Th. Skolem. Two remarks on set theory. Mathematica Scandinavica, vol. 5 (1957), pp. 40–46. - John H. Harris. On a problem of Th. Skolem. Notre Dame Journal of formal logic, vol. 11 (1970), pp. 372–374.

1971 ◽  
Vol 36 (4) ◽  
pp. 680-680
Author(s):  
Jean E. Rubin
Keyword(s):  
Set Theory ◽  
Formal Logic

scholarly journals WEIGHTED AUTOMATA AS COALGEBRAS IN CATEGORIES OF MATRICES

2013 ◽  
Vol 24 (06) ◽  
pp. 709-728 ◽  
Author(s):  
JOSÉ N. OLIVEIRA
Keyword(s):  
Computer Science ◽  
Linear Algebra ◽  
Set Theory ◽  
Matrix Algebra ◽  
Quantitative Methods ◽  
Qualitative Reasoning ◽  
Formal Logic ◽  
Quantitative Reasoning ◽  
Weighted Automata ◽  
Conventional Notation

The evolution from non-deterministic to weighted automata represents a shift from qualitative to quantitative methods in computer science. The trend calls for a language able to reconcile quantitative reasoning with formal logic and set theory, which have for so many years supported qualitative reasoning. Such a lingua franca should be typed, polymorphic, diagrammatic, calculational and easy to blend with conventional notation. This paper puts forward typed linear algebra as a candidate notation for such a unifying role. This notation, which emerges from regarding matrices as morphisms of suitable categories, is put at work in describing weighted automata as coalgebras in such categories. Some attention is paid to the interface between the index-free (categorial) language of matrix algebra and the corresponding index-wise, set-theoretic notation.

A homogeneous system for formal logic

1943 ◽  
Vol 8 (1) ◽  
pp. 1-23 ◽  
Author(s):  
R. M. Martin
Keyword(s):  
Set Theory ◽  
General Type ◽  
Homogeneous System ◽  
Formal Logic ◽  
Strict Sense ◽  
Standard Methods ◽  
Logical Construction ◽  
Logical Type ◽  
Classical Mathematics ◽  
Axiomatic Set Theory

Two more or less standard methods exist for the systematic, logical construction of classical mathematics, the so-called theory of types, due in the main to Russell, and the Zermelo axiomatic set theory. In systems based upon either of these, the connective of membership, “ε”, plays a fundamental role. Usually although not always it figures as a primitive or undefined symbol.Following the familiar simplification of Russell's theory, let us mean by a logical type in the strict sense any one of the following: (i) the totality consisting exclusively of individuals, (ii) the totality consisting exclusively of classes whose members are exclusively individuals, (iii) the totality consisting exclusively of classes whose members are exclusively classes whose members in turn are exclusively individuals, and so on. Any entity from (ii) is said to be one type higher than any entity from (i), any entity from (iii), one type higher than any entity from (ii), and so on. In systems based upon this simplified theory of types, the only significant atomic formulae involving “ε” are those asserting the membership of an entity in an entity one type higher. Thus any expression of the form “(x∈y)” is meaningless except where “y” denotes an entity of just one type higher than the type of the entity denoted by “x” It is by means of general type restrictions of this kind that the Russell and other paradoxes are avoided.

scholarly journals Graphic Deduction Based on Set

2020 ◽  
Vol 26 (10) ◽  
pp. 1331-1342
Author(s):  
Xia He ◽  
Guoping Du ◽  
Long Hong
Keyword(s):  
Artificial Intelligence ◽  
Set Theory ◽  
Representation Theorem ◽  
Basic Concept ◽  
Formal Language ◽  
Deductive Reasoning ◽  
Graphical Representation ◽  
Formal Logic ◽  
New Method ◽  
Symbolic Logic

Based on basic concept of symbolic logic and set theory, this paper focuses on judgments and attempts to provide a new method for the study of logic. It establishes the formal language of the extension of judgment J*, and formally describes a, e, i, o judgment, and thus gives set theory representation and graphical representation that can distinguish between universal judgments and particular judgments. According to the content of non-modal deductive reasoning in formal logic, it gives weakening theorem, strengthening theorem and a number of typical graphical representation theorem (graphic theorem), where graphic deduction is carried out. Graphic deduction will be beneficial to the research of artificial intelligence, which is closely related to judgment and deduction in logic.

On the quantificational logic of intuitionistic set theory

1986 ◽  
Vol 99 (1) ◽  
pp. 5-10 ◽  
Author(s):  
Harvey M. Friedman ◽  
Andrej Ščedrov
Keyword(s):  
Set Theory ◽  
Propositional Logic ◽  
Propositional Calculus ◽  
Formal Logic ◽  
Predicate Calculus ◽  
Intuitionistic Propositional Calculus ◽  
Quantificational Logic ◽  
Law Of Excluded Middle ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

Formal propositional logic describing the laws of constructive (intuitionistic) reasoning was first proposed in 1930 by Heyting. It is obtained from classical pro-positional calculus by deleting the Law of Excluded Middle, and it is usually referred to as Heyting's (intuitionistic) propositional calculus ([9], §§23, 19) (we write HPP in short). Formal logic involving predicates and quantifiers based on HPP is called Heyting's (intuitionistic) predicate calculus ([9], §§31, 19) (we write HPR in short).

scholarly journals A Poor Concept Script

2004 ◽  
Vol 2 ◽  
Author(s):  
Hartley Slater
Keyword(s):  
Natural Language ◽  
Set Theory ◽  
Basic Problem ◽  
Lambda Calculus ◽  
Higher Order ◽  
Formal Logic ◽  
Formal Structure ◽  
Elementary Level

The formal structure of Frege’s ‘concept script’ has been widely adopted in logic text books since his time, even though its rather elaborate symbols have been abandoned for more convenient ones. But there are major difficulties with its formalisation of pronouns, predicates, and propositions, which infect the whole of the tradition which has followed Frege. It is shown first in this paper that these difficulties are what has led to many of the most notable paradoxes associated with this tradition; the paper then goes on to indicate the lines on which formal logic—and also the lambda calculus and set theory—needs to be restructured, to remove the difficulties. Throughout the study of what have come to be known as first-, second-, and higher-order languages, what has been primarily overlooked is that these languages are abstractions. Many well known paradoxes, we shall see, arose because of the elementary level of simplification which has been involved in the abstract languages studied. Straightforward resolutions of the paradoxes immediately appear merely through attention to languages of greater sophistication, notably natural language, of course. The basic problem has been exclusive attention to a theory in place of what it is a theory of, leading to a focus on mathematical manipulation, which ‘brackets off ’ any natural language reading.

A Course on Set Theory

2009 ◽  
Author(s):  
Ernest Schimmerling
Keyword(s):  
Set Theory

Set Theory

2016 ◽  
Author(s):  
Daniel W. Cunningham
Keyword(s):  
Set Theory
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