New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Understand

Intuitionistic logic, in which the double negation law not-not-P = P fails, is dominant in categorical logic, notably in topos theory.

• This paper follows a different direction in which double negation does hold.
• The algebraic notions of effect algebra/module that emerged in theoretical physics form the cornerstone.
• It is shown that under mild conditions on a category, its maps of the form X -> 1+1 carry such effect module structure, and can be used as predicates.