Truth-values as labels: a general recipe for labelled deduction (original) (raw)
2003, Journal of Applied Non-Classical Logics
We introduce a general recipe for presenting non-classical logics in a modular and uniform way as labelled deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truth-values. More specifically, the main idea underlying our approach is the use of algebras of truth-values, whose operators reflect the semantics we have in mind, as the labelling algebras of our labelled deduction systems. The "truth-values as labels" approach allows us to give generalized systems for multiplevalued logics within the same formalism: since we can take multiple-valued logics as meaning not only finitely or infinitely many-valued logics but also power-set logics, i.e. logics for which the denotation of a formula can be seen as a set of worlds, our recipe allows us to capture also logics such as modal, intuitionistic and relevance logics, thus providing a first step towards the fibring of these logics with many-valued ones.
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Truth-value constants in multi-valued logics
In: Peter Schroeder-Heister on Proof-Theoretic Semantics, Springer, 2021
In some presentations of classical and intuitionistic logics, the object-language is assumed to contain (two) truth-value constants: verum and falsum, that are, respectively, true and false under every bivalent valuation. We are interested to de ne and study analogical constants that in an arbitrary multi-valued logic over truth-values have the truth-value v_i under every (multi-valued) valuation. We Definition ne such constants proof-theoretically via their associated I/E-rules in a natural-deduction proof system. As is well known, the absence or presence of such constants has a significant deductive impact on the logics studied. In particular, we propose a generalization of the notions of contradiction and explosiveness of a logic to the context of multi-valued logics.
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