Proper n-valued ukasiewicz algebras are obtained by adding some binary operators, fulfilling some simple equations, to the fundamental operations of n-valued ukasiewicz algebras. They are the s-algebras corresponding to an axiomatization of ukasiewicz n-valued propositional calculus that is an extention of the intuitionistic calculus.

ABSTRACTThe articles Maximality and Refutability Skura [. Maximality and refutability. Notre Dame Journal of Formal Logic, 45, 65–72] and Three-valued Maximal Paraconsistent Logics Skura and Tuziak [. Three-valued maximal paraconsistent logics. In Logika. Wydawnictwo Uniwersytetu Wrocławskiego] introduced a simple method of proving maximality of a given paraconsistent matrix. This method stemmed from the so-called refutation calculus, where the focus in on rejecting rather than accepting formulas. The article A Generalisation of a Refutation-related Method in Paraconsistent Logics Trybus (...) [. A generalisation of a refutation-related method in paraconsistent logics. Logic and Logical Philosophy, 27. doi:10.12775/llp.2018.002] was a first step towards generalising the method. In it, a number of 3-valued paraconsistent matrices were shown maximal. In this article we extend these results to cover a number of n-valued paraconsistent matrices using the same method. (shrink)

The method of calculating the degree of completeness presented below differs from methods discussed in [1] and [2] and is explicitly based on elementary properties of Lukasiewiczian valuations and reveals the character of possible enlargements of given n-valued Lukasiewicz logic. Above all it seems to be especially convenient for didactic purposes.

Propositional canonical Gentzen-type systems, introduced in [2], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. [2] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). [23] extends these (...) results to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n, k)-ary quantifiers, which bind k distinct variables and connect n formulas, and show that the coherence criterion remains constructive for such systems. Then we focus on the case of k ∈ {0, 1} and show that the following statements concerning a canonical calculus G are equivalent: (i) G is coherent, (ii) G has a strongly characteristic 2Nmatrix, and (iii) G admits strong cut-elimination. We also show that coherence is not a necessary condition for standard cut-elimination, and then characterize a subclass of canonical systems for which this property does hold. (shrink)

This study analyzes the relationship between CEO values, leadership style and ethical practices in organizations. The ethical practices of formal statement of ethics and diversity training are included in the study, as well as four categories of values based on Rokeach's (1973) typology including personal, social, competency-based and morality-based. Results indicate that all four types of values are positively and significantly related to transformational leadership, with transactional leadership positively related to morality-based and personal values, and laissez-faire leadership negatively related (...) to competency-based values. When size of company and values are controlled, transformational leadership explains a significant amount of change in formal statement of ethics, and transactional leadership explains a significant amount of change in diversity training. (shrink)

We introduce Boolean-like algebras of dimension n ($$n{\mathrm {BA}}$$ n BA s) having n constants $${{{\mathsf {e}}}}_1,\ldots,{{{\mathsf {e}}}}_n$$ e 1, …, e n, and an $$(n+1)$$ ( n + 1 ) -ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of $$n{\mathrm {BA}}$$ n BA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The $$n{\mathrm {BA}}$$ n BA s provide the (...) algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, $$n{\mathrm {CL}}$$ n CL, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in $$n{\mathrm {CL}}$$ n CL, and, via the embeddings, in all the finite tabular logics. (shrink)

We express here the statement » The probability of a given b equals r « symbolically by » p = r «. A formal axiomatic calculus can be constructed comprising all the well‐known laws of probability theory. This calculus can be interpreted in various ways. The present paper is a criticism of the subjective interpretation; that is to say, of any interpretation which assumes that probability expresses degrees of incomplete knowledge: a is the statement incompletely known, (...) b is our total knowledge, and p is the degree to which a is entailed by b. The subjective interpretation has often been proposed as an explanation and even a sharpening of the various objective theories . It is shown in the paper that this proposal cannot be realised because the subjective theory cannot lead to results which are compatible with the objective theory. This is due to various reasons, the most important of which is that the objective theory interprets » b « in » p « as a statement of the objective conditions of an experiment and » a « as one of its possible results. The subjective theory on the other hand interprets » b « as our total relevant knowledge which will in general include some knowledge of previous results of the experiment. It is shown that this must lead to incompatibility owing to the fact that this knowledge of previous results must influence the value of the probability.It is shown, in this way, that any probabilistic theory of the process of learning from experience—that is to say, any probabilistic theory of induction—must lead to contradictions.RésuméL'énoncé: » La probabilité de a, si b est donné, est égale à r « est ici exprimé symboliquement par » p = r «. On peut construire un système axiomatique formel qui contient toutes les règles bien connues du calcul des probabilités. Ce calcul formel peut ětre interprété de manières très différentes. L'article contient une critique des interprétations subjectives, c'est‐à‐dire des interprétations qui admettent que la probabilité exprime des degrés de connaissance incomplète: a est un énoncé qu'on ne connaǐt que partiellement, b est notre connaissance totale, et p exprime le degré auquel a est implicitement contenu dans b. L'interprétation subjective a souvent été proposée comme une explication et un renforcement des différentes théories objectives . Dans cet article, l'auteur montre que ce projet n'est pas réalisable, car la théorie subjective mène à des résultats qui contredisent la théorie objective. Plusieurs circonstances sont responsables de cette situation; la plus importante, c'est que la théorie interprète le » b « dans » p « comme une description des conditions objectives d'une expérience . D'autre part, la théorie subjective interprète » b « comme notre savoir total , et ce savoir inclura en général des informations sur les résultats obtenus antérieurement pour cette expérience. L'auteur montre que ce fait conduit à une contradiction entre les deux interprétations, car cette connaissance doit avoir une influence sur la valeur subjective de la probabilité.A l'aide de ce raisonnement, on peut montrer que la théorie des probabilités appliquée au processus inductif tirant une expérience du domaine expérimental conduit à des contradictions.ZusammenfassungDie Aussage » Die Wahrscheinlichkeit von a, wenn b gegeben ist, ist gleich r « wird hier symbolisch durch » p = r « ausgedrückt. Ein formales Axiomensystem kann konstruiert werden, das alle die wohlbekannten Regeln der Wahrscheinlichkeitsrechnung enthält. Dieser formale Kalkül kann in sehr verschiedener Weise interpretiert werden. Der Artikel enthält eine Kritik der subjektiven Interpretationen; das heisst, jener Interpretationen die annehmen, dass die Wahrscheinlichkeit Grade von unvollständigem Wissen ausdrückt: a ist ein Satz der unvollständig gewusst wird, b ist unser Gesamtwissen, und p drückt den Grad aus zu dem a in b implizit enthalten ist. Die subjektive Interpretation wurde oft als eine Erklärung und Verschärfung der verschiedenen objektiven Theorien vorgeschlagen. In dem Artikel wird gezeigt, dass der Vorschlag undurchführbar ist, da die subjektive Theorie zu Resultaten führt, die der objektiven Theorie widersprechen. Verschiedene Umstände tragen zu dieser Situation bei. Der wichtigste dieser Umstände ist, dass die objektive Theorie das » b « in » p « als eine Beschreibung der objektiven Bedingungen eines Experimentes interpretiert . Andererseits interpretiert die subjektive Theorie » b « als unser Gesamtwissen , und dieses wird im allgemeinen Informationen bezüglich früherer Resultate dieses Experiments einschliessen. Es wird gezeigt, dass das zu einem Widerspruch zwischen den Interpretationen führt, da dieses Wissen Einfluss auf den subjektiven Wahrscheinlichkeitswert haben muss.Mit Hilfe dieses Gedankenganges wird gezeigt, dass die Wahrscheinlichkeitstheorie des induktiven Lernens aus der Erfahrung zu Widersprüchen führt. (shrink)

This book was designed primarily as a textbook; though the author hopes that it will prove to be of interests to others beside logic students. Part I of this book covers the fundamentals of the subject the propositional calculus and the theory of quantification. Part II deals with the traditional formal logic and with the developments which have taken that as their starting-point. Part III deals with modal, three-valued, and extensional systems.

In this paper I am merely analysing the meaning of Phaedo 99–107 without discussing either its historical significance or its value as a contribution to the logic of science. The paper attempts in fact to be little more than a paraphrase of the Greek, which aims only at accuracy of statement, but I am adding in the third section a note on the relation of this passage to the discussion of method in Republic VI, mainly in order to minimize (...) its relevance. The first section of the paper deals with the linguistic analysis of the text, the second, very sketchily, with the general doctrine. (shrink)

A contribution to the sixth installment of the Common Knowledge symposium “Apology for Quietism,” this article compares the worldview of Qoheleth (Ecclesiastes) and the quietism that it presumably entails to the early Wittgenstein's worldview and his quietism. The first section of the article treats a relevant paradox in the worldview of the early Wittgenstein: his positive exhortations for certain types of speech and silence, for certain types of action and inaction, seem in conflict with his statement that, in the (...) world, “there is no value—and if there were, it would be of no value.” The second section examines an internal paradox in the worldview of Qoheleth, who commends the enjoyment of a world that he considers meaningless and vain. The third and final section of the article compares these paradoxes as constitutive of the texts in which they appear. Conveying no univocal message, while both writers exhibit quietism, they also gesture toward kinds of speech and action. (shrink)

I consider here the issue of whether and to what extent moral truths are absolute. My aim is to raise some new considerations in favor of moral relativism: the thesis that some moral statements can vary in truth-value depending on the moral standards at issue.1 2 This paper has three major components. First, I describe a new puzzle concerning the possibility of moral knowledge in light of expert disagreement. I argue that the best solution to this puzzle requires moral relativism. (...) Second, I develop a notion of a moral standard that incorporates recent developments in moral psychology, including work by.. (shrink)

In his (several times reprinted) article How Wide Is the Gap Between Facts and Values? N. Rescher aspires to clarify the long-lasting discussion on the ‘is-ought’ (‘fact-value’) gap by providing a framework in which the related arguments can be perspicuously articulated. He then argues that even if the logical gap may bereal, the transition from factual premises to value conclusions is smoothly mediated by trivially true value statements. We scrutinize Rescher’s argumentation and show defects in the presented lines of reasoning. (...) In particular, we argue that his tripartite division of statements (factual, evaluative, hybrid), as well as his emphasis on the role of evaluative truisms, introduce serious problems that escape Rescher’s attention. We thus conclude that he failed to achieve his goals. (shrink)

Mathematical methods in biology occupy a somewhat peculiar position, and the attitude of many biologists toward them is similar to that of many practical engineers toward what is called pure scientific research. The modern progressive engineer recognizes the value of pure science, which seeks for truth regardless of any possibility of practical applications; yet he still frequently shows a definite dislike towards such investigations. The whole history of civilization demonstrates that discoveries which, at the time they were made, did not (...) show any signs of the possibility of being turned into practical use, now lie at the basis of many useful engineering devices. As Professor R. A. Millikan has said, we owe our present-day civilization to Galileo. Every educated person, be he pure scientist or engineer, recognizes this fact. And yet the frame of mind of the “practical” man is not too receptive to investigations which do not show at least some remote prospects of practical applications. Unless he can see at least hypothetically some practical results of such research, he will not give it his whole-hearted enthusiasm. The ultimate criterion in the evaluation of any research still remains its practical use. Those pure scientists who have had close contact with what is called “industrial research” know only too well the truth of the above statement. (shrink)

The primary objective of this paper, which is an addendum to the author’s [8], is to apply the general study of the latter to Łukasiewicz’s n-valued logics [4]. The paper provides an analytical expression of a 2(n−1)-place sequent calculus (in the sense of [10, 9]) with the cut-elimination property and a strong completeness with respect to the logic involved which is most compact among similar calculi in the sense of a complexity of systems of premises of introduction rules. (...) This together with a quite effective procedure of construction of an equality determinant (in the sense of [5]) for the logics involved to be extracted from the constructive proof of Proposition 6.10 of [6] yields an equally effective procedure of construction of both Gentzen-style [2] (i.e., 2-place) and Tait-style [11] (i.e., 1-place) minimal sequent calculi following the method of translations described in Subsection 4.2 of [7]. (shrink)

This chapter contains sections titled: Introduction Qualitative Hedonism Objections to Mill's Qualitative Hedonism: Internal Inconsistency and Value Pluralism The Judgment of Competent Agents: Self‐Development and Value Measurement Self‐Development and Virtue Ethics Further Reading.

This is another volume in the continuing series published under the auspices of Tulane University. It contains eight articles. Ramona Cormier’s article "The Concept of Isolation in Contemporary Aesthetic Theory" uses the term isolation in accordance with Langer’s definition. In order to develop her point Cormier distinguishes between the historicity of an art work and the historiography of the work. On the basis of this she discusses briefly the attitudes of Jerome Schiller, Clive Bell, T. S. Eliot, Jerome Stolnitz, Roger (...) Fry, George Dickie. Shannon DuBose remarks in "Poesis and Cosmos" that "... views advanced [on art] in The Republic loom larger than need be in discussion of Plato’s theory of art; they should be balanced with suggestions advanced in other dialogues, particularly The Timaeus." One cannot agree more with this need for balance. Professor Feibleman’s article "The Art of the Philosophy of Art" is not so much a discussion of the philosophy of art as it is a discussion of the artist, the role of art in society, the relation of art to other areas of a culture in which art specifically exists. John Glenn, Jr. has written on "Kierkegaard on the Unity of Comedy and Tragedy." I do not agree with all his interpretations of Kierkegaard but his article is interesting. Harold N. Lee in "Action, Perception and Art" rethinks a position he held in the original edition of his Perception and Aesthetic Value. He now holds that "... the two views of the nature of art can be shown to be essentially related and not only externally so...." Marian L. Paulson writes an article, "Structures in Art Media," where she suggests that perhaps the problem of structural relations among art media may find a solution if one takes account of the Jungian notion of the racial unconscious, especially in the resolution of the tensions between anima and animus and between the shadow and transcendence. The article "Truth in Art" by Louise N. Roberts discusses one of the many ways in which the expression "truth in art" may be considered. The author in this case shows how paintings may present a truth in a way quite different from verbal statements. The book concludes with John Sallis’ commentary on Nietzsche’s The Birth of Tragedy.—P. N. (shrink)

The theory of mental models and its computer implementations have led to crucial experiments showing that no standard logic—the sentential calculus and all logics that include it—can underlie human reasoning. The theory replaces the logical concept of validity (the conclusion is true in all cases in which the premises are true) with necessity (conclusions describe no more than possibilities to which the premises refer). Many inferences are both necessary and valid. But experiments show that individuals make necessary inferences that (...) are invalid, e.g., Few people ate steak or sole; therefore, few people ate steak. Other crucial experiments show that individuals reject inferences that are not necessary but valid, e.g., He had the anesthetic or felt pain, but not both; therefore, he had the anesthetic or felt pain, or both. Nothing in logic can justify the rejection of a valid inference: a denial of its conclusion is inconsistent with its premises, and inconsistencies yield valid inferences of any conclusions whatsoever including the one denied. So inconsistencies are catastrophic in logic. In contrast, the model theory treats all inferences as defeasible (nonmonotonic), and inconsistencies have the null model, which yields only the null model in conjunction with any other premises. So inconsistences are local. Which allows truth values in natural languages to be much richer than those that occur in the semantics of standard logics; and individuals verify assertions on the basis of both facts and possibilities that did not occur. (shrink)

The primary purpose of this paper is to argue that biologists should stop citing Karl Popper on what a genuinely scientific theory is. Various ways in which biologists cite Popper on this matter are surveyed, including the use of Popper to settle debates on methodology in phylogenetic systematics. It is then argued that the received view on Popper—namely, that a genuinely scientific theory is an empirically falsifiable one—is seriously mistaken, that Popper’s real view was that genuinely scientific theories have the (...) form of statements of laws of nature. It is then argued that biology arguably has no genuine laws of its own. In place of Popperian falsifiability, it is suggested that a cluster class epistemic values approach (which subsumes empirical falsifiability) is the best solution to the demarcation problem between genuine science and pseudo- or non-science. (shrink)

The author presents an epistemic logic with quantification, formulated in terms of consistency in model systems. To support results like "a knows p, and p implies q, therefore a knows q," consistency of a set of statements is reinterpreted as defensibility, or immunity to criticism. Perspicuous informal analyses, such as the remarks on the role of pronouns, augment the book's philosophical value.--N. A. G.

. The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the (...) general structure of proofs in the original calculus in a way ensuring preservation of the weak cut elimination theorem under the transformation. The described transformation metod is illustrated on several concrete examples of many-valued logics, including a new application to information sources logics. (shrink)

Donald J. Munro's essay, "When Science Is in Defense of Value-Linked Facts," takes a stand against the fact-value dichotomy which has been heavily pronounced within the Greco-European philosophical canon. As Munro also points out, the continuing persistence of the fact-value dichotomy is traceable to Moore's discussion of the "naturalistic fallacy" and Hume's discussion of the is-ought problem. In opposition to these two views, classical Confucian thinkers present us with descriptive statements about human commonalities, including their inborn affects....

A diverse collection of essays, which reflect the breadth of Judith Jarvis Thomson's philosophical work. The diversity of topics discussed in this book reflects the breadth of Judith Jarvis Thomson's philosophical work. Throughout her long career at MIT, Thomson's straightforward approach and emphasis on problem-solving have shaped philosophy in significant ways. Some of the book's contributions discuss specific moral and political issues such as abortion, self-defense, the rights and obligations of prospective fathers, and political campaign finance. Other contributions concern the (...) foundations of moral theory, focusing on hedonism, virtue ethics, the nature of nonconsequentialism, and the objectivity of moral claims. Finally, contributions in metaphysics and epistemology discuss the existence of sets, the structures reflected in conditional statements, and the commitments of testimony. Contributors Jonathan Bennett, Richard L. Cartwright, Joshua Cohen, N. Ann Davis, Catherine Z. Elgin, Gilbert Harman, Barbara Herman, Frances Myrna Kamm, Claudia Mills, T.M. Scanlon, Ernest Sosa. (shrink)

Quantifiers in Frege's Grundgesetze like are not well-defined because the part Fx & Gx stands for a concept but the yoking conjunction is horizontalised and must stand for a truth-value. This standard interpretation is rejected in favor of a substitutional reading that, it is argued, both conforms better to the text and is well-defined. The theory of the horizontal is investigated in detail and the composite reading of Frege's connectives as made up of horizontals is rejected. The sense in which (...) the Grundgesetze has a many-valued but classical logic is explained by proving that its propositional fragment (under the standard interpretation) and Bochvar's 3-valued logic are instances of the same metatheoretic methods. Historically, the paper argues for a more naive but well-defined reading of the text. Theoretically, it provides a formally adequate statement of that semantics, as well as developing the abstract metatheory which embraces Bochvar's language and the Fregean fragment. (shrink)

The serious formal investigation of n-valued systems of logic for n>2 dates back to Post's 1921 doctoral dissertation. The primary use for such structures, however, has been as model-theoretic devices in the investigation of systems of lower order. Ackermann's short book now comes as a welcome addition to the literature dealing with the formal properties and applications of n-valued systems in their own right. Ackermann begins with a general discussion of implicational calculi in which fundamental ideas of validity, (...) well-formedness, and other basic notions are set forth elegantly and clearly. While assuming the classical implicational calculus as an underlying substructure, Ackermann next motivates a consideration of possible modifications of the formal system in the direction of greater generality by means of the standard Lewis objections to the reading of "⊃" as "implies." This leads to the well known story of strict entailment, introduction of modal operators, and then a four-valued matrix as an underlying semantics for the expanded system. There then follows a good general discussion of the properties of many-valued logics, various nonstandard propositional calculi, and an admirable presentation of the Lukasiewicz-Tarski propositional calculus, accompanied by a number of important metatheorems. Ackermann also includes chapters on alternative multivalued [[sic]] systems and a glance at their applications. One would like to see, however, some discussion of such controversial matters as n-valued quantificational theory and the well-known objections to the entire program of nonstandard logistic theory. There is a good bibliography of important books and articles in the field.—H. P. K. (shrink)

The purpose of this brief paper is to show that the primary problem of technology, like all problems related to possibilities and actions, is the conceptual adequacy of the intentions and values it implies, and not, as many critics have suggested, its social effects. Presupposed for this statement and evaluation is an interpretation of experience called here experiential idealism. On the basis of this position some suggestions are made about the meaning of technology and its correlative possibilities and constraints.

Kantian ethics can at times appear to defend the position that there is a unique sort of value that plays a foundational role in morality. For instance, Kant's most well known work in ethics, the Groundwork of the Metaphysic of Morals, begins by trying to establish that a good will is good without qualification' and then ends with a first statement of the fundamental principle that divides right from wrong, the Categorical Imperative.1 This presentation can make it seems as (...) if Kant believes the authority carried by the Categorical Imperative is somehow supposed to be grounded in the value of a good will. Again, the humanity formulation of the Categorical Imperative, the formulation that tell us we must respect the humanity in ourselves and others by treating it as an end in itself, appears to allude to a special value possessed by some feature of persons, their humanity, and then explain the authority of moral obligation by way of that value.2 This extolling of the value of humanity and the dramatic refrain about the unique value of a good will both appear to portray Kant as telling us to notice the peculiar value that they possess and see that this value demands that we adjust our deliberation and actions. We appear to be told that the good will and humanity are bits of metaphysical glitter, jewels carrying their... (shrink)

At first blush, values such as diversity appear to be worth striving for. The question is whether or not such values—which have become increasingly prevalent in university mission statements—are values as such, which is to ask whether they are things of moral worth (Value, n.d.), or are something else altogether. My unpopular suspicion leans toward the latter. Personal opinions, of course, are hardly a justification for an impassioned critique, however, my opinions mirror those held by moderate and conservative witnesses to (...) the sociopolitical climate of academia (Pew Research Center, 2017) as well as individuals who would typically be considered classically liberal such as Peter Boghossian (2021). Because of this concordance and in order to understand the constitutive relations of academic production, it becomes necessary to critically examine the nature of institutional values, separate from polemical rhetoric that either seeks to construct a "straw man" of them in order to lay them low or insulate them from outside reproach altogether Since "diversity" seems to be the chief concern among interested groups both in the academic setting and the world at large, this article's point of access to the relations of production in academia will be anchored by a Marxian assessment of diversity as a product of social labour in the academic environment. (shrink)

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is (...) a natural dual of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)

This volume offers to the English-speaking world a collection of important works by the eminent twentieth century logician, Jan Lukasiewicz, many of which are here translated into English for the first time. This edition differs significantly from the Polish edition which appeared in 1961—containing ten logic papers not appearing there and omitting articles primarily of interest to the Polish reader. In addition to writing in Polish, Lukasiewicz also published works in French, English, and notably in German, and sometimes translated his (...) own works from one language to another. One of the most valuable works on the history of logic is Lukasiewicz’ paper, "On the History of the Logic of Propositions". Lukasiewicz points out the difference between the two basic areas of formal logic, the logic of propositions and the logic of terms, undifferentiated before the development of modern mathematical logic. The understanding of this distinction leads Lukasiewicz to trace the history of propositional logic back to its original development by the Stoics, its further development by the medieval Scholastics, and its axiomatization by Gottlob Frege. The status of mathematical logic is discussed in various works. The most basic system is propositional logic, upon which depend the other logical disciplines and also mathematics. Mathematical logic, also called logistic, is independent of philosophy and espouses no philosophical viewpoint. Early in his career, Lukasiewicz refers to mathematical logic as the logic of algebra and uses mathematical symbolism to represent logical propositions. Later he introduces what we now call "Polish notation," in particular, "Cpq" for "if p, then q" and "Np" for "it is not the case that p" for the primitive functions, eliminating the need for punctuation. He adopts the symbols Π and Σ for quantification from Charles S. Peirce, and also brings to light the almost unknown fact that it was Peirce who invented the matrix method in 1885. The paper, "Investigations into the Sentential Calculus", written by Lukasiewicz and Tarski embodies the results of a decade of research on the sentential calculus initiated by Lukasiewicz at the University of Warsaw systematically compiling the contributions of five logicians: Lindenbaum, Sobocinski, Wajsberg, Tarski, and Lukasiewicz. The approach is metalogical and depends heavily upon set theoretic concepts; it covers both the matrix and axiomatic methods. One of the problems that preoccupied Lukasiewicz was that of determinism, which led to his development of three-valued logic. A proposition such as "I shall be in Warsaw at noon on 21 December of next year" is neither true nor false; a third truth value is needed, one which Lukasiewicz calls "the possible." Another area to which Lukasiewicz contributed is modal logic, proving that modal logic cannot be two-valued. Lukasiewicz’ analysis and axiomatization of Aristotle’s syllogistic is not included in the present volume since an English edition by Lukasiewicz on this topic is available. We have here briefly touched upon a few of Lukasiewicz’ numerous achievements in logic; the best means of appreciating them is to read his works.—T. G. N. (shrink)

This work is an introductory textbook for deductive logic being primarily concerned with truth-functional logic, but also containing an introduction to syllogisms with the application of Venn diagrams, an introduction to quantification theory, and a brief discussion of axiom systems. Harrison employs six logical operators in his truth-functional calculus, including both inclusive and exclusive disjunction. The six operators are initially defined by truth tables, but in the natural deduction presentation negation and conjunction are taken as primitive and the other (...) connectives are defined in terms of these two. The conditional and indirect methods of proof are included with the approach being essentially the same as that given in Copi’s Symbolic Logic. Categorical statements and syllogisms are analyzed from both hypothetical and existential viewpoints. The treatment of quantification theory includes two-place predicates and employs the four standard rules for generalization and instantiation. The book contains an abundance of explanations, examples, and exercises. Selected answers, usually for the odd numbered problems, are given in an appendix.—T. G. N. (shrink)

This study investigates what professional caregivers working in nursing homes consider to be a good life for residents suffering from dementia. Ten caregivers were interviewed; special attention was paid to the way in which they deal with conflicting values. Transcripts of the interviews were analysed qualitatively according to the method of grounded theory. The results were compared with those from a similar, earlier study on ideals found in mission statements of nursing homes. The concepts that were mentioned by most interviewed (...) participants as important for a good life were ‘peace and quiet’, ‘going along with subjective experience’ and ‘no enforcement: the way the resident wants it’. A considerable overlap was found between the interviews and the mission statements; however, when compared with the mission statements, the interviews put less emphasis on individuality and on giving meaning, and more on offering residents pleasant activities. When faced with conflicting values, caregivers tend to make pragmatic and more or less intuitive decisions. Although this has its merits, it may be desirable to stimulate conscious reflection regarding conflict between different values. (shrink)

The tetravalent modal logic is one of the two logics defined by Font and Rius :481–518, 2000) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the algebras is (...) not so good as in $${{\mathcal {TML}}}^N$$ TML N, but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus :481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al., we provide a sequent calculus for $${\mathcal {TML}}$$ TML with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic. (shrink)

In, Marker and Steinhorn characterized models of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models for which all 1‐types over M realized in N are definable has already the desired property. Although it is true that if M is an algebraically closed valued field such that all 1‐types over M are (...) definable then all types over M are definable, we build a counterexample for the relative statement, i.e., we show for any that there is a pair of algebraically closed valued fields such that all n‐types over M realized in N are definable but there is an ‐type over M realized in N which is not definable. (shrink)

A semantics of vagueness should reject the principle that every statement has a truth-value yet retain the classical tautologies. A many-value, non-truth-functional semantics and a semantics of super-valuations each have this result. According to the super-valuation approach, 'if a man with n hairs on his head is bald, then a man with n plus one hairs on his head is also bald' is false because it comes out false no matter how the vague predicate 'is bald' is appropriately made (...) precise. But why should a sentence in which components actually remain imprecise be regarded as actually false just because it would be false if its components were precise? On one of the alternative treatments of quantification allowed by the many-value approach, the sentence in question is assigned an intermediate value closer to 'false' than to 'true'. Despite the elegance of the super-valuation approach, there are reasons to prefer the many-value approach. (shrink)

The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particular, we obtain completeness theorems for Moisil calculus, n-valued Łukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods.

The diversity of topics discussed in this book reflects the breadth of Judith Jarvis Thomson's philosophical work. Throughout her long career at MIT, Thomson's straightforward approach and emphasis on problem-solving have shaped philosophy in significant ways. Some of the book's contributions discuss specific moral and political issues such as abortion, self-defense, the rights and obligations of prospective fathers, and political campaign finance. Other contributions concern the foundations of moral theory, focusing on hedonism, virtue ethics, the nature of nonconsequentialism, and the (...) objectivity of moral claims. Finally, contributions in metaphysics and epistemology discuss the existence of sets, the structures reflected in conditional statements, and the commitments of testimony. Contributors: Jonathan Bennett, Richard L. Cartwright, Joshua Cohen, N. Ann Davis, Catherine Z. Elgin, Gilbert Harman, Barbara Herman, Frances Myrna Kamm, Claudia Mills, T.M. Scanlon, Ernest Sosa. (shrink)

In recent research, some of the present authors introduced the concept of an $n$-dimensional Boolean algebra and its corresponding propositional logic $n\textrm{CL}$, generalizing the Boolean propositional calculus to $n\geq 2$ perfectly symmetric truth values. This paper presents a sound and complete sequent calculus for $n\textrm{CL}$, named $n\textrm{LK}$. We provide two proofs of completeness: one syntactic and one semantic. The former implies as a corollary that $n\textrm{LK}$ enjoys the cut admissibility property. The latter relies on the generalization to the (...) $n$-ary case of the classical proof based on the Lindenbaum algebra of formulas and Boolean ultrafilters. (shrink)

The paper is concerned with the problem of characterization of strengthenings of the so-called Łukasiewicz-like sentential calculi. The calculi under consideration are determined by n-valued Łukasiewicz matrices with superdesignated logical values. In general, Łukasiewicz-like sentential calculi are not implicative in the sense of [7]. Despite of this fact, in our considerations we use matrices analogous to S-algebras of Rasiowa. The main result of the paper says that the degree of maximality of any n-valued Łukasiewicz-like sentential calculus is (...) finite and equal to the degree of maximality of the corresponding n-valued Łukasiewicz calculus. (shrink)

In 1944 Hans Reichenbach developed a three-valued propositional logic (RQML) in order to account for certain causal anomalies in quantum mechanics. In this logic, the truth-value _indeterminate_ is assigned to those statements describing physical phenomena that cannot be understood in causal terms. However, Reichenbach did not develop a deductive calculus for this logic. The aim of this paper is to develop such a calculus by means of First Degree Entailment logic (FDE) and to prove it sound and (...) complete with respect to RQML semantics. In Section 1 we explain the main physical and philosophical motivations of RQML. Next, in Sections 2 and 3, respectively, we present RQML and FDE syntax and semantics and explain the relation between both logics. Section 4 introduces \(\varvec{\mathcal {Q}}\) calculus, an FDE-based tableaux calculus for RQML. In Section 5 we prove that \(\varvec{\mathcal {Q}}\) calculus is sound and complete with respect to RQML three-valued semantics. Finally, in Section 6 we consider some of the main advantages of \(\varvec{\mathcal {Q}}\) calculus and we apply it to Reichenbach’s analysis of causal anomalies. (shrink)

In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using $n$-sided sequent calculus. These systems are understood as a (...) further development of bilateralism—call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion and, with Mark Twain, that denial is more than a river in Egypt. (shrink)

Involutive Stone algebras were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, (...) and hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic that can be determined by a finite number of matrices. Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. (shrink)

A three-valued propositional logic is presented, within which the three values are read as ?true?, ?false? and ?nonsense?. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1−x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free n is the algebra of [0,1]-valued functions over the n-cube [0,1] n generated by the coordinate functions ξ i ,i=1, . . . ,n, with pointwise operations. Any such function f is (...) a McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] n →[0,1] are in Free n . The elements of Free n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of Łukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free n. (shrink)

The adoption of codes of ethics or values statements is intended to guide everyday decisions, as well as to influence the perceptions of external stakeholders. Questions have emerged in the literature about whether the effort to substantively direct decision-making in an organization is marginalized by the more obvious symbolic role of values statements. Here the perceived impact of values statements on decision-making in organizations is explored, and a number of positive effects observed. Respondents report that values statements create positive externalities (...) providing guidelines for decision-making, increasing accountability, and clarifying expectations. Yet, both cynicism and perceived management hypocrisy emerged in the survey, which together had strong negative effects on the perceived decision-making impact of values statements. Finally, positive external effects are almost never mentioned by respondents who give their firms high marks on the quality of values statement development, training, and implementation. Yet, such external effects get significantly greater representation in the comments of respondents who report less substance in their firms’ values statement development and implementation processes. In all, the results suggest that the substantive and symbolic roles of values statements are not independent and that external symbolism without internal legitimacy may in the long-run be problematic. (shrink)

C. L. Stevenson, in “The Scientist’s Role and the Aims of Education,” has recently recommended that scientists join in the activity of making educational value statements, and that educationists have their value attitudes “straightened out under the guidance of beliefs that are well verified.” The argument underlying his recommendation is that since scientific knowledge is in some way relevant to questions of value in education, this knowledge must be utilized, otherwise some rather absurd educational judgments will be made, perhaps with (...) disastrous results. As he puts it, if the scientist “won’t risk making value judgments, then I’d like to ask who is going to make them. There’s always the possibility that those who aren’t reluctant to make them will turn out to be … complete asses.”. (shrink)

We review the definition of the Full Bayesian Significance Test (FBST), and summarize its main statistical and epistemological characteristics. We review also the Abstract Belief Calculus (ABC) of Darwiche and Ginsberg, and use it to analyze the FBST’s value of evidence. This analysis helps us understand the FBST properties and interpretation. The definition of value of evidence against a sharp hypothesis, in the FBST setup, was motivated by applications of Bayesian statistical reasoning to legal matters where the sharp hypotheses (...) were defendants statements, to be judged according to the Onus Probandi juridical principle. (shrink)