Extending Intuitionistic Logic with Subtraction

This paper is an exercise in formal and philosophical logic. I will show how intuitionistic propositional logic can be extended with a new two-place connective, not expressible in the traditional language of intuitionistic logic (the language of conjunction, disjunction, negation and implication). The new system will be shown to be a conservative extension of intuitionistic logic. After examining the formal properties of this extension, the task will be to consider whether this is an `acceptable' extension of intuitionistic logic. It will turn out that on some intuitionistic considerations the extension is acceptable, and on others it is not. In this paper I presume that the reader has some idea both of the formal properties of intuitionistic logic, and some motivating philosophical principles which inform the development of intuitionistic logic. Readers wanting such an introduction can do no better than look at some of the excellent, extensive literature on intuitionism [3, 5, 8, 12]. Other work has been done on extending intuitionistic propositional logic with new connectives. Gabbay [6] considers extending the logic with propositional connectives. Our new connective is not one he considers. De Jongh too considers extending intuitionistic logic by adding arbitrary conjunction and disjunction. In his interesting paper [2] he shows that once you make such an addition to the basic logic, you end up with a proper class of propositional connectives expressible. In what follows we will consider adding just one new binary connective to the language of intuitintionistic propositional logic.