Normalisation and subformula property for a system of classical logic with Tarski’s rule

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for implication. The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic.

Negation, Structure, Transformation: Alain Badiou and the New Metaphysics

In this article, I discuss Alain Badiou’s 2008 address titled “The Three Negations.” Though the text was originally presented in a symposium concerning the relationship of law to Badiou’s theory of the event, I discuss the way this brief address offers an introduction to the broad sweep of Badiou’s metaphysics, outlining his accounts of being, appearing, and transformation. To do so, Badiou calls on the resources of three paradigms of negation: from classical Aristotelian logic, from Brouwer’s intuitionist logic, and in paraconsistent logics developed by DaCosta. I explain Badiou’s use of negation in the three primary areas of his metaphysics, as well as to diagnose the degrees of transformation that may have occurred in a situation. My analysis of Badiou’s use of negation in this text is aided by examples from his broader ontological oeuvre. I also explain the underlying requirement in Badiou’s work that formal considerations - mathematical or logical - get their sense by being tethered to readily-identifiable political, aesthetic, scientific, or interpersonal concerns. I conclude by addressing the foundation Badiou’s work establishes for pursuing a new metaphysics, and by discussing certain of the liabilities that remain in the wake of his account.

Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic

In An Introduction to Non-Classical Logic: From If to Is Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.

Causality is Logically Definable

This is the conclusion chapter. Bertrand Russell’s view on logic and mathematics is briefly reviewed. An enjoyable debate on bipolarity and isomorphism is presented. Some historical facts related to YinYang are discussed. Distinctions are drawn between BDL from established logical paradigms including Boolean logic, fuzzy logic, multiple-valued logic, truth-based dynamic logic, intuitionist logic, paraconsistent logic, and other systems. Some major comments from critics on related works are answered. A list of major research topics is enumerated. The ubiquitous effects of YinYang bipolar quantum entanglement are summarized. Limitations of this work are identified. Some conclusions are drawn.

RAZONANDO CON COLORES (Una aproximación a la lógica intuicionista)

We present an intuitive approach to Intuitionist logic based on the notion of n-paintingswhich generalizes the notion of sets; using several tonalities of a color, we constructoperations between n-paintings that can be interpreted in terms of propositions andlogical reasoning based on Heyting algebras.