Tomasz Bigaj

This is an attempt to defend Field's nominalistic program from the criticism raised by K. Wójtowicz in his article. The author argues for the following theses: (a) that Wójtowicz uses the notion of „mathematical theory” broader than Field does it; (b) that he misinterprets the conception of the „abstract counterparts” of nominalistic statements; (c) and that his general evaluation of Field's program is based on too high methodological standards which he applies to the possible nominalistic versions of empirical theories. The second part of this paper contains an attempt to generalize the results of Field's analysis. The following fact is proved: every open sentence expressible in the language of an empirical theory and being empirically contentful is implicitly definable by the set of certain qualitative predicates. In the case of first-order language this result can be strengthened via Beth's definability theorem, to the theorem stating that every open sentence fulfilling conditions formulated above is definable explicitly with the help of certain nominalistic formula. The philosophical significance of this result is that each mathematized empirical theory for which representation theorem is true, can be translated into purely qualitative version.

In the article the possibility of breaking the eigenvalue-eigenstate link in quantum mechanics is considered. An argument is presented to the effect that there are some non-maximal observables for which the implication from eigenstates to eigenvalues is not valid, i.e. such that although the probability of revealing certain value upon measurement is one, they don't possess this value before the measurement. It is shown that the existence of such observables leads to contextuality, i.e. the thesis that one Hermitean operator can represent more than one physical observable. Finally, contextuality brought about by these considerations is compared with contextuality suggested by the Kochen-Specker paradox.