Australasian Journal of Logic 15 (2):381-406 (2018)

Thomas Ferguson
City University of New York
The Routley star, an involutive function between possible worlds or set-ups against which negation is evaluated, is a hallmark feature of Richard Sylvan and Val Plumwood's set-up semantics for the logic of first-degree entailment. Less frequently acknowledged is the weaker mate function described by Sylvan and his collaborators, which results from stripping the requirement of involutivity from the Routley star. Between the mate function and the Routley star, however, lies an broad field of intermediate semantical conditions characterizing an infinite number of consequence relations closely related to first-degree entailment. In this paper, we consider the semantics and proof theory for deductive systems corresponding to set-up models in which the mate function is cyclical. We describe modifications to Anderson and Belnap's consecution calculus LE_fde2 that correspond to these constraints, for which we prove soundness and completeness with respect to the set-up semantics. Finally, we show that a number of familiar metalogical properties are coordinated with the parity of a mate function's period, including refined versions of the variable-sharing property and the property of gentle explosiveness.
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DOI 10.26686/ajl.v15i2.4071
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References found in this work BETA

The Non-Involutive Routley Star: Relevant Logics Without Weak Double Negation.Gemma Robles - 2010 - Teorema: International Journal of Philosophy 29 (3):103-116.
A Note on the Non-Involutive Routley Star.Gemma Robles - 2008 - Bulletin of the Section of Logic 37 (1):19-27.

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