scholarly journals Conservative Intensional Extension of Tarski's Semantics

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Zoran Majkić
Keyword(s):  
Free Algebra ◽  
Possible Worlds ◽  
Order Logic ◽  
Point Of View ◽  
Relational Algebra ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
Possible World ◽  
Logic Formula ◽  
First Order

We considered an extension of the first-order logic (FOL) by Bealer's intensional abstraction operator. Contemporary use of the term “intension” derives from the traditional logical Frege-Russell doctrine that an idea (logic formula) has both an extension and an intension. Although there is divergence in formulation, it is accepted that the “extension” of an idea consists of the subjects to which the idea applies, and the “intension” consists of the attributes implied by the idea. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In the case of standard FOL, we obtain a commutative homomorphic diagram, which is valid in each given possible world of an intensional FOL: from a free algebra of the FOL syntax, into its intensional algebra of concepts, and, successively, into an extensional relational algebra (different from Cylindric algebras). Then we show that this composition corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.

A First-Order Logic of Limited Belief Based on Possible Worlds

2020 ◽  
Author(s):  
Gerhard Lakemeyer ◽  
Hector J. Levesque
Keyword(s):  
Knowledge Base ◽  
Epistemic Logic ◽  
Possible Worlds ◽  
Order Logic ◽  
First Order Logic ◽  
Possible World ◽  
Possible World Semantics ◽  
First Order ◽  
Epistemic States

In a recent paper Lakemeyer and Levesque proposed a first-order logic of limited belief to characterize the beliefs of a knowledge base (\KB). Among other things, they show that their model of belief is expressive, eventually complete, and tractable. This means, roughly, that a \KB\ may consist of arbitrary first-order sentences, that any sentence which is logically entailed by the \KB\ is eventually believed, given enough reasoning effort, and that reasoning is tractable under reasonable assumptions. One downside of the proposal is that epistemic states are defined in terms of sets of clauses, possibly containing variables, giving the logic a distinct syntactic flavour compared to the more traditional possible-world semantics found in the literature on epistemic logic. In this paper we show that the same properties as above can be obtained by defining epistemic states as sets of three-valued possible worlds. This way we are able to shed new light on those properties by recasting them using the more familiar notion of truth over possible worlds.

scholarly journals The Road to Modern Logic—An Interpretation

2001 ◽  
Vol 7 (4) ◽  
pp. 441-484 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Order Logic ◽  
Point Of View ◽  
First Order Logic ◽  
Theoretical Point ◽  
Primary Role ◽  
Philosophical Discussion ◽  
Modern Logic ◽  
Core System ◽  
First Order ◽  
The Road

AbstractThis paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort)

Logical Tools

2021 ◽  
pp. 14-52
Author(s):  
Cian Dorr ◽  
John Hawthorne ◽  
Juhani Yli-Vakkuri
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
System P

This chapter presents the system of classical higher-order modal logic which will be employed throughout this book. Nothing more than a passing familiarity with classical first-order logic and standard systems of modal logic is presupposed. We offer some general remarks about the kind of commitment involved in endorsing this logic, and motivate some of its more non-standard features. We also discuss how talk about possible worlds can be represented within the system.

On the number of generators of cylindric algebras

1985 ◽  
Vol 50 (4) ◽  
pp. 865-873
Author(s):  
H. Andréka ◽  
I. Németi
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Single Element ◽  
Abstract Model ◽  
Finitely Generated ◽  
Cylindric Algebras ◽  
Abstract Model Theory ◽  
First Order ◽  
Cylindric Set Algebras

The theory of cylindric algebras (CA's) is the algebraic theory of first order logics. Several ideas about logic are easier to formulate in the frame of CA-theory. Such are e.g. some concepts of abstract model theory (cf. [1] and [10]–[12]) as well as ideas about relationships between several axiomatic theories of different similarity types (cf. [4] and [10]). In contrast with the relationship between Boolean algebras and classical propositional logic, CA's correspond not only to classical first order logic but also to several other ones. Hence CA-theoretic results contain more information than their counterparts in first order logic. For more about this see [1], [3], [5], [9], [10] and [12].Here we shall use the notation and concepts of the monographs Henkin-Monk-Tarski [7] and [8]. ω denotes the set of natural numbers. CAα denotes the class of all cylindric algebras of dimension α; by “a CAα” we shall understand an element of the class CAα. The class Dcα ⊆ CAα was defined in [7]. Note that Dcα = 0 for α ∈ ω. The classes Wsα, and Csα were defined in 1.1.1 of [8], p. 4. They are called the classes of all weak cylindric set algebras, regular cylindric set algebras and cylindric set algebras respectively. It is proved in [8] (I.7.13, I.1.9) that ⊆ CAα. (These inclusions are proper by 7.3.7, 1.4.3 and 1.5.3 of [8].)It was proved in 2.3.22 and 2.3.23 of [7] that every simple, finitely generated Dcα is generated by a single element. This is the algebraic counterpart of a property of first order logics (cf. 2.3.23 of [7]). The question arose: for which simple CAα's does “finitely generated” imply “generated by a single element” (see p. 291 and Problem 2.3 in [7]). In terms of abstract model theory this amounts to asking the question: For which logics does the property described in 2.3.23 of [7] hold? This property is roughly the following. In any maximal theory any finite set of concepts is definable in terms of a single concept. The connection with CA-theory is that maximal theories correspond to simple CA's (the elements of which are the concepts of the original logic) and definability corresponds to generation.

Logical Methods for Self-Configuration of Network Devices

2011 ◽  
pp. 189-216 ◽  
Author(s):  
Sylvain Hallé ◽  
Roger Villemaire ◽  
Omar Cherkaoui
Keyword(s):  
Mathematical Logic ◽  
Order Logic ◽  
Point Of View ◽  
First Order Logic ◽  
Declarative Languages ◽  
First Order ◽  
User Intervention

The goal of self-configuration consists of providing appropriate values for parameters that modulate the behaviour of a device. In this chapter, self-configuration is studied from a mathematical logic point of view. In contrast with imperative means of generating configurations, characterized by scripts and templates, the use of declarative languages such as propositional or first-order logic is argued. In that setting, device configurations become models of particular logical formulæ, which can be generated using constraint solvers without any rigid scripting or user intervention.

scholarly journals Encoding hybridized institutions into first-order logic

2014 ◽  
Vol 26 (5) ◽  
pp. 745-788 ◽  
Author(s):  
RĂZVAN DIACONESCU ◽  
ALEXANDRE MADEIRA
Keyword(s):  
Possible Worlds ◽  
Sufficient Conditions ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Institution Theory ◽  
Wide Range ◽  
Characteristic Features ◽  
Verification Process ◽  
Base Logic

A ‘hybridization’ of a logic, referred to as the base logic, consists of developing the characteristic features of hybrid logic on top of the respective base logic, both at the level of syntax (i.e. modalities, nominals, etc.) and of the semantics (i.e. possible worlds). By ‘hybridized institutions’ we mean the result of this process when logics are treated abstractly as institutions (in the sense of the institution theory of Goguen and Burstall). This work develops encodings of hybridized institutions into (many-sorted) first-order logic (abbreviated $\mathcal{FOL}$) as a ‘hybridization’ process of abstract encodings of institutions into $\mathcal{FOL}$, which may be seen as an abstraction of the well-known standard translation of modal logic into $\mathcal{FOL}$. The concept of encoding employed by our work is that of comorphism from institution theory, which is a rather comprehensive concept of encoding as it features encodings both of the syntax and of the semantics of logics/institutions. Moreover, we consider the so-called theoroidal version of comorphisms that encode signatures to theories, a feature that accommodates a wide range of concrete applications. Our theory is also general enough to accommodate various constraints on the possible worlds semantics as well a wide variety of quantifications. We also provide pragmatic sufficient conditions for the conservativity of the encodings to be preserved through the hybridization process, which provides the possibility to shift a formal verification process from the hybridized institution to $\mathcal{FOL}$.

Omitting types for finite variable fragments and complete representations of algebras

2008 ◽  
Vol 73 (1) ◽  
pp. 65-89 ◽  
Author(s):  
Hajnal Andréka ◽  
István Németi ◽  
Tarek Sayed Ahmed
Keyword(s):  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
First Order ◽  
Complete Representations ◽  
Atomic Relation ◽  
Representations Of Algebras ◽  
Omitting Types ◽  
Binary Relation Symbol

AbstractWe give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic.

scholarly journals Possibility Semantics for Intuitionistic Logic

2004 ◽  
Vol 2 ◽  
Author(s):  
M. J. Cresswell
Keyword(s):  
Intuitionistic Logic ◽  
Possible Worlds ◽  
Predicate Logic ◽  
Order Logic ◽  
Semantic Interpretation ◽  
First Order Logic ◽  
First Order ◽  
Partial Indices ◽  
Closure Conditions ◽  
First Order Predicate Logic

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

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