The Rule of Global Necessitation

TL;DR

This paper explores a systematic weakening of the necessitation rule in modal logic to address classical paradoxes and problems, demonstrating its application in various logical and philosophical contexts.

Contribution

It introduces a systematic approach to weakening necessitation, enabling solutions to longstanding paradoxes and problems in modal logic and epistemology.

Findings

Consistent S4 with arithmetic achieved

Resolution of the surprise examination paradox

Construction of a self-aware knowing machine

Abstract

For half a century, authors have weakened the rule of necessitation in various more or less ad hoc ways in order to make inconsistent systems consistent. More recently, necessitation was weakened in a systematic way, not for the purpose of resolving paradoxes but rather to salvage the deduction theorem for modal logic. We show how this systematic weakening can be applied to the older problem of paradox resolution. Four examples are given: a predicate symbol S4 consistent with arithmetic; a resolution of the surprise examination paradox; a resolution of Fitch's paradox; and finally, the construction of a knowing machine which knows its own code. We discuss a technique for possibly finding answers to a question of P. \'Egr\'e and J. van Benthem.