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Logic—The Big Picture

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New Directions in Paraconsistent Logic

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 152))

Abstract

The big picture is my big picture, as I see it, based on a lifetime of research into logic. We will cover a reasonably wide range of topics, with some level of author focus. The paper builds on my earlier work [9, 11, 19] (with Rush). Indeed, it can also be seen as an update of the approach to logic taken in [9]. We start with the issue of what logic is about, identifying two inference concepts, one of meaning containment (a connective) and one of deductive argument in general (a rule). Examining the other connectives, we point out the difference between disjunction, as understood in proof-theoretic systems, as opposed to that understood in standard semantics, and show why distribution is not an instance of meaning containment. Negation is judged as being incompletely captured, due to the non-recursive nature of deductive systems in general, but with Boolean negation being the intended concept. We then focus on the logic MC of meaning containment, setting out its axiomatization, content semantics and metavaluation. Quantification is added in a standard way, based on the connectives. We finally deal with applications, focusing on set theory and arithmetic.

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Notes

  1. 1.

    This paper was presented to the Logic Seminar at the University of Melbourne, on 30th November, 2012. I would like to thank those present for their useful comments and discussion, in particular Greg Restall, Lloyd Humberstone and Rohan French. I also wish to thank Che-Ping Su for his help in getting the paper ready for presentation.

  2. 2.

    However, Stephen Read, in his book [25], claims that relevance is already contained in truth-preservation, rather than an added extra. Here, we follow the standard approach of separating the two concepts.

  3. 3.

    I wish to thank Greg Restall for making the point that content semantics does not behave like model theory in this respect and that this specialized content semantics requires an understanding of what these contents are.

  4. 4.

    One could try to base content semantics on metavaluations with a tight correspondence between the semantic postulates of the content semantics and each of the metavaluations, substituting the latter for the standard Hilbert-style axioms and rules. This metavaluation-driven content semantics, however, has never been tried, to the author’s knowledge.

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Correspondence to Ross T. Brady .

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Brady, R.T. (2015). Logic—The Big Picture. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_17

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