Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square and of the hexagon in terms of acceptability. Then, we show how to construct probabilistic versions of the square and of the hexagon of opposition by forming suitable (...) tripartitions of the set of all coherent assessments. Finally, as an application, we present new versions of the square and of the hexagon involving generalized quantifiers. (shrink)

squares are fundamental features of cities, so nowing more about them as social arenas which is enabling contact between different groups is necessary. In fact, they represent sites of sociability, face to face interaction and at the same time their quality is commonly perceived to be a measurement for social quality of urban life. The concern of this research is how to ensure that public open places use their potential for enhancing social sustainability.

In this paper, I will show to what extent we can use our modern understanding of the Square of Opposition in order to make sense of Kant 's double standard solution to the cosmological antinomies. Notoriously, for Kant, both theses and antitheses of the mathematical antinomies are false, while both theses and antitheses of the dynamical antinomies are true. Kantian philosophers and interpreters have criticized Kant 's solution as artificial and prejudicial. In the paper, I do not dispute such (...) claims, but I show that our modern understanding of the Square of Opposition enables us to more naturally deliver the result Kant was aiming at. Accordingly, the paper does not pretend to be exegetically accurate. It is an attempt to revise the antinomies with the help of standard classical logic. And although such a revision entails some re-interpretation, in the end, it will actually help to unveil some of Kant 's thoughts. (shrink)

The historiography on American science and technology in the 1970s is still small, yet there are already three distinct strands of work: studies of countercultural scientists, portrayed as enacting or advocating ‘groovy’ research; studies of the politically polarized debate pitting conservative and libertarian ‘cornucopianists’ against environmentalists and modelers forecasting resource scarcity; and studies of the early commercialization of technoscience (e.g., biotechnology) that took off in the 1980s. Left out, I argue, are a class of ‘square scientists’ with little sympathy (...) for the counterculture, and yet open to (even eager for) a new kind of science oriented to the same problems activists said they wanted science to solve: pollution, mass transit, housing, biomedicine, disability technologies, pedagogical machines, etc. Square scientists at places like NASA and Texas Instruments adapted military-industrial-academic templates to a wide variety of socially ‘relevant’ topics in the 1970s. Yet square scientists still looked to the military-industrial complex for allies, rather than to countercultural colleagues. This potential middle ground remained excluded – contributing, in large part, to the failure of schemes to reorient US R&D to civilian social problems, and to the invisibility of the squares in today's historical accounts. (shrink)

This article shows that there are square circles in the sense that there are mathematical objects that are at the same time both perfectly circular and perfectly square. The philosophical significance of this is discussed, especially in view of philosophy's widespread use of “square circle” as a typical example of an impossibility. In particular, the focus is on what the existence of square circles means for the possibility of conceptual analysis, and more generally what we can (...) learn about the nature of non-formal concepts and their use in philosophy. (shrink)

. In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual (resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give some examples.

In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus. We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.

The square of opposition is a diagram related to a theory of oppositions that goes back to Aristotle. Both the diagram and the theory have been discussed throughout the history of logic. Initially, the diagram was employed to present the Aristotelian theory of quantification, but extensions and criticisms of this theory have resulted in various other diagrams. The strength of the theory is that it is at the same time fairly simple and quite rich. The theory of oppositions has (...) recently become a topic of intense interest due to the development of a general geometry of opposition (polygons and polyhedra) with many applications. A congress on the square with an interdisciplinary character has been organized on a regular basis (Montreux 2007, Corsica 2010, Beirut 2012, Vatican 2014, Rapa Nui 2016). The volume at hand is a sequel to two successful books: The Square of Opposition - A General Framework of Cognition, ed. by J.-Y. Béziau & G. Payette, as well as Around and beyond the Square of Opposition, ed. by J.-Y. Béziau & D. Jacquette, and, like those, a collection of selected peer-reviewed papers. The idea of this new volume is to maintain a good equilibrium between history, technical developments and applications. The volume is likely to attract a wide spectrum of readers, mathematicians, philosophers, linguists, psychologists and computer scientists, who may range from undergraduate students to advanced researchers. (shrink)

We define , a square principle in the context of , and prove its consistency relative to ZFC by a directed-closed forcing and hence that it is consistent to have hold when κ is supercompact, whereas □κ is known to fail under this condition. The new principle is then extended to produce a principle with a non-reflection property. Another variation on is also considered, this one based on a family of club subsets of . Finally, a new square (...) principle for cardinals, denoted , is introduced. This principle is proved consistent with κ being supercompact. It is shown to yield a non-reflection result similar to that given by □κ. (shrink)

In this paper, we introduce a Hilbert style axiomatic calculus for intutionistic logic with strong negation. This calculus is a preservative extension of intuitionistic logic, but it can express that some falsity are constructive. We show that the introduction of strong negation allows us to define a square of opposition based on quantification on possible worlds.

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

After explaining the interdisciplinary aspect of the series of events organized around the square of opposition since 2007, we discuss papers related to the 4th World Congress on the Square of Opposition which was organized in the Vatican at the Pontifical Lateran University in 2014. We distinguish three categories of work: those dealing with the evolution and development of the theory of opposition, those using the square as a metalogical tool to give a better understanding of various (...) systems of logic and those related with applications of the theory of opposition to conceptual analysis and pedagogy. (shrink)

The paper discusses various relationships between the concepts mentioned in the title. In Section 1 Todorcevic functions are shown to arise from both morasses and square. In Section 2 the theme is of supplements to morasses which have some of the flavour of square. Distinctions are drawn between differing concepts. In Section 3 forcing axioms related to the ideas in Section 2 are discussed.

We present a construction of a global square sequence in extender models with λ-indexing.

This research discusses the history of a certain space in the capital of Hungary as the physical concretization of the Soviet Union’s ideological impact on the country. Even though this 360 meters × 85 meters territory has had a very short lifetime of circa sixty years, it has been the location of many political and cultural events of nationwide importance. After the territorial and chronological contextualization, this article introduces the story of all the planned, established, demolished or removed public buildings (...) and statues that were located there from the mid-20th century until 2011. The text contains art analysis and comparative research methodologies to point out the conscious allusions and unconscious correlations of different regimes and their aims and ideologies in time. The third section provides possible explanations and analyzing tools to understand the relevance and the possible future options of the square. Lastly, the paper points out further possible research directions to investigate the broader topic of the connection between an urban location and the representational will of the contemporary leadership from different angels. (shrink)

We prove that in all Mitchell-Steel core models, □ κ holds for all κ. (See Theorem 2.). From this we obtain new consistency strength lower bounds for the failure of □ κ if κ is either singular and countably closed, weakly compact, or measurable. (Corallaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □ κ holds (...) iff κ is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.). (shrink)

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, ♦ δ holds for every regular uncountable cardinal δ, and below the least supercompact cardinal κ, □ δ holds on a stationary subset of κ. There are no restrictions in our model on the structure of the class of supercompact cardinals.

In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic. We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition.

In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition is (...) represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square. (shrink)

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find (...) suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

Pelkey’s anchoring of the semiotic square in embodiment is excellent news for cognitive literary theory, a dynamic field still in search of itself. However, his validation of the square, though theoretically unexceptionable, suffers in the execution, for his interpretation of the country song “Follow your Arrow” is less successful. The present article benefits from Pelkey’s validation as it organizes a tool of cultural-semantic analysis (CS-tool) as a ‘deviant’ semiotic square. The article then shows how this particular semiotic (...) square allows us to analyze the song in terms which build on Pelkey’s analysis, but also arrive at more satisfying results. Where Pelkey sees liberation in the song and the square, the tool uncovers manipulation in the former and closure in the latter. The article then assesses the complementarity of and differences between the two squares: Pelkey works on a local sentence-level through direct implicature, thus following the narrative/authorial voice of the poem. The CS-tool starts from a position of higher abstraction requiring a less defined, but still sufficient and more wide-ranging, three-step implicature. This allows the tool to step back from the song’s authorial voice and uncover its manipulations. The article closes by discussing the deviant features of the present square. (shrink)

. This papers examines formal properties of logical squares and their generalizations in the form of hexagons and octagons. Then, several applications of these constructions in philosophical analysis are elaborated. They concern contingency (accidentality), possibility, permission, axiological concepts (bonum and malum), the generalized Hume thesis (deontic and epistemic modalities), determinism, truth and consistency (in various senses. It is shown that relations between notions used in various branches of philosophy fall into the same formal scheme.

The square of a language L is the set of all words pp where p ∈ L. The square of a regular language may be regular too or context-free or none of both. We give characterizations for each of these cases and show that it is decidable whether a regular language has one of these properties.

In this paper, we propose the introduction of a neutrosophic chi-square-test for consistency, incorporating neutrosophic statistics. Our aim is to modify the existing chi-square -test for consistency in order to analyze imprecise data. We present a novel test statistic for the neutrosophic chi-square -test for consistency, which accounts for the uncertainties inherent in the data. To evaluate the performance of the proposed test, we compare it with the traditional chi-square -test for consistency based on classical statistics. (...) By conducting a comparative analysis, we assess the efficiency and effectiveness of our proposed neutrosophic chi-square-test for consistency. Furthermore, we illustrate the application of the proposed test through a numerical example, demonstrating how it can be utilized in practical scenarios. Through this implementation, we aim to provide empirical evidence of the improved performance of our proposed test when compared to the traditional chi-square-test for consistency based on classical statistics. We anticipate that the proposed neutrosophic chi-square-test for consistency will outperform its classical counterpart, offering enhanced accuracy and reliability when dealing with imprecise data. This advancement has the potential to contribute significantly to the field of statistical analysis, particularly in situations where data uncertainty and imprecision are prevalent. (shrink)

. Each predicate of the Aristotelian square of opposition includes the word “is”. Through a twofold interpretation of this word the square includes both classical logic and non-classical logic. All theses embodied by the square of opposition are preserved by the new interpretation, except for contradictories, which are substituted by incommensurabilities. Indeed, the new interpretation of the square of opposition concerns the relationships among entire theories, each represented by means of a characteristic predicate. A generalization of (...) the square of opposition is achieved by not adjoining, according to two Leibniz’ suggestions about human mind, one more choice about the kind of infinity; i.e., a choice which was unknown by Greek’s culture, but which played a decisive role for the birth and then the development of modern science. This essential innovation of modern scientific culture explains why in modern times the Aristotelian square of opposition was disregarded. (shrink)

The square of opposition has never been drawn by classical Arabic logicians, such as al-Fārābī and Avicenna. However, in some later writings, we do find squares, which their authors call rather ‘tables’ (sing. _lawḥ_). These authors are Shihāb al-Dīn al-Suhrawardī and Muhammed b. Yūsuf al-Sanūsī. They do not pertain to the same geographic area, but they both provide squares of opposition. The aim of this paper is to analyse these two squares, to compare them with each other and with (...) the traditional square of opposition drawn by Apuleius and Boethius, with regard to their particular structure and their vertices and to evaluate their validity. We will show that both squares are different from each other and from the traditional square both with regard to their vertices, and to the arrangements of the oppositional relations. In addition, while al-Suhrawardī uses the notion of matter modalities to define the oppositions, al-Sanūsī relies rather on al-Rāzī’s and al-Khūnajī’s distinctions. Both authors agree, however, on the fact that affirmatives have an import while negatives do not. For this reason, their squares are valid from a logical viewpoint. As diagrams, these squares are nevertheless entirely original, due to the particular arrangements of their oppositional relations. Our analysis shows also that the later author does not seem to know the square drawn by the earlier one and that both authors do not seem to know Apuleius’ or Boethius’ squares. (shrink)

Papers... "selected from a larger number of contributions most of them based on talks presented at the First World Congress on the Square of Opposition organized in Montreux in June 2007"--Preface, p. 12.

In this article, I offer the notion of “skeptimentality” as a framework for thinking about the strikingly transmuted character of the noble moral sentiments (sympathy, empathy, benevolence, compassion, care, and pity) in the privilege-sensitive public culture of contemporary Scandinavia. Skeptimentality is my term for the sense that there is something morally embarrassing about the moral sentiments. I bring into play insights from feminist studies of sentimental sympathy as mediating factor in gender, race, and class-relations in order to highlight the extent (...) to which skeptimentality differs from sentimentality and the aesthetic of sympathy we associate with it. The latter part of the essay develops further the notion of skeptimentality through an analysis of Swedish director Ruben Östlund’s award-winning feature film The Square (2017). As an aesthetic mode, skeptimentality has its own tropes, which often take an explicitly critical position on sentimental ones; it is, I submit, an aesthetic not of sympathy, but of complicity. (shrink)

Nation states in the twenty‐first century confront new challenges to their political legitimacy. Borders are more porous and less secure. Infectious disease epidemics, climate change, financial fraud, terrorism, and cybersecurity all involve cross‐border flows of material, human bodies, and information that threaten to overwhelm state power and expert knowledge. Concurrently, doubts have multiplied about whether citizens, subject to manipulation through the internet, have lost the critical capacity to hold rulers accountable for their expert decisions. I argue that the primary threat (...) to democracy is not the public's epistemic incompetence but a slow dissolution of the deliberative practices that are essential for self‐rule. We need a radical reimagining of the sites, forms, and performances of democratic deliberation. For this purpose, the American state needs to reconstitute a public square open to citizens who are deemed to be epistemically competent and capable of informed judgment. (shrink)

This paper investigates the principles □λ,δta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{{\rm ta}}}_{\lambda,\delta}}$$\end{document}, weakenings of □λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_\lambda}$$\end{document} which allow δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta}$$\end{document} many clubs at each level but require them to agree on a tail-end. First, we prove that □λ,<ωta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{\rm {ta}}}_{\lambda,< \omega}}$$\end{document} implies □λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...) \begin{document}$${\square_\lambda}$$\end{document}. Then, by forcing from a model with a measurable cardinal, we show that □λ,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\lambda,2}}$$\end{document} does not imply □λ,δta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{\rm{ta}}}_{\lambda,\delta}}$$\end{document} for regular λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document}, and □δ+,δta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{\rm{ta}}}_{\delta^+,\delta}}$$\end{document} does not imply □δ+,<δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\delta^+,< \delta}}$$\end{document}. With a supercompact cardinal the former result can be extended to singular λ, and the latter can be improved to show that □λ,δta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square^{{\rm {ta}}}_{\lambda,\delta}}$$\end{document} does not imply □λ,<δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\lambda,< \delta}}$$\end{document} for δ<λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta < \lambda}$$\end{document}. (shrink)

During the last decades, the psychology of reasoning has identified experimentally many fallacies committed by spontaneous reasoners. Given these experimental results, some theories have been developed about this phenomenon, mainly algorithmic theories. This paper develops instead a computational modelling of these current fallacies which appear as simplifications in the treatment of information that do not respect the formal rules of classical propositional logic. These fallacies are explained as crushes in the Klein group structure and so, in squares of opposition. These (...) crushes are an effect of not to take into account the dual of the binary operator at work in the major premise of the inference. This analysis allows predictions on fallacies not identified before, which are fallacies produced when reasoning with incompatibilities. The paper concludes with consequences of the analysis on pedagogical strategies for the teaching of logic. (shrink)

Walt Disney is largely responsible for popularizing the princess story in American culture. These stories are the centerpieces of the Disney collection and their flagship theme parks. Indeed, Cinderella's castle itself is at the heart of Disney's Magic Kingdom. The first of Disney's theme parks, the Magic Kingdom was intended to capture the magic and imagination of the Disney movies, and bring to life the settings of Disney stories. Epcot was the second of four parks built at the Walt Disney (...) World Resort location in Florida. While Liberty Square in the Magic Kingdom celebrates the independence of those who live in liberal democracies, the World Showcase at Epcot celebrates multiculturalism, and the possibility of appreciating and learning from a multitude of cultures. The chapter provides a better understanding of the philosophical tensions in Disney's theme park by taking a tour of the underlying political ideas lurking inside the Epcot and Magic Kingdom theme parks. (shrink)

We show that MRP + MA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . This generalizes a result by Weiß who showed that PFA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA (...) with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω 2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω 1 ) for all λ ≥ ω 2. (shrink)

Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof and subsequent work with Sharon, he isolated two reflection principles, CP and S, which, under certain circumstances, are satisfied by all covering matrices of a certain shape. Using square sequences, we construct covering matrices for which CP and S fail. This leads naturally to an investigation of square principles intermediate between □κ and □ for a regular cardinal κ. We provide (...) a detailed picture of the implications between these square principles. (shrink)

We are pleased to present this special issue of the journal History and Philosophy of Logic dedicated to the square of opposition.The square of opposition is a diagram and a theory of opposition re...

This entry traces the historical development of the Square of Opposition, a collection of logical relationships traditionally embodied in a square diagram. This body of doctrine provided a foundation for work in logic for over two millenia. For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms. Certain additional (...) principles ("contraposition" and "obversion") were sometimes adopted along with the Square, and they genuinely yielded inconsistency. By the nineteenth century an inconsistent set of doctrines was widely adopted. Strawson's 1952 attempt to rehabilitate the Square does not apply to the traditional doctrine; it does salvage the nineteenth century version but at the cost of yielding inferences that lead from truth to falsity when strung together. (shrink)

This essay explores the validity of Gregory Boyd’s open theistic account of the nature of the future. In particular, it is an investigation into whether Boyd’s logical square of opposition for future contingents provides a model of reality for free will theists that can preserve both bivalence and a classical conception of omniscience. In what follows, I argue that it can.

Wolfgang Stegmüller, the leading German philosopher of science, considers the status of scientific revolutions the central issue in the field ever since "the famous Popper-Lakatos-Kuhn discussion" of a decade and a half ago, comments on "almost all contributions to this problem", and offers his alternative solutions in a series of papers culminating with, and summarized in, his recent "A Combined Approach to Dynamics of Theories. How To Improve Historical Interpretations of Theory Change By Applying Set Theoretical Structures", published in Gerard (...) Radnitzky and Gunnar Andersson, editors, The Structure and Development of Science, issued in the Boston Studies in the Philosophy of Science series, volume 59, 1979, pp. 151-86. Popper views scientific revolutions as rational and due to empirical refutations, but there are no refutations in science. Lakatos agrees and assumes that research programs are refutable and their replacements to be revolutions, but the same arguments he launches against Popper apply to him; moreover, applying his philosophy to itself makes it collapse anyway . Kuhn's view was interpreted to be one of scientific revolutions as quite irrational and as arbitrary as mob action. Stegmüller presents revolution in another interpretation of Kuhn - as non-rational, as based on hopes and value judgment but not on facts. He thinks there are big and small revolutions. And he uses his own modifications of J. D. Sneed's famous formal analysis of scientific theory to make his point. After presenting a summary of Stegmüller's ideas in our own way, which seems to us a clarification of presentation with no change of content, especially due to our stressing all differences of opinion, we apply Stegmüller's idea to itself, the way Stegmüller has done with the view of Lakatos, and with similar results. (shrink)

In this paper I propose a set-theoretical interpretation of the logical square of opposition, in the perspective opened by generalized quantifier theory. Generalized quantifiers allow us to account for the semantics of quantificational Noun Phrases, and of other natural language expressions, in a coherent and uniform way. I suggest that in the analysis of the meaning of Noun Phrases and Determiners the square of opposition may help representing some semantic features responsible to different logical properties of these expressions. (...) I will conclude with some consideration on scope interactions between quantifiers. (shrink)