In a previous study, using experimental metapopulations of the flour beetle, Tribolium castaneum, we investigated phase III of Wright's shifting balance process (Wade and Griesemer 1998). We experimentally modeled migration of varying amounts from demes of high mean fitness into demes of lower mean fitness (as in Wright's characterization of phase III) as well as the reciprocal (the opposite of phase III). We estimated the meta-populational heritability for this level of selection by regression of offspring deme means on the weighted (...) parental deme means.Here we develop a Punnett Square representation of the inheritance of the group mean to place our empirical findings in a conceptual context similar to Mendelian inheritance of individual traits. The comparison of Punnett Squares for individual and group inheritance shows how the latter concept can be rigorously defined and extended despite the lack of explicitly formulated, simple Mendelian laws of inheritance at the group level. Whereas Wright's phase III combines both interdemic selection and meta-populational inheritance, our formulation separates the issue of meta-populational heritability from that of interdemic selection. We use this conceptual context to discuss the controversies over the levels of selection and the units of inheritance. (shrink)

A square tabular array was introduced by R. C. Punnett in (1907) to visualize systematically and economically the combination of gametes to make genotypes according to Mendel’s theory. This mode of representation evolved and rapidly became standardized as the canonical way of representing like problems in genetics. Its advantages over other contemporary methods are discussed, as are ways in which it evolved to increase its power and efficiency, and responded to changing theoretical perspectives. It provided a natural visual decomposition (...) of a complex problem into a number of inter-related stages. This explains its computational and conceptual power, for one could simply “read off” answers to a wide variety of questions simply from the “right” visual representation of the problem, and represent multiple problems, and multiple layers of problems in the same diagram. I relate it to prior work on the evolution of Weismann diagrams by Griesemer and Wimsatt (What Philosophy of Biology Is, Martinus-Nijhoff, the Hague, 1989), and discuss a crucial change in how it was interpreted that midwifed its success. (shrink)

The origin and development of Punnett’s Square for the enumeration and display of genotypes arising in a cross in Mendelian genetics is described. Due to R. C. Punnett, the idea evolved through the work of the ‘Cambridge geneticists’, including Punnett’s colleagues William Bateson, E. R. Saunders and R. H. Lock, soon after the rediscovery of Mendel’s paper in 1900. These geneticists were thoroughly familiar with Mendel’s paper, which itself contained a similar square diagram. A previously-unpublished three-factor diagram (...) by Sir Francis Galton existing in the Bateson correspondence in Cambridge University Library is then described. Finally the connection between Punnett’s Square and Venn Diagrams is emphasized, and it is pointed out that Punnett, Lock and John Venn overlapped as Fellows of Gonville and Caius College, Cambridge. Copious illustrations are given. (shrink)

This is a collection of new investigations and discoveries on the theory of opposition (square, hexagon, octagon, polyhedra of opposition) by the best specialists from all over the world. The papers range from historical considerations to new mathematical developments of the theory of opposition including applications to theology, theory of argumentation and metalogic.

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)

In logic, Aristotelian diagrams are almost always assumed to be closed under negation, and are thus highly symmetric in nature. In linguistics, by contrast, these diagrams are used to study lexicalization, which is notoriously not closed under negation, thus yielding more asymmetric diagrams. This paper studies the interplay between logical symmetry and linguistic asymmetry in Aristotelian diagrams. I discuss two major symmetric Aristotelian diagrams, viz. the square and the hexagon of opposition, and show how linguistic considerations yield various asymmetric versions (...) of these diagrams. I then discuss a pentagon of opposition, which occupies an uneasy position between the square and the hexagon. Although this pentagon belongs neither to the symmetric realm of logic nor to the asymmetric realm of linguistics, it occurs several times in the literature. The oldest known occurrence can be found in the cosmological work of the 14th-century author Nicole Oresme. (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)

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)

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)

The theory of the square of opposition has been worked out many centuries ago as a part of Aristotelian logic of terms.In spite of its inexactness (for instance it is not possible to decide whether the termsquare of opposition is a logical or a metalogical term) this theory is included without any changes in the usual elementary course of logic.The author defines the square of opposition in the language of the logic of propositions (see Def. 1.000) and derives from this (...) definition the usual laws of the square of opposition and several new theorems. (shrink)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then introduce (...) two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

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)

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 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.

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.

Early in the 20th century Bateson, Doncaster and Punnett formed a cooperative collective to share research findings on the chromosome theory of heredity (CTH). They cross-bred plants and animals to correlate behaviour of chromosomes and heredity of individual traits. Doncaster was the most enthusiastic proponent of the new theory and worked for years to convince Bateson and Punnett on its relevance to their own research. The two younger biologists collaborated with Bateson, the preeminent geneticist in England. As their (...) own reputations developed, their research findings allied with the consensus on the importance of the CTH by the broader scientific community. After Doncaster’s tragic death in 1920, major objections to the theory had been resolved; Bateson and Punnett then utilized the CTH to construct chromosome maps detailing locations of specific genes on particular chromosomes in several different species. The marriage of heredity and cytology enhanced confidence that the theory was an accurate mechanism to explain inheritance in both plants and animals. (shrink)

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.

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)

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.

We prove that every weakly square compact cardinal is a strong limit cardinal, and therefore weakly compact. We also study Aronszajn trees with no uncountable finitely splitting subtrees, characterizing them in terms of being Lindelöf with respect to a particular topology. We prove that the class of such trees is consistently non-empty and lies between the classes of Suslin and Aronszajn trees.

Russell derived many of his logical symbols from the pioneering notation of Giuseppe Peano. Principia Mathematica (1910–13) made these “Peanese” symbols (and others) famous. Here I focus on one of the more peculiar notational derivatives from Peano, namely, Principia ’s dual use of a squared dot or dots for both conjunction and scope. As Dirk Schlimm has noted, Peano always had circular dots and only used them to symbolize scope distinctions. In contrast, Principia has squared dots and conventions such that (...) some dots mark scope distinctions while others symbolize conjunction. How did this come to pass? In this paper I trace a genealogy of Principia ’s square dots back to Russell’s appropriation of Peano’s use of circular dots. Russell never explicitly justifies appropriating Peano’s notations to symbolize two distinct notions, but below I explain why Russell deployed Peano’s dot notations in this manner. Further, I argue that it was Cambridge University Press who squared the circular dots. (shrink)

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.

Review of “The Mad Square: Modernity in German Art 1910-37”, National Gallery of Victoria, Melbourne, Australia, November 25, 2011-March 4, 2012. A version of this essay appeared in the Appendices of Gavin Keeney, Not-I/Thou: The Other Subject of Art and Architecture (CSP, 2014), pp. 153-57.

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.

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)

The article expands the traditional system of concepts used in deontic logic, in order to allow the inclusion of supererogatory behaviour. This requires the development of a deontic decagon. In addition, it is shown how this decagon can be used to interpret deontic terms, e.g. in Islamic Law.

Can an appeal to the difference between contrary and contradictory statements, generated by a non-uniform behaviour of negation, deal adequately with paradoxical cases like the sorites or the liar? This paper offers a negative answer to the question. This is done by considering alternative ways of trying to construe and justify in a useful way (in this context) the distinction between contraries and contradictories by appealing to the behaviour of negation only. There are mainly two ways to try to do (...) so: i) by considering differences in the scope of negation, ii) by considering the possibility that negation is semantically ambiguous. Both alternatives are shown to be inapt to handle the problematic cases. In each case, it is shown that the available alternatives for motivating or grounding the distinction, in a way useful to deal with the paradoxes, are either inapplicable, or produce new versions of the paradoxes, or both. (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)

We study the partition relation $X@>{\rm w}>>[Y]_{p}^{2}$ that is a weakening of the usual partition relation $X\rightarrow [Y]_{p}^{2}$ . Our main result asserts that if κ is an uncountable strongly compact cardinal and $\germ{d}_{\kappa}\leq \lambda ^{<\kappa}$ , then $I_{\kappa,\lambda}^{+}@>{\rm w}>>[I_{\kappa,\lambda}^{+}]_{\lambda <\kappa}^{2}$ does not hold.

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.

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)

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)

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...