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.

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)

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)

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)

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)

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)

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)

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)

The proof of God’s existence, known as Ratio Anselmi, is being analyzed. Four first-order theories are constructed to mirror versions of Anselm’s reasoning. God’s existence is shown to be provable in all of them. A traditional objection to the employment of a concept of God is overruled. And yet, Anselm’s proof is eventually found to be incorrect. The error attributed to Anselm consists in an illegitimate use of the words “greater‘ and “conceivable‘, and is identified as quaternio terminorum or petitio (...) principii, depending on circumstances. It is claimed that there is no direct way to improve the argument. (shrink)