The propositional logic of ordinary discourse

Inquiry 11 (1-4):295 – 320 (1968)
The logical properties of the 'if-then' connective of ordinary English differ markedly from the logical properties of the material conditional of classical, two-valued logic. This becomes apparent upon examination of arguments in conversational English which involve (noncounterfactual) usages of if-then'. A nonclassical system of propositional logic is presented, whose conditional connective has logical properties approximating those of 'if-then'. This proposed system reduces, in a sense, to the classical logic. Moreover, because it is equivalent to a certain nonstandard three-valued logic, its decision procedure is almost as efficient as that of the classical logic. It therefore provides a rational and convenient system in which to formalize English arguments.
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DOI 10.1080/00201746808601531
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Lloyd Humberstone (2000). The Revival of Rejective Negation. Journal of Philosophical Logic 29 (4):331-381.

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