The first part of the paper aims at showing that the notion of an Aristotelian square may be seen as a special case of a variety of different more general notions: the one of a subAristotelian square, the one of a semiAristotelian square, the one of an Aristotelian cube, which is a construction made up of six semiAristotelian squares, two of which are Aristotelian. Furthermore, if the standard Aristotelian square is seen as a special ordered 4-tuple of formulas, there (...) are 4-tuples describing rotations of the original square which are non-standard Aristotelian squares. The second part of the paper focuses on the notion of a composition of squares. After a discussion of possible alternative definitions, a privileged notion of composition of squares is identified, thus opening the road to introducing and discussing the wider notion of composition of cubes. (shrink)
We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two kinds of (...) import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle’s system appears then as a fragment of a broader system which can be developed by using FOL. (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)
The aim of this paper is to elucidate the relationship between Aristotelian conceptual oppositions, commutative diagrams of relational structures, and Galois connections.This is done by investigating in detail some examples of Aristotelian conceptual oppositions arising from topological spaces and similarity structures. The main technical device for this endeavor is the notion of Galois connections of order structures.
With this paper I aim to demonstrate that a look beyond the Aristotelian square of opposition, and a related non-conservative view on logical determiners, contributes to both the understanding of Aristotelian syllogistics as well as to the study of quantificational structures in natural language.
After giving a short summary of the traditional theory of the syllogism, it is shown how the square of opposition reappears in the much more powerful concept logic of Leibniz. Within Leibniz’s algebra of concepts, the categorical forms are formalized straightforwardly by means of the relation of concept-containment plus the operator of concept-negation as ‘S contains P’ and ‘S contains Not-P’, ‘S doesn’t contain P’ and ‘S doesn’t contain Not-P’, respectively. Next we consider Leibniz’s version of the so-called Quantification of (...) the Predicate which consists in the introduction of four additional forms ‘Every S is every P’, ‘Some S is every P’, ‘Every S isn’t some P’, and ‘Some S isn’t some P’. Given the logical interpretation suggested by Leibniz, these unorthodox propositions also form a Square of Opposition which, when added to the traditional Square, yields a “Cube of Opposition”. Finally it is shown that besides the categorical forms, also the non-categorical forms can be formalized within an extension of Leibniz’s logic where “indefinite concepts” X, Y, Z\ function as quantifiers and where individual concepts are introduced as maximally consistent concepts. (shrink)
In 1995 Slater argued both against Priest’s paraconsistent system LP (1979) and against paraconsistency in general, invoking the fundamental opposition relations ruling the classical logical square. Around 2002 Béziau constructed a double defence of paraconsistency (logical and philosophical), relying, in its philosophical part, on Sesmat’s (1951) and Blanche’s (1953) “logical hexagon”, a geometrical, conservative extension of the logical square, and proposing a new (tridimensional) “solid of opposition”, meant to shed new light on the point raised by Slater. By using n-opposition (...) theory (NOT) we analyse Beziau’s anti-Slater move and show both its right intuitions and its technical limits. Moreover, we suggest that Slater’s criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics. This last criticism has been addressed by Malinowski (1990) and Shramko and Wansing (2005), who developed a family of tenable logical counter-examples to it: trans-Suszkian systems are radically many-valued. This family of new logics has some strange logical features, essentially: each system has more than one consequence operator. We show that a new, deeper part of the aforementioned geometry of logical oppositions (NOT), the “logical poly-simplexes of dimension m”, generates new logical-geometrical structures, essentially many-valued, which could be a very natural (and intuitive) geometrical counterpart to the “strange”, new, non-Suszkian logics of Malinowski, Shramko and Wansing. By a similar move, the geometry of opposition therefore sheds light both on the foundations of paraconsistent logics and on those of many-valued logics. (shrink)
Although logical consistency is desirable in scientific research, standard statistical hypothesis tests are typically logically inconsistent. To address this issue, previous work introduced agnostic hypothesis tests and proved that they can be logically consistent while retaining statistical optimality properties. This article characterizes the credal modalities in agnostic hypothesis tests and uses the hexagon of oppositions to explain the logical relations between these modalities. Geometric solids that are composed of hexagons of oppositions illustrate the conditions for these modalities to (...) be logically consistent. Prisms composed of hexagons of oppositions show how the credal modalities obtained from two agnostic tests vary according to their threshold values. Nested hexagons of oppositions summarize logical relations between the credal modalities in these tests and prove new relations. (shrink)
A general theory of logical oppositions is proposed by abstracting these from the Aristotelian background of quantified sentences. Opposition is a relation that goes beyond incompatibility (not being true together), and a question-answer semantics is devised to investigate the features of oppositions and opposites within a functional calculus. Finally, several theoretical problems about its applicability are considered.
So called “shapes of opposition”—like the classical square of opposition and its extensions—can be seen as graphical representations of the ways in which types of statements constrain each other in their possible truth values. As such, they can be used as a novel way of analysing the subject matter of disputes. While there have been great refinements and extensions of this logico-topological tool in the last years, the broad range of shapes of opposition are not widely known outside of a (...) circle of specialists. This ignorance may lead to the presumption that the classical square of opposition fits all disputes. A broader view, which takes expanded shapes of opposition into account, may come to a more nuanced appraisal of possible disputes. Once we take other shapes of opposition into account, some alleged disputes may turn out to be Scheindisputes. In order to do the wide range of linguistic expressions justice and to differentiate Scheindisputes from real ones, a broader view is advised. To illustrate this point, I discuss the notion of “introspective disputes”. These are commonly reconstructed as obeying the square, but are more aptly reconstructed with a more complex octagon. If we reconstruct these disputes based on Buridan’s octagon, it becomes obvious that “introspective disputes” are likely Scheindisputes. (shrink)
The solution attributed to Archytas for the problem of doubling the cube is a landmark of the pre-Euclidean mathematics. This paper offers textual arguments for a new reading of the text of Archytas’ solution for doubling the cube, and an approach to the solution which fits closely with the new reading. The paper also reviews modern attempts to explain the text, which are as complicated as the original, and its connections with some xvi-century mathematical results, without any documented (...) relation to Archytas’ doubling the cube. (shrink)
Roy Bhaskar's Social Cube model based on critical realist philosophy has not been dealt with in theory of decision-making at any length, nor has it raised any notable debate in social theory in general. The model demonstrates that decision-making is regulated and transformed by a constantly evolving complexity of mechanisms emerging from physical, mental, material, human and social levels of reality. With the help of this device, Graham Allison's argument against the Rational Actor Model that decisions are not so (...) much acts of unitary decision-makers but more outputs of large organisations can be elaborated and converted into a statement that decisions are products of a complexity of knowledge-producing mechanisms. The openness of social systems revealed by the Social Cube therefore requires methodological pluralism, for example the application of psychological theories in order to explain political phenomena. By applying the Social Cube model and by drawing new empirical evidence from the Linda Melvern Rwanda Genocide Archive, this article demonstrates that the Somalia effect on the UN's failure in Rwanda was more complicated than the existing literature claims; it was “multi-layered”, generated not only by intentional political calculations, as previous studies argue, but also by a multiplicity of other mechanisms operative at various levels, particularly cognitive dissonance in organisational learning at the unintentional or subconscious level. (shrink)
Cube Living 221A is the most recent iteration of the Cube Living project, initiated in 2008. It appropriates the language, media and social practices of real estate development campaigns to engage in speculation about spatial ontologies, examining how social, legal and financial conventions determine the creation of space in our cities.This paper describes the staging and production process by which Cube Living 221A performs the creation of a spatial commodity. Drawing on the concepts presented in Žižek’s 2009 (...) talk “Architectural Parallax: Spandrels and Other Phenomena of Class Struggle”, it goes on to speculate that urban real estate development results in an architectural parallax: real estate property must function simultaneously as a financial asset and also as a residence. Similarly, the property owner must play simultaneous but irreconcilable roles as both investor and citizen. The Cube object can be regarded as a spandrel at the hinge of this parallax. (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.
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.
. This paper aims to highlight some peculiarities of the semiotic square, whose creation is due in particular to Greimas’ works. The starting point is the semiotic notion of complex term, which I regard as one of the main differences between Greimas’ square and Blanché’s hexagon. The remarks on the complex terms make room for a historical survey in Aristotle’s texts, where one can find the philosophical roots of the idea of middle term between two contraries and its relation to (...) notions such as difference, position and motion. In the Stagirite’s non-logical works, the theory of the intermediate, or middle term, represents an important link between opposition issues and ethics: this becomes a privileged perspective from which to reconsider the semiotic use of the square, i.e., its inclusion in the semio-narrative structures articulating the sense of texts. (shrink)
Universality of generalized Alexandroff's cube plays essential role in theory of absolute retracts for the category of , -closure spaces. Alexandroff's cube. is an , -closure space generated by the family of all complete filters. in a lattice of all subsets of a set of power .Condition P(, , ) says that is a closure space of all , -filters in the lattice ( ).
The main result of this paper were announced in Kosheleva — Kreinovich [7, 8]; for other algorithmic aspects of Hilbert's Third Problem see Kosheleva . The authors are greatly thankful to Alexandr D. Alexandrov , Vladimir G. Boltianskii and Patrick Suppes for valuable discussions, and to the anonymous referee for important suggestions. This work was partially supported by an NSF grant No. CDA-9015006.
In the paper  the following theorem is shown: Theorem (Th. 3,5, ), If =0 or = or , then a closure space X is an absolute extensor for the category of , -closure spaces iff a contraction of X is the closure space of all , -filters in an , -semidistributive lattice.In the case when = and =, this theorem becomes Scott's theorem.