Journal of Philosophical Logic 39 (5):511 - 529 (2010)

Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of meaning containment. DW is the logic focussed upon, but the results extend to MC. The semantics is called 'free semantics' since it is disjunctively and existentially free in that no disjunctive or existential witnesses are produced, unlike in truth-functional semantics. Such 'witnesses' are only assumed in generality and are not necessarily actual. The paper sets up the free semantics in a truth-functional style and gives a natural deduction interpetation of the meta-logical connectives. We then set out a familiar tableau-style system, but based on natural deduction proof rather than truth-functional semantics. A proof of soundness and completeness is given for a reductio system, which is a transform of the tableau system. The reductio system has positive and negative rules in place of the elimination and introduction rules of Brady's normalized natural deduction system for DW. The elimination-introduction turning points become closures of threads of proof, which are at the points of contradiction for the reductio system.
Keywords Relevant logic  Normalized natural deduction  Entailment logic  Tableau system  Reductio system
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DOI 10.1007/s10992-010-9129-3
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References found in this work BETA

Universal Logic.Ross Brady - 2006 - CSLI Publications.
Models for Entailment.Kit Fine - 1974 - Journal of Philosophical Logic 3 (4):347 - 372.

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Citations of this work BETA

Some Concerns Regarding Ternary-Relation Semantics and Truth-Theoretic Semantics in General.Ross T. Brady - 2017 - IfCoLog Journal of Logics and Their Applications 4 (3):755--781.
Recent Work in Relevant Logic.Mark Jago - 2013 - Analysis 73 (3):526-541.
Four Basic Logical Issues.Ross Brady & Penelope Rush - 2009 - Review of Symbolic Logic 2 (3):488-508.

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