Logic in AI
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Humans and intelligent computer programs must often jump to the conclusion that the objects they can determine to have certain properties or relations are the only objects that do. Circumscription formalizes such conjectural reasoning.
Throughout the orthodox mainsteam of the development of logic in the West, the prevailing view was that every proposition is either true or else false (although which of these is the case may well neither be necessary as regards the matter itself nor determinable as regards our knowledge of it). This thesis, commonly called the ‘law of bivalence’ — constituting one key articulation of the ‘law of the excluded middle’ — was, however, already questioned in antiquity. In Ch. IX of his treatise On Interpretation (De interpretatione), Aristotle discussed of the truth-status of alternatives regarding ‘future-contingent’ matters, whose occurrence — like that of the sea-battle tomorrow — is not yet determinable by us, and may indeed actually be undetermined. His views on the matter are still disputed, but many commentators, both in antiquity and later, held him to assert that propositions about future contingents, like that asserting the occurrence of the sea-battle, are neither actually true nor actually false, but potentially either, thus having — at least prior to the event — a third, indeterminate truth-status. The acceptance of the principle of bivalence was, in antiquity, closely bound up with the doctrine of determinism. The Epicureans, who were indeterminists, rejected the law of bivalence; the Stoics (and above all Chrysippus) who were rigid determinists, insisted upon it.1
A textbook on modal logic, intended for readers already acquainted with the elements of formal logic, containing nearly 500 exercises. Brian F. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. Illustrative chapters focus on deontic logic and conditionality. Modality is a rapidly expanding branch of logic, and familiarity with the subject is now regarded as a necessary part of every philosopher's technical equipment. Chellas here offers an up-to-date and reliable guide essential for the student.