8   Challenging the paradigm

In this chapter we step back a little to broaden our view of logical theory, leading us to re-examine some of its most basic features. We consider some arguments to the effect that "classical" logic is flawed in that it is based on an over-hasty and over-simplified account of what arguments are. It is at least interesting to look at alternative views and the kinds of logic to which they might give rise. A full investigation of non-classical logic is far beyond the scope of these notes, but the existence of other paradigms needs to be noted and some of the flavour of such "deviant" logics needs to be brought out.

We note three non-classical propositional logics, and conclude with some notes on their semantics. While we do not detail the semantic theories in full, we do look briefly at a semantic technique for showing classically valid sequents to be invalid by the lights of some of the alternative systems.

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• Paradoxes of Implication Relating logic and language: the case of relevant logic
• Vagueness Thoughts on the sorites paradox and fuzzy logic
• A constructive view of truth A quick look at intuitionistic logic
• Modelling non-classical logic Using 3-valued semantics to disprove sequents
• Examples Proofs in the non-classical logics R and F*
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  Non-classical proof exercises Sample problems with solutions