We study here extensions of the Artemov's logic of proofs in the language with quantifiers on proof variables. Since the provability operator □ A could be expressed in this language by the formula u[u]A, the corresponding logic naturally extends the well-known modal provability logic GL. Besides, the presence of quantifiers on proofs allows us to study some properties of provability not covered by the propositional logics.In this paper we study the arithmetical complexity of the provability logic with quantifiers on proofs (...) for a given arithmetical theory T and a class of proof predicates.In the last section we define Kripke style semantics for the logics corresponding to the standard Gödel proof predicate and its multiple conclusion version. (shrink)

Since the mid-90’s the figure of Peter Lombard and his Book of Sentences has regained the importance in scholarly world and been studied from both historical-theological and historical-philosophical perspectives. But some aspects of his thinking, encapsulated in the written form, which was to become the material basis for the thirteenth- through the fifteenth-century theological projects, remained somewhat insufficiently researched. Therefore this article analyzes the select parts of the Book of Sentences with the purpose of looking at how Peter Lombard handled (...) the issue of God’s knowledge. The article shows that for Peter Lombard God’s knowledge is God’s awareness of everything knowable. It has no causal power which belongs to the divine will. Nevertheless, this knowledge is able to function in two different modes: it can be either a purely cognitive act as awareness alone, or a double cognitive and voluntary act as awareness and simultaneous volition in the form of approbation. Hence, God’s knowledge in general is not causative, but God’s knowledge of the good must be causative because he simultaneously knows and wills what is good. The article reasonably suggests that Lombard’s logic implies the compatibility of God’s (fore)knowledge and voluntary activity, on the one hand, and the contingency of the created order and the rational creatures’ free will, on the other hand. But the details of this conception remain unrevealed as Lombard’s presentation of the problem is to be declared underdeveloped. (shrink)

Łu logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of Łu logic by adding a new connective which expresses multiplication. The resulting logic, PŁ, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of PŁ logic is introduced and developed too.

The global Peter Lombard research reinaugurated in 1990s has resulted in a number of recent publications, but the Master of the Sentences’ theology proper is partially underresearched. In particular, a more detailed exposition of the distinctions 35-41 of his Book of Sentences is needed in order to clarify his doctrine of God’s knowledge and its relation to the human free will. The article builds on the earlier established evidence that, for Peter Lombard in distinctions 35-38, God’s knowledge, in general, is (...) not causative, although some causative power has to be ascribed to God’s knowledge of the good. The last part of distinction 38 and the content of distinction 39 further analyze the capacities and functionalities of the divine omniscience and explain how it interacts with acts of human will. The key question here deals with the problem of alternative states of affairs: whether something may be otherwise than God foreknew. As it is shown, Master Peter agrees that it is possible for created things and events to be otherwise than they are, but insists that God’s knowledge must be in any case exhaustive and infallible. He uses a number of logical tools to defend the thesis about God’s perfect knowledge and the possibility of things happening otherwise, but lacks a strict definition of the notion of “possibility” used here. The study concludes that in few cases Lombard’s posse could mean a potency or a simple logical possibility, or the diachronic contingency, but the overall theological statement is clear: potentially, God’s knowledge can be different or include alternative state of affairs but it cannot change. (shrink)

It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL-chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL-chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation (...) defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness. (shrink)

The purpose of this study is to discover, present and analyze the key ideas of Anselm of Canterbury concerning the notions of knowledge, will and mode of divine-human relations in the context of this “knowledge-will” framework which is important due to (a) somewhat insufficient attention to the medieval insights on the issue and (b) the peculiarity that Anselm’s intuitions have. More specifically, the object of the given paper is Anselmian understanding of relations between God’s foreknowledge and will, on the one (...) side, and human free will, on the other side, as it is presented in the work entitled De Concordia. In this treatise Anselm of Canterbury partially uses and further elaborates some ideas of Augustine and Boethius, while integrating, updating and synthesizing them in a creative manner, and partially develops a number of thoughts of his own. As a result, there is a set of the well-formulated and comprehensive theses concerning (i) different types of (metaphysical) necessity, (ii) a proper definition of freedom of choice and will, (iii) the nature of will per se, and, finally, (iv) unique relation of “simultaneity” that exists between God’s knowledge, God’s will and the creature’s rational will. These ideas are presented and explicated in the article along with detailed exposition and analytical examination of the main line of argumentation found in the De Concordia. (shrink)

Abstract.ΠMTL is a schematic extension of the monoidal t-norm based logic (MTL) by the characteristic axioms of product logic. In this paper we prove that ΠMTL satisfies the standard completeness theorem. From the algebraic point of view, we show that the class of ΠMTL-algebras (bounded commutative cancellative residuated l-monoids) in the real unit interval [0,1] generates the variety of all ΠMTL-algebras.

The last thirty years of scholarship in western medieval philosophical historiography have seen a number of reflections on the methodological paradigms, schools, trends, and dominant approaches in the field. As a contribution to this ongoing assessment of the existing methods of studies in medieval philosophy and theology and a supplement to classifications offered by M. Colish, J. Inglis, C. König-Pralong, J. Marenbon, A. de Libera, and others, the article offers another explanatory tool. Here is a description of an imaginary system (...) of methodological coordinates that systematizes the current tendencies by placing them in a three-dimensional system of axes. Every axis corresponds to a certain aspect of the historical and systematic research in medieval thought and symbolizes a possible movement between two extremes representing opposite methodological values and directions. The methods and approaches practiced in recent studies in medieval philosophy and theology might be schematically located inside this general system of argumentational, focal (or objectival), and (con)textual axes with their intersection identified with what some scholars call the “integral” model of study. This explanatory tool allows one to see how current approaches and methods form a panoply of axes that belong together in one complex grid and helps to visualize the tapestry of existing approaches in medieval philosophical historiography. (shrink)

The article treats the doctrine of transubstantiation or the Eucharistic change as formulated by Thomas Aquinas in his Summa theologiae, Question 75, against its double conceptual (Christian religious vs. Aristotelian philosophical), as well as double linguistic (Latin vs. translated Greek), background. The doctrine is presented and analyzed as a philosophical-theological theory that can be explicated and assessed using the concept of philosophical untranslatable(s), recently discovered and brought to the fore by the proponents of the “translational turn” in continental philosophy. It (...) is argued on the basis of careful study of Aquinas’ text that the notion of transubstantiation should be identified as a conceptual untranslatable. -/- Here the term “untranslatable” means a unique reinterpretation of Aristotelian language of being and change, which results in a conceptual innovation that has some inherent problems as well as creative and promising inventions. The usage of such a concept is justified by the fact that the reconceptualization that Aquinas did to a number of Peripatetic ideas creates a few untranslatable notions, which do not have much meaning outside the peculiarly Thomistic Latin vocabulary. Despite some criticisms they are not necessarily to be regarded as contradictions, but rather as thomistic untranslatables. (shrink)

This paper investigates a quasi-variety of representable integral commutative residuated lattices axiomatized by the quasi-identity resulting from the well-known Wajsberg identity → q ≤ → p if it is written as a quasi-identity, i. e., → q ≈ 1 ⇒ → p ≈ 1. We prove that this quasi-identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi-variety is in fact a variety and provide an axiomatization. The obtained results shed some light (...) on the structure of Archimedean integral commutative residuated chains. Further, they can be applied to various subvarieties of MTL-algebras, for instance we answer negatively Hájek's question asking whether the variety of ΠMTL-algebras is generated by its Archimedean members. (shrink)

We solve several open problems on the cardinality of atoms in the subvariety lattice of residuated lattices and FL-algebras [4, Problems 17—19, pp. 437]. Namely, we prove that the subvariety lattice of residuated lattices contains continuum many 4-potent commutative representable atoms. Analogous results apply also to atoms in the subvariety lattice of FL i -algebras and FL o -algebras. On the other hand, we show that the subvariety lattice of residuated lattices contains only five 3-potent commutative representable atoms and two (...) integral commutative representable atoms. Inspired by the construction of atoms, we are also able to prove that the variety of integral commutative representable residuated lattices is generated by its 1-generated finite members. (shrink)

The goal of the lecture is to introduce Aesthetics as a theory of art critics. Seen in this way, the subject matter of Aesthetics is to be an analysis of art ctritics statements. Key concepts are presented: aesthetic judgment or art critics judgment, and supervenience. The subject matter of Aesthetics is – keeping with tradition – judgment about beauty, above all. A structured concept of beauty is presented. A boundary between nonaesthetic and aesthetic properties is said to be as an (...) essential one – mutual dependence of both is shortly mentioned. Theories of Hume, Beardsley a Sibley are seen as basic ones for such kind of theorizing. (shrink)

The goal of this article is a general inspection of indexicals as a specific phenomenon of natural language on the one side, and a consideration of "semantization" of indexicals on the other side. The term "semantization" represents a criterion on the basis of which various approaches to relevant solutions are valuated: if, according to Frege´s view, all language expressions should express their senses, what kind of senses could be expressed in the case of indexicals? First, indexicals are explicated as a (...) specific sort of language expressions. Second, how some contemporary philosophers deal with indexical puzzles is briefly propounded. It takes for granted that a comparison of various philosophers´ approaches makes their pro et contra clear. By that way the article can acquire a parameter of basic synopsis. In certain passages of the text, summaries of the issues dealt with are included. The article is divided into three parts, dividing idexicals into temporal, individual and spatial, and demonstratives. In the first part of this article temporal indexicals, , as well as grammatical tensed sentences are put to test. Different approaches of Tichý, Quine, Russell, Reichenbach, and Wittgenstein are interpreted and evaluated from the "semantization" point of view. In the case of temporal indexicals, knowledge of possible referents plays an important part, and ontology of time is approximated in connection with McTaggart´s argument. Pavel Materna´s analysis of "now" is a special subject of brief comment. (shrink)

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP -based. This presentation is then used to obtain, in a uniform way applicable to most substructural logics, a form (...) of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics. Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved. (shrink)

Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider (...) scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction. (shrink)