Exogenous Quantum Logic

Most of the work on quantum logic (since the seminal paper [4]) has continued to adopt the lattice of closed subspaces of a Hilbert space as the basis for its semantics [11, 8]. Here we take a quite different approach, what we call the exogenous approach. The key idea is to keep the models of the classical logic (say propositional logic) as they are, to produce models for the envisaged quantum logic as superpositions of classical models, and, finally, to design a suitable language for constraining such superpositions. The exogenous approach is a variation of the possible worlds approach originally proposed by Kripke [14] for modal logic, and it is also akin to the society semantics introduced in [7] for many-valued logic and to the possible translations semantics proposed in [6] for paraconsistent logic. In fact, Kripke structures can be described as binary relations between classical models, the models in [7] are just collections of classical models, and the models in [6] are obtained using translation maps into the original logic(s). The possible worlds approach was also used in [19, 20] for probabilistic logic1: the models turn out to be probability spaces of classical models, as first recognized in [10]. The difference between the possible worlds approach and the exogenous approach is subtle but full of consequences. The exogenous approach is used in [16] to develop a probabilistic version of any given logic where, as in [10], each model is a probability space of the original models, but where the connectives are different from those of the logic being probabilized. As explained in Section 2, the global semantics of the new connectives arises naturally when using the new models. Note that the endogenous approach to probabilistic logic is also useful and, actually, widely used. By endogenous approach we mean that we tinker with the classical models in order to make them suitable for a specific type of probabilistic reasoning. For instance, if we want a logic for reasoning about probabilistic transition systems (probabilistic automata) we can modify the Kripke models of dynamic logic by labelling the transition pairs (pairs of the accessibility relation) with probabilities [12, 15]. As another example of the endogenous approach, consider the probabilization of first-order logic obtained by enriching