Quantum conditional and quantum conditional probability

Abstract

It is well known that in the classical, elementary propositional logic one can define the classical conditional connective A → B = A ^⊥ ∨ B. This conditional, which is also known as “material implication”, or “horseshoe”, describes certain features of the inference “if A ... then B” In this chapter the basic properties of the so-called “quantum conditional” are described. The quantum conditional is the analogue of the classical conditional connective, it is defined in the Hilbert lattice P(H) (or more generally in any orthomodular lattice) in terms of the lattice operations ∧, ∨ and ⊥ in P(H), and it reflects certain features of “if...then” inference. The semantic content of the quantum conditional is given by the “minimal implicative criteria”, which the quantum conditional is required to satisfy. Section 8.1 describes these minimal implicative criteria. It turns out that there exist three conditionals satisfying the minimal implicative criteria. Each of these conditionals is a natural non-commutative generalization of the classical horseshoe, and each of them violates some conditions that the classical conditional satisfies (Propositions 8.2 and 8.4). Proposition 8.3 is the main result in Section 8.1, it tells us that the orthomodularity of P(H) is equivalent to the existence of a unique quantum conditional that satisfies the natural weakening of the classical implication condition. A further characteristic property of the quantum conditional is that it is a counterfactual conditional (Proposition 8.5).