This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to (...) one carrying a dssi. We prove that the set theory BIST+Coll is sound and complete relative to forcing interpretations in toposes with natural numbers object . Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijkʼs Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation. (shrink)

This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to (...) one carrying a dssi. We prove that the set theory BIST+Coll is sound and complete relative to forcing interpretations in toposes with natural numbers object. Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijkʼs Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation. (shrink)

We study the non-Fregean propositional logic with propositional quantifiers, denoted by $\mathsf{SCI}_{\mathsf{Q}}$. We prove that $\mathsf{SCI}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in $\mathsf{SCI}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of $\mathsf{SCI}_{\mathsf{Q}}$-sentences. Finally, we present a translation of $\mathsf{SCI}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove some model-theoretic properties of (...) $\mathsf{SCI}_{\mathsf{Q}}$. (shrink)

The paper concerns Grzegorczyk’s non-Fregean logics that are intended to be a formal representation of the equimeaning relation defined on descriptions. We argue that the main Grzegorczyk logics discussed in the literature are too strong and we propose a new logical system, \, which satisfies Grzegorczyk’s fundamental requirements. We present a sound and complete semantics for \ and we prove that it is decidable. Finally, we show that many non-classical logics are extensions of \, which makes it a generic non-Fregean (...) logic. (shrink)

We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems (...) are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other. (shrink)

We show that there are continuum many different extensions of SCI (the basic theory of non-Fregean propositional logic) that lie below WF (the Fregean extension) and are closed under substitution. Moreover, continuum many of them are independent from WB (the Boolean extension), continuum many lie above WB and are independent from WH (the Boolean extension with only two values for the equality relation), and only countably many lie between WH and WF.

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and its dual (...) tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object. (shrink)

The non-Fregean logic SCI is obtained from the classical sentential calculus by adding a new identity connective = and axioms which say ?a = ß' means ?a is identical to ß'. We present complete and sound proof system for SCI in the style of Rasiowa-Sikorski. It provides a natural deduction-style method of reasoning for the non-Fregean sentential logic SCI.

Qualitative description of movements can be very important for representation and reasoning about dynamic systems which are complex in structure or whenever numerical data are incomplete or inaccessible. For this reason, we present a hybrid approach based on the combination of qualitative reasoning, quantitative data and logical methods. In this article, we introduce a new propositional dynamic logic QM for representation and reasoning with relative movements of objects. In this way, we can infer additional information about movements by using axioms (...) and the logic apparatus. We present a sound and complete deduction system in dual tableaux style for the logic QM. The system can be used for verification of validity of formulas of the logic in question. (shrink)

We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and completeness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems. -/- .

It is known that various complexity-theoretical problems can be translated into some special spectra problems. Thus, questions about complexity classes are translated into questions about the expressive power of some languages. In this paper we investigate the spectra of some logics with Henkin quantifiers in the empty vocabulary.

The book presents logical foundations of dual tableaux together with a number of their applications both to logics traditionally dealt with in mathematics and philosophy (such as modal, intuitionistic, relevant, and many-valued logics) and to various applied theories of computational logic (such as temporal reasoning, spatial reasoning, fuzzy-set-based reasoning, rough-set-based reasoning, order-of magnitude reasoning, reasoning about programs, threshold logics, logics of conditional decisions). The distinguishing feature of most of these applications is that the corresponding dual tableaux are built in a (...) relational language which provides useful means of presentation of the theories. In this way modularity of dual tableaux is ensured. We do not need to develop and implement each dual tableau from scratch, we should only extend the relational core common to many theories with the rules specific for a particular theory. (shrink)

We present a dual tableau system, RLK, which is itself a deterministic decision procedure verifying validity of K-formulas. The system is constructed in the framework of the original methodology of relational proof systems, determined only by axioms and inference rules, without any external techniques. Furthermore, we describe an implementation of the system RLK in Prolog, and we show some of its advantages.

We present a dual tableau system, RLK, which is itself a deterministic decision procedure verifying validity of K-formulas. The system is constructed in the framework of the original methodology of relational proof systems, determined only by axioms and inference rules, without any external techniques. Furthermore, we describe an implementation of the system in Prolog, and we show some of its advantages.

Qualitative description of the movement of objects can be very important when there are large quantity of data or incomplete information, such as in positioning technologies and movement of robots. We present a first step in the combination of fuzzy qualitative reasoning and quantitative data obtained by human interaction and external devices as GPS, in order to update and correct the qualitative information. We consider a Propositional Dynamic Logic which deals with qualitative velocity and enables us to represent some reasoning (...) tasks about qualitative properties. The use of logic provides a general framework which improves the capacity of reasoning. In this way, we can infer additional information by using axioms and the logic apparatus. In this paper we present sound and complete relational dual tableau that can be used for verification of validity of formulas of the logic in question. (shrink)

We introduce an Automatic Theorem Prover (ATP) of a dual tableau system for a relational logic for order of magnitude qualitative reasoning, which allows us to deal with relations such as negligibility, non-closeness and distance. Dual tableau systems are validity checkers that can serve as a tool for verification of a variety of tasks in order of magnitude reasoning, such as the use of qualitative sum of some classes of numbers. In the design of our ATP, we have introduced some (...) heuristics, such as the so called phantom variables, which improve the efficiency of the selection of variables used un the proof. (shrink)

In 2011, Andrzej Grzegorczyk formulated Logic of Descriptions, a new logical system in which the classical equivalence has been replaced with the descriptive equivalence. Two sentences are descriptively equivalent whenever they describe the same state of affairs. Grzegorczyk’s logic LD is built from the ground up by revising the axioms of classical propositional logic and rejecting those that do not correspond to the intended interpretation of the descriptive equivalence as the connective expressing equimeaning relations between sentences. Grzegorczyk’s last paper, which (...) introduced in detail philosophical motivations of LD and its axiomatization, has become an inspiration for investigating the properties of LD and its various modifications. In this paper we present the basics of Grzegorczyk’s logic LD and then we survey the recent results on LD that have shed light on mysterious properties of the Grzegorczyk’s descriptive equivalence connective. (shrink)

A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this article. Modal logics could be very helpful in modelling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently the Connectionist Modal Logics has been presented, which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness and the fact that the system itself is a decision (...) procedure are proved in this article. The main advantages of this approach are: first, the system is deterministic, i.e. it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula ; and third, the language of deduction and the language of a logic coincide. Some of these advantages are compared with other classical approaches. (shrink)

We present a relational proof system in the style of dual tableaux for the relational logic associated with a multimodal propositional logic for order of magnitude qualitative reasoning with a bidirectional relation of negligibility. We study soundness and completeness of the proof system and we show how it can be used for verification of validity of formulas of the logic.

Monoidal triangular norm logic MTL is the logic of left-continuous triangular norms. In the paper we present a relational formalization of the logic MTL and then we introduce relational dual tableau that can be used for verification of validity of MTL-formulas. We prove soundness and completeness of the system.

We present several classes of logics for reasoning with information stored in information systems. The logics enable us to cope with the phenomena of incompleteness of information and uncertainty of knowledge derived from such an information. Relational inference systems for these logics are developed in the style of dual tableaux.

We show that there are continuum many different non-Fregean sentential logics that have adequate models. The proof is based on the construction of a special class of models of the power of the continuum.

Qualitative Reasoning (QR) is an area of research within Artificial Intelligence that automates reasoning and problem solving about the physical world. QR research aims to deal with representation and reasoning about continuous aspects of entities without the kind of precise quantitative information needed by conventional numerical analysis techniques. Order-of-magnitude Reasoning (OMR) is an approach in QR concerned with the analysis of physical systems in terms of relative magnitudes. In this paper we consider the logic OMR_N for order-of-magnitude reasoning with the (...) bidirectional negligibility relation. It is a multi-modal logic given by a Hilbert-style axiomatization that reflects properties and interactions of two basic accessibility relations (strict linear order and bidirectional negligibility). Although the logic was studied in many papers, nothing was known about its decidability. In the paper we prove decidability of OMR N by showing that the logic has the strong finite model property. (shrink)

Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.

We present relational proof systems for the four groups of theories of spatial reasoning: contact relation algebras, Boolean algebras with a contact relation, lattice-based spatial theories, spatial theories based on a proximity relation.

This chapter provides a concise overview of Ewa Orłowska’s research contributions and the content of the volume.

The chapter is a transcription of editors’ discussion with Ewa Orłowska. It reveals some extracurricular flavors of Ewa Orłowska’s biography, brings to light a difficult historical context of her academic career and life, and shows how much internal fortitude she demonstrated while overcoming these difficulties.

This book is a tribute to Professor Ewa Orłowska, a Polish logician who was celebrating the 60th year of her scientific career in 2017. It offers a collection of contributed papers by different authors and covers the most important areas of her research. Prof. Orłowska made significant contributions to many fields of logic, such as proof theory, algebraic methods in logic and knowledge representation, and her work has been published in 3 monographs and over 100 articles in internationally acclaimed journals (...) and conference proceedings. The book also includes Prof. Orłowska’s autobiography, bibliography and a trialogue between her and the editors of the volume, as well as contributors' biographical notes, and is suitable for scholars and students of logic who are interested in understanding more about Prof. Orłowska’s work. (shrink)

Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations between intervals and/or (...) various time orderings. (shrink)

An automatic theorem prover for a proof system in the style of dual tableaux for the relational logic associated with modal logic K has been introduced. Although there are many well-known implementations of provers for modal logic, as far as we know, it is the first implementation of a specific relational prover for a standard modal logic. There are two main contributions in this paper. First, the implementation of new rules, called (k1) and (k2), which substitute the classical relational rules (...) for composition and negation of composition in order to guarantee not only that every proof tree is finite but also to decrease the number of applied rules in dual tableaux. Second, the implementation of an order of application of the rules which ensures that the proof tree obtained is unique.As a consequence, we have implemented a decision procedure for modal logic K. Moreover, this work would be the basis for successive extensions of this logic, such as T, B and S4. (shrink)

It is known that various complexity-theoretical problems can be translated into some special spectra problems (see e.g. Fagin [Fa74] or Blass and Gurevich, [Bl-Gu86]). So questions about complexity classes are translated into questions about the expressive power of some languages. In this paper we investigate the spectra of some logics with Henkin quanti fiers in the empty vocabulary. This problem has been investigated fi rstly by Krynicki and Mostowski in [Kr-Mo 92] and [Kr- Mo 95]. All presented results can be (...) also treated as results about the expressive power of certain languages in fi nite models. (shrink)

Recenzja książki Teresy Hołówki "Błędy, spory, argumenty".

In this paper, I argue that the Hole Argument can be formulated without using the notion of isomorphism, and for this reason it is not threatened by the criticism of Halvorson and Manchak (Br J Philos Sci, 2022. https://doi.org/10.1086/719193). Following Earman and Norton (Br J Philos Sci 38, pp. 515–525, 1987), I divide the Hole Argument into two steps: the proof of the Gauge Theorem and the transition from the Gauge Theorem to the conclusion of radical indeterminism. In the analaysis (...) of the first step, I argue that the Gauge Theorem does not rely on the notion of isomorphism but on the notion of the diffeomorphism-invariance of the equations of local spacetime theories; however, for this approach to work, the definition of local spacetime theories needs certain amendments with respect to Earman and Norton’s formulation. In the analysis of the second step, I postulate that we should use the notion of radical indeterminism instead of indeterminism simpliciter and that we should not decide in advance what kind of maps are to be used in comparing models. Instead, we can tentatively choose some kind of maps for this purpose and check whether a given choice leads to radical indeterminism involving empirically indistinguishable models. In this way, the use of the notion of isomorphism is also avoided in the second step of the Hole Argument. A general picture is that physical equivalence can be established by means of an iterative procedure in which we examine various candidate classes of maps, and, depending on the outcomes, we need to broaden or narrow these classes. The Hole Argument can be viewed as a particular instance of this procedure. (shrink)

A scientifically proven method to overcome obstacles and make choices that lead us closer to our goals. WITH A FOREWORD BY MARTINA NAVRATILOVA What do weight gain, poor employee engagement, and climate change all have in common? All three are persistent problems for which solutions are known and readily available. Yet, on an individual and collective level, we continually make choices that lead us not closer to but further away from our stated objectives. Whether we choose the burger and fries (...) over the salad, top-down management over collaboration, or leniency over rigor in combatting climate change, it all comes down to the same thing. The Choice Point. The moment at which we each choose to take an action that we know will undermine our success, but we do it anyway. The Choice Point explains how we experience thousands of thoughts every day: 80 percent are negative, while only 5 percent are new thoughts. We make most of our decisions within two seconds, using mental autopilot systems that we've built through years of experiences, and some of our thoughts (like quitting during a challenge) can compromise our goals. But there's a strategy to change this: Functional Imagery Training (FIT). FIT teaches us how to pick the thoughts that deserve our attention, and how to stop the other, negative thoughts from impacting our actions. Through FIT, we can pause decisions that would typically conflict with our goals, lengthen our Choice Point, and go from passenger to driver of our own minds. Initially developed in 2005 by academics in the UK and Australia, with formal principles emerging through research in 2018, FIT blends motivational interviewing with imagery into a user-friendly model so we can learn, adapt, and adjust our behavior to go beyond what we think is possible. (shrink)

Provides character education through different scenarios that demonstrate children being responsible.

Life typically becomes an object of reflection when it is seen to be under threat. In particular, humans have a tendency to engage in thinking about life (instead of just continuing to live it) when being confronted with the prospect of death: be it the death of individuals due to illness, accident or old age; the death of whole ethnic or national groups in wars and other forms of armed conflict; but also of whole populations, be they human or nonhuman. (...) Even though Minimal Ethics for the Anthropocene is first and foremost concerned with life--understood as both a biological and social phenomenon--it is the narrative about the impending death of the human population (i.e., about the extinction of the human species), that provides a context for its argument. "Anthropocene" names a geo-historical period in which humans are said to have become the biggest threat to life on earth. However, rather than as a scientific descriptor, the term serves here primarily as an ethical injunction to think critically about human and nonhuman agency in the universe. Restrained in tone yet ambitious in scope, the book takes some steps towards outlining a minimal ethics thought on a universal scale. The task of such minimal ethics is to consider how humans can assume responsibility for various occurrences in the universe, across different scales, and how they can respond to the tangled mesh of connections and relations unfolding in it. Its goal is not so much to tell us how to live but rather to allow us to rethink "life" and what we can do with it, in whatever time we have left. The book embraces a speculative mode of thinking that is more akin to the artist's method; it also includes a photographic project by the author."--Publisher's description. (shrink)

Artykuł ma na celu przybliżenie specyfiki aktorstwa radiowego. Wskazuje na podobieństwa i różnice z teatralną i telewizyjną formą sztuki aktorskiej, określa cechy wyróżniające grę w teatrze radiowym w stosunku do innych mediów. Omawia także metody pracy reżyserów Teatru Polskiego Radia z aktorami podczas prób i nagrań słuchowisk.

Everyone knows what is feels like to be in pain. Scraped knees, toothaches, migraines, giving birth, cancer, heart attacks, and heartaches: pain permeates our entire lives. We also witness other people - loved ones - suffering, and we 'feel with' them. It is easy to assume this is the end of the story: 'pain-is-pain-is-pain', and that is all there is to say. But it is not. In fact, the way in which people respond to what they describe as 'painful' has (...) changed considerably over time. In the eighteenth and nineteenth centuries, for example, people believed that pain served a specific (and positive) function - it was a message from God or Nature; it would perfect the spirit. 'Suffer in this life and you wouldn't suffer in the next one'. Submission to pain was required. Nothing could be more removed from twentieth and twenty-first century understandings, where pain is regarded as an unremitting evil to be 'fought'. Focusing on the English-speaking world, this book tells the story of pain since the eighteenth century, addressing fundamental questions about the experience and nature of suffering over the last three centuries. How have those in pain interpreted their suffering - and how have these interpretations changed over time? How have people learnt to conduct themselves when suffering? How do friends and family react? And what about medical professionals: should they immerse themselves in the suffering person or is the best response a kind of professional detachment? (shrink)

For her entire childhood, Joanna's father, Bob, had a ritual: wake up at dawn, walk the dog, and sit down at the kitchen table with a blank pad of paper and plenty of colored markers to craft notes for his two children. Over the years, word games and puzzles for five-year-olds morphed into thoughtful guidance and reflections for his teenagers approaching adulthood. Now, with more than 3,500 of her father's colorful notes in hand, Joanna has decided that the (...) lessons tucked inside are worth sharing. Folded Wisdom highlights the collection of Bob's notes, telling a story filled with universal values that encourages meaningful self-reflection about how we all face successes and failures; express happiness and sadness; and communicate frustration, praise, and love to one another. Heartfelt and full of possibility for the future, a father's folded notes and drawings are timeless reminders of love. (shrink)

The question of whether AI systems such as robots can or should be afforded moral agency or patiency is not one amenable either to discovery or simple reasoning, because we as societies constantly reconstruct our artefacts, including our ethical systems. Consequently, the place of AI systems in society is a matter of normative, not descriptive ethics. Here I start from a functionalist assumption, that ethics is the set of behaviour that maintains a society. This assumption allows me to exploit the (...) theoretical biology of sociality and autonomy to explain our moral intuitions. From this grounding I extend to consider possible ethics for maintaining either human- or of artefact-centred societies. I conclude that while constructing AI systems as either moral agents or patients is possible, neither is desirable. In particular, I argue that we are unlikely to construct a coherent ethics in which it it is ethical to afford AI moral subjectivity. We are therefore obliged not to build AI we are obliged to. (shrink)