What is a non-truth-functional logic?

Studia Logica 92 (2):215 - 240 (2009)
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
Alfred Tarski (2002). On the Concept of Following Logically. History and Philosophy of Logic 23 (3):155-196.
Newton C. A. Da Costa (1974). On the Theory of Inconsistent Formal Systems. Notre Dame Journal of Formal Logic 15 (4):497-510.
Timothy Smiley (1996). Rejection. Analysis 56 (1):1–9.

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