This paper contributes to the principled construction of tableau-based decision procedures for hybrid logic with global, difference, and converse modalities. We also consider reflexive and transitive relations. For converse-free formulas we present a terminating control that does not rely on the usual chain-based blocking scheme. Our tableau systems are based on a new model existence theorem.

This paper, we propose a modal logic satisfying minimal requirements for reasoning about diagrams via collection of sets and relations between them, following Harel's proposal. We first give an axiomatics of such a theory and then provide its Kripke semantics. Then we extend previous works of ours in order to obtain a decision procedure based on tableaux for this logic. Beside soundness and completeness of our tableaux, we manage to define a strategy of rule application ensuring termination by extending the (...) usual loop test of modal logic S4 to whole sub-structures of the model being computed. (shrink)

In this paper I investigate the relation between physics and metaphysics in Plato’s participation theory. I show that the logic shoring up Plato’s metaphysics in paraconsistent, as had been suggested already by Graham Priest. The transformation of the paradoxical One-and-Many of the pre-Socratics into a paraconsistent Great-and-Small bridges the abyss between archaic rationality and the world of classical logic based ultimately on the principle of contradiction. Indeed, language is an organ of perception, not simply a means of communication. J. Jaynes, (...) Origin of Consciousness. (shrink)

way on the information available in the contexts in which they are used, it’s not surprising that there is a minor but growing industry of work in semantics and the philosophy of language concerned with the precise nature of the context-dependency of epistemically modalized sentences. Take, for instance, an epistemic might-claim like..

The topological semantics for modal logic interprets a standard modal propositional language in topological spaces rather than Kripke frames: the most general logic of topological spaces becomes S4. But other modal logics can be given a topological semantics by restricting attention to subclasses of topological spaces: in particular, S5 is logic of the class of almost discrete topological spaces, and also of trivial topological spaces. Dynamic Topological Logic interprets a modal language enriched with two unary temporal connectives, next and henceforth. (...) DTL interprets the extended language in dynamic topological systems: a DTS is a topological space together with a continuous function used to interpret the temporal connectives. In this paper, we axiomatize four conservative extensions of S5, and show them to be the logic of continuous functions on almost discrete spaces, of homeomorphisms on almost discrete spaces, of continuous functions on trivial spaces and of homeomorphisms on trivial spaces. (shrink)

This paper uses Popper's treatment of probability and an epistemic constraint on probability assignments to conditionals to extend the Bayesian representation of rational belief so that revision of previously accepted evidence is allowed for. Results of this extension include an epistemic semantics for Lewis' theory of counterfactual conditionals and a representation for one kind of conceptual change.

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions'', ``substances'' or ``substantial forms''. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances'' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae N of formulae of a normal modal theory Tm based on (...) T, such that the counterpart `i' of a the modal variable `xi' of L(Tm) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T). (shrink)

This paper tries to extend Hintikka's inductive logic so that we can confirm a causally necessary statement. For this purpose, a joint system of inductive logic and logic of causal modalities is constructed. This system can offer a plausible explication of the distinction between nomic and accidental universality, as well as a good formulation of a causal law. And the transition from actuality to causal necessity is construed, in this system, as essentially probabilistic; i.e. no statements about actuality can entail (...) a necessary statement, but certain statements about actuality can single out a most probable causal law. (shrink)

In his published work and even more in conversations, Tarski emphasized what he thought were important philosophical aspects of his work. The English translation of his more philosophical papers [56m] was dedicated to his teacher Tadeusz Kotarbinski, and in informal discussions of philosophy he often referred to the influence of Kotarbinski. Also, the influence of Leiniewski, his dissertation adviser, is evident in his early papers. Moreover, some of his important papers of the 1930s were initially given to philosophical audiences. For (...) example, the famous monograph on the concepotf truth ([33”], [35b]) was first given as twol ectures to the Logic Section of the Philosophical Society in Warsaw in 1930. Second, his paper [33], which introduced the concepts of co-consistency and co-completeness as well as the rule of infinite induction: was first given at the Second Conference of the Polish Philosophical Society in Warsaw in 1927. Also [35c] was based upon an address given in 1934 to the conference for the Unity of Science in Prague; C361 and [36a] summarize an address given at the International Congress of Scientific Philosophy in Paris in 1935. The article [44a] was published in a philosophical journal and widely reprinted in philosophical texts. This list is of course not exhaustiveb ut only representative of Tarski’s philosophical interactions as reflected in lectures given to philosophical audiences, which were later embodied in substantial papers. After 1945 almost all of Tarski’s publications and presentations are mathematical in character with one or two minor exceptions. This division, occurring about 1945, does not, however, indicate al oss of interest in philosophical questionbsu t is a result of Tarski’s moving to the Department of Mathematics at Berkeley. There he assumed an important role in the development of logic within mathematics in the United States. (shrink)

The modal logician's notion of possible world and the computer scientist's notion of state of a machine provide a point of commonality which can form the foundation of a logic of action. Extending ordinary modal logic with the calculus of binary relations leads to a very natural logic for describing the behavior of computer programs.

Summary This paper is intended to discuss the problems occurring in the relation between the notion of truth and the question of self-reference. To do this, we shall review Tarski's (T) convention and its related terminology. We shall clarify the relation between truth and extension in order to lead into the question of semantic paradoxes appearing in the theoretical models concerned with truth. Subsequently, we shall review the logical system which develops in the reformulation of the modal proposal of the (...) (T) convention. In closing, we shall critically examine Kripke's interpretation from the proposals made by Tarski. (shrink)

In spite of the philosophical significance of higher-order modal logic, the modal logician's main concern has been with sentential logic. In this paper we do not intend to go into philosophical details, but we only remark that higher-order modal logic has a close relationship with Montague's well-known idea of “universal grammar”, which is an ambitious attempt to build a logical theory of natural languages with exact syntax and semantics, comparable with the artificial languages of mathematical logic. For this matter, the (...) reader can consult, e.g., Montague [8], [9] and Gallin [2]. (shrink)

It is shown that the normal modal logic of two reflexive points jointed with a symmetric binary relation splits the lattice of normal extensions of the logic KTB. By this fact, it is easily seen that there exists the third largest logic in the class of all normal extensions of KTB.

The claim that truth is mind dependent has some initial plausibility only if truth bearers are taken to be mind dependent entities such as beliefs or statements. Even on that assumption, however, the claim is not uncontroversial. If it is spelled out as the thesis that “in a world devoid of mind nothing would be true”, then everything depends on how the phrase ‘true in world w’ is interpreted. If ‘A is true in w’ is interpreted as ‘A is true (...) of w’ (i.e. ‘w satisfies A’s truth conditions’, the claim need not be true. If on the other hand it is interpreted as ‘A is true of w and exists in w’ then the claim is trivially true, though devoid of any antirealistic efficacy. Philosophers like Heidegger and Rorty, who hold that truth is mind dependent but reality is not, must regard such principles as “A if and only if it is true that A” as only contingently true, which may be a good reason to reject the mind dependence of truth anyway. (shrink)

The first example of an intuitively meaningful propositional logic without the finite model property, and still the simplest one in the literature. The question of its decidability appears still to be open.

An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping f satisfying the following: (1) f associates with each formula A of DPL a sentence f(A) of Peano arithmetic (PA) and with each program α a formula f(α) of PA with one free variable describing formally a supertheory of PA; (2) f commutes with logical connectives; (3) f([α] A) is the sentence saying that f(A) is provable in the theory f(α); (4) for each axiom A of DPL, f(A) is (...) provable in PA (and consequently, for each A provable in DPL, f(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretation f, f(A) is provable in PA. Various modifications of this result are considered. (shrink)

In 1960, E. P. Wigner, a joint winner of the 1963 Nobel Prize for Physics, published a paper titled On the Unreasonable Effectiveness of Mathematics in the Natural Sciences [61]. This paper can be construed as an examination and affirmation of Galileo's tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effectiveness of mathematics in accurately describing physical phenomena. Wigner viewed these examples as (...) illustrations of what he called the empirical law of epistemology, which asserts that the mathematical formulation of the laws of nature is both appropriate and accurate, and that mathematics is actually the correct language for formulating the laws of nature. At the same time, Wigner pointed out that the reasons for the success of mathematics in the natural sciences are not completely understood; in fact, he went as far as asserting that “… the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.”. (shrink)

We do a quantitative analysis of modal logic. For example, for each Kripke structure M, we study the least ordinal μ such that for each state of M, the beliefs of up to level μ characterize the agents' beliefs (that is, there is only one way to extend these beliefs to higher levels). As another example, we show the equivalence of three conditions, that on the face of it look quite different, for what it means to say that the agents' (...) beliefs have a countable description, or putting it another way, have a "countable amount of information". The first condition says that the beliefs of the agents are those at a state of a countable Kripke structure. The second condition says that the beliefs of the agents can be described in an infinitary language, where conjunctions of arbitrary countable sets of formulas are allowed. The third condition says that countably many levels of belief are sufficient to capture all of the uncertainty of the agents (along with a technical condition). The fact that all of these conditions are equivalent shows the robustness of the concept of the agents' beliefs having a "countable description". (shrink)

The basic bimodal systemK/K can be interpreted as an analysis of the logic of ability developed in [1]. Where in [1] we would express the claimI can bring it about that P using the formula, with its non-normal operator, we will now use the formula. Here is a normal alethic possibilitation operator.is a normal necessitation operator, but it is independent of, and not subject to an alethic interpretation. Rather, is interpreted to meanI bring it about that P. The result is (...) a simplification and clarification of a combined logic of ability and action like that in [2], but employing only normal operators.A number of extensions of the basic systemK/K are constructed, first by strengthening the two normal sublogics independently and then by linking the two sublogics via axiom schemata involving both operators. The result is a series of increasingly strong systems which more and more adequately fulfill our expectations for a satisfactory logic of action and ability. (shrink)

Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for (...) an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic. (shrink)

Since belief revision deals with the interaction of belief and information over time, branching-time temporal logic seems a natural setting for a theory of belief change. We propose two extensions of a modal logic that, besides the next-time temporal operator, contains a belief operator and an information operator. The first logic is shown to provide an axiomatic characterization of the first six postulates of the AGM theory of belief revision, while the second, stronger, logic provides an axiomatic characterization of the (...) full set of AGM postulates. © 2006 Elsevier B.V. All rights reserved. (shrink)

We consider strategic-form games with ordinal payoffs and provide a syntactic analysis of common belief/knowledge of rationality, which we define axiomatically. Two axioms are considered. The first says that a player is irrational if she chooses a particular strategy while believing that another strategy is better. We show that common belief of this weak notion of rationality characterizes the iterated deletion of pure strategies that are strictly dominated by pure strategies. The second axiom says that a player is irrational if (...) she chooses a particular strategy while believing that a different strategy is at least as good and she considers it possible that this alternative strategy is actually better than the chosen one. We show that common knowledge of this stronger notion of rationality characterizes the restriction to pure strategies of the iterated deletion procedure introduced by Stalnaker (1994). Frame characterization results are also provided. (shrink)

Kai kurios šiuolaikinio kognityvinio mokslo teorijos teigia, kad žmogaus prote veikia dvi sistemos: sistema, vykdanti greitą intuityvų mąstymą, bei sistema, vadovaujanti lėtam logiškam mąstymui. Būtų galima manyti, kad šių sistemų egzistavimą patikrinti sudėtinga. Šiame straipsnyje pateikiamas būdas palaipsniui patvirtinti šių dviejų sistemų egzistavimą. Pasitelkiamas dviejų sistemų, pasireiškiančių per mentalinių modelių teoriją, principas. Be to, laikantis Carnapo redukcijos idėjos, straipsnyje aprašomos dvi procedūros, kuriomis hipotezė patvirtinama palaipsniui. Viena iš jų tyrinėja, kaip žmogaus protas nagrinėja žmogaus proto veiklą darant išvedimus pagal modus (...) tollendo tollens. Kita hipotezė tyrinėja išvedimus iš griežtosios disjunkcijos su trimis disjunktais. (shrink)