Categories of frames for modal logic

1975 ◽  
Vol 40 (3) ◽  
pp. 439-442 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Boolean Algebra ◽  
Binary Relation ◽  
Polynomial Identity ◽  
Category Theory ◽  
Kripke Frame ◽  
Modal Algebra ◽  
Modal Formula ◽  
Unary Operator ◽  
Atomic Boolean Algebra

§1. A complete atomic modal algebra (CAMA) is a complete atomic Boolean algebra with an additional completely additive unary operator. A (Kripke) frame is just a binary relation on a nonempty set. If is a frame, then is a CAMA, where mX = {y ∣ (∃x)(y < x Є X)}; and if is a CAMA then is a frame, where is the set of atoms of and b1 < b2 ⇔ b1 ∩ mb2 ≠∅.Now , and the validity of a modal formula on is equivalent to the satisfaction of a modal algebra polynomial identity by and conversely, so the validity-preserving constructions on frames ought to be in some sense equivalent to the identity-preserving constructions on CAMA's. The former are important for modal logic, and many of the results of universal algebra apply to the latter, so it is worthwhile to fix precisely the sense of the equivalence.The most important identity-preserving constructions on CAMA's can be described in terms of homomorphisms and complete homomorphisms. Let and be the categories of CAMA's with homomorphisms and complete homomorphisms, respectively. We shall define categories and of frames with appropriate morphisms, and show them to be dual respectively to and . Then we shall consider certain identity-preserving constructions on CAMA's and attempt to describe the corresponding validity-preserving constructions on frames.The proofs of duality involve some rather detailed calculations, which have been omitted. All the category theory a reader needs to know is in the first twenty pages of [7].

An Outline of a Theory of Propositional Vagueness

2018 ◽  
Author(s):  
Andrew Bacon
Keyword(s):  
Boolean Algebra ◽  
Total Evidence ◽  
Coarse Grained ◽  
Atomic Boolean Algebra

This chapter presents a series questions in the philosophy of vagueness that will constitute the primary subjects of this book. The stance this book takes on these questions is outlined, and some preliminary ramifications are explored. These include the idea that (i) propositional vagueness is more fundamental than linguistic vagueness; (ii) propositions are not themselves sentence-like; they are coarse grained, and form a complete atomic Boolean algebra; (iii) vague propositions are, moreover, not simply linguistic constructions either such as sets of world-precisification pairs; and (iv) propositional vagueness is to be understood by its role in thought. Specific theses relating to the last idea include the thesis that one’s total evidence can be vague, and that there are vague propositions occupying every evidential role, that disagreements about the vague ultimately boil down to disagreements in the precise, and that one should not care intrinsically about vague matters.

scholarly journals Object-Free Definition of Categories

2013 ◽  
Vol 21 (3) ◽  
pp. 193-205
Author(s):  
Marco Riccardi
Keyword(s):  
Binary Relation ◽  
Category Theory ◽  
Final Part ◽  
One To One ◽  
Definition Of

Summary Category theory was formalized in Mizar with two different approaches [7], [18] that correspond to those most commonly used [16], [5]. Since there is a one-to-one correspondence between objects and identity morphisms, some authors have used an approach that does not refer to objects as elements of the theory, and are usually indicated as object-free category [1] or as arrowsonly category [16]. In this article is proposed a new definition of an object-free category, introducing the two properties: left composable and right composable, and a simplification of the notation through a symbol, a binary relation between morphisms, that indicates whether the composition is defined. In the final part we define two functions that allow to switch from the two definitions, with and without objects, and it is shown that their composition produces isomorphic categories.

An algebraic characterization of power set in countable standard models of ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 167-170
Author(s):  
George Metakides ◽  
J. M. Plotkin
Keyword(s):  
Standard Model ◽  
Boolean Algebra ◽  
Classical Result ◽  
Algebraic Characterization ◽  
Categorical Theory ◽  
Power Set ◽  
Standard Models ◽  
Image Position ◽  
Atomic Boolean Algebra

The following is a classical result:Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ0-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈PN(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such thatIt is clear that a pseudo-power set is atomic.

scholarly journals Orthomodular generalizations of homogeneous Boolean algebras

1973 ◽  
Vol 15 (1) ◽  
pp. 94-104 ◽  
Author(s):  
C. H. Randall ◽  
M. F. Janowitz ◽  
D. J. Foulis
Keyword(s):  
Boolean Algebra ◽  
Dense Subset ◽  
Unit Interval ◽  
Boolean Algebras ◽  
Borel Sets ◽  
History Of ◽  
Atomic Boolean Algebra

It is well known that the so-called reduced Borel algebra, that is, the Boolean algebra of all Borel subsets of the unit interval modulo the meager Borel sets of this interval, can be abstractly characterized as a complete, totally non-atomic Boolean algebra containing a countable join dense subset. (For an indication of the history of this result, see, for example, ([2], p. 483, footnote 12.) From this characterization, it easily follows that the reduced Borel algebra B is “homogeneous” in the sense that every non-trivial in B is isomorphic to B.

scholarly journals On the order scale of a uniform space

1983 ◽  
Vol 34 (2) ◽  
pp. 248-257 ◽  
Author(s):  
D. C. Kent
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Closed Subspace ◽  
Uniform Space ◽  
Order Topology ◽  
Totally Bounded ◽  
Samuel Compactification ◽  
Closed Image ◽  
Atomic Boolean Algebra

AbstractThe order topology is compact and T2 in both the scale and retracted scale of any uniform space (S, U). if (S, U) is T2 and totally bounded, the Samuel compactification associated with (S, U) can be obtained by uniformly embedding (S, U) in its order retracted scale (that is, the retracted scale with its order topology). This implies that every compact T2 space is both a closed subspace of a complete, infinitely distributive lattice in its order topology, and also a continuous, closed image of a closed subspace of a complete atomic Boolean algebra in its order topology.

First-order definability in modal logic

1975 ◽  
Vol 40 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. I. Goldblatt
Keyword(s):  
Modal Logic ◽  
Binary Relation ◽  
Second Order ◽  
Order Conditions ◽  
Kripke Models ◽  
Order Condition ◽  
First Order ◽  
Axiom Systems ◽  
Set Of Points ◽  
First Order Definability

In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp.Definition 1. A modal frame ℱ = 〈W, R〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).

NEITHER CATEGORICAL NOR SET-THEORETIC FOUNDATIONS

2012 ◽  
Vol 6 (1) ◽  
pp. 16-23
Author(s):  
GEOFFREY HELLMAN
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Category Theory ◽  
Top Down ◽  
Ongoing Debate ◽  
Modest Proposal ◽  
Categories And Functors ◽  
Theoretic Foundations

AbstractFirst we review highlights of the ongoing debate about foundations of category theory, beginning with Feferman’s important article of 1977, then moving to our own paper of 2003, contrasting replies by McLarty and Awodey, and our own rejoinders to them. Then we offer a modest proposal for reformulating a theory of category of categories that would actually meet the objections of Feferman and Hellman and provide a genuine alternative to set theory as a foundation for mathematics. This proposal is more modest than that of our (2003) in omitting modal logic and in permitting a more “top-down” approach, where particular categories and functors need not be explicitly defined. Possible reasons for resisting the proposal are offered and countered. The upshot is to sustain a pluralism of foundations along lines actually foreseen by Feferman (1977), something that should be welcomed as a way of resolving this long-standing debate.

The rise and fall of typed sentences

2000 ◽  
Vol 65 (4) ◽  
pp. 1858-1862
Author(s):  
Marcel Crabbé
Keyword(s):  
Boolean Algebra ◽  
Atomic Boolean Algebra

AbstractWe characterize the 3-stratiflable theorems of NF as a 3-stratifiable extension of NF3: and show that NF is equiconsistent with TT plus raising type axioms for sentences asserting the existence of some predicate over an atomic Boolean algebra.

scholarly journals The modal logic of affine planes is not finitely axiomatisable

2008 ◽  
Vol 73 (3) ◽  
pp. 940-952
Author(s):  
Ian Hodkinson ◽  
Altaf Hussain
Keyword(s):  
Modal Logic ◽  
Affine Plane ◽  
Kripke Frame ◽  
Affine Planes ◽  
Modal Language ◽  
Line Point ◽  
Point Line

AbstractWe consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable.

scholarly journals Tableau-based translation from first-order logic to modal logic

2021 ◽  
Vol 56 ◽  
pp. 57-74
Author(s):  
Tin Perkov ◽  
Luka Mikec
Keyword(s):  
Modal Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Modal Formula ◽  
First Order ◽  
Starting Point ◽  
Invariance Test

We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

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