http://hdl.handle.net/11089/53268 Mon, 06 Apr 2026 23:40:34 GMT 2026-04-06T23:40:34Z https://dspace.uni.lodz.pl:443/bitstream/id/ed35e0ed-b2f3-4b94-a325-c4a7867e690f/ http://hdl.handle.net/11089/53268 http://hdl.handle.net/11089/53273 About Logically Probable Sentences Olszewski, Adam The starting point of this paper is the empirically determined ability to reason in natural language by employing probable sentences. A sentence is understood to be logically probable if its schema, expressed as a formula in the language of classical propositional calculus, takes the logical value of truth for the majority of Boolean valuations, i.e., as a logically probable formula. Then, the formal system P is developed to encode the set of these logically probable formulas. Based on natural semantics, a strong completeness theorem for P is proved. Alternative notions of consequence for logically probable sentences are also considered. Tue, 23 Apr 2024 00:00:00 GMT http://hdl.handle.net/11089/53273 2024-04-23T00:00:00Z http://hdl.handle.net/11089/53274 Free Spectra of Equivalential Algebras with Conjunction on Dense Elements Przybyło, Sławomir; Słomczyńska, Katarzyna We construct free algebras in the variety generated by the equivalential algebra with conjunction on dense elements and compute the formula for the free spectrum of this variety. Moreover, we describe the decomposition of free algebras into directly indecomposable factors. Mon, 20 May 2024 00:00:00 GMT http://hdl.handle.net/11089/53274 2024-05-20T00:00:00Z http://hdl.handle.net/11089/53272 On Pre-Hilbert and Positive Implicative Pre-Hilbert Algebras Walendziak, Andrzej In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin’s Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown. Mon, 20 May 2024 00:00:00 GMT http://hdl.handle.net/11089/53272 2024-05-20T00:00:00Z http://hdl.handle.net/11089/53271 On Combining Intuitionistic and S4 Modal Logic Rasga, João; Sernadas, Cristina We address the problem of combining intuitionistic and S4 modal logic in a non-collapsing way inspired by the recent works in combining intuitionistic and classical logic. The combined language includes the shared constructors of both logics namely conjunction, disjunction and falsum as well as the intuitionistic implication, the classical implication and the necessity modality. We present a Gentzen calculus for the combined logic defined over a Gentzen calculus for the host S4 modal logic. The semantics is provided by Kripke structures. The calculus is proved to be sound and complete with respect to this semantics. We also show that the combined logic is a conservative extension of each component. Finally we establish that the Gentzen calculus for the combined logic enjoys cut elimination. Wed, 05 Jun 2024 00:00:00 GMT http://hdl.handle.net/11089/53271 2024-06-05T00:00:00Z