The paper introduces a concept of logic applied to a formalization of the so-called inferences preserving degrees of truth. Semantical and syntactical characterizations of three kinds of logics preserving degrees of truth are provided. The other approach than in [3] and [9] to the problem of expressing that a sentence is less true than a sentence is presented.

In the paper, tracing the traditional Hilbert-style syntactic account of logics, a syntactic characteristic of a closure operation defined on a complete lattice follows. The approach is based on observation that the role of rule of inference for a given consequence operation may be played by an ordinary binary relation on the complete lattice on which the closure operation is defined.

Formalization of reasoning which accepts rules of inference leading to conclusions whose logical values are not smaller than the logical value of the “weakest” premise leads to the concept of consequence operation preserving degrees of truth. Several examples of such consequence operation have already been considered . In the present paper we give a general notion of the consequence operation preserving degrees of truth and its characterization in terms of projective generation and selfextensionality.

The paper deals with a generalization of the notion of partition for wider classes of binary relations than equivalences: for quasiorders and tolerance relations. The counterpart of partition for the quasiorders is based on a generalization of the notion of equivalence class while it is shown that such a generalization does not work in case of tolerances. Some results from [5] are proved in a much more simple way. The third kind of “partition” corresponding to tolerances, not occurring in [5], (...) is introduced. (shrink)

There are well-known isomorphisms between the complete lattice of all partitions of a given set A and the lattice of all equivalence relations on A. In the paper the notion of partition is generalized in order to work correctly for wider classes of binary relations than equivalence ones such as quasiorders or tolerance relations. Some others classes of binary relations and corresponding counterparts of partitions are considered.

Our purpose is to formulate a complete logic of propositions that takes into account the fact that propositions are both senses provided with truth values and contents of conceptual thoughts. In our formalization, propositions are more complex entities than simple functions from possible worlds into truth values. They have a structure of constituents (a content) in addition to truth conditions. The formalization is adequate for the purposes of the logic of speech acts. It imposes a stronger criterion of propositional identity (...) than strict equivalence. Two propositions P and Q are identical if and only if, for any illocutionary force F, it is not possible to perform with success a speech act of the form F(P) without also performing with success a speech act of the form F(Q). Unlike hyperintensional logic, our logic of propositions is compatible with the classical Boolean laws of propositional identity such as the symmetry and the associativity of conjunction and the reduction of double negation. (shrink)

Two examples of Galois connections and their dual forms are considered. One of them is applied to formulate a criterion when a given subset of a complete lattice forms a complete lattice. The second, closely related to the first, is used to prove in a short way the Knaster-Tarski’s fixed point theorem.

Two different kinds of multiple-conclusion consequence relations taken from Shoesmith and Smiley and Galatos and Tsinakis or Nowak, called here disjunctive and conjunctive, respectively, defined on a formal language, are considered. They are transferred into a bounded lattice and a complete lattice, respectively. The properties of such abstract consequence relations are presented.

The concept of multiple-conclusion consequence relation from [8] and [7] is considered. The closure operation C assigning to any binary relation r the least multiple-conclusion consequence relation containing r, is dened on the grounds of a natural Galois connection. It is shown that the very closure C is an isomorphism from the power set algebra of a simple binary relation to the Boolean algebra of all multiple-conclusion consequence relations.

The paper deals with seven propositional paraconsistent logics. Four of them are based on intuitionistic positive logic: the minimal Johansson’s logic, some two its weakenings and its extension by the law of excluded middle. The remaining three ones are their counterparts having the classical positive base. For all logics the Kripke-style semantics is provided.

Celem pracy jest analiza formalna pewnych ogólnych własności charakteryzujących wnioskowania. Autorzy twierdzą, że posiadanie tych własności jest warunkiem koniecznym, aby wnioskowanie było logicznie wartościowe. Wprowadzają aksjomatycznie pojęcie "uogólnionej operacji inferencji", które formalnie ujmuje wnioskowanie mające owe cechy. Następnie reprezentują uogólnioną operację inferencji przy użyciu relacji binarnych określonych na podzbiorach języka. Podają również taką reprezentację dla logicznej operacji konsekwencji (każda logiczna operacja konsekwencji jest uogólnioną operacją inferencji).

The aim of this article is to evaluate critically possibilities of interpreting Łukasiewicz logic by means of J.Słupecki s method presented in reference (1). In paragraph 1, we are giving a formal construction on which the method of partial interpretation Ƚ₃ is based. Paragraph 2 contains initial assumptions through which prof. J. Słupecki obtains the partial interpretation Ƚ₃ In paragraph 3, we prove that the way of obtaining this interpretation is really based on the construction from paragraph 1. Finally, in (...) paragraph 4, we present our opinion on the interpretation Ƚ₃. We prove that this interpretation cannot be obtained by means of the applied method. (shrink)

Theological Faculty of the Adam Mickiewicz University in Poznań organized a series of sessions devoted to problems of dialogue. Professor Baniak, the main organizer of those meetings, invited philosophers, theologians, pedagogues, psychologists and other intellectuals/scientists, whose area of interest was dialogue. The first conference took place in June 2001, the last in June 2007, and organizers have a hope that the endeavor would be continued. Lectures given at conferences were devoted to many subjects—classical philosophy of dialogue, dialogue in theological thought, (...) interreligious dialogue etc. (shrink)

The paper contains two concepts of implication that can be called analytic in the sense of Parry [5] or Fine [2]. Contrary to the Parry's approach, these implications do not involve S4 strict implication but the classical one. This fact refers to the similar notion of so-called demodalized analytic implication of Dunn [1] and strong implication of Vanderveken, cf. [4, 6]. The features of analytic classical implications are presented in the form of two propositional logics on the pure implicational language.

W artykule analizuje się pojęcie „proposition" (sądu w sensie logicznym) wprowadzone w pracy D. Vandervekena "What is a Proposition", stosując metody algebraiczne. Analiza ta umożliwia głębsze zrozumienie tego pojęcia, prowadzi m.in. do uogólnienia pojęcia „mocnej implikacji" (§5), jej głównym rezultatem jest pewna reprezentacja pojęcia „proposition” (§ 6).