Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed‐point logic. Inclusion logic is not axiomatisable in full, but its first order consequences can be axiomatized. In this paper, we provide such an explicit partial axiomatization by introducing a system of natural deduction for inclusion logic that is sound and complete for first order consequences in inclusion logic.

Classical first-order logic \ is commonly used to study logical connections between statements, that is sentences that in every context have an associated truth-value. Inquisitive first-order logic \ is a conservative extension of \ which captures not only connections between statements, but also between questions. In this paper we prove the disjunction and existence properties for \ relative to inquisitive disjunction Open image in new window and inquisitive existential quantifier \. Moreover we extend these results to several families of theories, (...) among which the one in the language of \. To this end, we initiate a model-theoretic approach to the study of \. In particular, we develop a toolkit of basic constructions in order to transform and combine models of \. (shrink)

In the paper, original formal-logical conception of syntactic and semantic: intensional and extensional senses of expressions of any language L is outlined. Syntax and bi-level intensional and extensional semantics of language L are characterized categorically: in the spirit of some Husserl’s ideas of pure grammar, Leśniewski-Ajukiewicz’s theory syntactic/semantic categories and in accordance with Frege’s ontological canons, Bocheński’s famous motto—syntax mirrors ontology and some ideas of Suszko: language should be a linguistic scheme of ontological reality and simultaneously a tool of its (...) cognition. In the logical conception of language L, its expressions should satisfy some general conditions of language adequacy. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic, and semantic compatibility, correspondence guaranteed by the acceptance of a postulate of categorial compatibility syntactic and semantic categories of expressions of L. From this postulate, three principles of compositionality follow: one syntactic and two semantic already known to Frege. They are treated as conditions of homomorphism partial algebra of L into algebraic models of L: syntactic, intensional, and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, of course, an idealization, but only expressions with high degrees of precision of their senses, after due justification, may become theorems of science. (shrink)

We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless Boolean algebras. This establishes interesting connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov :957–1012, 2013), the former case of independence can be seen as the discrete version of the latter.

This is a review of Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena, written by Jon Barwise and Lawrence Moss and published by CSLI Publications in 1996.

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

In this paper we study connections between game theoretical concepts and results, and features of IF-predicate logic, extending observations from J. van Benthem (2001) for IF-propositional logic. We highlight how both characteristics of perfect recall can fail in the semantic games for IF-formulas, and we discuss the four Thompson transformations in relation with IF-logic. Many (strong) equivalence schemes for IF-logic correspond to one or more of the transformations. However, we also find one equivalence that does not fit in this picture, (...) by the type of imperfect recall involved. We point out that the connection between the transformations and logical equivalence schemes is less direct in IF-first order logic than in the propositional case. The transformations do not generate a reduced normal form for IF-logic, because the IF-language is not flexible enough. (shrink)

The paper concentrates on the problem of adequate reflection of fragments of reality via expressions of language and inter-subjective knowledge about these fragments, called here, in brief, language adequacy. This problem is formulated in several aspects, the most being: the compatibility of language syntax with its bi-level semantics: intensional and extensional. In this paper, various aspects of language adequacy find their logical explication on the ground of the formal-logical theory T of any categorial language L generated by the so-called classical (...) categorial grammar, and also on the ground of its extension to the bi-level, intensional and extensional semantic-pragmatic theory ST for L. In T, according to the token-type distinction of Ch.S. Peirce, L is characterized first as a language of well-formed expression-tokens (wfe-tokens) - material, concrete objects - and then as a language of wfe-types - abstract objects, classes of wfe-tokens. In ST the semantic-pragmatic notions of meaning and interpretation for wfe-types of L of intensional semantics and the notion of denotation of extensional semanics for wfe-types and constituents of knowledge are formalized. These notions allow formulating a postulate (an axiom of categorial adequacy) from which follow all the most important conditions of the language adequacy, including the above, and a structural one connected with three principles of compositionality. (shrink)

Typical applications of Hintikka’s game-theoretical semantics give rise to semantic attributes—truth, falsity—expressible in the $\Sigma^{1}_{1}$-fragment of second-order logic. Actually a much more general notion of semantic attribute is motivated by strategic considerations. When identifying such a generalization, the notion of classical negation plays a crucial role. We study two languages, $L_{1}$ and $L_{2}$, in both of which two negation signs are available: $\rightharpoondown $ and $\sim$. The latter is the usual GTS negation which transposes the players’ roles, while the former (...) will be interpreted via the notion of mode. Logic $L_{1}$ extends independence-friendly logic; $\rightharpoondown $ behaves as classical negation in $L_{1}$. Logic $L_{2}$ extends $L_{1}$, and it is shown to capture the $\Sigma^{2}_{1}$-fragment of third-order logic. Consequently the classical negation remains inexpressible in $L_{2}$. (shrink)

The paper analyses what is said and what is presupposed by thePrinciple of Compositionality for semantics, as it is commonly stated. ThePrinciple of Compositionality is an axiom which some semantics satisfy andsome don’t. It says essentially that if two expressions have the same meaning then they make the same contribution to the meanings of expressionscontaining them. This is a sensible axiom only if one combines it with aconverse, that if two expressions make the same contribution to the meanings of sentences (...) containing them, then they have the same meaning;and some assumption that two expressions which can’t meaningfully besubstituted for each other have different meanings. as a full abstraction principle, and as ‘Husserl’s principle’.) Moreoverthe Principle of Compositionality speaks only about when two expressionshave the same meaning; it adds nothing whatever about what that meaningmight be . Some recent discussions by writers in linguistics and logic are assessed. The paper finishes by reviewing thehistory of the notion of compositionality. (shrink)

The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar (...) from ML; the resulting asymmetric simulation concept is used to prove that IFML s is not closed under complementation. In fact we obtain a much stronger result: the only IFML s formulas admitting their classical negation to be expressed in IFML s itself are those whose truth-condition is in fact expressible in ML. (shrink)

We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindström, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in fact definably equivalent (...) to the independence atom recently introduced by Väänänen and Grädel. (shrink)

The hybrid logic and the independence friendly modal logic IFML are compared for their expressive powers. We introduce a logic IFML c having a non-standard syntax and a compositional semantics; in terms of this logic a syntactic fragment of IFML is singled out, denoted IFML c . (In the Appendix it is shown that the game-theoretic semantics of IFML c coincides with the compositional semantics of IFML c .) The hybrid logic is proven to be strictly more expressive than IFML (...) c . By contrast, and the full IFML are shown to be incomparable for their expressive powers. Building on earlier research (Tulenheimo and Sevenster 2006), a PSPACE -decidable fragment of the undecidable logic is disclosed. This fragment is not translatable into the hybrid logic and has not been studied previously in connection with hybrid logics. In the Appendix IFML c is shown to lack the property of ‘quasi-positionality’ but proven to enjoy the weaker property of ‘ bounded quasi-positionality’. The latter fact provides from the IFML internal perspective an account of what makes the compositional semantics of IFML c possible. (shrink)

In this paper, we introduce a logic based on team semantics, called $\mathbf {FOT} $, whose expressive power is elementary, i.e., coincides with first-order logic both on the level of sentences and (possibly open) formulas, and we also show that a sublogic of $\mathbf {FOT} $, called $\mathbf {FOT}^{\downarrow } $, captures exactly downward closed elementary (or first-order) team properties. We axiomatize completely the logic $\mathbf {FOT} $, and also extend the known partial axiomatization of dependence logic to dependence logic (...) enriched with the logical constants in $\mathbf {FOT}^{\downarrow } $. (shrink)

Team semantics is a highly general framework for logics which describe dependencies and independencies among variables. Typically, the dependencies considered in this context are properties of sets of configurations or data records. We show how team semantics can be further generalized to support languages for the discussion of interventionist counterfactuals and causal dependencies, such as those that arise in manipulationist theories of causation. We show that the “causal teams” we introduce in the present paper can be used for modelling some (...) classical counterfactual scenarios which are not captured by the usual causal models. We then analyse the basic properties of our counterfactual languages and discuss extensively the differences with respect to the Lewisian tradition. (shrink)

The paper deals with the problem of logical adequacy of language knowledge with cognition of reality. A logical explication of the concept of language knowledge conceived of as a kind of codified knowledge is taken into account in the paper. Formal considerations regarding the notions of meta-knowledge (logical knowledge about language knowledge) and truth are developed in the spirit of some ideas presented in the author’s earlier papers (1991, 1998, 2001a,b, 2007a,b,c) treating about the notions of meaning, denotation and truthfulness (...) of well-formed expressions (wfes) of any given categorial language. Three aspects connected with knowledge codified in language are considered, including: 1) syntax and two kinds of semantics: intensional and extensional, 2) three kinds of non-standard language models and 3) three notions of truthfulness of wfes. Adequacy of language knowledge to cognitive objects is understood as an agreement of truthfulness of sentences in these three models. (shrink)

A dichotomy result of Sevenster (2014) [29] completely classified the quantifier prefixes of regular Independence-Friendly (IF) logic according to the patterns of quantifier dependence they contain. On one hand, prefixes that contain “Henkin” or “signalling” patterns were shown to characterize fragments of IF logic that capture NP-complete problems; all the remaining prefixes were shown instead to be essentially first-order. In the present paper we develop the machinery which is needed in order to extend the results of Sevenster to non-prenex, regular (...) IF sentences. This involves shifting attention from quantifier prefixes to a (rather general) class of syntactical tree prefixes. We partially classify the fragments of regular IF logic that are thus determined by syntactical trees; in particular, a) we identify four classes of tree prefixes that are neither signaling nor Henkin, and yet express NP-complete problems and other second-order concepts; and b) we give more general criteria for checking the first-orderness of an IF sentence. (shrink)

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. The theme of this book lies at the interface of many disciplines. Logic is the main methodology, but the various chapters cross easily between mathematics, computer science, philosophy, linguistics, cognitive and social sciences, while also ranging from pure theory to empirical work. Accordingly, the authors of this book represent a wide variety of original thinkers from different research communities. And their interconnected themes challenge at the same time how we think of logic, philosophy and computation. Thus, very much in line with van Benthem's work over many decades, the volume shows how all these disciplines form a natural unity in the perspective of dynamic logicians (broadly conceived) exploring their new themes today. And at the same time, in doing so, it offers a broader conception of logic with a certain grandeur, moving its horizons beyond the traditional study of consequence relations. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

The problem of intentional identity, as originally offered by Peter Geach, says that there can be an anaphoric link between an indefinite term and a pronoun across a sentential boundary and across propositional attitude contexts, where the actual existence of an individual for the indefinite term is not presupposed. In this paper, a semantic resolution to this elusive puzzle is suggested, based on a new quantified intensional logic and game-theoretic semantics (GTS) of imperfect information. This constellation leads to an expressive (...) intensional language with a property of informational independence, argued to produce a purely semantic explication to intentional identity statements. One consequence is that various extra-logical and pragmatic factors become of secondary concern; it is possible to solve the puzzle by logico-semantic methods, albeit somewhat radically renewed ones. (shrink)

Prioritized bases, i.e., weakly ordered sets of sentences, have been used for specifying an agent’s ‘basic’ or ‘explicit’ beliefs, or alternatively for compactly encoding an agent’s belief state without the claim that the elements of a base are in any sense basic. This paper focuses on the second interpretation and shows how a shifting of priorities in prioritized bases can be used for a simple, constructive and intuitive way of representing a large variety of methods for the change of belief (...) states—methods that have usually been characterized semantically by a system-of-spheres modeling. Among the methods represented are ‘radical’, ‘conservative’ and ‘moderate’ revision, ‘revision by comparison’ in its raising and lowering variants, as well as various constructions for belief expansion and contraction. Importantly, none of these methods makes any use of numbers. (shrink)

The initial goal of the present paper is to reveal a mistake committed by Hintikka in a recent paper on the foundations of mathematics. His claim that independence-friendly logic is the real logic of mathematics is supported in that article by an argument relying on uniformity concepts taken from real analysis. I show that the central point of his argument is a simple logical mistake. Second and more generally, I conclude, based on the previous remarks and on another standard fact (...) of IFL, that first-order logic can adequately express uniformity concepts in real analysis, whereas IFL cannot. This not only radically contradicts Hintikka’s particular claim in that article, but also undermines his whole enterprise of founding mathematics on his logic system. (shrink)

In this paper we consider truth as a vague predicate and inquire into the relation between truth and definite truth. We use some tools from modal logic to clarify this distinction, as done in McGee . Finally, we consider the question whether some of the results given by McGee can be transferred to the case in which the underlying logic is stronger than first-order logic. The result will be seen to be negative.

The properties of the ${\forall^{1}}$ quantifier defined by Kontinen and Väänänen in [13] are studied, and its definition is generalized to that of a family of quantifiers ${\forall^{n}}$ . Furthermore, some epistemic operators δ n for Dependence Logic are also introduced, and the relationship between these ${\forall^{n}}$ quantifiers and the δ n operators are investigated.The Game Theoretic Semantics for Dependence Logic and the corresponding Ehrenfeucht- Fraissé game are then adapted to these new connectives.Finally, it is proved that the ${\forall^{1}}$ quantifier (...) is not uniformly definable in Dependence Logic, thus answering a question posed by Kontinen and Väänänen in the above mentioned paper. (shrink)

We make a proposal for formalizing simultaneous games at the abstraction level of player's powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of 'concurrent game logic' CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.

The paper offers a new analysis of the difficulties involved in the construction of a general and substantive correspondence theory of truth and delineates a solution to these difficulties in the form of a new methodology. The central argument is inspired by Kant, and the proposed methodology is explained and justified both in general philosophical terms and by reference to a particular variant of Tarski's theory. The paper begins with general considerations on truth and correspondence and concludes with a brief (...) outlook on the "family" of theories of truth generated by the new methodology. (shrink)

The dynamic nature ofGame-Theoretical Semantics is emphasized. The role of strategic meaning in accounting for linguistic competence is examined. The semantics of epistemic possibility is shown to involve a dynamic ingredient. Update semantics has been designed to capture it. The paper focuses on the interplay betweenlogical and linguistic competences indiscourse understanding.

We analyze the expressive resources of \ logic that do not stem from Henkin quantification. When one restricts attention to regular \ sentences, this amounts to the study of the fragment of \ logic which is individuated by the game-theoretical property of action recall. We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of (...) Henkin or signalling patterns. We also study irregular IF logic and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models. (shrink)

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous (...) dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games. (shrink)

In this paper, we axiomatize the negatable consequences in dependence and independence logic by extending the systems of natural deduction of the logics given in [22] and [11]. We prove a characterization theorem for negatable formulas in independence logic and negatable sentences in dependence logic, and identify an interesting class of formulas that are negatable in independence logic. Dependence and independence atoms, first-order formulas belong to this class. We also demonstrate our extended system of independence logic by giving explicit derivations (...) for Armstrong's Axioms and the Geiger-Paz-Pearl axioms of dependence and independence atoms. (shrink)

We study the expressive power of independence-friendly quantifier prefixes composed of universal$\left$, existential$\left$, and majority quantifiers$\left$. We provide four quantifier prefixes that can express NP hard properties and show that all quantifier prefixes capable of expressing NP-hard properties embed at least one of these four quantifier prefixes. As for the quantifier prefixes that do not embed any of these four quantifier prefixes, we show that they are equivalent to a first-order quantifier prefix composed of$\forall x$,$\exists x$, and Mx. In unison, (...) our results imply a dichotomy result: every independence-friendly quantifier prefix is either decidable in LOGSPACE or NP hard. (shrink)

We investigate extensions of dependence logic with generalized quantifiers. We also introduce and investigate the notion of a generalized atom. We define a system of semantics that can accommodate variants of dependence logic, possibly extended with generalized quantifiers and generalized atoms, under the same umbrella framework. The semantics is based on pairs of teams, or double teams. We also devise a game-theoretic semantics equivalent to the double team semantics. We make use of the double team semantics by defining a logic (...) $$\hbox {DC}^2$$ DC 2 which canonically fuses together two-variable dependence logic $$\hbox {D}^2$$ D 2 and two-variable logic with counting quantifiers $$\hbox {FOC}^2$$ FOC 2 . We establish that the satisfiability and finite satisfiability problems of $$\hbox {DC}^2$$ DC 2 are complete for $$\hbox {NEXPTIME}$$ NEXPTIME. (shrink)

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot be (...) so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games. (shrink)

Tarski's model-theoretic truth definition of the 1950s differs from his 1930s truth definition by allowing the language to have a set of parameters that are interpreted by means of structures. The paper traces how the model-theoretic theorems that Tarski and others were proving in the period between these two truth definitions became increasingly difficult to fit into the framework of the earlier truth definition, making the later one more or less inevitable. The paper also maintains that neither recursiveness nor satisfaction (...) are essential features of the truth definition . The recursive form was a ‘practical’ step towards the explicit definition that was Tarski's target. There are model-theoretic languages for which satisfaction is provably too crude a notion for expressing those properties of formulas that are needed for determining the truth of sentences. (shrink)