The square of opposition in orthomodular logic

TL;DR

This paper explores how the classical square of opposition, used in Aristotelian logic, can be represented and generalized within orthomodular logic frameworks, connecting modal operators and algebraic structures.

Contribution

It introduces a novel algebraic representation of the square of opposition in orthomodular logic, extending the classical modal square to non-Boolean structures.

Findings

Representation of the square of opposition using orthomodular logic

Generalizations of monadic first-order logic through algebraic structures

New interpretations of the square in non-Boolean contexts

Abstract

In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square.