Set theory as modal logic

Studia Logica ◽  
1980 ◽  
Vol 39 (4) ◽  
pp. 335-345
Author(s):  
Herman Dishkant
Keyword(s):  
Modal Logic ◽  
Set Theory

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.

Structuralism

2009 ◽  
Author(s):  
Geoffrey Hellman
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Model Theory ◽  
Second Order ◽  
Mathematical Framework ◽  
Topos Theory ◽  
Mathematical Structuralism

The main types of mathematical structuralism that have been proposed and developed to the point of permitting systematic and instructive comparison are four: structuralism based on model theory, carried out formally in set theory (e.g., first- or second-order Zermelo–Fraenkel set theory), referred to as STS (for set-theoretic structuralism); the approach of philosophers such as Shapiro and Resnik of taking structures to be sui generis universals, patterns, or structures in an ante rem sense (explained in this article), referred to as SGS (for sui generis structuralism); an approach based on category and topos theory, proposed as an alternative to set theory as an overarching mathematical framework, referred to as CTS (for category-theoretic structuralism); and a kind of eliminative, quasi-nominalist structuralism employing modal logic, referred to as MS (for modal-structuralism). This article takes these up in turn, guided by few questions, with the aim of understanding their relative merits and the choices they present.

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

1999 ◽  
Vol 10 (01) ◽  
pp. 29-62 ◽  
Author(s):  
GERMANO RESCONI ◽  
GEORGE J. KLIR ◽  
ELIANO PESSA
Keyword(s):  
Quantum Mechanics ◽  
Modal Logic ◽  
Set Theory ◽  
Possible Worlds ◽  
Evidence Theory ◽  
Foundations Of Quantum Mechanics ◽  
Interpretation Of Quantum Mechanics ◽  
Many Worlds ◽  
New Interpretation ◽  
Quantum Phenomena

Recognizing that syntactic and semantic structures of classical logic are not sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence theory. The connection between these two theories is obtained through a new language, quantum set theory, built on a suggestion by J. Bell. Further, we give a modal logic interpretation of quantum mechanics and quantum set theory by using Kripke's semantics of modal logic based on the concept of possible worlds. This is grounded on previous work of a number of researchers (Resconi, Klir, Harmanec) who showed how to represent evidence theory and other uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in terms of Kripke's semantics. We thus show how three different theories — quantum mechanics, evidence theory, and modal logic — are interrelated. This opens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the possibility of building new generalizations of quantum mechanics itself.

scholarly journals The interpretability logic of Peano arithmetic

1990 ◽  
Vol 55 (3) ◽  
pp. 1059-1089 ◽  
Author(s):  
Alessandro Berarducci
Keyword(s):  
Modal Logic ◽  
Modal Analysis ◽  
Set Theory ◽  
Decision Procedure ◽  
Peano Arithmetic ◽  
Relevant Modal Logic ◽  
Binary Operator ◽  
Base Theory ◽  
Interpretability Logic ◽  
Work Done

AbstractPA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.

A complete and consistent modal set theory

1967 ◽  
Vol 32 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Continuous Functions ◽  
Subsequent Paper ◽  
Object Language ◽  
Real Numbers ◽  
Consistent System ◽  
Detailed Treatment ◽  
The Way

1.1. The aim of this paper is the construction of a demonstrably consistent system of set theory that (1) contains roughly the same amount of mathematics as the writer's system K′ [3], including a theory of continuous functions of real numbers, and (2) provides a way for expressing in the object language various propositions which, in the case of K′, could be expressed only in the metalanguage, for example, general propositions about all real numbers. It was not originally intended that the desired system should be a modal logic, but the modal character of the system appears to be a natural outgrowth of the way it is constructed. A detailed treatment of the natural, rational, and real numbers is left for a subsequent paper.

On systems containing Aristotle's thesis

1968 ◽  
Vol 33 (1) ◽  
pp. 82-96 ◽  
Author(s):  
R. Routley ◽  
H. Montgomery
Keyword(s):  
Modal Logic ◽  
Philosophy Of Science ◽  
Set Theory ◽  
Good Reason ◽  
Quantification Theory ◽  
New Approach

“Apart from merits or defects of PA1, however, its existence demonstrates the feasibility of a new approach to the logic of propositions involving the principle of subjunctive contrariety. We thus have good reason to investigate the effect this principle, and a concept of conditionality compatible with it, might exert if introduced into standard quantification theory, into set theory, into modal logic and into epistemology and the philosophy of science.”

Wilson's Laws and Other Worlds

Dialogue ◽  
1989 ◽  
Vol 28 (2) ◽  
pp. 321-328 ◽  
Author(s):  
Robert M. Martin
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Possible Worlds ◽  
David Lewis ◽  
Nelson Goodman ◽  
Constant Conjunction

I like the idea behind this book. It is an empiricist alternative to the powerful and influential contemporary rationalist position on scientific law, causality, and counterfactuals, associated with Chisholm and David Lewis, which “takes its inspiration from modal logic and set theory [and] involves … an assumption of non-truth-functional connectives to define unanalyzable natural necessities, or defines such connectives and such necessities in terms of an ontology of possible worlds”. Wilson's views, springing from Hume and Nelson Goodman, involve no unobservables such as natural necessities or possible worlds, but are based instead on good old Humean laws of constant conjunction. How lovely if Wilson could pull this off!

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