On A Neglected Path to Intuitionism

According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard (Heyting) semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the dispute between classicists and intuitionists. I argue for this by presenting an account of consequence, and a cognate semantic theory for the language of the propositional calculus, which (a) respects the meanings of the connectives as embodied in the familiar classical truth-tables, (b) does not presuppose Bivalence, and with respect to which (c) the rules of the intuitionist propositional calculus are sound and complete. Thus the disagreement between classicists and intuitionists, at least, need not stem from their attaching different senses to the connectives; one may deny the doctrine without changing the subject. The basic notion of my semantic theory is truth at a possibility, where a possibility is a way that (some) things might be, but which differs from a possible world in that the way in question need not be fully specific or determinate. I compare my approach with a previous theory of truth at a possibility due to Lloyd Humberstone, and with a previous attempt to refute Quine’s thesis due to John McDowell.     

Reply to Øystein Linnebo and Stewart Shapiro

In reply to Linnebo, I defend my analysis of Tait's argument against the use of classical logic in set theory, and make some preliminary comments on Linnebo's new argument for the same conclusion. I then turn to Shapiro's discussion of intuitionistic analysis and of Smooth Infinitesimal Analysis (SIA). I contend that we can make sense of intuitionistic analysis, but only by attaching deviant meanings to the connectives. Whether anyone can make sense of SIA is open to doubt: doing so would involve making sense of mathematical quantities (infinitesimals) whose relationship to zero and to one another is inherently indeterminate.     

Intuitionism and the Modal Logic of Vagueness

Journal of Philosophical Logic - Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics...