Since quantum physics has been developed, we must deal with the problem of a physical, empirical logic, just as relativity has given us the problem of a physical, empirical geometry. As stressed by Finkelstein, logic is indeed empirical and it constitutes a dynamical element of a physical theory, itself evolving conditionally to processes. Thus, there is need to turn upside down the hierarchy on which physics is built, that, starting from mathematical logic and throughout set theory, geometry and dynamics, goes up to quantization, so there is a breakdown of such a hierarchy and of its epistemic, foundationalist presuppositions. On the other hand, the existence of undecidable enunciates in the set universe and the plurality of the possible set theories have definitively determined the breakdown of the foundation role of the set universe as the unique universe of all the (mathematical) concepts, and furtherly, with the contraposition among classical logic and other "deviant" logics, as the intuitionistic-constructive ones, they have raised the possibility to construct different mathematics. Then, it is the formulation of logic by the algebraic theory of categories, that, recognizing the irreducible plurality of the universes of discourses through the so called elementary "topoi", allows for an internalization of logic, which is determined by the algebra of the considered processes: it determines also an overcoming of the idealistic nature of the set foundation and of the general ambiguity of the foundationalist paradigm. So it is the dynamical algebra of "quantum-relativistic" physical processes that determines a quantum-relativistic logic and a quantum-relativistic mathematics (set theory, topology, geometry), eluding any foundation and realizing an actual "physics of logic" as well as a "physics of mathematics", and so a "physics of knowledge and of epistemology".
Following such a perspective of a physics of logic", in this paper, we analyse a problem concerning the definition itself of a quantum logic, that is if a truth value can be assigned or it does not to the microphysical enunciates.
We consider the proposal of Karel Lambert to apply a "free logic" system to quantum mechanics: this proposal was formulated just in an aprioristic way, starting by a new definition of logical truth (given by van Fraassen) and solving the quantum logic dilemma as it points out only the need for a new semantics of the old classical logic.
On the contrary, we show that a free quantum logic is needed just "to save microphysical phenomena" and to no aprioristic claim and furthermore such a quantum logic, though it maintains a classical code, represents an epistemological fracture in regard to classical mathematical logic, being irreducible to a mere new semantics.
Indeed, turning back to Reichenbach's approach and its physical motivations, from which Lambert was started, we can conclude that microphysical enunciates, related either to conjugate variable simultaneous measure operations or to elementary quantum physical processes, considered apart from a measure operation, actually present "truth-holes", and we must use non-denotating singular terms and free logic.
Elementary quantum processes cannot be eliminated by a mere syntactic-operational cut, because without them the measure processes themselves cannot have any meaning.
The enunciates related to possible measure operations have no "truth" other than the one derived by either tautology or contradiction from the "atomic" enunciates related to the elementary quantum process, and, therefore, this "truth" must be considered only as a mere preservation of truth-holes, of a fundamental lack of truth: indeed, there is a breakdown of denotability as well as a breakdown of the concepts of truth and of semantics.
The unique universe of discourse that is logically definable is that one related to the algebra of measure processes; on the contrary, elementary physical processes do not constitute a domain of classical objects and do not constitute a universe of discourse, because they can be associated only to non-referential singular terms, just as considered apart from measures (that only can determine an actual denotation or an actual semantical valuation) they are indeterminate, that is "empty" of semantical or ontological determinations and they cannot be associated to any object in the universe of discourse of measure processes.