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2 results

  • 6 - Theories and Closure

  • Edwin Mares, Victoria University of Wellington
  • Book:
    The Logic of Entailment and its History
    Published online:
    08 February 2024
    Print publication:
    15 February 2024, pp 119-135

  • Summary

    The semantical framework for the positive view of this book is one in which entailment is understood primarily in terms of theory closure. This chapter outlines both the history of the notion, beginning with Alfred Tarski’s theory of closure operators, and the relationship between closure operators and the entailment connective. At the end of the chapter, it is shown how closure operators can be used to model a simple logic, Graham Priest’s logic N4.

  • CLOSURE OPERATORS, FRAMES AND NEATEST REPRESENTATIONS

  • Part of
  • ROB EGROT
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 96 / Issue 3 / December 2017
    Published online by Cambridge University Press:
    21 June 2017, pp. 361-373
    Print publication:
    December 2017
    • Article
      • You have access
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  • Given a poset $P$ and a standard closure operator $\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$, we give a necessary and sufficient condition for the lattice of $\unicode[STIX]{x1D6E4}$-closed sets of ${\wp}(P)$ to be a frame in terms of the recursive construction of the $\unicode[STIX]{x1D6E4}$-closure of sets. We use this condition to show that, given a set ${\mathcal{U}}$ of distinguished joins from $P$, the lattice of ${\mathcal{U}}$-ideals of $P$ fails to be a frame if and only if it fails to be $\unicode[STIX]{x1D70E}$-distributive, with $\unicode[STIX]{x1D70E}$ depending on the cardinalities of sets in ${\mathcal{U}}$. From this we deduce that if a poset has the property that whenever $a\wedge (b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$, then it has an $(\unicode[STIX]{x1D714},3)$-representation.