This paper is a contribution to the general study of consequence relations which contain (definable) connective of “disjunction”. Our work is centered around the “proof by cases property”, we present several of its equivalent definitions, and show some interesting applications, namely in constructing axiomatic systems for intersections of logics and recognizing weakly implicative fuzzy logics among the weakly implicative ones.

This book is an edited collection on one of Plato’s most dramatic as well as most complex dialogues, where a defence of the philosopher’s way of life is carried out against the background of interconnected rhetorical and political stances of the time.

A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguen's implication) and the corresponding negation (Gödel's negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying "x is true" and satisfying the "dequotation schema" $\varphi \equiv \text{Tr}(\bar{\varphi})$ for all sentences φ? This problem is investigated in the frame of Lukasiewicz infinitely valued logic.

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying “xis true” and satisfying the “dequotation schema”for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.

Rational Pavelka logic extends Lukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0, 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

Axiomatic set theory with full comprehension is known to be consistent in Łukasiewicz fuzzy predicate logic. But we cannot assume the existence of natural numbers satisfying a simple schema of induction; this extension is shown to be inconsistent.

We show that the modal prepositional logicILM (interpretability logic with Montagna's principle), which has been shown sound and complete as the interpretability logic of Peano arithmetic PA (by Berarducci and Savrukov), is sound and complete as the logic ofπ 1-conservativity over eachbE 1-sound axiomatized theory containingI⌆ 1 (PA with induction restricted tobE 1-formulas). Furthermore, we extend this result to a systemILMR obtained fromILM by adding witness comparisons in the style of Guaspari's and Solovay's logicR (this will be done in a (...) separate continuation of the present paper). (shrink)

Fuzzy logic is understood as a logic with a comparative and truth-functional notion of truth. Arithmetical complexity of sets of tautologies and satisfiable sentences as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated.

A very simple many-valued predicate calculus is presented; a completeness theorem is proved and the arithmetical complexity of some notions concerning provability is determined.

Witnessed models of fuzzy predicate logic are models in which each quantified formula is witnessed, i.e. the truth value of a universally quantified formula is the minimum of the values of its instances and similarly for existential quantification. Systematic theory of known fuzzy logics endowed with this semantics is developed with special attention paid to problems of arithmetical complexity of sets of tautologies and of satisfiable formulas.

Some aspects of vagueness as presented in Shapiro’s book Vagueness in Context [23] are analyzed from the point of fuzzy logic. Presented are some generalizations of Shapiro’s formal apparatus.

First the expansion of the Łukasiewicz logic by the unary connectives of dividing by any natural number is studied; it is shown that in the predicate case the expansion is conservative w.r.t. witnessed standard 1-tautologies. This result is used to prove that the set of witnessed standard 1-tautologies of the predicate product logic is Π2-hard.

The influence and reputation of Pierre Menard, Author of the Quixote, is easily comparable to the impact of groundbreaking theoretical texts. Numerous philosophers, aestheticians and theorists of literature, music, or visual arts have been induced by this short story by J.L. Borges to reconsider the status of the literary work of art, to rethink the relationship between work and text. The essays collected here move from analyses of the identity of the literary work of art, as it is explicitly established (...) by Borges? narrator, to arguments that simply employ the Menard case as an opportunity for discussing broader issues of literary studies and the philosophy of literature. Selected essays even abandon the field of literature altogether, moving on to analogous issues in the theory of visual arts and music. (shrink)

It is shown that the propositional modal logic IRM (interpretability logic with Montagna's principle and with witness comparisons in the style of Guaspari's and Solovay's logicR) is sound and complete as the logic ofII 1-conservativity over each∑ 1-sound axiomatized theory containingI∑ 1. The exact statement of the result uses the notion of standard proof predicate. This paper is an immediate continuation of our paper [HM]. Knowledge of [HM] is presupposed. We define a modal logic, called IRM, which includes both ILM (...) andR (the logic of [GS]) and prove an arithmetical completeness theorem in the style of [GS], thus showing that IRM is the logic ofII 1-conservativity with witness comparisons. The reader is recommended to have Smoryński's book [Sm] at his/her disposal. (shrink)

Results on arithmetical complexity of important sets of formulas of several fuzzy predicate logics are surveyed and some new results are proven.

Advances in artificial intelligence and robotics stand to change many aspects of our lives, including our values. If trends continue as expected, many industries will undergo automation in the near future, calling into question whether we can still value the sense of identity and security our occupations once provided us with. Likewise, the advent of social robots driven by AI, appears to be shifting the meaning of numerous, long-standing values associated with interpersonal relationships, like friendship. Furthermore, powerful actors’ and institutions’ (...) increasing reliance on AI to make decisions that may affect how people live their lives may have a significant impact on privacy while also raising issues about algorithmic transparency and human control. In this paper, building and expanding on previous works, we will look at how the deployment of Artificial Intelligence technology may lead to changes in identity, security, and other crucial values. We will discuss what challenges we may face in the process, while critically reflecting on whether such changes may be desirable. Finally, drawing on a series of considerations underlying virtue ethics, we will formulate a set of preliminary suggestions, which—we hope—can be used to more carefully guide the future roll out of AI technologies for human flourishing; that is, for social and moral good. (shrink)

Each closed (i.e. variable free) formula of interpretability logic is equivalent in ILF to a closed formula of the provability logic G, thus to a Boolean combination of formulas of the form n.

Two variants of monadic fuzzy predicate logic are analyzed and compared with the full fuzzy predicate logic with respect to finite model property (properties) and arithmetical complexity of sets of tautologies, satisfiable formulas and of analogous notion restricted to finite models.

In [10] it is claimed that the set of predicate tautologies of all complete BL-chains and the set of all standard tautologies coincide. As noticed in [11], this claim is wrong. In this paper we show that a complete BL-chain B satisfies all standard BL-tautologies iff for any transfinite sequence of elements of B, the condition ∧i ∈ I = 2 holds in B.

In this paper we explore the following question: how weak can a logic be for Rosser’s essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much (...) weaker arithmetic theory, a version of Robinson’s R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic. (shrink)

The concept of mimesis has been central to philosophical aesthetics from Aristotle to Kendall Walton: in plain terms, it highlights the links between a fictional world or a representational practice on the one hand and the real world on the other. The present collection of essays includes discussions of its general viability and pertinence and of its historical origins, as well as detailed analyses of various relevant issues regarding literature, film, theatre, images and computer games. The individual papers offer new (...) arguments for the specialist, yet in their sum also provide a solid and helpful survey of the current state of the debate. Contributions by P. Alward, G. Currie, D. Davies, L. Dole el, J. Hamilton, T. Kobli ek, P. Kot'atko, A. Kuzmicova, J. Levinson, A. Meskin, A. Pettersson, M. Pokorny, J. Robson, G. Rossholm, R. M. Sainsbury, F. Stjernberg, E. Terrone, K. Thein, A. Voltolini. (shrink)

It is shown that the low basis theorem is meaningful and provable in $I\sum_1$ and that the priority-free solution to Post's problem formalizes in this theory.

It is shown that the low basis theorem is meaningful and provable in I∑ 1 and that the priority-free solution to Post's problem formalizes in this theory.

Gödel logics with truth sets being countable closed subsets of the unit real interval containing 0 and 1 are studied under their usual semantics and under the witnessed semantics, the latter admitting only models in which the truth value of each universally quantified formula is the minimum of truth values of its instances and dually for existential quantification and maximum. An infinite system of such truth sets is constructed such that under the usual semantics the corresponding logics have pairwise different (...) sets of tautologies, all these sets being non-arithmetical, whereas under the witnessed semantics all the logics have the same set of tautologies and it is Π2-complete. Further similar results are obtained. (shrink)

Pavel Materna proposed valuable explications of concept and conceptual system. After their introduction, we contrast conceptual systems with (a novel notion of) derivation systems. Derivation systems differ from conceptual systems especially in including derivation rules. This enables us to show close connections among the realms of objects, their concepts, and reasoning with concepts.

ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.

Philosophical essays are a collection of occasional reflections that have arisen over the course of ten years, from 2002-2011.

В работе сделана попытка рассмотреть с позиции библейской, святоотеческой и научной антропологии общие аспекты психической жизни человека. Работа адресована широкому кругу читателей.

This chapter contains sections titled: Origin Many‐Valued Logic Fuzzy Logic in a Broad and Narrow Sense The Basic Fuzzy Propositional Calculus The Basic Fuzzy Predicate Calculus Similarity The Liar and Dequotation Very True Probability Conclusion.

This book collects most of the invited papers presented at the 12th International Congress of Logic, Methodology and Philosophy of Science in Oviedo, August 2003. It contains state of the art accounts of ongoing work by a selection of the most renowned researchers in the field. The papers in the Logic section deal with topics in mathematical logic, as well as philosophical logic, and the area of logic and computation. The section on General Methodology contains articles on models, theories, probability, (...) induction, causation, and other topics. A number of papers discuss Philosophical Issues of Particular Sciences, such as mathematics, physics, linguistics, psychology, biology, and medicine. There is also a section on Ethics of Science, and papers from a special symposium on the Emergence of Scientific Medicine in the 19th-20th Century. (shrink)

In fuzzy predicate logic, assignment of truth values may be partial, i.e. the truth value of a formula in an interpretation may be undefined . A logic is supersound if each provable formula is true in each interpretation in which the truth value of is defined. It is shown that among the logics given by continuous t-norms, Gödel logic is the only one that is supersound; all others are not supersound. This supports the view that the usual restriction of semantics (...) to safe interpretations is very natural. (shrink)

A t-tautology is a propositional formula which is a tautology in all fuzzy logics defined by continuous triangular norms. In this paper we show that the problem of recognizing t-tautologies is coNP complete, and thus decidable.

Rational Pavelka logic extends Lukasiewicz infinitely valued logic $by adding truth constants \bar{r} for rationals in [0, 1].$ We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping f satisfying the following: (1) f associates with each formula A of DPL a sentence f(A) of Peano arithmetic (PA) and with each program α a formula f(α) of PA with one free variable describing formally a supertheory of PA; (2) f commutes with logical connectives; (3) f([α] A) is the sentence saying that f(A) is provable in the theory f(α); (4) for each axiom A of DPL, f(A) is (...) provable in PA (and consequently, for each A provable in DPL, f(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretation f, f(A) is provable in PA. Various modifications of this result are considered. (shrink)