What is a non-truth-functional logic?

Studia Logica 92 (2):215 - 240 (2009)
Abstract
What is the fundamental insight behind truth-functionality ? When is a logic interpretable by way of a truth-functional semantics? To address such questions in a satisfactory way, a formal definition of truth-functionality from the point of view of abstract logics is clearly called for. As a matter of fact, such a definition has been available at least since the 70s, though to this day it still remains not very widely well-known. A clear distinction can be drawn between logics characterizable through: (1) genuinely finite-valued truth-tabular semantics; (2) no finite-valued but only an infinite-valued truthtabular semantics; (3) no truth-tabular semantics at all. Any of those logics, however, can in principle be characterized through non-truth-functional valuation semantics, at least as soon as their associated consequence relations respect the usual tarskian postulates. So, paradoxical as that might seem at first, it turns out that truth-functional logics may be adequately characterized by non-truth-functional semantics . Now, what feature of a given logic would guarantee it to dwell in class (1) or in class (2), irrespective of its circumstantial semantic characterization?
Keywords Abstract logics  formal semantics  truth-functionality
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References found in this work BETA
Gary M. Hardegree (2005). Completeness and Super-Valuations. Journal of Philosophical Logic 34 (1):81 - 95.
Kenton F. Machina (1976). Truth, Belief, and Vagueness. Journal of Philosophical Logic 5 (1):47-78.

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