Abstract logics

Abstract

Abstract There are many formal logics- classical logic (already encountered in Chapter 2), intuitionistic logic, various modal and temporal logics, and numerous logics arising in various ways from applications such as the logic of automata and the logic of quantum mechanics. Moreover, there are many structures-such as context-free languages, combinatorial systems, etc.-which are sufficiently logic-like to deserve the name logic. Many logics are primarily the embodiment of some notion of truth. They are thereby mainly concerned with the generating theories-a corpus of theorems or ‘universal truths’--thus tending to obscure the view of formal logic as the science of valid inference. In Chapter 1, §1.3.2, there arose the problem of treating neutrality as if it were a truth-value. What has to be done to it to make a logic out of truth, falsehood and neutrality’? Truth-values alone do not constitute a logic. Our strategy must clearly be guided by the various extant logics. In the brief treatment of classical logic in Chapter 2 the notion of logical consequence played some part, though it was subordinate to the idea of theoremhood. But in the discussion of scientific theories in Chapter 1 special significance was given to the deductive postulate. On grounds of generality, it can be claimed that logical consequence is more fundamental than theoremhood, as all that needs to be said about theoremhood can also be stated in terms of logical consequence.