Set Theory and Logic

   

Rule of Necessitation: a Non-Contingent Truthity, But not a Tautology

Authors: Colin James III

1. The axiom or rule of necessitation N states that if p is a theorem, then necessarily p is a theorem: If ⊢ p then ⊢ ◻p. We show this is non-contingent (a truthity), but not tautologous (a proof). We evaluate axioms (in bold) of N, K, T, 4, B, D, 5 to derive systems (in italics) of K, M, T, S4, S5, D. We conclude that N the axiom or rule of necessitation is not tautologous. Because system M as derived and rendered is not tautologous, system G-M also not tautologous. What follows is that systems derived from using M are tainted, regardless of the tautological status of the result so masking the defect, such as systems S4, B, and S5. We also find that Gentzen-sequent proof is suspicious, perhaps due to its non bi-valent lattice basis in a vector space.

Comments: 4 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

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[v1] 2018-11-23 13:43:15

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