The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. Cut elimination is (...) proved in a constructive way for sequent calculi adequate for all logics under consideration. (shrink)

In several applications of sequent calculi going beyond pure logic, an introduction of suitably defined rules seems to be more profitable than addition of extra axiomatic sequents. A program of formalization of mathematical theories via rules of special sort was developed successfully by Negri and von Plato. In this paper a general theorem on possible ways of transforming axiomatic sequents into rules in sequent calculi is proved. We discuss its possible applications and provide some case studies for illustration.

Hypersequent calculus, developed by A. Avron, is one of the most interesting proof systems suitable for nonclassical logics. Although HC has rather simple form, it increases significantly the expressive power of standard sequent calculi. In particular, HC proved to be very useful in the field of proof theory of various nonclassical logics. It may seem surprising that it was not applied to temporal logics so far. In what follows, we discuss different approaches to formalization of logics of linear frames and (...) provide a cut-free HC formalization ofKt4.3, the minimal temporal logic of linear frames, and some of its extensions. The novelty of our approach is that hypersequents are defined not as finite sets but as finite lists of ordinary sequents. Such a solution allows both linearity of time flow, and symmetry of past and future, to be incorporated by means of six temporal rules. Extensions of the basic calculus with simple structural rules cover logics of serial and dense frames. Completeness is proved by Schütte/Hintikka-style argument using models built from saturated hypersequents. (shrink)

The paper is a brief survey of some sequent calculi which do not follow strictly the shape of sequent calculus introduced by Gentzen. We propose the following rough classification of all SC: Systems which are based on some deviations from the ordinary notion of a sequent are called generalised; remaining ones are called ordinary. Among the latter we distinguish three types according to the proportion between the number of primitive sequents and rules. In particular, in one of these types, called (...) Gentzen’s type, we have a subtype of standard SC due to Gentzen. Hence by nonstandard ones we mean all these ordinary SC where other kinds of rules are applied than those admitted in standard Gentzen’s sequent calculi. We describe briefly some of the most interesting or important nonstandard SC belonging to the three abovementioned types. (shrink)

Hypersequent calculi can formalize various non-classical logics. In [9] we presented a non-commutative variant of HC for the weakest temporal logic of linear frames Kt4.3 and some its extensions for dense and serial flow of time. The system was proved to be cut-free HC formalization of respective temporal logics by means of Schütte/Hintikka-style semantical argument using models built from saturated hypersequents. In this paper we present a variant of this calculus for Kt4.3 with a constructive syntactical proof of cut elimination.

This is a sequel article to [10] where a hypersequent calculus for some temporal logics of linear frames includingKt4.3and its extensions for dense and serial flow of time was investigated in detail. A distinctive feature of this approach is that hypersequents are noncommutative, i.e., they are finite lists of sequents in contrast to other hypersequent approaches using sets or multisets. Such a system in [10] was proved to be cut-free HC formalization of respective logics by means of semantical argument. In (...) this article we present an equivalent variant of this calculus for which a constructive syntactical proof of cut elimination is provided. (shrink)

This is an extended version of the lectures given during the 12-thConference on Applications of Logic in Philosophy and in the Foundationsof Mathematics in Szklarska Poręba. It contains a surveyof modal hybrid logic, one of the branches of contemporary modal logic. Inthe first part a variety of hybrid languages and logics is presented with adiscussion of expressivity matters. The second part is devoted to thoroughexposition of proof methods for hybrid logics. The main point is to showthat application of hybrid logics (...) may remarkably improve the situation inmodal proof theory. (shrink)

In the paper a decision procedure for S5 is presented which uses a cut-free sequent calculus with additional rules allowing a reduction to normal modal forms. It utilizes the fact that in S5 every formula is equivalent to some 1-degree formula, i.e. a modally-flat formula with modal functors having only boolean formulas in its scope. In contrast to many sequent calculi for S5 the presented system does not introduce any extra devices. Thus it is a standard version of SC but (...) with some additional simple rewrite rules. The procedure combines the proces of saturation of sequents with reduction of their elements to some normal modal form. (shrink)

The paper is devoted to the concise description of some Natural Deduction System (ND for short) for Linear Temporal Logic. The system's distinctive feature is that it is labelled and analytical. Labels convey necessary semantic information connected with the rules for temporal functors while the analytical character of the rules lets the system work as a decision procedure. It makes it more similar to Labelled Tableau Systems than to standard Natural Deduction. In fact, our solution of linearity representation is rather (...) independent of the underlying proof method, provided that some form of (analytic) cut is admissible. We will also discuss some generalisations of the system and compare it with other formalizations of linearity. (shrink)

The equivalence connective in ukasiewicz logic has its algebraic counterpart which is the distance function d(x,y) =|x–y| of a positive cone of a commutative -group. We make some observations on logically motivated algebraic structures involving the distance function.

SummaryStarting from late Classical-early Hellenistic age a series of witty, lighthearted and irreverent funerary verse-inscriptions aiming to produce some effect of amusement or laughter appeared on a number of monuments, reaching their apogee during Greek-Roman era. Most of them originated in Asia Minor and Rome. Some earliest examples were related to widespread hedonistic exhortations on tombs. Their later ramifications, consisting of ironical or playful expressions, amusing puns and instances of black humour, were written in a more satirical vein, except with (...) inscriptions dedicated for animals that were rife with sentimental motifs. Remarkably diverse as they were, such verse-inscriptions cannot be defined in terms of a distinctly separate, continuous tradition, but they shared some common features. Lacking – for the most part – conventional and formulaic elements, they struck us as heavily individualised, which sets them apart from the mainstream tradition of funerary poetry. This in turn might shed some light on social standing or/and mentality of individuals who opted for such expressive ways of remembering the dead. (shrink)

The subject of this paper is the controversy between cognitivist philosophy and phenomenology concerning the nature of emotions. This article has a limited scope restricted to the domain of the aesthetic emotions usually dependent on the works of art and literature. The author raises the criticism against the cognitivist stance, this criticism based chiefly on the principles of Max Scheler's phenomenology of emotions.

Philosophy in this text is considered under the model of mise en abyme, which is used in particular in the art of fiction narration. A novel about the process of novel writing is an example of mise en abyme. A film about movie directing is also an example of mise en abyme. A painting showing the artist working on the work of art also has features of mise en abyme. Likewise, mise en abyme philosophy is philosophy that discusses how philosophy (...) is created. Philosophy in such a shape is no longer understood as an endless space “inhabited” by situations, concepts, assertions, philosophical theses which are, however, perceived as if from the inside of that space; by contrast, it starts to be perceived from the outside, from the perspective of activities that form its frame, that frame it. For example, two such framing activities are considered – refining situations which is responsible for the “splendor of truth” that shines forth in them and defying the passage of time when constructing situations which is decisive for their isolation from the stream of life. Philosophy that notices its own frame is anchored in the foundations of everyday life which is something irrefutable in relation to the very inside of philosophy; as a consequence, the idea of epistemological foundationalism may be renewed and recovered for philosophy. (shrink)

Social inequalities abound worldwide. However, those social inequalities that encompass the lack of access to resources, including primary healthcare, are more prevalent at older ages in low- and middle-income countries, which is the focus of this chapter. This challenge has become critical due to the increasing population of older age. The design and implementation of key policy measures, including primary healthcare in low- and middle-income countries, is essential in reducing such social disparities. Access to primary healthcare is perhaps the principal (...) determinant of social equality and/or equity for older people, which in turn supports achieving Sustainable Development Goals and comprehensive healthcare access. (shrink)

We present a detailed proof of the admissibility of cut in sequent calculus for some congruent modal logics. The result was announced much earlier during the Trends in Logic Conference, Toruń 2006 and the proof for monotonic modal logics was provided already in Indrzejczak [5]. Also some tableau and natural deduction formalizations presented in Indrzejczak [6] and Indrzejczak [7] were based on this result but the proof itself was not published so far. In this paper we are going to fill (...) this gap. The delay was partly due to the fact that the author from time to time was trying to improve the result and extend it to some additional logics by testing other methods of proving cut elimination. Unfortunately all these attempts failed and cut elimination holds only for these logics which were proved to satisfy this property already in 2005. (shrink)

The paper presents a comparison of two generalised sequent calculi for temporal logics. In both cases the main technical solution is the multiplication of the sorts of sequents and, additionally, the application of some kind of labelling to formulae. The first approach was proposed by Kaziemierz Trzęsicki at the 1980s. The second, called Multiple Sequent Calculus (MSC), was proposed in the beginning of the present century. Both approaches are examples of the family of multisequent calculi.

Jan Łukasiewicz distinguished three various formulations of the law of contradiction in Aristotle's considerations concerning axiomatic foundations of philosophia prima in the book Γ of Methaphysics. Łukasiewicz referred to these formulations as „ontological”, „logical”, and „psychological”, respectively. The author focuses his attention on the last of them, namely to the so called psychological approach. He finds this approach to be an inadequate interpretation of Aristotle's views and tries to show that the most appropriate interpretation is pragmatic-logical.

The status of the equality predicate as a logical constant is problematic. In the paper we look at the problem from the proof-theoretic standpoint and survey several ways of treating equality in formal systems of different sorts. In particular, we focus on the framework of sequent calculus and examine equality in the light of criteria of logicality proposed by Hacking and Došen. Both attempts were formulated in terms of sequent calculus rules, although in the case of Došen it has a (...) nonstandard character. It will be shown that equality can be characterised in a way which satisfies Došen’s criteria of logicality. In the case of Hacking’s approach the fully satisfying result can be obtained only for languages with a nonempty, finite set of predicate constants other than equality. Otherwise, cut elimination theorem fails to hold. (shrink)

Name der Zeitschrift: Kierkegaard Studies Yearbook Jahrgang: 23 Heft: 1 Seiten: 179-203.

The problem of precise characterisation of traditional forms of reasoning applied in mathematics was independently investigated and successfully resolved by Jaśkowski and Gentzen in 1934. However, there are traces of earlier interests in this field exhibited by the members of the Lvov-Warsaw School. We focus on the results obtained by Jaśkowski and Leśniewski. Jaśkowski provided the first formal system of natural deduction in 1926. Leśniewski also demonstrated in some of his papers how to construct proofs in accordance with intuitively correct (...) principles. (shrink)

We present a uniform characterisation of three-valued logics by means of a bisequent calculus (BSC). It is a generalised form of a sequent calculus (SC) where rules operate on the ordered pairs of ordinary sequents. BSC may be treated as the weakest kind of system in the rich family of generalised SC operating on items being some collections of ordinary sequents, like hypersequent and nested sequent calculi. It seems that for many non-classical logics, including some many-valued, paraconsistent and modal logics, (...) the reasonably modest generalisation of standard SC offered by BSC is sufficient. In this paper, we examine a variety of three-valued logics and show how they can be formalised in the framework of BSC. We present a constructive syntactic proof that these systems are cut-free, satisfy the subformula property, and allow one to prove the interpolation theorem in many cases. (shrink)

The paper provides a proof theoretic characterization of the Russellian theory of definite descriptions (RDD) as characterized by Kalish, Montague and Mar (KMM). To this effect three sequent calculi are introduced: LKID0, LKID1 and LKID2. LKID0 is an auxiliary system which is easily shown to be equivalent to KMM. The main research is devoted to LKID1 and LKID2. The former is simpler in the sense of having smaller number of rules and, after small change, satisfies cut elimination but fails to (...) satisfy the subformula property. In LKID2 an additional analysis of different kinds of identities leads to proliferation of rules but yields the subformula property. This refined proof theoretic analysis leading to fully analytic calculus with constructive proof of cut elimination is the main contribution of the paper. (shrink)

We classify ultrafilters on ω with respect to sequential contours (see [4].[5]) of different ranks. In this way we obtain an ω1 sequence {Pα}1≤α≤ω1 of disjoint classes. We prove that non-emptiness of Pα for successor α ≥ 2 is equivalent to the existence of P-point. We investigate relations between P-hierarchy and ordinal ultrafilters (introduced by J. E. Baumgartner in [1]), we prove that it is relatively consistent with ZFC that the successor classes (for α ≥ 2) of P-hierarchy and ordinal (...) ultrafilters intersect but are not the same. (shrink)

We provide a version of first-order hybrid tense logic with predicate abstracts and definite descriptions as the only non-rigid terms. It is formalised by means of a tableau calculus working on sat-formulas. A particular theory of DD exploited here is essentially based on the approach of Russell, but with descriptions treated as genuine terms. However, the reductionist aspect of the Russellian approach is retained in several ways. Moreover, a special form of tense definite descriptions is formally developed. A constructive proof (...) of the interpolation theorem for this calculus is given, which is an extension of the result provided by Blackburn and Marx. (shrink)

Hybrid logics are extensions of standard modal logics, which significantly increase the expressive power of the latter. Since most of hybrid logics are known to be decidable, decision procedures for them is a widely investigated field of research. So far, several tableau calculi for hybrid logics have been presented in the literature. In this paper we introduce a sound, complete and terminating tableau calculus T H(@,E,D, ♦ −) for hybrid logics with the satisfaction operators, the universal modality, the difference modality (...) and the inverse modality as well as the corresponding sequent calculus S H(@,E,D, ♦ −) . They not only uniformly cover relatively wide range of various hybrid logics but they are also conceptually simple and enable effective search for a minimal model for a satisfiable formula. The main novelty is the exploitation of the unrestricted blocking mechanism introduced as an explicit, sound tableau rule. (shrink)

The central - and philosophically important - question connected with selectivity is the question of the nature of this phenomenon as a specific feature of human cognitive system. It is revealed in acts of two types. Acts of the first type are selective extraction of concrete as well as symbolic informations derived from the external world, and contained in sensoric representations. Acts of the second type are selective actualization of informations preserved in an individual permanent memory. The hypothesis of existing (...) general mental disposition common to acts of both the types is very attractive. The author analyzes some psychological (empirical) and philosophical aspects of these problems. (shrink)

In 1934 Jaśkowski and Gentzen independently published their groundbreaking works on Natural Deduction. The aim of this paper is to provide some criteria for division of the diversity of existing systems on some natural subcategories and to show that despite the differences all these systems are descendants of original systems of Jaśkowski and Gentzen. Three criteria are discussed:The kind of items which are building-blocks of the proof.The format of proof.The kind of rules.The first leads to the division of ND into (...) two main classes: F-systems working on formulas and S-systems working on sequents. The second distinguishes between T-systems with tree-proofs and L-systems with linear proofs. Finally, the third leads to several minor divisions in the main categories. (shrink)

The paper is a brief survey of the most important semantic constructions founded on the concept of possible world. It is impossible to capture in one short paper the whole variety of the problems connected with manifold applications of possible worlds. Hence, after a brief explanation of some philosophical matters I take a look at possible worlds from rather technical standpoint of logic and focus on the applications in formal semantics. In particular, I would like to focus on the fruitful (...) marriage of possible world semantics and algebra and its evolution leading to very general construction of Wójcicki called referential semantics and some of its refinements. The presentation is informal and sketchy; the main purpose is to put in one place a short, and readable I hope, description of the most important constructions and to point out the main sources of these solutions. (shrink)

We discuss the methods for providing sequent calculi for Suszko’s basic non-Fregean Logic with sentential identity SCI. After examination of possible strategies and already proposed systems we focus on the new calculus and its modification. It does not satisfy full cut elimination but a slightly generalised form of the subformula property holds for it. It is also standard in the sense of satisfying several conditions on rules formulated by Gentzen and his followers. We examine also the problem of providing a (...) constructive proof of the interpolation theorem in the setting of this sequent calculus. (shrink)