In this paper we consider possibilities for applying p-adic valued logic BL to the task of designing an unconventional computer based on the medium of slime mould, the giant amoebozoa that looks for attractants and reaches them by means of propagating complex networks. If it is assumed that at any time step t of propagation the slime mould can discover and reach not more than attractants, then this behaviour can be coded in terms of p-adic numbers. As a result, this (...) behaviour implements some p-adic valued arithmetic circuits and can verify p-adic valued logical propositions. (shrink)

In this article, the author attempts to explicate the notion of the best known Talmudic inference rule called qal wa-omer. He claims that this rule assumes a massive-parallel deduction, and for formalizing it, he builds up a case of massive-parallel proof theory, the proof-theoretic cellular automata, where he draws conclusions without using axioms.

In the paper we build up the ontology of Leśniewski’s type for formalizing synthetic propositions. We claim that for these propositions an unconventional square of opposition holds, where a, i are contrary, a, o (resp. e, i) are contradictory, e, o are subcontrary, a, e (resp. i, o) are said to stand in the subalternation. Further, we construct a non-Archimedean extension of Boolean algebra and show that in this algebra just two squares of opposition are formalized: conventional and the square (...) that we invented. As a result, we can claim that there are only two basic squares of opposition. All basic constructions of the paper (the new square of opposition, the formalization of synthetic propositions within ontology of Leśniewski’s type, the non-Archimedean explanation of square of opposition) are introduced for the first time. (shrink)

We relate Popper functions to regular and perfectly additive such non-Archimedean probability functions by means of a representation theorem: every such non-Archimedean probability function is infinitesimally close to some Popper function, and vice versa. We also show that regular and perfectly additive non-Archimedean probability functions can be given a lexicographic representation. Thus Popper functions, a specific kind of non-Archimedean probability functions, and lexicographic probability functions triangulate to the same place: they are in a good sense interchangeable.

I characterize the deductivist ideal of justification and, following to a great extent Toulmin’s work The Uses of Argument, I try to explain why this ideal is erroneous. Then I offer an alternative model of justification capable of making our claims to knowledge about substantial matters sound and reasonable. This model of justification will be based on a conception of justification as the result of good argumentation, and on a model of argumentation which is a pragmatic linguistic reconstruction of Toulmin’s (...) model of argument. (shrink)