First-Order Classical Modal Logic

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.     

First-Order da Costa Logic

Priest (2009) formulates a propositional logic which, by employing the worldsemantics for intuitionist logic, has the same positive part but dualises the negation, to produce a paraconsistent logic which it calls ‘Da Costa Logic’. This paper extends matters to the first-order case. The paper establishes various connections between first order da Costa logic, da Costa’s own C_ω, and classical logic. Tableau and natural deductions systems are provided and proved sound and complete.     

Idealization in Applied First-Order Logic

Applying first-order logic to derive the consequences of laws that are only approximately true of empirical phenomena involves idealization of a kind that is akin to applying arithmetic to calculate the area of a rectangular surface from approximate measures of the lengths of its sides. Errors in the data, in the exactness of the lengths in one case and in the exactness of the laws in the other, are in some measure transmitted to the consequences deduced from them, and the aim of a theory of idealization is to describe this process. The present paper makes a start on this in the case of applied first-order logic, and relates it to Plato's picture of a world or model of 'appearances' in which laws are only approximately true, but which to some extent resembles an ideal world or model in which they are exactly true. This revised version was published online in June 2006 with corrections to the Cover Date.     

Loosely Guarded Fragment of First-Order Logic has the Finite Model Property

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.     

A Temporal Semantics for Basic Logic

In the context of truth-functional propositional many-valued logics, Hájek’s Basic Fuzzy Logic BL [14] plays a major rôle. The completeness theorem proved in [7] shows that BL is the logic of all continuous t-norms and their residua. This result, however, does not directly yield any meaningful interpretation of the truth values in BL per se. In an attempt to address this issue, in this paper we introduce a complete temporal semantics for BL. Specifically, we show that BL formulas can be interpreted as modal formulas over a flow of time, where the logic of each instant is ?ukasiewicz, with a finite or infinite number of truth values. As a main result, we obtain validity with respect to all flows of times that are non-branching to the future, and completeness with respect to all finite linear flows of time, or to an appropriate single infinite linear flow of time. It may be argued that this reduces the problem of establishing a meaningful interpretation of the truth values in BL logic to the analogous problem for ?ukasiewicz logic.     

Equality and Monodic First-Order Temporal Logic

It has been shown recently that monodic first-order temporal logic without functional symbols but with equality is incomplete, i.e., the set of the valid formulae of this logic is not recursively enumerable. In this paper we show that an even simpler fragment consisting of monodic monadic two-variable formulae is not recursively enumerable.     

Interpolation in Extensions of First-Order Logic

We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.

     

Completeness and Cut-Elimination for First-Order Ideal Paraconsistent Four-Valued Logic

Studia Logica - In this study, we prove the completeness and cut-elimination theorems for a first-order extension F4CC of Arieli, Avron, and Zamansky’s ideal paraconsistent four-valued logic...     

Lottery Semantics: A Compositional Semantics for Probabilistic First-Order Logic with Imperfect Information

We present a compositional semantics for first-order logic with imperfect information that is equivalent to Sevenster and Sandu’s equilibrium semantics (under which the truth value of a sentence in a finite model is equal to the minimax value of its semantic game). Our semantics is a generalization of an earlier semantics developed by the first author that was based on behavioral strategies, rather than mixed strategies.     

The Basic Constructive Logic for a Weak Sense of Consistency defined with a Propositional Falsity Constant

The logic B_Kc1 is the basic constructive logic in the ternaryrelational semantics (without a set of designated points) adequateto consistency understood as the absence of the negation ofany theorem. Negation is introduced in B_Kc1 with a negationconnective. The aim of this paper is to define the logic B_Kc1F.In this logic negation is introduced via a propositional falsityconstant. We prove that B_Kc1 and B_Kc1F are definitionally equivalent.     

A Translation of Intuitionistic Predicate Logic into Basic Predicate Logic

Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.     

工作家庭间跨界增益:逻辑主线与模型

已有工作家庭界面研究主要立足于冲突视角,近年来部分研究开始从积极视角揭示工作家庭关系的本质。工作家庭增益强调角色资源的跨界渗溢有助提升个体另一角色表现和生活品质的程度。工作家庭资源会增强个体心理资源,进而影响体验工作家庭间的跨界增益,资源的累积循环与工作家庭增益互为持续保障,跨界增益的程度受到工作家庭边界特征的调节影响。论证和提出了"角色资源累积-个体心理因素改变-跨界增益-角色评价与投入"的逻辑主线,以此构建工作家庭间角色资源跨界增益模型。     

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation

In preference aggregation a set of individuals express preferences over a set of alternatives, and these preferences have to be aggregated into a collective preference. When preferences are represented as orders, aggregation procedures are called social welfare functions. Classical results in social choice theory state that it is impossible to aggregate the preferences of a set of individuals under different natural sets of axiomatic conditions. We define a first-order language for social welfare functions and we give a complete axiomatisation for this class, without having the number of individuals or alternatives specified in the language. We are able to express classical axiomatic requirements in our first-order language, giving formal axioms for three classical theorems of preference aggregation by Arrow, by Sen, and by Kirman and Sondermann. We explore to what extent such theorems can be formally derived from our axiomatisations, obtaining positive results for Sen’s Theorem and the Kirman-Sondermann Theorem. For the case of Arrow’s Theorem, which does not apply in the case of infinite societies, we have to resort to fixing the number of individuals with an additional axiom. In the long run, we hope that our approach to formalisation can serve as the basis for a fully automated proof of classical and new theorems in social choice theory.     

Establishing Connections between Aristotle's Natural Deduction and First-Order Logic

This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T _RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T _RD that bear upon the same properties of the natural deduction counterpart – that is, Corcoran's system. Moreover, the first-order logic framework that we work with allows us to understand how complicated the semantics of the syllogistic is in providing us with examples of bizarre, unexpected interpretations of the syllogistic rules. Finally, we provide a first attempt at finding the structure of that semantics, reducing the search to the characterization of the class of models of T _RD.     

A Defense of the Kripkean Account of Logical Truth in First-Order Modal Logic

This paper responds to criticism of the Kripkean account of logical truth in first-order modal logic. The criticism, largely ignored in the literature, claims that when the box and diamond are interpreted as the logical modality operators, the Kripkean account is extensionally incorrect because it fails to reflect the fact that all sentences stating truths about what is logically possible are themselves logically necessary. I defend the Kripkean account by arguing that some true sentences about logical possibility are not logically necessary.