Abstract

I argue that there is a common core to the negation operators found in classical logic, quantum logic and some versions of three-valued logic--logics I refer to an 'valent' logics. In particular, each of these operators adheres to the following three constraints: A proposition and its negation cannot intersect in a non-trivial way, The law of contraposition holds, and Negation maps propositions one-to-one and onto propositions. I further show that a logic consisting of an operator defined by just these constraints, i.e., a minimal negation operator, along with a standard disjunctive operator, is decidable. Finally, I argue that intuitionistic logic does not contain an operator which meets the constraints outlined above, but that it does contain an operator which provides a different form of negation. ;Analyzing these results, I suggest that the difference between negation in intuitionistic logic and in valent logics can best be explained by examining the way in which the latter, unlike the former, aim at describing relations between real states of affairs. I argue that the constraints placed on the negation operator in valent logics arise naturally from the assumption that the logic is "talking about" some aspect of reality. I show that the most natural interpretation of valent logic requires that they have more than one semantic value whereas intuitionistic logic contains but a single semantic value. Michael Dummett has claimed that the division between realist and non-realist logic ought to be made on the basis of whether the logic contains the law of bivalence. I criticize his view and suggest that this division might more plausibly be made based on the type of negation operator the logic contains. ;A final chapter considers a possible logic for vague predicates using fuzzy set theory. I show that such a logic contains a negation operator that meets the valent constraints. I use this result and the conclusions drawn previously to argue--against Putnam--that vagueness can be accommodated within a realist semantic structure