We introduce the concept of partial event as a pair of disjoint sets, respectively the favorable and the unfavorable cases. Partial events can be seen as a De Morgan algebra with a single fixed point for the complement. We introduce the concept of a measure of partial probability, based on a set of axioms resembling Kolmogoroff’s. Finally we define a concept of conditional probability for partial events and apply this concept to the analysis of the two-slit experiment in quantum mechanics.

The sorites as a formal theory. The sorites paradox has been understood as a failure of classical logic and a way out can be found by choosing a non classical logic (fuzzy logic, for instance). We understand the sorites not as a problem of logic but of ontology, so we preserve classical logic and submit the premisses of the argument to a deeper analysis. As a result we obtain two theories for soritical arguments and enlighten the kind of structure requested (...) by these arguments. The former theory has a lattice theoretic flavor and is fitted for paradoxes like “the heap” and “the bald man”; the latter is group theoretic in character and can manage paradoxes based on observational predicates like “tall”, “red” and so on. In both cases the contradiction is derived from a larger set of hypotheses than in usual analysis and some of these hypotheses make assertions whose empirical meaning can be questioned. (shrink)

The sorites as a formal theory. The sorites paradox has been understood as a failure of classical logic and a way out can be found by choosing a non classical logic (fuzzy logic, for instance). We understand the sorites not as a problem of logic but of ontology, so we preserve classical logic and submit the premisses of the argument to a deeper analysis. As a result we obtain two theories for soritical arguments and enlighten the kind of structure requested (...) by these arguments. The former theory has a lattice theoretic flavor and is fitted for paradoxes like “the heap” and “the bald man”; the latter is group theoretic in character and can manage paradoxes based on observational predicates like “tall”, “red” and so on. In both cases the contradiction is derived from a larger set of hypotheses than in usual analysis and some of these hypotheses make assertions whose empirical meaning can be questioned. (shrink)

In this work we show that every structure [MATHEMATICAL SCRIPT CAPITAL A] can be expanded to a partial structure [MATHEMATICAL SCRIPT CAPITAL A]* with universal functions for the class of polynomials on [MATHEMATICAL SCRIPT CAPITAL A]*. We can embed [MATHEMATICAL SCRIPT CAPITAL A]* monomorphically in a total structure [MATHEMATICAL SCRIPT CAPITAL A]º that preserves universal functions of [MATHEMATICAL SCRIPT CAPITAL A]* and that is universal among such structures, i.e. [MATHEMATICAL SCRIPT CAPITAL A]º can be homomorphically embedded in every total structure (...) that preserves universal functions of [MATHEMATICAL SCRIPT CAPITAL A]*. Universal functions are the starting point for developing recursion theoretic tools in an [MATHEMATICAL SCRIPT CAPITAL A]* that satisfies some simple additional conditions. (shrink)