Three-Valued Temporal Logic <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">t</Emphasis></Subscript> and Future Contingents

Prior’s three-valued modal logic Q was developed as a philosophically interesting modal logic. Thus, we should be able to modify Q as a temporal logic. Although a temporal version of Q was suggested by Prior, the subject has not been fully explored in the literature. In this paper, we develop a three-valued temporal logic Q _t and give its axiomatization and semantics. We also argue that Q _t provides a smooth solution to the problem of future contingents. Presented by Daniele Mundici     

Non-Adjunctive Inference and Classical Modalities

The article focuses on representing different forms of non-adjunctive inference as sub-Kripkean systems of classical modal logic, where the inference from □A and □B to □A∧B fails. In particular we prove a completeness result showing that the modal system that Schotch and Jennings derive from a form of non-adjunctive inference in (Schotch and Jennings, 1980) is a classical system strictly stronger than EMN and weaker than K (following the notation for classical modalities presented in Chellas, 1980). The unified semantical characterization in terms of neighborhoods permits comparisons between different forms of non-adjunctive inference. For example, we show that the non-adjunctive logic proposed in (Schotch and Jennings, 1980) is not adequate in general for representing the logic of high probability operators. An alternative interpretation of the forcing relation of Schotch and Jennings is derived from the proposed unified semantics and utilized in order to propose a more fine-grained measure of epistemic coherence than the one presented in (Schotch and Jennings, 1980). Finally we propose a syntactic translation of the purely implicative part of Jaśkowski's system D₂ into a classical system preserving all the theorems (and non-theorems) explicilty mentioned in (Jaśkowski, 1969). The translation method can be used in order to develop epistemic semantics for a larger class of non-adjunctive (discursive) logics than the ones historically investigated by Jaśkowski.     

A Duality for the Algebras of a Łukasiewicz <Emphasis Type="Italic">n</Emphasis> + 1-valued Modal System

In this paper, we develop a duality for the varieties of a Łukasiewicz n + 1-valued modal system. This duality is an extension of Stone duality for modal algebras. Some logical consequences (such as completeness results, correspondence theory...) are then derived and we propose some ideas for future research. Presented by Daniele Mundici     

Subcellular interactions between parallel fibre and climbing fibre signals in purkinje cells predict sensitivity of classical conditioning to interstimulus interval

Classical conditioning of the nictitating membrane response requires a specific temporal interval between conditioned stimulus and unconditioned stimulus, and produces an incrase in Protein Kinase C (PKC) activation in Purkinje cells. To evaluate whether biochemical interactions within the Purkinje cell may explain the temporal sensitivity, a model of PKC activation byCa ²⁺, diacylglycerol (DAG), and arachidonic acid (AA) is developed.Ca ²⁺ elevation is due to CF stimulation and IP3 inducedCa ²⁺ release (IICR). DAG andIP ₃ result from PF stimulation, while AA results from phospholipase A2 (PLA ₂). Simulations predict increased PKC activation when PF stimulation precedes CF stimulation by 0.1 to 3 s. The sensitivity of IICR to the temporal relation between PF and CF stimulation, together with the buffering system of Purkinje cells, significantly contribute to the temporal sensitivity.     

Finitistic and Frequentistic Approximation of Probability Measures with or without <Emphasis Type="Italic">σ</Emphasis>-Additivity

In this paper a theory of finitistic and frequentistic approximations — in short: f-approximations — of probability measures P over a countably infinite outcome space N is developed. The family of subsets of N for which f-approximations converge to a frequency limit forms a pre-Dynkin system . The limiting probability measure over D can always be extended to a probability measure over , but this measure is not always σ-additive. We conclude that probability measures can be regarded as idealizations of limiting frequencies if and only if σ-additivity is not assumed as a necessary axiom for probabilities. We prove that σ-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full f-approximations. We also show that every non-σ-additive probability measure is f-approximable, though neither canonically nor fully f-approximable. Finally, we transfer our results to probability measures on open or closed formulas of first-order languages.     

Necessity and Relative Contingency

The paper introduces a contingential language extended with a propositional constant τ axiomatized in a system named KΔτ , which receives a semantical analysis via relational models. A definition of the necessity operator in terms of Δ and τ allows proving (i) that KΔτ is equivalent to a modal system named K□τ (ii) that both KΔτ and K□τ are tableau-decidable and complete with respect to the defined relational semantics (iii) that the modal τ -free fragment of KΔτ is exactly the deontic system KD. In §4 it is proved that the modal τ -free fragment of a system KΔτw weaker than KΔτ is exactly the minimal normal system K. Presented by Daniele Mundici     

Electrical conductivity of Nd and Ce co-doped Gd2Zr2O7 ceramics

In this study, (Gd_{1? x} Nd _x )₂(Zr_{1? x} Ce _x )₂O₇ (0 ≤ x ≤ 0.5) ceramics have been prepared by pressureless sintering at 1973 K to investigate the influence of Nd and Ce co-doping on their electrical conductivity. The electrical conductivity of the ceramics was investigated by impedance spectroscopy measurements from 723 to 1173 K over the frequency range of 20 Hz to 2 MHz in air. The measured values obey the Arrhenius relation. For each composition, the grain conductivity gradually increases with increasing temperature from 723 to 1173 K. At a given temperature, it gradually decreases with increasing neodymium and cerium contents from x = 0 to 0.3; thereafter, the grain conductivity exhibits a slight increase with further increasing neodymium and cerium contents up to x = 0.5.     

Constructive Logic with Strong Negation is a Substructural Logic. I

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL _ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL _ew . Presented by Heinrich Wansing     

Syntactical results on the arithmetical completeness of modal logic

In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS of a modal sequentS, which allow the transformation of PA-proofs ofS into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of the form p _i,m _i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic.     

FOIL Axiomatized

In an earlier paper, [5], I gave semantics and tableau rules for a simple firstorder intensional logic called FOIL, in which both objects and intensions are explicitly present and can be quantified over. Intensions, being non-rigid, are represented in FOIL as (partial) functions from states to objects. Scoping machinery, predicate abstraction, is present to disambiguate sentences like that asserting the necessary identity of the morning and the evening star, which is true in one sense and not true in another.In this paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [2] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view.This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.     

Free Modal Lattices via Priestley Duality

A Priestley duality is developed for the variety j of all modal lattices. This is achieved by restricting to j a known Priestley duality for the variety of all bounded distributive lattices with a meet-homomorphism. The variety j was first studied by R. Beazer in 1986.The dual spaces of free modal lattices are constructed, paralleling P.R. Halmos' construction of the dual spaces of free monadic Boolean algebras and its generalization, by R. Cignoli, to distributive lattices with a quantifier.     

First-order Expressivity for S5-models: Modal vs. Two-sorted Languages

Standard models for model predicate logic consist of a Kripke frame whose worlds come equipped with relational structures. Both modal and two-sorted predicate logic are natural languages for speaking about such models. In this paper we compare their expressivity. We determine a fragment of the two-sorted language for which the modal language is expressively complete on S5-models. Decidable criteria for modal definability are presented.     

Game logic and its applications I

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II.The authors thank Hiroakira Ono for helpful discussions and encouragements from the early stage of this research project, and Philippe Mongin, Mitio Takano and a referee of this journal for comments on earlier versions of this paper. The first and second authors are partially supported, respectively, by Tokyo Center of Economic Research and Grant-in-Aids for Scientific Research 04640215, Ministry of Education, Science and Culture.Presented by H. Ono     

Completeness of S4 for the Lebesgue Measure Algebra

We prove completeness of the propositional modal logic S4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, ^[¯]\Box and \Diamond\Diamond. Propositional modal formulae are assigned to Lebesgue-measurable subsets of the real interval [0, 1], modulo sets of measure zero. Equivalence classes of Lebesgue-measurable subsets form a measure algebra, M\mathcal M, and we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in M\mathcal M with an open representative. We prove completeness of the modal logic S4 for the algebra M\mathcal M. A corollary to the main result is that non-theorems of S4 can be falsified at each point in a subset of the real interval [0, 1] of measure arbitrarily close to 1. A second corollary is that Intuitionistic propositional logic (IPC) is complete for the frame of open elements in M\mathcal M.     

Non-Commutative Topology and Quantales

The relationship between q-spaces (c.f. [9]) and quantum spaces (c.f. [5]) is studied, proving that both models coincide in the case of Spec A, the spectrum of a non-commutative C*-algebra A. It is shown that a sober T ₁ quantum space is a classical topological space. This difficulty is circumvented through a new definition of point in a quantale. With this new definition, it is proved that Lid A has enough points. A notion of orthogonality in quantum spaces is introduced, which permits us to express the usual topological properties of separation. The notion of stalks of sheaves over quantales is introduced, and some results in categorial model theory are obtained.     

The ontology of intentionality I: the dependence ontological account of order: mediate and immediate moments and pieces of dependent and independent objects

This is the first of three essays which use Edmund Husserl's dependence ontology to formulate a non-Diodorean and non-Kantian temporal semantics for two-valued, first-order predicate modal languages suitable for expressing ontologies of experience (like physics and cognitive science). This essay's primary desideratum is to formulate an adequate dependence-ontological account of order. To do so it uses primitive (proper) part and (weak) foundation relations to formulate seven axioms and 28 definitions as a basis for Husserl's dependence ontological theory of relating moments. The essay distinguishes between dependence v. independence, pieces v. moments, mediate v. immediate pieces and moments, maximal v. non-maximal pieces, founded v. unfounded qualities, integrative v. disintegrative dependence, and defines the concepts of the completion of an object, the adumbrational equivalence relation of objects, moments of unity which unify objects, and relating moments which relate objects. The eight theorems [CUT90]-[CUT97] show that relating moments of unity provide an adequate account of order in terms of primitive (proper) part and (weak) foundation relations.     

Interfacial precipitation of the M5B3-type boride in Ni-based superalloys

Microstructural characteristics of an experimental Ni-based superalloy with boron addition subjected to a long-term ageing treatment were systematically investigated by various kinds of transmission electron microscopy technique. Based on detailed electron diffraction analyses, we found that there are many nanosized M₅B₃ precipitates in our long-term ageing alloys, which keeps a good orientation relationship with the γ/γ′ matrix. Furthermore, the precipitation characteristics of M₅B₃ phase were clarified. It is found that the M₅B₃-type boride prefers to precipitate at the γ/γ′ interfaces and low-angle grain boundaries. These interfacial nanosized precipitates can play the role of pinning effect and are expected to be advantageous for postponing the γ′ rafting and low-angle grain boundary migration to some extent at high temperature.     

Electrical properties of thin films of a-Ga x Te100− x composed of nanoparticles

Thin films of Ga _x Te_{100? x} (x?=?3, 6, 9 and 12) have been synthesized by thermal evaporation. From SEM images, it is observed that all the films contain nanoparticles of sizes varying from 100 to 200?nm. The dc electrical conductivity of the as-deposited films of Ga _x Te_100?x nanoparticles is measured as a function of temperature range from 298 to 383?K, and increases exponentially with temperature. The value of the activation energy, calculated from the slope of ln?σ _dc versus 1000/T plots, is found to decrease with increase in the Ga content. On the basis of the value of the pre-exponential factor σ _o, it is suggested that the conduction is due to thermally assisted tunneling of carriers in localized states near the band edges. The optical measurements suggest an indirect optical band gap in this system. The value of the optical band gap decreases on increasing the Ga concentration.     

On Modal <Emphasis Type="Italic">μ</Emphasis>-Calculus and Gödel-Löb Logic

We show that the modal μ-calculus over GL collapses to the modal fragment by showing that the fixpoint formula is reached after two iterations and answer to a question posed by van Benthem in [4]. Further, we introduce the modal μ ^~-calculus by allowing fixpoint constructors for any formula where the fixpoint variable appears guarded but not necessarily positive and show that this calculus over GL collapses to the modal fragment, too. The latter result allows us a new proof of the de Jongh, Sambin Theorem and provides a simple algorithm to construct the fixpoint formula. Presented by Melvin Fitting