Malinowski Strong versus weak quantum consequence operations

This paper is a study of similarities and differences between strong and weak quantum consequence operations determined by a given class of ortholattices. We prove that the only strong orthologics which admits the deduction theorem (the only strong orthologics with algebraic semantics, the only equivalential strong orthologics, respectively) is the classical logic. In the papers concerning quantum logics, there exist two different notions of logical consequences determined by a class of ortholattices. Historically first there was the notion of logical consequence created by G. Kalmbach [9]. A sentence α is a (weak) logical consequence of the set X of sentences iff in every model and every valuation in which every sentence of the set X has the unit of certain ortholattice as its logical value, the sentence α has the unit as its logical value, too. The weak consequence operation determined by the class of orthomodular lattices was intensively studied. [10] contains the monograph of this approach. From the algebraic point of view the weak orthologics (this means the weak consequence operations determined by any class of ortholattices) and especially the weak orthomodular logics possess numerous desirable properties. In particular the weak orthologics have strongly adequate algebraic semantics, moreover the weak orthomodular logics are implicative in the sense of [14]. However no weak orthomodular orthologic admits the deduction theorem [12]. H. Dishkant in [5] has given a Kripke-style semantics for the set of all sentences which hold true in all ortholattices. Kripke-style semantics treated as a consequence operation semantics determines another kind of orthologics, namely following strong orthologics: A sentence α is the (strong) logical consequence of the set of sentences X if and only if for any ortholattice A from a given class of ortholattices and any valuation v, v(β) ≤ v(α) for every β ∈ X (the symbol ≤ denotes the lattice order of A). This approach is due to R. Goldblatt’s paper [6]. The review of results concerning strong orthologics can be find in [7].