We consider the results of adding to a basic modal system T0the axioms G1.CLpp;Pn.CLnpLn+1p; Bn.CpLnMp, where n ≧ 11, in all combinations. The method of Meredith's [7] will be extended to get models of these systems in lower predicate calculus (LPC) with a constant binary relation,U. Most of the results were already obtained in [1]–[6], though systems as in (i) and (ii) below were not investigated, except that S40in (ii) was mentioned in [1]. However some repetition (...) may be excused in view of the simplicity with which the results are obtained by the present method. (shrink)

This approach does not define a probability measure by syntactical structures. It reveals a link between modal logic and mathematical probability theory. This is shown (1) by adding an operator (and two further connectives and constants) to a system of lower predicate calculus and (2) regarding the models of that extended system. These models are models of the modal system S₅ (without the Barcan formula), where a usual probability measure is defined on their set of (...) possible worlds. Mathematical probability models can be seen as models of S₅. (shrink)

In order to avoid the use of individual variables in predicate calculus, several authors proposed language whose expressions can be interpreted, in general, as denotations of predicates . The present author also proposed a language of this kind [5]. The absence of individual variables makes all these languages rather different from the traditional language of predicate calculus and from the usual language of mathematics. The translation procedures from the ordinary predicate languages into the predicate (...) languages without individual variables and vice versa have a technical character. It is desirable to make possible carrying out such procedures by performing some transformations in a suitable new language which comprises both a sublanguage similar to the ordinary predicate languages and another one not using individual variables. We shall propose now a language of the desired kind. The expressions of this language could be interpreted, in general, as denoting predicates which depend on parameters on a given object domain – this predicate has one argument and depends on one parameter). The proposed language will have the modal features noted in [3] and [5]. It is worthy of mention that the language proposed at the end of [4] although containing expressions interpreted as predicates and expressions with parameters does not allow simultaneous availability of argument places and parameters. (shrink)

The modal object calculus is the system of logic which houses the (proper) axiomatic theory of abstract objects. The calculus has some rather interesting features in and of itself, independent of the proper theory. The most sophisticated, type-theoretic incarnation of the calculus can be used to analyze the intensional contexts of natural language and so constitutes an intensional logic. However, the simpler second-order version of the calculus couches a theory of fine-grained properties, relations and propositions (...) and serves as a framework for defining situations, possible worlds, stories, and fictional characters, among other things. In the present paper, we focus on the second-order calculus. The second-order modal object calculus is so-called to distinguish it from the second-order modal predicate calculus. Though the differences are slight, the extra expressive power of the object calculus significantly enhances its ability to resolve logical and philosophical concepts and problems. (shrink)

Global properties of canonical derivability predicates in Peano Arithmetic) are studied here by means of a suitable propositional modal logic GL. A whole book [1] has appeared on GL and we refer to it for more information and a bibliography on GL. Here we propose a sequent calculus for GL and, by exhibiting a good proof procedure, prove that such calculus admits the elimination of cuts. Most of standard results on GL are then easy consequences: completeness, decidability, (...) finite model property, interpolation and the fixed point theorem. (shrink)

This paper treats a kind of a modal logic based on the intuitionistic propositional logic which arose from the provability predicate in the first order arithmetic. The semantics of this calculus is presented in both a relational and an algebraic way.Completeness theorems, existence of a characteristic model and of a characteristic frame, properties of FMP and FFP and decidability are proved.

This paper is devoted to present Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method for multi-attribute group decision making under rough neutrosophic environment. The concept of rough neutrosophic set is a powerful mathematical tool to deal with uncertainty, indeterminacy and inconsistency. In this paper, a new approach for multi-attribute group decision making problems is proposed by extending the TOPSIS method under rough neutrosophic environment. Rough neutrosophic set is characterized by the upper and lower approximation operators and (...) the pair of neutrosophic sets that are characterized by truth-membership degree, indeterminacy membership degree, and falsity membership degree. In the decision situation, ratings of alternatives with respect to each attribute are characterized by rough neutrosophic sets that reflect the decision makers’ opinion. Rough neutrosophic weighted averaging operator has been used to aggregate the individual decision maker’s opinion into group opinion for rating the importance of attributes and alternatives. Finally, a numerical example has been provided to demonstrate the applicability and effectiveness of the proposed approach. (shrink)

I discuss in this paper a criticism of modal logic due to Donald Davidson and John Wallace. They have claimed that, to quote Wallace, “modal predicate calculus does not provide a reasonable standpoint from which to interpret a language” (1970, p. 147). The aim of this paper is to present and evaluate their argument for this claim.

I develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system (...) built on these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language. (shrink)

In this paper the language of first-order modal logic is enriched with an operator @ ('actually') such that, in any model, the evaluation of a formula @A at a possible world depends on the evaluation of A at the actual world. The models have world-variable domains. All the logics that are discussed extend the classical predicate calculus, with or without identity, and conform to the philosophical principle known as serious actualism. The basic logic relies on the system (...) K, whereas others correspond to various properties that the actual world may have. All the logics are axiomatized. (shrink)

We present the design goals and metatheory of the Monadic Hybrid Calculus, a new formal system that has the same power as the Monadic Predicate Calculus. MHC allows quantification, including relative quantification, in a straightforward way without the use of bound variables, using a simple adaptation of modal logic notation. Thus “all Greeks are mortal” can be written as [G]M. MHC is also ‘hybrid’ in that it has individual constants, which allow us to formulate statements about (...) particular individuals. Thus “Socrates is Athenian and mortal” can be formalised as s.For our proof system, we use a simple adaptation of Beth-style tableaus. The availability of individual constants eliminates the need for labelled deduction.We discuss first the pragmatic and pedagogical advantages of MHC. Then we present the metatheory: formal syntax, semantics, proof rules, soundness, completeness and expressive equivalence with the Monadic Predicate Calculus. (shrink)

We offer an analysis of future morphemes as epistemic operators. The main empirical motivation comes from the fact that future morphemes have systematic purely epistemic readings—not only in Greek and Italian, but also in Dutch, German, and English will. The existence of epistemic readings suggests that the future expressions quantify over epistemic, not metaphysical alternatives. We provide a unified analysis for epistemic and predictive readings as epistemic necessity, and the shift between the two is determined compositionally by the lower (...) tense. Our account thus acknowledges a systematic interaction between modality and tense—but the future itself is a pure modal, not a mixed temporal/modal operator. We show that the modal base of the future is nonveridical, i.e. it includes p and ¬p worlds, parallel to epistemic modals such as must, and present arguments that future morphemes are a category that stands in between epistemic modals and predicates of personal taste. We identify, finally, a subclass of epistemic futures which are ratificational, and argue that will is a member of this class. (shrink)

The paper presents and applies Hilbert's Epsilon Calculus, first describing its standard proof theory, and giving it an intensional semantics. These are contrasted with the proof theory of Fregean Predicate Logic, and the traditional (extensional) choice function semantics for the calculus. The semantics provided show that epsilon terms are referring terms in Donnellan's sense, enabling the symbolisation and validation of argument forms involving E-type pronouns, both in extensional and intensional contexts. By providing for transparency in intensional constructions (...) they support a Model Conceptualism to contrast with traditional intensional logic's Modal Realism. But epsilon-terms also symbolise fictions, and through their difference from iota terms enable the solution of a number of outstanding puzzles about Direct Reference and de re beliefs. (shrink)

The paper presents and applies Hilbert's Epsilon Calculus, first describing its standard proof theory, and giving it an intensional semantics. These are contrasted with the proof theory of Fregean Predicate Logic, and the traditional (extensional) choice function semantics for the calculus. The semantics provided show that epsilon terms are referring terms in Donnellan's sense, enabling the symbolisation and validation of argument forms involving E-type pronouns, both in extensional and intensional contexts. By providing for transparency in intensional constructions (...) they support a Model Conceptualism to contrast with traditional intensional logic's Modal Realism. But epsilon-terms also symbolise fictions, and through their difference from iota terms enable the solution of a number of outstanding puzzles about Direct Reference and de re beliefs. (shrink)

We observe that removing contraction from a standard sequent calculus for first-order predicate logic preserves completeness for the modal fragment.

We define a tableau calculus for the logic of only knowing and knowing at most ON, which is an extension of Levesque's logic of only knowing O. The method is based on the possible-world semantics of the logic ON, and can be considered as an extension of known tableau calculi for modal logic K45. From the technical viewpoint, the main features of such an extension are the explicit representation of "unreachable" worlds in the tableau, and an additional branch (...) closure condition implementing the property that each world must be either reachable or unreachable. The calculus allows for establishing the computational complexity of reasoning about only knowing and knowing at most. Moreover, we prove that the method matches the worst-case complexity lower bound of the satisfiability problem for both ON and O. With respect to [22], in which the tableau calculus was originally presented, in this paper we both provide a formal proof of soundness and completeness of the calculus, and prove the complexity results for the logic ON. (shrink)

We establish a completeness theorem for first-order basic predicate logic BQC, a proper subsystem of intuitionistic predicate logic IQC, using Kripke models with transitive underlying frames. We develop the notion of functional well-formed theory as the right notion of theory over BQC for which strong completeness theorems are possible. We also derive the undecidability of basic arithmetic, the basic logic equivalent of intuitionistic Heyting Arithmetic and classical Peano Arithmetic.

We introduce a Gentzen-style modal predicate logic and prove the cut-elimination theorem for it. This sequent calculus of cut-free proofs is chosen as a proxy to develop the proof-theory of the logics introduced in [14, 15, 4]. We present syntactic proofs for all the metatheoretical results that were proved model-theoretically in loc. cit. and moreover prove that the form of weak reflection proved in these papers is as strong as possible.

The main purpose of this paper is to give alternative proofs of syntactical and semantical properties, i.e. the subformula property and the nite model property, of the sequent calculi for the modal logics K4.3, KD4.3, and S4.3. The application of the inference rules is said to be acceptable, if all the formulas in the upper sequents are subformula of the formulas in lower sequent. For some modal logics, Takano analyzed the relationships between the acceptable inference rules and (...) semantical properties by constructing models. By using these relationships, he showed Kripke completeness and subformula property. However, his method is difficult to apply to inference rules for the sequent calculi for K4.3, KD4.3, and S4.3. Lookinglosely at Takano's proof, we nd that his method can be modied to construct nite models based on the sequent calculus for K4.3, if the calculus has and all the applications of the inference rules are acceptable. Similarly, we can apply our results to the calculi for KD4.3 and S4.3. This leads not only to Kripke completeness and subformula property, but also to finite model property of these logics simultaneously. (shrink)

Two subject-predicate calculi with equality,SP = and its extensionUSP =, are presented as systems of natural deduction. Both the calculi are systems of free logic. Their presentation is preceded by an intuitive motivation.It is shown that Aristotle's syllogistics without the laws of identitySaP andSiP is definable withinSP =, and that the first-order predicate logic is definable withinUSP =.

The essay introduces a non-Diodorean, non-Kantian temporal modal semantics based on part-whole, rather than class, theory. Formalizing Edmund Husserl’s theory of inner time consciousness, §3 uses his protention and retention concepts to define a relation of self-awareness on intentional events. §4 introduces a syntax and two-valued semantics for modal first-order predicate object-languages, defines semantic assignments for variables and predicates, and truth for formulae in terms of the axiomatic version of Edmund Husserl’s dependence ontology (viz. the Calculus (...) [CU] of Urelements) introduced by The Ontology of Intentionality I & II. It then uses the §3 results to define the modalities of truth, and §5 extends the semantics to identity claims. §6 defines and contrasts synthetic a priori truths to analytic a priori truths, and §7 compares Brentano School noetic semantic and Leibnizian possible-world semantic perspectives on modality. The essay argues that the modal logics it defines semantically are two-valued, first-order versions of the type of language which Husserl viewed as the language of any ontology of experience (i.e. of any science), and conceived as the logic of intentionality. (shrink)

Predicates of personal taste (fun, tasty) and epistemic modals (might, must) share a similar analytical difficulty in determining whose taste or knowledge is being expressed. Accordingly, they have parallel behavior in attitude reports and in a certain kind of disagreement. On the other hand, they differ in how freely they can be linked to a contextually salient individual, with epistemic modals being much more restricted in this respect. I propose an account of both classes using Lasersohn’s (Linguistics and Philosophy 28: (...) 643–686, 2005) “judge” parameter, at the same time arguing for crucial changes to Lasersohn’s view in order to allow the extension to epistemic modals and address empirical problems faced by his account. (shrink)

The purpose of this paper is to outline an alternative approach to introductory logic courses. Traditional logic courses usually focus on the method of natural deduction or introduce predicate calculus as a system. These approaches complicate the process of learning different techniques for dealing with categorical and hypothetical syllogisms such as alternate notations or alternate forms of analyzing syllogisms. The author's approach takes up observations made by Dijkstrata and assimilates them into a reasoning process based on modified notations. (...) The author's model adopts a notation that addresses the essentials of a problem while remaining easily manipulated to serve other analytic frameworks. The author also discusses the pedagogical benefits of incorporating the model into introductory logic classes for topics ranging from syllogisms to predicate calculus. Since this method emphasizes the development of a clear and manipulable notation, students can worry less about issues of translation, can spend more energy solving problems in the terms in which they are expressed, and are better able to think in abstract terms. (shrink)

In this paper I shall attempt to enter part of the way into the microstructure of the account of truth in the Parmenidean fragment 8, and to reveal that account as a dialectical sequence of affirmation and denial involving various kinds of modal utterance. The sequence will then be put into parallel with the first four hypotheses of the second half of Plato’s Parmenides as well as with Zeno and some of the later tradition.

The Modal Predicate Calculus gives rise to issues surrounding the Barcan formulas, their converses, and necessary existence. I examine these issues by means of the Quantified Argument Calculus, a recently developed, powerful formal logic system. Quarc is closer in syntax and logical properties to Natural Language than is the Predicate Calculus, a fact that lends additional interest to this examination, as Quarc might offer a better representation of our modal concepts. The validity of (...) the Barcan formulas and their converses is shown by Quarc to be a result of the specific incorporation of quantification in the Predicate Calculus, and not as reflecting a feature of the interaction of quantification and modality more generally. Necessary existence is shown to follow from the identification, in the Predicate Calculus on its canonical interpretation, of particular quantification, ascription of existence and the ‘there is’ construction, three constructions which are distinguished in both Quarc and Natural Language. The issues surrounding the Barcan formulas, their converses and necessary existence are thus shown to be an artefact of a specific logic system, not an essential feature of our relevant modal concepts or of formal logic. (shrink)

Mark Jago’s four arguments against Takashi Yagisawa’s extended modal realism are examined and shown to be ineffective. Yagisawa’s device of modal tense renders three of Jago’s arguments harmless, and the correct understanding of predications of modal properties of world stages blocks the fourth one.

I have attempted to show that many attributive adjectives can be dealt with within the framework of first-order predicate calculus by the method suggested in this paper. I've also supplied independent reasons for the claim that attributive adjectives that are not responsive to this method require a formal treatment different from the one that the adjectives successfully dealt with by that method require. Thus, if the method I've argued for is sound, then the scope of first-order predicate (...) calculus was shown to be wider than assumed by several logicians. This I take to be of interest from a logical point of view. (shrink)

Motivated by semantic inferentialism and logical expressivism proposed by Robert Brandom, in this paper, I submit a nonmonotonic modal relevant sequent calculus equipped with special operators, □ and R. The base level of this calculus consists of two different types of atomic axioms: material and relevant. The material base contains, along with all the flat atomic sequents (e.g., Γ0, p |~0 p), some non-flat, defeasible atomic sequents (e.g., Γ0, p |~0 q); whereas the relevant base consists of (...) the local region of such a material base that is sensitive to relevance. The rules of the calculus uniquely and conservatively extend these two types of nonmonotonic bases into logically complex material/relevant consequence relations and incoherence properties, while preserving Containment in the material base and Reflexivity in the relevant base. The material extension is supra-intuitionistic, whereas the relevant extension is stronger than a logic slightly weaker than R. The relevant extension also avoids the fallacies of relevance. Although the extended material consequence relation is defeasible and insensitive to relevance, it has local regions of indefeasibility and relevance (the latter of which is marked by the relevant extension). The newly introduced operators, □ and R, codify these local regions within the same extended material consequence relation. (shrink)

The most basic laws and principles of the Predicate Calculus, also known as Quantification Theory, are stated, as clearly and concisely as possible.