Three-valued logic, indeterminacy and quantum mechanics

The paper consists of two parts. The first part begins with the problem of whether the original three-valued calculus, invented by J. Lukasiewicz, rea lly conforms to his philosophical and semantic intuitions. I claim that one of the basic semantic assumptions underlying Lukasiewicz's three-valued lo gic should be that if under any possible circumstances a sentence of the fo rm "x will be the case at time t' is true (resp. false) at time t, then thi s sentence must be already true (resp. false) at present. However, it is ea sy to see that this principles is violated in Lukasiewicz's original calcul us (as the cases of the law of excluded middle and the law of contradiction show). Nevertheless it is possible to construct (either with the help of t he notion of 'supervaluation', or purely incorporate Lukasiewicz's initial intuitions. Algebraically, this calculus has the ordinary Boolean structure , and therefore it retains all classically valid formulas. Yet because poss ible valuations are no longer represented by ultrafilters, but by filters ( not necessarily maximal), the new calculus displays certain non-classical m etalogical features (like, for example, non-extensionality and the lack of the metalogical rule enabling one to derive 'p is true or q is true' from p \/ q is true'). The second part analyses whether the proposed calculus cou ld be useful in formalizing inferences in situations, when for some reason (epistemological or ontological) our knowledge of certain facts is subject to limitation. Special attention should be paid to the possibility of emplo ying this calculus to the case of quantum mechanics. I am going to compare it with standard non-Boolean quantum logic (in the Jauch-Prion approach), a nd to show that certain shortcomings of the latter can be avoided in the fo rmer. For example, I will argue that in order to properly account for quant um features of microphysics, we do not need to drop the law of distributivi ty. Also the idea of 'reading off' the logical structure of propositions fr om the structure of Hilbert space leads to some conceptual troubles, which I am going to point out. The thesis of the paper is that all we need to spe ak about quantum reality can be acquired by dropping the principle of bival ence and extensionality, while accepting all classically valid formulas.