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.
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