Refined common knowledge logics or logics of common information

 In terms of formal deductive systems and multi-dimensional Kripke frames we study logical operations know, informed, common knowledge and common information. Based on [6] we introduce formal axiomatic systems for common information logics and prove that these systems are sound and complete. Analyzing the common information operation we show that it can be understood as greatest open fixed points for knowledge formulas. Using obtained results we explore monotonicity, omniscience problem, and inward monotonocity, describe their connections and give dividing examples. Also we find algorithms recognizing these properties for some particular cases. Received: 21 October 2000 / Published online: 2 September 2002 Key words or phrases: Multi-agent systems – Non-standard logic – Knowledge representation – Common knowledge – Common information – Fixed points, Kripke models – Modal logic     

“Setting” n-Opposition

Our aim is to show that translating the modal graphs of Moretti’s “n-opposition theory” (2004) into set theory by a suited device, through identifying logical modal formulas with appropriate subsets of a characteristic set, one can, in a constructive and exhaustive way, by means of a simple recurring combinatory, exhibit all so-called “logical bi-simplexes of dimension n” (or n-oppositional figures, that is the logical squares, logical hexagons, logical cubes, etc.) contained in the logic produced by any given modal graph (an exhaustiveness which was not possible before). In this paper we shall handle explicitly the classical case of the so-called 3(3)-modal graph (which is, among others, the one of S5), getting to a very elegant tetraicosahedronal geometrisation of this logic.      

From the Logical Square to Blanché’s Hexagon: Formalization, Applicability and the Idea of the Normative Structure of Thought

The square of opposition and many other geometrical logical figures have increasingly proven to be applicable to different fields of knowledge. This paper seeks to show how Blanché generalizes the classical theory of oppositions of propositions and extends it to the structure of opposition of concepts. Furthermore, it considers how Blanché restructures the Apuleian square by transforming it into a hexagon. After presenting G. Kalinowski??s formalization of Blanché??s hexagonal theory, an illustration of its applicability to mathematics, to modal logic, and to the logic of norms is depicted. The paper concludes by criticizing Blanché??s claim according to which, his logical hexagon can be considered as the objective basis of the structure of the organisation of concepts, and as the formal structure of thought in general. It is maintained that within the frame of diagrammatic reasoning Blanché??s hexagon keeps its privileged place as a ??nice?? and useful tool, but not necessarily as a norm of thought.     

Logical omniscience as infeasibility

Logical theories for representing knowledge are often plagued by the so-called Logical Omniscience Problem. The problem stems from the clash between the desire to model rational agents, which should be capable of simple logical inferences, and the fact that any logical inference, however complex, almost inevitably consists of inference steps that are simple enough. This contradiction points to the fruitlessness of trying to solve the Logical Omniscience Problem qualitatively if the rationality of agents is to be maintained. We provide a quantitative solution to the problem compatible with the two important facets of the reasoning agent: rationality and resource boundedness. More precisely, we provide a test for the logical omniscience problem in a given formal theory of knowledge. The quantitative measures we use are inspired by the complexity theory. We illustrate our framework with a number of examples ranging from the traditional implicit representation of knowledge in modal logic to the language of justification logic, which is capable of spelling out the internal inference process. We use these examples to divide representations of knowledge into logically omniscient and not logically omniscient, thus trying to determine how much information about the reasoning process needs to be present in a theory to avoid logical omniscience.     

On an assumption of geometric foundation of numbers

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying.     

Polymodal Lattices and Polymodal Logic

A polymodal lattice is a distributive lattice carrying an n-place operator preserving top elements and certain finite meets. After exploring some of the basic properties of such structures, we investigate their freely generated instances and apply the results to the corresponding logical systems — polymodal logics — which constitute natural generalizations of the usual systems of modal logic familiar from the literature. We conclude by formulating an extension of Kripke semantics to classical polymodal logic and proving soundness and completeness theorems. Mathematics Subject Classification: 03G10, 06D99, 03B45.     

Logic in representations of groups

Let K be a commutative and associative ring with unit. We consider representations of groups over K from the viewpoint of some logic. In particular, different logical invariants of representations, as well as relations between the various representations corresponding to the invariants, are studied. One of the basic relations is isotypeness. Here we use the concept of a type adopted in model theory. This paper adjoins [11], where similar results were derived for one-sorted algebras.     

On specifications,subset types and interpretation of proposition in type theory

In this paper we discuss some practical aspects of using type theory as a programming and specification language, where the viewpoint is to use it not only as a basis for program synthesis but also as a programming language with a programming logic allowing us to do ordinary verification.The subset type has been added to type theory in order to avoid irrelevant information in programs. We give an example of a proof which illustrates the problems that may occur if the subset type is used in specifications when we have the standard interpretation of propositions as types. Harrop-formulas and Squash are then discussed as solutions to these problems. It is argued that they are not acceptable from a practical point of view.An extension of the theory to include the two new judgment forms:A is a proposition, andA is true, is then given and explained in terms of the old theory. The logical constants are no longer identified with the corresponding type theoretical constants, but propositions are interpreted as Gödel formulas, which allow us to introduce and justify logical rules similar to rules for classical logic. The interpretation is extended to include predicates defined by using reflections of the ordinary definition of Gödel formulas in a type of small propositions.The programming example is then revisited and stronger elimination rules are discussed.     

On arithmetic complexity of certain constructive logics

A constructive arithmetical theory is an arbitrary set of closed arithmetical formulas that is closed with respect to derivability in an intuitionsitic arithmetic with the Markov principle and the formal Church thesis. For each arithmetical theory T there is a corresponding logic L(T) consisting of closed predicate formulas in which all arithmetic instances belong to T. For so-called internally enumerable constructive arithmetical theories with the property of existentiality, it is proved that the logic L(T) is II₁ ^T -@#@ complete. This implies, for example, that the logic of traditional constructivism is II₂ ⁰-complete.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 94–104, July, 1992.     

The Geometry of Standard Deontic Logic

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).      

Logical Extensions of Aristotle’s Square

We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle’s square under its modal form has the following four vertices: A is □α, E is , I is and O is , where α is a logical formula and □ is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether is involutive or not) modal logic. [3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of by the logical operations , under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle’s square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures.      

Why the Logical Hexagon?

The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the ??logical square?? (or ??square of opposition??), of which it is a much more general and powerful ??relative??. The discovery of the former did not raise interest, neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, ??sentenced to death?? by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with ??oppositions??) has appeared, ??oppositional geometry?? (also called ??n-opposition theory??, ??NOT??), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of ??logical bi-simplexes of dimension m??, itself just one term of the more general infinite series (of series) of the ??logical poly-simplexes of dimension m??. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of ??hybrid logical hexagon??, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change.     

New Logics for Quantum Non-individuals?

According to a very widespread interpretation of the metaphysical nature of quantum entities—the so-called Received View on quantum non-individuality—, quantum entities are non-individuals. Still according to this understanding, non-individuals are entities for which identity is restricted or else does not apply at all. As a consequence, it is said, such approach to quantum mechanics would require that classical logic be revised, given that it is somehow committed with the unrestricted validity of identity. In this paper we examine the arguments to the inadequacy of classical logic to deal with non-individuals, as previously defined, and argue that they fail to make a good case for logical revision. In fact, classical logic may accommodate non-individuals in that specific sense too. What is more pressing for the Received View, it seems, is not a revision of logic, but rather a more adequate metaphysical characterization of non-individuals.     

Omitting types theorem in hybrid dynamic first-order logic with rigid symbols

In the present contribution, we prove an Omitting Types Theorem (OTT) for an arbitrary fragment of hybrid dynamic first-order logic with rigid symbols (i.e. symbols with fixed interpretations across worlds) closed under negation and retrieve. The logical framework can be regarded as a parameter and it is instantiated by some well-known hybrid and/or dynamic logics from the literature. We develop a forcing technique and then we study a forcing property based on local satisfiability, which lead to a refined proof of the OTT. For uncountable signatures, the result requires compactness, while for countable signatures, compactness is not necessary. We apply the OTT to obtain upwards and downwards Löwenheim-Skolem theorems for our logic, as well as a completeness theorem for its constructor-based variant.     

N.A. Vasil’ev’s Logical Ideas and the Categorical Semantics of Many-Valued Logic

Here we suggest a formal using of N.A. Vasil’ev’s logical ideas in categorical logic: the idea of “accidental” assertion is formalized with topoi and the idea of the notion of nonclassical negation, that is not based on incompatibility, is formalized in special cases of monoidal categories. For these cases, the variant of the law of “excluded n-th” suggested by Vasil’ev instead of the tertium non datur is obtained in some special cases of these categories. The paraconsistent law suggested by Vasil’ev is also demonstrated with linear and tensor logics but in a form weaker than he supposed. As we have, in fact, many truth-values in linear logic and topos logic, the admissibility of the traditional notion of inference in the categorical interpretation of linear and intuitionistic proof theory is discussed.     

Theories and Theories of Truth

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.     

NP-containment for the coherence test of assessments of conditional probability: a fuzzy logical approach

In this paper we investigate the problem of testing the coherence of an assessment of conditional probability following a purely logical setting. In particular we will prove that the coherence of an assessment of conditional probability χ can be characterized by means of the logical consistency of a suitable theory T _χ defined on the modal-fuzzy logic FP _k (RŁ_Δ) built up over the many-valued logic RŁ_Δ. Such modal-fuzzy logic was previously introduced in Flaminio (Lecture Notes in Computer Science, vol. 3571, 2005) in order to treat conditional probability by means of a list of simple probabilities following the well known (smart) ideas exposed by Halpern (Proceedings of the eighth conference on theoretical aspects of rationality and knowledge, pp 17–30, 2001) and by Coletti and Scozzafava (Trends Logic 15, 2002). Roughly speaking, such logic is obtained by adding to the language of RŁ_Δ a list of k modalities for “probably” and axioms reflecting the properties of simple probability measures. Moreover we prove that the satisfiability problem for modal formulas of FP _k (RŁ_Δ) is NP-complete. Finally, as main result of this paper, we prove FP _k (RŁ_Δ) in order to prove that the problem of establishing the coherence of rational assessments of conditional probability is NP-complete.      

Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices

Formal concept analysis (FCA) associates a binary relation between a set of objects and a set of properties to a lattice of formal concepts defined through a Galois connection. This relation is called a formal context, and a formal concept is then defined by a pair made of a subset of objects and a subset of properties that are put in mutual correspondence by the connection. Several fuzzy logic approaches have been proposed for inducing fuzzy formal concepts from L-contexts based on antitone L-Galois connections. Besides, a possibility-theoretic reading of FCA which has been recently proposed allows us to consider four derivation powerset operators, namely sufficiency, possibility, necessity and dual sufficiency (rather than one in standard FCA). Classically, fuzzy FCA uses a residuated algebra for maintaining the closure property of the composition of sufficiency operators. In this paper, we enlarge this framework and provide sound minimal requirements of a fuzzy algebra w.r.t. the closure and opening properties of antitone L-Galois connections as well as the closure and opening properties of isotone L-Galois connections. We apply these results to particular compositions of the four derivation operators. We also give some noticeable properties which may be useful for building the corresponding associated lattices.     

A common generalization for MV-algebras and Łukasiewicz–Moisil algebras

We introduce the notion of n-nuanced MV-algebra by performing a Łukasiewicz–Moisil nuancing construction on top of MV-algebras. These structures extend both MV-algebras and Łukasiewicz–Moisil algebras, thus unifying two important types of structures in the algebra of logic. On a logical level, n-nuanced MV-algebras amalgamate two distinct approaches to many valuedness: that of the infinitely valued Łukasiewicz logic, more related in spirit to the fuzzy approach, and that of Moisil n-nuanced logic, which is more concerned with nuances of truth rather than truth degree. We study n-nuanced MV-algebras mainly from the algebraic and categorical points of view, and also consider some basic model-theoretic aspects. The relationship with a suitable notion of n-nuanced ordered group via an extension of the Γ construction is also analyzed.     

Logics that define their own semantics

The capability of logical systems to express their own satisfaction relation is a key issue in mathematical logic. Our notion of self definability is based on encodings of pairs of the type (structure, formula) into single structures wherein the two components can be clearly distinguished. Hence, the ambiguity between structures and formulas, forming the basis for many classical results, is avoided. We restrict ourselves to countable, regular, logics over finite vocabularies. Our main theorem states that self definability, in this framework, is equivalent to the existence of complete problems under quantifier free reductions. Whereas this holds true for arbitrary structures, we focus on examples from Finite Model Theory. Here, the theorem sheds a new light on nesting hierarchies for certain generalized quantifiers. They can be interpreted as failure of self definability in the according extensions of first order logic. As a further application we study the possibility of the existence of recursive logics for PTIME. We restate a result of Dawar concluding from recursive logics to complete problems. We show that for the model checking Turing machines associated with a recursive logic, it makes no difference whether or not they may use built in clocks. Received: 7 February 1997