Paradox, truth and logic part I: Paradox and truth
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The discussion of the semantics of inconsistent truth theories now comes to a pause. The preceding is of course but a sketch; many interesting questions remain to be answered. The second part of this essay, however, will not seek to answer them. Rather, I will turn to the discussion of the proof theory of truth theory: the local and global logic of truth.
Under the first heading, I show how to replace the inductive construction of models with an appropriate infinitary proof theory, and relate this on the one hand to the so-called “dependence” approach to inductive truth theories (Davis, 1979; Yablo, 1982) and on the other to van Fraassen's “fact” semantics for relevance logic.
Under the second heading, I offer formals systems which capture the inferences valid in all approximate models. Not surprisingly, these turn out to be relevant logics.
With formalism in hand, I discuss finally the extent to which the gap and/or glut approach can in fact be said to “solve” the paradoxes; that is, to allow us to say that the very language we are speaking is of the sort described in our theory.
KeywordsFormal System Approximate Model Truth Theory Proof Theory Relevance Logic
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