In this paper it is argued that the theory of truth approximation should be pursued in the framework of some kind of geometry of logic. More specifically it is shown that the theory of interval structures provides a general framework for dealing with matters of truth approximation. The qualitative and the quantitative accounts of truthlikeness turn out to be special cases of the interval account. This suggests that there is no principled gap between the qualitative and quantitative approach. (...) Rather, there is a connected spectrum of ways of measuring truthlikeness depending on the specifics of the context in which it takes place. (shrink)

In order to measure the degree of dissimilarity between elements of a Boolean algebra, the author’s proposed to use pseudometrics satisfying generalizations of the usual axioms for identity. The proposal is extended, as far as is feasible, from Boolean algebras to Brouwerian algebras. The relation between Boolean and Brouwerian geometries of logic turns out to resemble in a curious way the relation between Euclidean and non-Euclidean geometries of physical space. The paper ends with a brief consideration of the problem (...) of the metrization of the algebra of theories. (shrink)

After sketching the historical development of “emergence” and noting several recent problems relating to “emergent properties”, this essay proposes that properties may be either “emergent” or “mergent” and either “intrinsic” or “extrinsic”. These two distinctions define four basic types of change: stagnation, permanence, flux, and evolution. To illustrate how emergence can operate in a purely logical system, the Geometry of Logic is introduced. This new method of analyzing conceptual systems involves the mapping of logical relations onto geometrical figures, (...) following either an analytic or a synthetic pattern (or both together). Evolution is portrayed as a form of discontinuous change characterized by emergent properties that take on an intrinsic quality with respect to the object(s) or proposition(s) involved. Causal leaps, not continuous development, characterize the evolution of human life in a developing foetus, of a thought out of certain brain states, of a new idea (or insight) out of ordinary thoughts, and of a great person out of a set of historical experiences. The tendency to assume that understanding evolutionary change requires a step-by-step explanation of the historical development that led to the appearance of a certain emergent property is thereby discredited. (shrink)

The paper is concerned with topological and geometrical characteristics of ultrafilter space which is widely employed in mathematical logic.Some philosophical applications are offeredtogether with visulisations that reveal the beauty of logical constructions.

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) (...) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

We establish the Robinson joint consistency theorem for the infinite-valued propositional logic of Łukasiewicz. As a corollary we easily obtain the amalgamation property for MV-algebras—the algebras of Łukasiewicz logic: all pre-existing proofs of this latter result make essential use of the Pierce amalgamation theorem for abelian lattice-ordered groups together with the categorical equivalence Γ between these groups and MV-algebras. Our main tools are elementary and geometric.

In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's (...) multiple conclusion systems for classical logic and Girard's proofnets for linear logic. (shrink)

Both Kant's architectonic and the Yijing can be structured as four perspectival levels: 0 + 4 + 12 + = 64. The first, unknowable level is unrepresentable. The geometry of logic provides well‐structured maps for levels two to four. Level two consists of four basic gua , corresponding to Kant's category‐headings . Level three's twelve gua, derived logically from the initial four, correspond to Kant's twelve categories. Level four correlates the remaining 48 gua to Kant's theory of the (...) four university faculties , and to four categorially organized domains comprising his philosophical system. (shrink)

Both Kant’s architectonic and the Yijing can be structured as four perspectival levels: 0 + 4 + 12 + = 64. The first, unknowable level is unrepresentable. The geometry of logic provides well-structured maps for levels two to four. Level two consists of four basic gua, corresponding to Kant’s category-headings. Level three’s twelve gua, derived logically from the initial four, correspond to Kant’s twelve categories. Level four correlates the remaining 48 gua to Kant’s theory of the four university (...) faculties, and to four categorially organized domains comprising his philosophical system. (shrink)

The words "analysis" and "synthesis" are among the most widely used and misused terms in the history of philosophy. They were originally used in geometrical reasoning during the age of Euclid to describe two opposing, but complementary, methods of arguing (roughly equivalent to deduction and induction). Since then philosophers have used them not only in this way, but also to refer to distinctions of various sorts between types of judgment or classes of propositions. To some they are regarded as defining (...) differences of kind, while others regard them as defining differences of degree. Moreover, they have been connected in numerous different ways with other distinctions, such as "a priori vs. a posteriori" or "necessary vs. contingent". Some philosophers have become so frustrated at the ambiguity attached to the various uses of the terms "analytic" and "synthetic" that they have given up all hope of assigning a coherent meaning to this distinction. (shrink)

In this paper we provide an analysis of the logical relations within the conceptual or lexical field of angles in 2D geometry. The basic tripartition into acute/right/obtuse angles is extended in two steps: first zero and straight angles are added, and secondly reflex and full angles are added, in both cases extending the logical space of angles. Within the framework of logical geometry, the resulting partitions of these logical spaces yield bitstring semantics of increasing complexity. These bitstring analyses (...) allow a straightforward account of the Aristotelian relations between angular concepts. In addition, also relational concepts such as complementary and supplementary angles receive a natural bitstring analysis. (shrink)

The Fifth International Congress of Logic, Methodology and Philosophy of Science was held at the University of Western Ontario, London, Canada, 27 August to 2 September 1975. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, and was sponsored by the National Research Council of Canada and the University of Western Ontario. As those associated closely with the work of the Division over (...) the years know well, the work undertaken by its members varies greatly and spans a number of fields not always obviously related. In addition, the volume of work done by first rate scholars and scientists in the various fields of the Division has risen enormously. For these and related reasons it seemed to the editors chosen by the Divisional officers that the usual format of publishing the proceedings of the Congress be abandoned in favour of a somewhat more flexible, and hopefully acceptable, method of pre sentation. Accordingly, the work of the invited participants to the Congress has been divided into four volumes appearing in the University of Western Ontario Series in Philosophy of Science. The volumes are entitled, Logic, Foundations of Mathematics and Computability Theory, Foun dational Problems in the Special Sciences, Basic Problems in Methodol ogy and Linguistics, and Historical and Philosophical Dimensions of Logic, Methodology and Philosophy of Science. (shrink)

The main result of this paper were announced in Kosheleva — Kreinovich [7, 8]; for other algorithmic aspects of Hilbert's Third Problem see Kosheleva [6]. 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.

Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let $M \models T$ be uncountable and saturated, H = p(M). We say $D \subset H$ is locally modular if for all $X, Y \subset D$ with $X = \operatorname{acl}(X) \cap D, Y = \operatorname{acl}(Y) \cap D$ and $X \cap Y \neq \varnothing$ , dim(X ∪ Y) + dim(X ∩ Y) = dim(X) + dim(Y). Theorem 1. Let p ∈ S(A) be weakly (...) minimal and D the realizations of $\operatorname{stp}(a/A)$ for some a realizing p. Then D is locally modular or p has Morley rank 1. Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all $a \in H\backslash\operatorname{acl}(A), b \in G\operatorname{acl}(A)$ there are a' ∈ H, b' ∈ G such that $a' \in \operatorname{acl}(abb' A)\backslash\operatorname{acl}(aA)$ . Similarly when H and G are the realizations of complete types or strong types over A. (shrink)

We investigate the geometry of forking for SU-rank 2 elements in supersimple ω-categorical theories and prove stable forking and some structural properties for such elements. We extend this analysis to the case of SU-rank 3 elements.

The most widely used attractive logical account of knowledge uses standard epistemic models, i.e., graphs whose edges are indistinguishability relations for agents. In this paper, we discuss more general topological models for a multi-agent epistemic language, whose main uses so far have been in reasoning about space. We show that this more geometrical perspective affords greater powers of distinction in the study of common knowledge, defining new collective agents, and merging information for groups of agents.

Customary interpretations state that Tractarian thoughts are pictures, and, a fortiori, facts. I argue that important difficulties are unavoidable if we assume this standard view, and I propose a reading of the concept taking advantage of an analogy that Wittgenstein introduces, namely, the analogy between thoughts and projective geometry. I claim that thoughts should be understood neither as pictures nor as facts, but as acts of geometric projection in logical space. The interpretation I propose thus removes the root of (...) the identified difficulties. Moreover, it allows important clarification concerning some central aspects of the Tractarian theory of representation, and it yields a unifying elucidation regarding Wittgenstein’s remarks on the solipsistic thesis. (shrink)

Following ideas elaborated by Hering in his celebrated analysis of color, the psychologist and gestalt theorist Otto Selz developed in the 1930s a theory of “natural space”, i.e., space as it is conceived by us. Selz’s thesis is that the geometric laws of natural space describe how the points of this space are related to each other by directions which are ordered in the same way as the points on a sphere. At the end of one of his articles, Selz (...) tries to derive within his framework two of Hilbert’s axioms for Euclidean geometry. Such derivations would, according to Selz, disclose the psychological origin and meaning of the geometric axioms and would thus contribute to a clarification of their epistemological status. In the present article Selz’s theory is explained and analyzed, his basic principles are amended, and implicit assumptions are made explicit. It is shown that the resulting system is one of ordered affine geometry in which all of Hilbert’s axioms except the axioms of congruence and those of continuity are derivable. The primary aim of the present paper is to make explicit the basic principles behind Selz’s geometry of natural space. The question of the logical independence of these principle is not investigated. (shrink)

The concept of geometry may evoke a world of pure platonic shapes, such as spheres and cubes, but a deeper understanding of visual experience demands insight into the perceptual organization of naturalistic form. Japanese gardens excel as designed environments where the complex fractal geometry of nature has been simplified to a structural core that retains the essential properties of the natural landscape, thereby presenting an ideal opportunity for investigating the geometry and perceptual significance of such naturalistic characteristics. (...) Here, fronto-parallel perspective, asymmetrical structuring of the ground plane, spatial arrangement of garden elements, tuning of textural qualities and choice of naturalistic form, are presented as a set of physical features that facilitate a systematic analysis of Japanese garden design per se, as well as the geometry of the particular naturalistic features that it aims to enhance. Comparison with Western landscape design before and after contact between Western and Eastern hemispheres illustrates the degree of naturalness achieved in the Japanese garden, and suggests how classical Western landscape design generally differs in this regard. It further reveals how modern Western gardens culminate in a different naturalistic geometry, thus also a distinctly different vision of the natural landscape, even if these designs were greatly influenced by the gardens of China and Japan. (shrink)

There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are “the other way around”). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is understoood as (...) involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators—Post’s and Kleene’s—and we provide a simple group-theoretic argument demonstrating their generative power. (shrink)

The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.

Fix an algebraically closed field of characteristic zero and let G be its geometry of transcendence degree one extensions. Let X be a set of points of G. We show that X extends to a projective subgeometry of G exactly if the partial derivatives of the polynomials inducing dependence on its elements satisfy certain separability conditions. This analysis produces a concrete representation of the coordinatizing fields of maximal projective subgeometries of G.

We investigate the isomorphism types of combinatorial geometries arising from Hrushovski's flat strongly minimal structures and answer some questions from Hrushovski's original paper.

Assume T is a superstable theory with $ countable models. We prove that any *-algebraic type of M-rank > 0 is m-nonorthogonal to a *-algebraic type of M-rank 1. We study the geometry induced by m-dependence on a *-algebraic type p* of M-rank 1. We prove that after some localization this geometry becomes projective over a division ring F. Associated with p* is a meager type p. We prove that p is determined by p* up to nonorthogonality and (...) that F underlies also the geometry induced by forking dependence on any stationarization of p. Also we study some *-algebraic *-groups of M-rank 1 and prove that any *-algebraic *-group of M-rank 1 is abelian-by-finite. (shrink)

An intermediate stage in Hrushovski’s construction of flat strongly minimal structures in a relational language L produces ω-stable structures of rank ω. We analyze the pregeometries given by forking on the regular type of rank ω in these structures. We show that varying L can affect the isomorphism type of the pregeometry, but not its finite subpregeometries. A sequel will compare these to the pregeometries of the strongly minimal structures.

In this paper, we are concerned with the development of a general set theory using the single axiom version of Leśniewski’s mereology. The specification of mereology, and further of Tarski’s geometry of solids will rely on the Calculus of Inductive Constructions (CIC). In the first part, we provide a specification of Leśniewski’s mereology as a model for an atomless Boolean algebra using Clay’s ideas. In the second part, we interpret Leśniewski’s mereology in monadic second-order logic using names and (...) develop a full version of mereology referred to as CIC-based Monadic Mereology (λ-MM) allowing an expressive theory while involving only two axioms. In the third part, we propose a modeling of Tarski’s solid geometry relying on λ-MM. It is intended to serve as a basis for spatial reasoning. All parts have been proved using a translation in type theory. (shrink)

This note extends earlier results on geometrical interpretations of the logic KR to prove some additional results, including a simple undecidability proof for the four-variable fragment of KR.

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 recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.

In this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution Tarski's Postulate 4 (...) can be omitted, together with its versions 4' and 4". We also prove that the equivalence of postulates 4, 4' and 4" is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms. We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory. Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski. (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)

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.