabulary, axioms and/or rules aside, license a certain set of theorems and inferences. Systems which license different theorems and/or infer- ences are 'nonclassicaF or 'nonstandard.'
The convenient dichotomy of 'classical' versus 'non-classical' logics should not be allowed to suggest too simple a historical picture. Though many non- standard systems were developed, after the 'classical' logical apparatus had been formulated, by logicians who believed that that apparatus was inadequate or incorrect, the ideas motivating what are now called 'nonclassicaF logics have a long history. Some of the key ideas that were to motivate modal and many- valued logics, for example, were explored by Aristotle. And nonstandard systems developed alongside the formulation of the now standard apparatus: in 1908, two years before the publication of Russell and Whi- tehead's Principia Mathematica, Hugh MacColl was recalling in the pages of Mind that for nearly thirty years he had been arguing the inadequacy of (what is now called) material implication, and, besides offering a definition of (what is now called) strict implication, arguing the merits of a 'logic of three dimensions'; in 1909 C. S. Peirce, recognized as a pioneer of quanti- fication theory and the logic of relations, and the originator of truth-table semantics, gave truth-tables for a 'triadic' logic which, he claimed, is 'universally true.' What is known as 'classical logic' is simply that class of systems which emerged as the dominant, stan- dard approach.
Its critics see classical logic as inadequate, as failing to license theorems or inferences which ought to be licensed, or as incorrect, as licensingtheorems or infer- ences which ought not to be licensed—or, sometimes, as defective in both these ways. Correspondingly, non- standard systems may extend classical logic, by adding new vocabulary and licensing additional theorems and/or inferences essentially involving that vocabu- lary ('extended logics'), or restrict it, by repudiating classical theorems and/or inferences ('deviant logics'), or both. This distinction is however somewhat rough and ready, because the apparently simple contrast between adding new theorems or inferences and repu-
diating old ones turns out (Sect. 4 below) to be less straightforward than it may appear.
There are many different nonstandard logics: modal logic, epistemic logic, tense logic, many-valued logic, intuitionist logic, relevance logic, quantum logic, free logic, fuzzy logic, etc. In fact there are many different non-standard systems within each of these categories: a multiplicity of modal logics, many many-valued logics, rival relevance logics, more than one family of fuzzy logic, etc. And the plethora of motivations offered for this plurality of nonstandard systems resists any simple classification, for different systems are offered for the same purpose, and the same system is put to different purposes. Comprehensiveness is obviously impossible; in what follows, therefore, after a preliminary survey focusing primarily on deviant systems, the emphasis will be on those challenges to the correctness of classical logic which arise in one way or another from considerations of language.
2. Survey andGlossary
2.1 Modal, etc.,Logics
C. I. Lewis, the originator of modern modal systems, was primarily concerned to correct what he saw as the unacceptable weakness of the classical notion of implication. It is now more usual for modal logics to be presented simply as extending classical logic by adjoining the operators 'L' or 'D ,' meaning 'necess- arily,' 'M' or '<>,' meaning 'possibly' (and definable as 'not necessarily not'), and '-3,' representing strict implication (defined as necessity of material impli- cation); and as licensing new, nonclassical theorems and inferences essentially involving this modal vocabulary.
Closely modeled after modal logics are epistemic logics (which introduce 'Kap,' meaning 'a knows that p,' and 'Bap,' meaning 'a believes that p'); deontic logics (which introduce 'Op,' meaning 'it is obligatory that p' and 'Pp,' meaning 'it is permitted that p'); and tense logics (which, first, construe the sentence letters of classical prepositional calculus, not, as they are ordinarily construed, as tenseless, but as present- tensed, and then introduce 'Pp,' meaning 'it was the case that p,' and 'Fp,' meaning 'it will be the case that P').
2.2 Many-valued Logics
Classical prepositional calculus has a two-valued characteristic matrix, i.e., truth-tables can be given in terms of the two truth-values 'true' and 'false' such that all and only the theorems of classical prepo- sitional calculus uniformly take the value 'true,' and all and only the inferences valid in classical prepo- sitional calculus are invariably truth-preserving, on those truth-tables. 'Many-valued logic' refers to prepo- sitional calculi matrices which require three or more values; in these systems some classical principles fail.
Deviant Logics
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