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.     

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.     

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.

     

Impossibility Results for Infinite-Electorate Abstract Aggregation Rules

Following Lauwers and Van Liedekerke (1995), this paper explores in a model-theoretic framework the relation between Arrovian aggregation rules and ultraproducts, in order to investigate a source of impossibility results for the case of an infinite number of individuals and an aggregation rule based on a free ultrafilter of decisive coalitions.     

Agreement Theorems in Dynamic-Epistemic Logic

This paper introduces Agreement Theorems to dynamic-epistemic logic. We show first that common belief of posteriors is sufficient for agreement in epistemic-plausibility models, under common and well-founded priors. We do not restrict ourselves to the finite case, showing that in countable structures the results hold if and only if the underlying plausibility ordering is well-founded. We then show that neither well-foundedness nor common priors are expressible in the language commonly used to describe and reason about epistemic-plausibility models. The static agreement result is, however, finitely derivable in an extended modal logic. We provide the full derivation. We finally consider dynamic agreement results. We show they have a counterpart in epistemic-plausibility models, and provide a new form of agreements via public announcements.     

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.     

Logic and Aggregation

Paraconsistent logic is an area of philosophical logic that has yet to find acceptance from a wider audience. The area remains, in a word, disreputable. In this essay, we try to reassure potential consumers that it is not necessary to become a radical in order to use paraconsistent logic. According to the radicals, the problem is the absurd classical account of contradiction: Classically inconsistent sets explode only because bourgeois classical semantics holds, in the face of overwhelming evidence to the contrary, that both A and A cannot simultaneously be true! We suggest (more modestly) that there is, at least sometimes, something else worth preserving, even in an inconsistent, unsatisfiable premise set. In this paper we present, in a new guise, a very general version of this preservationist approach to paraconsistency.     

社会群体中的偏好逻辑

在许多多主体偏好逻辑系统中,主体之间是没有联系的,因而无法描述主体间的偏好互动。借鉴＂The Logic in the Community＂一文中称为＂群体压力＂的例子对偏好的影响,本文在＂The Logic in the Community＂所提出的系统构架上,加入偏好算子和两个动态算子,通过构造归约公理说明这些算子可以被无偿添加,并借助这些规约公理证明了系统的完全性。     

Some Embedding Theorems for Conditional Logic

We prove some embedding theorems for classical conditional logic, covering ‘finitely cumulative’ logics, ‘preferential’ logics and what we call ‘semi-monotonic’ logics. Technical tools called ‘partial frames’ and ‘frame morphisms’ in the context of neighborhood semantics are used in the proof.     

A Canonical Model for Constant Domain Basic First-Order Logic

Studia Logica - I build a canonical model for constant domain basic first-order logic ( $$\textsf {BQL}_{\textsf {CD}}$$ ), the constant domain first-order extension of Visser’s basic...     

On the Proof-Theory of two Formalisations of Modal First-Order Logic

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.     

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...     

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.     

一个带有事务管理的动态偏好逻辑

问题对偏好的变化会有影响。本文在偏好逻辑的背景下考虑事务管理并提出了一个新的带有事务管理的动态偏好逻辑。我们的逻辑表明它能把其他情况均同的偏好（ceteris paribus preference）与普通的偏好联系起来并将其动态化和显性化。并且,通过引入混合逻辑使得我们能在同一层次谈论命题和对象之间的偏好,除提问这个行动之外,我们还在逻辑系统的动态部分考虑了解决、建议和软公开宣告等行动。在对这四种行动的不同影响进行比较后,我们证明了这个动态系统的完全性。