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The Fact Semantics for Ramified Type Theory and the Axiom of Reducibility

Edwin D. Mares
2007 Notre Dame Journal of Formal Logic  
This paper uses an atomistic ontology of universals, individuals, and facts to provide a semantics for ramified type theory.  ...  It is shown that with some natural constraints on the sort of universals and facts admitted into a model, the axiom of reducibility is made valid.  ...  Acknowledgments I wish to thank Bernard Linsky, who got me interested in ramified type theory and read earlier drafts of this paper, and Allen Hazen who also read a draft.  ... 
doi:10.1305/ndjfl/1179323266 fatcat:ivltvy3b4bdn5i657v5kppeqhm

Russell's Zigzag Path to the Ramified Theory of Types

Alasdair Urquhart
1988 Russell: the Journal of Bertrand Russell Studies  
THE RAMIFIED THEORY of types was five years in the making. A curious aspect of the historical evolution of the theory is that Russell began and ended with a theory of types.  ...  I shall go into this question in some detail after discussing the first version of type theory, which is sketched in Appendix B of The Principles of Mathematics.  ...  ACKNOWLEDGMENTS I would like to thank the staff of the Russell Archives and in particular Kenneth Blackwell and Carl Spadoni for their very kind assistance.  ... 
doi:10.15173/russell.v8i1.1735 fatcat:ome4z7rffzd4djqubceaihdkua

Inquiry into RDF and OWL Semantics [chapter]

Seiji Koide, Hideaki Takeda
2016 Lecture Notes in Computer Science  
The purpose of this paper is to present the higher order formalization of RDF and OWL with setting up ontological meta-modeling criteria through the discussion of Russell's Ramified Type Theory, which  ...  This paper briefly summarize some of set theories, and reviews the RDF and OWL Semantics with higher order classes from the view of Russell's Principia Mathematica.  ...  Those criteria are actually derived from the axioms and principles introduced in Ramified Type Theory for the resolution of Russell paradox in Principia Mathematica (PM, Vol.1).  ... 
doi:10.1007/978-3-319-50112-3_2 fatcat:qf7i3m26qfd5filhzduran5sri

Page 118 of The Philosophical Review Vol. 67, Issue 1 [page]

1958 The Philosophical Review  
The remainder of the chapter discusses well ordering, axioms of infinity, predicative and ramified functional calculi, and axioms of reducibility ; the treatment is reminiscent of the theory of types.  ...  Types,”’ ‘“Axiomatic Set Theory,” and “‘Mathematical Intuitionism.”’  ... 

Page 664 of Mathematical Reviews Vol. 12, Issue 9 [page]

1951 Mathematical Reviews  
Spanish summary) This paper is concerned with the troublesome Axiom of Reducibility connected with Russell’s ramified theory of types.  ...  Development of the notion of set and of the axioms for sets. Synthése 7, 374-390 (1949). This paper is expository and historical.  ... 

An Essay on the Foundations of Our Knowledge

Philip P. Wiener, Antoine Augustin Cournot, Merritt H. Moore
1958 Philosophical Review  
The remainder of the chapter discusses well ordering, axioms of infinity, predicative and ramified functional calculi, and axioms of reducibility ; the treatment is reminiscent of the theory of types.  ...  Types,”’ ‘“Axiomatic Set Theory,” and “‘Mathematical Intuitionism.”’  ... 
doi:10.2307/2182780 fatcat:4nu5nwaf5reiphe6hyyunkasem

The 1910 *Principia*'s Theory of Functions and Classes and the Theory of Descriptions

William Demopoulos
2007 European Journal of Analytic Philosophy  
The present paper develops a reconstruction of Principia's theory of functions and classes that is based on Russell's epistemological applications of the method of contextual analysis.  ...  Such a reconstruction is not eliminativist—indeed, it explicitly assumes the existence of classes—and possesses certain advantages over the no–classes theory advocated by Whitehead and Russell.  ...  It is difficult to motivate the theory consisting of ramified types plus reducibility when it is separated from its epistemological point.  ... 
doaj:6087390beafc452994290b56f9e49957 fatcat:7l53d3bsq5bh3in4m7hjemidki

Was the Axiom of Reducibility a Principle of Logic?

Bernard Linsky
1990 Russell: the Journal of Bertrand Russell Studies  
OBJECTIONS TO THE AXIOM The Ramified Theory of Types of the first edition of Principia goes beyond the divisions of a "simple" theory between individuals, first-order propositional functions which apply  ...  I would like to thank the Center for the use of its facilities, and those affiliated with the Center for discussions which led to this paper.  ...  There he expresses doubts about whether the axiom The axiom of reducibility and logic 133 propositional function can depend is its members, and so the type theory cannot be ramified.  ... 
doi:10.15173/russell.v10i2.1775 fatcat:aik6nufrbzgfjezfn5ayoafnxa

Fitting the (Old) Pattern [review of George Roberts, ed., Bertrand Russell Memorial Volume]

Michael Byrd
1986 Russell: the Journal of Bertrand Russell Studies  
of individuals such that D(x, P) and not Px. But, in ramified type theory, there is no such type as the type: property of individuals.  ...  Russell's solution to this problem was, in 1910, the ramified theory of types.  ...  Far from conflating constituency and aboutness, Russell's theories of denoting in the years from 1900 to 1905 are an explicit attempt to come to grips with their distinctness.  ... 
doi:10.15173/russell.v6i2.1675 fatcat:ssmsgulonjcajkefjd6pf6trri

Substitution and the Theory of Types [review of Gregory Landini, Russell's Hidden Substitutional Theory]

Graham Stevens
2003 Russell: the Journal of Bertrand Russell Studies  
This explains how Landini thinks the ramified theory of types is to be understood as a semantic theory rather than a syntactic one, but we still need justification for the claim that the semantics is nominalistic  ...  Ramsey's extrication and expulsion of the order part of the ramified hierarchy, along with the offending axiom, placated some of the opponents of type-stratified logic, but dissatisfaction remained.  ... 
doi:10.15173/russell.v23i2.2048 fatcat:cpgwwuiwdrf5hn4zirnngoexbe

A Constructive Examination of a Russell-style Ramified Type Theory [article]

Erik Palmgren
2017 arXiv   pre-print
This is sufficient to make the type hierarchy usable for development of constructive mathematics. We present a ramified type theory suitable for this purpose.  ...  In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-L\"of type theory with an infinite sequence of universes.  ...  Ramified Type Theory Our version of the ramified type hierarchy is built from basic types 1 (the unit type) and N (the type of natural number) using the product type construction ×, and for each n = 0,  ... 
arXiv:1704.06812v1 fatcat:aoyh7lm23ngilpl4czel3waxti

Page 1923 of Mathematical Reviews Vol. , Issue 95d [page]

1995 Mathematical Reviews  
To avoid “an undesirably weak theory”, the author also adds axioms of reducibility to his ramified theory of types. N. B.  ...  To resolve the Epimenides and related paradoxes, the author extends the simple theory of types into a ramified theory with a primitive connective for the identity of propositions.  ... 

A paradox about sets of properties

Nathan Salmón
2021 Synthese  
Whitehead and Russell's ramified theory of types with axioms of reducibility is a logical apparatus tailor-made for theorizing about such things as propositions about propositions and properties that generalize  ...  There is a position worthy of consideration that broadens the combination of ramified type theory with lambda-abstraction (with axioms of reducibility), through a significant concession to simple type  ... 
doi:10.1007/s11229-021-03353-8 fatcat:ggfauxkrdzd55b6iabbwosax6e

Principia's Second Edition [review of Bernard Linsky, The Evolution of Principia Mathematica: Bertrand Russell's Manuscripts and Notes for the Second Edition]

Russell Wahl
2013 Russell: the Journal of Bertrand Russell Studies  
different formal systems for the second edition. propositional functions of the same order/type) and without the axiom of reducibility.  ...  Schütte (Myhill, " The Undefinability of the Set of Natural Numbers in the Ramified Principia", p. 21) and then adds his comprehension and extensionality axioms A and B in place of the axiom of reducibility  ... 
doi:10.15173/russell.v33i1.2241 fatcat:qdnljq3ulfhidjbtq6aax2xdfu

Page 1629 of Mathematical Reviews Vol. , Issue 81E [page]

1981 Mathematical Reviews  
Ramified type theory achieves predicativity by assigning to a set {x: (x) holds} not only a type (which exceeds that of its ele- ments x by one) but also an order that is larger than the orders of the  ...  Finally the author considers Russell’s axiom of reducibility, Red, which claims, essentially, that the order of a set can be lowered.  ... 
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