On Two Squares of Opposition: the Leśniewski’s Style Formalization of Synthetic Propositions

Abstract

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.

Go Games on Plasmodia of Physarum Polycephalum

“…In other words, we can draw more natural conclusions for protoplasmic tubes which are decentralized and have some cycles. The formal syllogistic system over propositions with such properties is constructed in [9], [10]. This system is called the performative syllogistic.…”

“…The formal properties of this axiomatic system are considered in [9], [10]. In the performative syllogistic we can analyze the collective dimension of behavior.…”

“…Thus, the performative syllogistic allows us to study different zones containing attractants for Physarum For more details on formal properties of performative syllogistic, please see [9], [10]. This syllogistic describes the logic of plasmodium propagation in all possible directions.…”

“…12 there are no universal affirmative propositions and particular affirmative propositions. We face only universal negative propositions and particular negative propositions such as 'No S' 4,9 are S' 4,10 ', where S' 4,9 are neighbors for S 4,10 differing from S 4,9 and S' 4,10 are neighbors for S 4,9 differing from S 4,10 , and 'Some S' 4,11 are not S' 5,11 ', where S' 4,11 are neighbors for S 5,11 differing from S 4,11 and S' 5,11 are neighbors for S 4,11 differing from S 5,11 .…”

“…Therefore it is better to focus just on games, where locations of black and white stones simulate spatial reasoning. In this paper we propose two logics in the universe of possible Go games: (1) Aristotelian syllogistic [3]; (2) performative syllogistic [9], [10].…”