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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 adifferent direction in which double negation does hold. The algebraic notionsof effect algebra/module that emerged in theoretical physics form thecornerstone. It is shown that under mild conditions on a category, its maps ofthe form X -> 1+1 carry such effect module structure, and can be used aspredicates. Predicates are identified in many different situations, and capturefor instance ordinary subsets, fuzzy predicates in a probabilistic setting,idempotents in a ring, and effects (positive elements below the unit) in aC*-algebra or Hilbert space. In quantum foundations the duality between statesand effects plays an important role. It appears here in the form of anadjunction, where we use maps 1 -> X as states. For such a state s and apredicate p, the validity probability s |= p is defined, as an abstract Bornrule. It captures many forms of (Boolean or probabilistic) validity known fromthe literature. Measurement from quantum mechanics is formalised categoricallyin terms of `instruments', using L\"uders rule in the quantum case. Theseinstruments are special maps associated with predicates (more generally, withtests), which perform the act of measurement and may have a side-effect thatdisturbs the system under observation. This abstract description ofside-effects is one of the main achievements of the current approach. It isshown that in the special case of C*-algebras, side-effect appear exclusivelyin the non-commutative case. Also, these instruments are used for testoperators in a dynamic logic that can be used for reasoning about quantumprograms/protocols. The paper describes four successive assumptions, towards acategorical axiomatisation of quantitative logic for probabilistic and quantumsystems.

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