We consider the problem of finding, in the ambit of modal logic, a minimal characterization for finite Kripke frames, i.e., a formula which, given a frame, axiomatizes its theory employing the lowest possible number of variables and implies the other axiomatizations. We show that every finite transitive frame admits a minimal characterization over K4, and that this result can not be extended to K.

In the paper we analyse the problem of axiomatizing the minimal variant of discussive logic denoted as $${\textsf {D}}_{\textsf {0}}$$ D 0. Our aim is to give its axiomatization that would correspond to a known axiomatization of the original discussive logic $${\textsf {D}}_{\textsf {2}}$$ D 2. The considered system is minimal in a class of discussive logics. It is defined similarly, as Jaśkowski’s logic $${\textsf {D}}_{\textsf {2}}$$ D 2 but with the help of (...) the deontic normal logic $$\textbf{D}$$ D. Although we focus on the smallest discussive logic and its correspondence to $${\textsf {D}}_{\textsf {2}}$$ D 2, we analyse to some extent also its formal aspects, in particular its behaviour with respect to rules that hold for classical logic. In the paper we propose a deductive system for the above recalled discussive logic. While formulating this system, we apply a method of Newton da Costa and Lech Dubikajtis—a modified version of Jerzy Kotas’s method used to axiomatize $${\textsf {D}}_{\textsf {2}}$$ D 2. Basically the difference manifests in the result—in the case of da Costa and Dubikajtis, the resulting axiomatization is pure modus ponens-style. In the case of $${\textsf {D}}_{\textsf {0}}$$ D 0, we have to use some rules, but they are mostly needed to express some aspects of positive logic. $${\textsf {D}}_{\textsf {0}}$$ D 0 understood as a set of theses is contained in $${\textsf {D}}_{\textsf {2}}$$ D 2. Additionally, any non-trivial discussive logic expressed by means of Jaśkowski’s model of discussion, applied to any regular modal logic of discussion, contains $${\textsf {D}}_{\textsf {0}}$$ D 0. (shrink)

We investigate under what conditions a given set of collective judgments can arise from a specific voting procedure. In order to answer this question, we introduce a language similar to modal logic for reasoning about judgment aggregation procedures. In this language, the formula expresses that is collectively accepted, or that is a group judgment based on voting. Different judgment aggregation procedures may be underlying the group decision making. Here we investigate majority voting, where holds if a majority of individuals accepts, (...) consensus voting, where holds if all individuals accept, and dictatorship. We provide complete axiomatizations for judgment sets arising from all three aggregation procedures. (shrink)

We define the concepts of minimal p-morphic image and basic p-morphism for transitive Kripke frames. These concepts are used to determine effectively the least number of variables necessary to axiomatize a tabular extension of K4, and to describe the covers and co-covers of such a logic in the lattice of the extensions of K4.

In Part I of this paper, an abstract analogue of the minimization problem for Boolean functions and of the notion of prime implicant is defined, so that this general problem can be solved in the same steps as in the classical case: 1) determination of the prime implicants; 2) determination of all the solutions made up of prime implicants. In Part II it is shown that the classical minimization problem, as well as certain set-theoretical and graphtheoretical problems are particular cases (...) of the general problem defined in Part I. (shrink)

In a recent paper Berto introduces a semantic system for a logic of imagination, intended as positive conceivability, and aboutness of imaginative acts. This system crucially adopts elements of both the semantics of conditionals and the semantics of analytical implications in order to account for the central logical traits of the notion of truth in an act of imagination based on an explicit input. The main problem left unsolved is to put forward a complete set of axioms for the proposed (...) system. In the present paper I offer a solution to this problem by providing a complete axiomatization of a generalization of the original semantics. The difficulty in proving completeness lies in the fact that the modalities that capture the notion of truth in an act of imagination are neither standard nor minimal, so that the construction of the canonical model and the proof of the truth lemma are to be substantially modified. (shrink)

We propose a minimal logic for interactive epistemology based on a qualitative representation of epistemic individual and group attitudes including knowledge, belief, strong belief, common knowledge and common belief. We show that our logic is sufficiently expressive to provide an epistemic foundation for various game-theoretic solution concepts including “1-round of deletion of weakly dominated strategies, followed by iterated deletion of strongly dominated strategies” ) and “2-rounds of deletion of weakly dominated strategies, followed by iterated deletion of strongly dominated strategies” (...) ). Axiomatization and complexity results for the logic are given in the paper. (shrink)

This paper proposes a minimal logic of fine-grained information dynamics via belief bases. The framework is shown to be able to accommodate explicit belief, implicit belief, awareness of and awareness that, where awareness of agents is not treated as a tacit premise. A sound and complete axiomatization of static logic is established, upon which a series of dynamic operations are defined. It is argued that these dynamics adapt different scenarios. Our logic is minimal because to each agent (...) we only attach two databases, from which a variety of epistemic attitudes are generated. (shrink)

This paper sketches the antilogism of Christine Ladd-Franklin and historical advancement about antilogism, mainly constructs an axiomatic system Atl based on first-order logic with equality and the wholly-exclusion and not-wholly-exclusion relations abstracted from the algebra of Ladd-Franklin, with soundness and completeness of Atl proved, providing a simple and convenient tool on syllogistic reasoning. Atl depicts the empty class and the whole class differently from normal set theories, e.g. ZFC, revealing another perspective on sets and set theories. Two series of Dotterer (...) and the theorem proved by Russinoff are re-characterized in antilogistic formulas corresponding to the square of opposition and twenty-four moods of syllogism. Then the restricted equivalence between two forms of antilogism and minimal inconsistency is built up, implying the internal connection between antilogisms and minimal inconsistency and leaving two related conjectures not proved. Besides, this paper provides a new explanation for the reason why contrariety, subcontrariety, and subalternation and the nine special moods require the non-empty assumption. (shrink)

In this paper we will investigate different axiomatic theories of truth that are minimal in some sense. One criterion for minimality will be conservativity over Peano Arithmetic. We will then give a more fine-grained characterization by investigating some interpretability relations. We will show that disquotational theories of truth, as well as compositional theories of truth with restricted induction are relatively interpretable in Peano Arithmetic. Furthermore, we will give an example of a theory of truth that is a conservative extension (...) of Peano Arithmetic but not interpretable in it. We will then use stricter versions of interpretations to compare weak theories of truth to subsystems of second-order arithmetic. (shrink)

This paper sketches the antilogism of Christine Ladd-Franklin and historical advancement about antilogism, mainly constructs an axiomatic system Atl based on first-order logic with equality and the wholly-exclusion and not-wholly-exclusion relations abstracted from the algebra of Ladd-Franklin, with soundness and completeness of Atl proved, providing a simple and convenient tool on syllogistic reasoning. Atl depicts the empty class and the whole class differently from normal set theories, e.g. ZFC, revealing another perspective on sets and set theories. Two series of Dotterer (...) and the theorem proved by Russinoff are re-characterized in antilogistic formulas corresponding to the square of opposition and twenty-four moods of syllogism. Then the restricted equivalence between two forms of antilogism and minimal inconsistency is built up, implying the internal connection between antilogisms and minimal inconsistency and leaving two related conjectures not proved. Besides, this paper provides a new explanation for the reason why contrariety, subcontrariety, and subalternation and the nine special moods require the non-empty assumption. (shrink)

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. Recently, Cluckers et al. developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this paper, we continue the study of Hensel minimality, in particular focusing on [math]-h-minimality and [math]-h-minimality, for [math] a positive integer. Our main results include an analytic criterion for [math]-h-minimality, preservation (...) of [math]-h-minimality under coarsening of the valuation and [math]-dimensional geometry. (shrink)

Recently, Cluckers et al. developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this paper, we continue the study of Hensel minimality, in particular focusing on [Formula: see text]-h-minimality and [Formula: see text]-h-minimality, for [Formula: see text] a positive integer. Our main results include an analytic criterion for [Formula: see text]-h-minimality, preservation of [Formula: (...) see text]-h-minimality under coarsening of the valuation and [Formula: see text]-dimensional geometry. (shrink)

This paper sketches the antilogism of Christine Ladd-Franklin and historical advancement about antilogism, mainly constructs an axiomatic system Atl based on first-order logic with equality and the wholly-exclusion and not-wholly-exclusion relations abstracted from the algebra of Ladd-Franklin, with soundness and completeness of Atl proved, providing a simple and convenient tool on syllogistic reasoning. Atl depicts the empty class and the whole class differently from normal set theories, e.g. ZFC, revealing another perspective on sets and set theories. Two series of Dotterer (...) and the theorem proved by Russinoff are re-characterized in antilogistic formulas corresponding to the square of opposition and twenty-four moods of syllogism. Then the restricted equivalence between two forms of antilogism and minimal inconsistency is built up, implying the internal connection between antilogisms and minimal inconsistency and leaving two related conjectures not proved. Besides, this paper provides a new explanation for the reason why contrariety, subcontrariety, and subalternation and the nine special moods require the non-empty assumption. (shrink)

This paper proposes a class of independence axioms for simple acts. By introducing the E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}$$\end{document}-cominimum independence axiom that is stronger than the comonotonic independence axiom but weaker than the independence axiom, we provide a new axiomatization theorem of simple acts within the framework of Choquet expected utility. Furthermore, in order to provide the axiomatization of simple acts, we generalize Kajii et al. into an infinite state space. Our (...) axiomatization theorem relates Choquet expected utility to multi-prior expected utility through the core of a capacity that is explicitly derived within our framework. Our result in this paper also derives Gilboa, Eichberger and Kelsey, and Rohde as a corollary. (shrink)

Simple finite axiomatizations are given for versions of the modal logics K and K4 with non-contingency (or contingency) as the sole modal primitive. This answers two questions of I. L. Humberstone.

Let R be an o-minimal field with a proper convex subring V. We axiomatize the class of all structures such that , the corresponding residue field with structure induced from R via the residue map, is o-minimal. More precisely, in Maříková [8] it was shown that certain first-order conditions on are sufficient for the o-minimality of . Here we prove that these conditions are also necessary.

This paper analyzes the notion of a minimal belief change that incorporates new information. I apply the fundamental decision-theoretic principle of Pareto-optimality to derive a notion of minimal belief change, for two different representations of belief: First, for beliefs represented by a theory – a deductively closed set of sentences or propositions – and second for beliefs represented by an axiomatic base for a theory. Three postulates exactly characterize Pareto-minimal revisions of theories, yielding a weaker set of (...) constraints than the standard AGM postulates. The Levi identity characterizes Pareto-minimal revisions of belief bases: a change of belief base is Pareto-minimal if and only if the change satisfies the Levi identity (for “maxichoice” contraction operators). Thus for belief bases, Pareto-minimality imposes constraints that the AGM postulates do not. The Ramsey test is a well-known way of establishing connections between belief revision postulates and axioms for conditionals (“if p, then q”). Pareto-minimal theory change corresponds exactly to three characteristic axioms of counterfactual systems: a theory revision operator that satisfies the Ramsey test validates these axioms if and only if the revision operator is Pareto-minimal. (shrink)

This paper analyzes the notion of a minimal belief change that incorporates new information. I apply the fundamental decisiontheoretic principle of Pareto-optimality to derive a notion of minimal belief change, for two different representations of belief: First, for beliefs represented by a theory –a deductively closed set of sentences or propositions–and second for beliefs represented by an axiomatic base for a theory. Three postulates exactly characterize Pareto-minimal revisions of theories, yielding a weaker set of constraints than the (...) standard AGM postulates. The Levi identity characterizes Pareto-minimal revisions of belief bases: a change of belief base is Pareto-minimal if and only if the change satisfies the Levi identity (for “maxichoice” contraction operators). Thus for belief bases, Pareto-minimality imposes constraints that the AGM postulates do not. Keywords: belief revision, decision theory.. (shrink)

What is the simplest and most natural axiomatic replacement for the set-theoretic definition of the minimal fixed point on the Kleene scheme in Kripke’s theory of truth? What is the simplest and most natural set of axioms and rules for truth whose adoption by a subject who had never heard the word "true" before would give that subject an understanding of truth for which the minimal fixed point on the Kleene scheme would be a good model? Several axiomatic (...) systems, old and new, are examined and evaluated as candidate answers to these questions, with results of Harvey Friedman playing a significant role in the examination. (shrink)

In the study of nonmonotonic reasoning the main emphasis has been on static (declarative) aspects. Only recently has there been interest in the dynamic aspects of reasoning processes, particularly in artificial intelligence. We study the dynamics of reasoning processes by using a temporal logic to specify them and to reason about their properties, just as is common in theoretical computer science. This logic is composed of a base temporal epistemic logic with a preference relation on models, and an associated nonmonotonic (...) inference relation, in the style of Shoham, to account for the nonmonotonicity. We present an axiomatic proof system for the base logic and study decidability and complexity for both the base logic and the nonmonotonic inference relation based on it. Then we look at an interesting class of formulas, prove a representation result for it, and provide a link with the rule of monotonicity. (shrink)

We show that is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model.

Let R be an o-minimal expansion of an ordered group R has no poles, R cannot define a real closed field with domain R and order R is eventually linear and every R -definable set is a finite union of cones. As a corollary we get that Th has quantifier elimination and universal axiomatization in the language with symbols for the ordered group operations, bounded R -definable sets and a symbol for each definable endomorphism of the group.

ABSTRACT A simple method of axiomatizing every finite intermediate propositional logic by a finite set of axioms with the minimal number of variables is proposed. The method is based on Jankov's characteristic formulas.

This article presents and compares various algebraic structures that arise in axiomatic unsharp quantum physics. We begin by stating some basic principles that such an algebraic structure should encompass. Following G. Mackey and G. Ludwig, we first consider a minimal state-effect-probability (minimal SEFP) structure. In order to include partial operations of sum and difference, an additional axiom is postulated and a SEFP structure is obtained. It is then shown that a SEFP structure is equivalent to an effect algebra (...) with an order determining set of states. We also consider σ-SEFP structures and show that these structures distinguish Hilbert space from incomplete inner product spaces. Various types of sharpness are discussed and under what conditions a Brouwer complementation can be defined to obtain a BZ-poset is investigated. In this case it is shown that every effect has a best lower and upper sharp approximation and that the set of all Brouwer sharp effects form an orthoalgebra. (shrink)

By pure calculus of names we mean a quantifier-free theory, based on the classical propositional calculus, which defines predicates known from Aristotle’s syllogistic and Leśniewski’s Ontology. For a large fragment of the theory decision procedures, defined by a combination of simple syntactic operations and models in two-membered domains, can be used. We compare the system which employs `ε’ as the only specific term with the system enriched with functors of Syllogistic. In the former, we do not need an empty name (...) in the model, so we are able to construct a 3-valued matrix, while for the latter, for which an empty name is necessary, the respective matrices are 4-valued. (shrink)

In the literature there are at least two models for probabilistic belief revision: Bayesian updating and imaging [Lewis, D. K. (1973), Counterfactuals, Blackwell, Oxford; Gärdenfors, P. (1988), Knowledge in flux: modeling the dynamics of epistemic states, MIT Press, Cambridge, MA]. In this paper we focus on imaging rules that can be described by the following procedure: (1) Identify every state with some real valued vector of characteristics, and accordingly identify every probabilistic belief with an expected vector of characteristics; (2) For (...) every initial belief and every piece of information, choose the revised belief which is compatible with this information and for which the expected vector of characteristics has minimal Euclidean distance to the expected vector of characteristics of the initial belief. This class of rules thus satisfies an intuitive notion of minimal belief revision. The main result in this paper is to provide an axiomatic characterization of this class of imaging rules. (shrink)

We continue our investigation of paraorthomodular BZ*-lattices PBZ*-lattices, started in Giuntini et al., Mureşan. We shed further light on the structure of the subvariety lattice of the variety PBZL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {PBZL}^{\mathbb {*}}$$\end{document} of PBZ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{*}$$\end{document}–lattices; in particular, we provide axiomatic bases for some of its members. Further, we show that some distributive subvarieties of PBZL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...) \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {PBZL}^{\mathbb {*}}$$\end{document} are term-equivalent to well-known varieties of expanded KleeneKleene, C. lattices or of nonclassical modal algebrasNonclassical modal algebras. By so doing, we somehow help the reader to locate PBZ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{*}$$\end{document}–lattices on the atlas of algebraic structures for nonclassical logics. (shrink)

This paper relates to a formal statement of the mechanisms that are thought minimally necessary to underpin consciousness. This is expressed in the form of axioms. We deem this to be useful if there is ever to be clarity in answering questions about whether this or the other organism is or is not conscious. As usual, axioms are ways of making formal statements of intuitive beliefs and looking, again formally, at the consequences of such beliefs. The use of this style (...) of exposition does not entail a claim to provide a mathematically rigorous formal deductive system. Conventional mathematical notation is used to achieve clarity, although this is elaborated with natural language in an attempt to reduce terseness. In our view, making the approach axiomatic is synonymous with building clear usable tests for consciousness and is therefore a central feature of the paper. The extended scope of this approach is to lay down some essential properties that should be considered when designing machines that could be said to be conscious. In the broader discussion about the nature of consciousness and its neurological mechanisms, it may seem to some that axiomatisation is premature and continues to beg many questions. However, the approach is meant to be open- ended so that others can build further axiomatic clarifications that address the very large number of questions which, in the search for a formal basis for consciousness, still remain to be answered. Of course, in discussions about consciousness many will also argue that the subject is not one that may ever be formally addressed by means of axioms. The view taken in this paper is 'let's try to do it and see how far it gets'. (shrink)

In the chapter on quantum logic in Volume 6 of Handbook of Philosophical Logic, Dalla Chiara and Giuntini make an interesting observation that there is a unified relational semantics underlying both the {¬,∧}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ {\lnot }, {\wedge } \}$$\end{document}-fragment of intuitionistic logic and ortho-logic. In this paper, we contribute to a systematic investigation of this relational semantics by providing an axiomatization of its logic.

Dynamic Topological Logic ( ) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems , which are pairs consisting of a topological space X and a continuous function f : X → X . The function f is seen as a change in one unit of time; within one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is (...) particularly interesting is that of minimal systems ; these are dynamic topological systems which admit no proper, closed, f -invariant subsystems. In such systems the orbit of every point is dense, which within translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. (shrink)

We give an example of two ordered structures $\mathcal {M},\mathcal {N}$ in the same language $\mathcal {L}$ with the same universe, the same order and admitting the same one-variable definable subsets such that $\mathcal {M}$ is a model of the common theory of o-minimal $\mathcal {L}$ -structures and $\mathcal {N}$ admits a definable, closed, bounded, and discrete subset and a definable injective self-mapping of that subset which is not surjective. This answers negatively two question by Schoutens; the first being (...) whether there is an axiomatization of the common theory of o-minimal structures in a given language by conditions on one-variable definable sets alone. The second being whether definable completeness and type completeness imply the pigeonhole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeonhole principle. (shrink)

Wilkie 5 397) proved a “theorem of the complement” which implies that in order to establish the o-minimality of an expansion of with C∞ functions it suffices to obtain uniform bounds on the number of connected components of quantifier free definable sets. He deduced that any expansion of with a family of Pfaffian functions is o-minimal. We prove an effective version of Wilkie's theorem of the complement, so in particular given an expansion of the ordered field with finitely many (...) C∞ functions, if there are uniform and computable upper bounds on the number of connected components of quantifier free definable sets, then there are uniform and computable bounds for all definable sets. In such a case the theory of the structure is effectively o-minimal: there is a recursively axiomatized subtheory such that each of its models is o-minimal. This implies the effective o-minimality of any expansion of with Pfaffian functions. We apply our results to the open problem of the decidability of the theory of the real field with the exponential function. We show that the decidability is implied by a positive answer to the following problem ): given a language L expanding the language of ordered rings, if an L-sentence is true in every L-structure expanding the ordered field of real numbers, then it is true in every o-minimal L-structure expanding any real closed field. (shrink)

Motivated by the analysis of a general optimal portfolio selection problem, which encompasses as special cases an optimal consumption and an optimal debt-arrangement problem, we are concerned with the questions of how a personality trait like risk-perception can be formalized and whether the two objectives of utility-maximization and risk-minimization can be both achieved simultaneously. We address these questions by developing an axiomatic foundation of preferences for which utility-maximization is equivalent to minimizing a utility-based shortfall risk measure. Our axiomatization hinges (...) on a novel axiom in decision theory, namely the risk-perception axiom. (shrink)

The paper expands upon the work by Wang [4], who proposes a new framework based on quantifier-free predicate language extended by a new modality ∃x□\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists x\Box$$\end{document} and axiomatizes the logic over S5 frames. This paper gives the logics over K, D, T, 4, S4 frames with increasing and constant domains. And we provide a general strategy for proving completeness theorems for logics w.r.t. the increasing domain and logics w.r.t. the (...) constant domain respectively. (shrink)

Without introducing quantifiers, minimal axiomatic systems have already been constructed for Peirce's triadic logics. The present paper constructs a dual pair of axiomatic systems which can be used to introduce quantifiers into Peirce's preferred system of triadic logic. It is assumed (on the basis of textual evidence) that Peirce would prefer a system which rejects the absurd but tolerates the absolutely undecidable. The systems which are introduced are shown to be absolutely consistent, deductively complete, and minimal. These dual (...) axiomatic systems reveal an interesting elegance, independent of their historical motivation. (shrink)

Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the Sofia school , the method remains little known. In our view this has deprived temporal logic of a valuable tool.The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the first technical, the second conceptual. First, we show that hybridization gives rise to well-behaved (...) logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full first-order expressive strength, is demonstrated for a weaker local language capable of defining the Until operator; we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths. (shrink)

In this paper the view is developed that classes should not be understood as individuals, but, rather, as "classes as many" of individuals. To correlate classes with individuals "labelling" and "colabelling" functions are introduced and sets identified with a certain subdomain of the classes on which the labelling and colabelling functions are mutually inverse. A minimal axiomatization of the resulting system is formulated and some of its extensions are related to various systems of set theory, including nonwellfounded set (...) theories. (shrink)

We give a complete axiomatization of the identities of the basic game algebra valid with respect to the abstract game board semantics. We also show that the additional conditions of termination and determinacy of game boards do not introduce new valid identities.En route we introduce a simple translation of game terms into plain modal logic and thus translate, while preserving validity both ways, game identities into modal formulae.The completeness proof is based on reduction of game terms to a certain (...) ‘minimal canonical form’, by using only the axiomatic identities, and on showing that the equivalence of two minimal canonical terms can be established from these identities. (shrink)

In the paper we investigate typed axiomatizations of the truth predicate in which the axioms of truth come with a built-in, minimal and self-sufficient machinery to talk about syntactic aspects of an arbitrary base theory. Expanding previous works of the author and building on recent works of Albert Visser and Richard Heck, we give a precise characterization of these systems by investigating the strict relationships occurring between them, arithmetized model constructions in weak arithmetical systems and suitable set existence axioms. (...) The framework considered will give rise to some methodological remarks on the construction of truth theories and provide us with a privileged point of view to analyze the notion of truth arising from compositional principles in a typed setting. (shrink)

In this paper we provide a quantifier-free, constructive axiomatization of metric-Euclidean and of rectangular planes . The languages in which the axiom systems are expressed contain three individual constants and two ternary operations. We also provide an axiom system in algorithmic logic for finite Euclidean planes, and for several minimal metric-Euclidean planes. The axiom systems proposed will be used in a sequel to this paper to provide ‘the simplest possible’ axiom systems for several fragments of plane Euclidean geometry.

This book is a contribution to the flourishing field of formal and philosophical work on truth and the semantic paradoxes. Our aim is to present several theories of truth, to investigate some of their model-theoretic, recursion-theoretic and proof-theoretic aspects, and to evaluate their philosophical significance. In Part I we first outline some motivations for studying formal theories of truth, fix some terminology, provide some background on Tarski’s and Kripke’s theories of truth, and then discuss the prospects of classical type-free truth. (...) In Chapter 4 we discuss some minimal adequacy conditions on a satisfactory theory of truth based on the function that the truth predicate is intended to fulfil on the deflationist account. We cast doubt on the adequacy of some non-classical theories of truth and argue in favor of classical theories of truth. Part II is devoted to grounded truth. In chapter 5 we introduce a game-theoretic semantics for Kripke’s theory of truth. Strategies in these games can be interpreted as reference-graphs of the sentences in question. Using that framework, we give a graph-theoretic analysis of the Kripke-paradoxical sentences. In chapter 6 we provide simultaneous axiomatizations of groundedness and truth, and analyze the proof-theoretic strength of the resulting theories. These range from conservative extensions of Peano arithmetic to theories that have the full strength of the impredicative system ID1. Part III investigates the relationship between truth and set-theoretic comprehen- sion. In chapter 7 we canonically associate extensions of the truth predicate with Henkin-models of second-order arithmetic. This relationship will be employed to determine the recursion-theoretic complexity of several theories of grounded truth and to show the consistency of the latter with principles of generalized induction. In chapter 8 it is shown that the sets definable over the standard model of the Tarskian hierarchy are exactly the hyperarithmetical sets. Finally, we try to apply a certain solution to the set-theoretic paradoxes to the case of truth, namely Quine’s idea of stratification. This will yield classical disquotational theories that interpret full second-order arithmetic without set parameters, Z2- . We also indicate a method to recover the parameters. An appendix provides some background on ordinal notations, recursion theory and graph theory. (shrink)

In this paper we are concerned with definably, with or without parameters, complete expansions of ordered fields, i. e. those with no definable gaps. We present several axiomatizations, like being definably connected, in each of the two cases. As a corollary, when parameters are allowed, expansions of ordered fields are o-minimal if and only if all their definable subsets are finite disjoint unions of definably connected subsets. We pay attention to how simply a definable gap in an expansion is (...) so. Next we prove that over parametrically definably complete expansions of ordered fields, all one-to-one definable continuous functions are monotone and open. Moreover, in both parameter and parameter-free cases again, definably complete expansions of ordered fields satisfy definable versions of the Heine-Borel and Extreme Value theorems and also Bounded Intersection Property for definable families of closed bounded subsets. (shrink)

This paper presents an axiomatization of a class of set-theoretic conditional operators using minimization of the geodesic distance defined as the shortest path generated by the accessibility relation on a frame. The objective of this modeling is to define conditioning based on a notion of similarity generated by degrees of indistinguishability.

This paper studies a decision maker who for each choice set selects a subset of (at most) two alternatives. We axiomatize three types of procedures: (i) The top two: the decision maker has in mind an ordering and chooses the two maximal alternatives. (ii) The two extremes: the decision maker has in mind an ordering and chooses the maximal and the minimal alternatives. (iii) The top and the top: the decision maker has in mind two orderings and he chooses (...) the maximal element from each. (shrink)

This three-part chapter explores a higher-order logic we call ‘Classicism’, which extends a minimal classical higher-order logic with further axioms which guarantee that provable coextensiveness is sufficient for identity. The first part presents several different ways of axiomatizing this theory and makes the case for its naturalness. The second part discusses two kinds of extensions of Classicism: some which take the view in the direction of coarseness of grain (whose endpoint is the maximally coarse-grained view that coextensiveness is sufficient (...) for identity), and some which take the view in the direction of fineness of grain (whose endpoint is the maximally fine-grained theory containing all distinctness claims compatible with Classicism). The third part introduces some techniques for constructing models of Classicism, and uses them to prove the consistency of many of the extensions of Classicism introduced in the second part. (shrink)

This paper studies a decision maker who tackles a choice problem by selecting a subset of (at most) two alternatives which he will consider further in the second stage of his deliberation. We focus on the first stage where he chooses the delebration set. We axiomatize three types of procedures: (i) The top two: the decision maker has in mind an ordering and chooses the two maximal alternatives. (ii) The two extremes: the decision maker has in mind an ordering and (...) chooses the maximal and the minimal alternatives. (iii) The top and the top: the decision maker has in mind two orderings and he chooses the maximal elements. (shrink)

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of (...) the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators. (shrink)