A formal language is positional if it involves a positional connecitve, i.e. a connective of realization to relate formulas to points of a kind, like points of realization or points of relativization. The connective in focus in this paper is the connective “R”, first introduced by Jerzy Łoś. Formulas [Rαφ] involve a singular name α and a formula φ to the effect that φ is satisfied relative to the position designated by α. In weak positional calculi no nested occurences of (...) the connective “R” are allowed. The distribution problem in weak positional logics is actually the problem of distributivity of the connective “R” over classical connectives, viz. the problem of relation between the occurences of classical connectives inside and outside the scope of the positional connective “R”. (shrink)

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, truth-functional 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.

According to Bocheński’s description of the history of formal logic, there clearly is some objective development, though far from being cumulative or linear. The history of logic throughout the world consists of three relatively short pinnacles preceded by also short periods of awakening and followed by periods of extensive commentary running into long periods of standstill and decadence, when nearly all achievements are consigned to oblivion.

The antinomy of future contingents is here understood as a trilemma whose horns are (a) the thesis of the closed past, (b) the thesis of the open future, and (c) the thesis that all events can be represented at any time. The latter thesis can take different forms, like the principle of bivalence or the thesis of divine foreknowledge. Different versions of (c) lead to different versions of the antinomy itself. The antinomy has been formalized. It hasbeen proven that the (...) theses (a), (b), and (c) make up an inconsistent set but are consistent with each other. Possible solutions have been considered. It has been argued that there are only two global solutions to the antinomy: radical determinism (fatalism) and retroactive causality. The latter solution has been recommended and developed. (shrink)

Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic £ are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E.J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatic is not independent: one axiom is derivable on the base of the remanent two. Several axiomatizations, based on three, two (...) or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “◊” and (d) the single specific axiom schema: ⬜A ∧ B → A ∧ ⬜B, called the jumping necessity axiom, constitute an elegant axiomatics of the system £. (shrink)

I argue that the Stoic logic is explosive. The claim applies to the Stoics' syllogistic in the strictest sense, because there is a provable syllogism which qualifies as a principle of explosion. It applies also to the general consequence operation, in the sense that every sentence is derivable from any pair containing both a sentence and the negation of the sentence. Finally, it applies to the connective of implication (conditional), in the sense that any conditional is derivable, providing its antecedent (...) is a conjunction of a sentence and the negation of the sentence. All three claims allow weakening, i.e., additions of extra premises to an inference or extra conjuncts to the antecedent of an implication, respectively. Consequently, no concept of relevance, let alone paraconsistency or connexivity is applicable to the Stoic logic; in particular, the Stoics' connective of implication is either material (Boolean) or formal (strict). (shrink)

Two questions concerning Anselm of Canterbury’s theistic argument provided in Proslogion Ch. 2 are asked and answered: is the argumentvalid? under what conditions could it be sound? In order to answer thequestions the argument is formalized as a first-order theory called AP2. Theargument turns out to be valid, although it contains a hidden premise. Theargument is also claimed not to be ontological one, but rather an a posteriori argument. One of the premises is found to be false, so the argument (...) isclaimed not to be sound and to fail to prove its conclusion. (shrink)

The paper contains a review and a discussion of Robert Trueman's book Properties and Propositions: The Metaphysics of Higher-Order Logic, Cambridge University Press, 2021, pp. xii + 227. ISBN 978-1-108-81410-2. The discussion is focused on the consistency of Truema's language-based ontology and on its value in defending higher-order logic.

część I. Natura logiki -- część II. Kultura logiczna.

A system HW of normal modal logic, developed by R. Bigelow & R. Pargetter is presented. Some formal issues concerning the system are examined, such as completeness, number of distinct modalities and relations to other systems. Some philosophical topics are also discussed. The Authors interpret the system HW as the system of physical (nomic) modalities. It is questioned, whether or not the system HW is justified to be claimed to be the logic of physical necessity. The answer seems to may (...) be negative. (shrink)

Przedmiotem dociekania jest filozofia religii pojmowana jako nauka o tak zwanych przekonaniach religijnych. Ponieważ pojęcie religii jest bardzo niejasne, filozofia religii nie ma dobrze określonego przedmiotu. Nie ma też określonej metody. Każde poważne osiągnięcie w tej dziedzinie stanowi bądź kryptoteologię, bądź kryptometafizykę. Studium przypadku stanowi analiza tezy „Bóg jest wszechmocny”. Okazuje się, że sens i sposób uzasadnienia tej tezy jest inny na gruncie teologii katolickiej i na gruncie metafizyki. Ponieważ nie widać miejsca na inny sposób rozumienia i uzasadniania tez tego (...) typu, nie ma miejsca na filozofię religii jako odrębną gałąź wiedzy. Uprawiając rzekomo filozofię religii, filozofowie tworzą — jak się wydaje — własne, prywatne religie, właściwie ogrywają role proroków. Przedmiotem ich analiz są w rzeczywistości takie prywatne religie. Wydaje się natomiast, że filozoficzna analiza pojęcia religii zasługiwałaby na miejsce wśród zagadnień filozoficznych. Religia w takim rozumieniu byłaby jednak raczej własnością ludzi niż przekonań. Byłaby to własność, przez którą człowiek jest religijny (wierny). (shrink)

Artykuł poświęcony jest charakterystyce kulturowego tła manifestu koła krakowskiego i jego zespołu, który stanowili: Józef Maria Bocheński OP, Jan Franciszek Drewnowski, ks. Jan Salamucha oraz Bolesław Sobociński, i który był wspierany przez Jana Łukasiewicza i ks. Konstantego Michalskiego. Manifestem koła była aplikacja programu szkoły lwowsko-warszawskiej do katolickiej teologii i filozofii. W tym przedsięwzięciu szczególną rolę odegrała Łukasiewicza wersja programu. Do zastosowanych środków należały: odnowienie języka tak, aby spełniał on wszystkie warunki właściwego dyskursu naukowego, recepcja logiki matematycznej, uwspółcześnienie semiotyki i metodologii (...) oraz wykorzystanie metod formalnych. Analizowane są cztery grupy okoliczności będące środowiskiem koła krakowskiego: esprit de l’époque, sytuacja w filozofii, sytuacja w logice i sytuacja w teologii – na początku XX wieku – w bezpośrednim związku z decydującymi działaniami czterech założycieli koła. (shrink)

Two questions concerning Anselm of Canterbury’s theistic argument provided in Proslogion Ch. 2 are asked and answered: is the argument valid? under what conditions could it be sound? In order to answer the questions the argument is formalized as a first-order theory called AP2. The argument turns out to be valid, although it contains a hidden premise. The argument is also claimed not to be ontological one, but rather an a posteriori argument. One of the premises is found to be (...) false, so the argument is claimed not to be sound and to fail to prove its conclusion. (shrink)

J. Bigelow and R. Pargetter in their work Science and Necessity put forward a theory of the laws of nature as statements objectively different with respect to their modal qualification both from the laws of logic and from contingent truths. Contrary to the latter ones all laws are characterized by necessity. However, there are various kinds of necessity. The laws of logic are characterized by logical necessity, and the laws of nature - by natural necessity. The objective basis for differentiating (...) modal qualification of statements belonging to the particular classes is that laws are truths about possibilities, also the ones that have not been actualized. The source of difference between logical and natural necessity is the differentiation between the range of possibilities described by respective laws. Hence, laws of nature prove to be - which is not mentioned by the authors - a posteriori necessary statements. The modal character has been the basis of the explanation of other considered properties of scientific laws: certain generality and the so-called range void. (shrink)

Temporal interpretation of modal logic consists in replacing possible worlds with temporal states of the world or any time determinates and the accessibility relation with a relation of passage of time. That issue has been raised by A. N. Prior, who was thinking of propositions as things which could change their truth-values (could become true or become false) with the passage of time. Under such interpretation Prior was reading a formula as: it (is and) will always be the case that (...) or: it (is and) has always been the case that . The formula should be read respectively. In the present paper the interpretation in question is examined. Its sources are presented and its consequences are analysed. It is claimed, the interpretation to be highly disputable because of its disagreement with physical meaning of temporal statements, established in the special relativity theory. (shrink)

Modal concepts - among them the concepts of logical, physical (nomic) and metaphysical necessity - used to be quite important for philosophy of science during centuries. However, in the XX c. most philosophers preferred not to recognize those concepts in science (especially the concept of physical necessity). They were wrong. Some patterns from history of physics are presented, showing the concept of physical necessity playing an important role in the scientific research of nature. And the nature of physically necessary statements (...) is different from both logically necessary statements - on the one hand - and contingent statements, on the other. Consequently three attempts to explain the nature of physical necessity are discussed. (1) Physical necessities are just relative necessities, logical consequences of physical laws. (2) Logical and physical necessities are of the same nature, although logically necessary statements are as well analytical, while those physically necessary are synthetic. Some serious difficulties arising in both theories are shown. Finally a third explanation is outlined. (3) The two kinds of necessary statements differ in their reference: they describe different kinds of real relationships (connection). (shrink)