Negation in weak positional calculi

Abstract

Abstract. Four weak positional calculi are constructed and examined. They refer to the use of the connective of negation within the scope of the positional connective "R" of realization. The connective of negation may be fully classical, partially analogical or independent from the classical, truthfunctional negation. It has been also proved that the strongest system, containing fully classical connective of negation, is deductively equivalent to the system MR from Jarmużek and Pietruszczak.

Distribution Laws in Weak Positional Logics

“…The system MR has been presented and examined by Tomasz Jarmużek and Andrzej Pietruszczak (2004), further examined by Marcin Tkaczyk (2013), by Jarmużek and Tkaczyk (2015) and by Anna Maria Karczewska (2018). It has been originally axiomatized by Jarmużek and Pietraszak (2004)…”

“…and the system thus constructed is exactly equal to the original version of MR (TKACZYK 2009(TKACZYK , 2013. This information may turn out to be of some use in analyses to come.…”

“…is the calculus based on the axioms: (1), (RE), (RH), (RK), (RP), (RQ) and the rule (MP). The top calculus, ABCDEFGHIJKMNPQ (A-Q for short), is deductively equivalent to the calculus MR from Jarmużek and Pietruszczak (TKACZYK 2009;2013). It turns out then that all the schemata (RA)-(RQ) are deductively perfectly independent, and hence separable from one another.…”

Maximality of the Minimal R-Logic

“…Alternative semantic structures, as well as different axiomatizations of the system MR have been defined and examined in [8] and [3]. In particular, it has been shown that the semantics presented here may be generalized, without affecting the set of formulas true in a model, to the cases in which the interpretation of quasi-formulas in positions non denoted by constants from IN need not behave classically [3, pp.…”

Book Reviews

“…The authors might be treated as continuers of Łoś's approach as series of their articles show: [2,3,4,13,14]. But there are also others logicians who find subject of propositional logic interesting (for example, [7,8,9,10]).…”