David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
History and Philosophy of Logic 32 (3):241-264 (2011)
In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ?no loss of generality? problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53?54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. The purpose of this article is to clarify Russell's claim and to solve the ?no loss of generality? problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (?ramif-types?). Then, comparing different important interpretations of Principia?s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other hand, makes it possible to give us a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the ?no loss of generality? problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
Charles S. Chihara (1973). Ontology and the Vicious-Circle Principle. Ithaca [N.Y.]Cornell University Press.
Alonzo Church (1976). Comparison of Russell's Resolution of the Semantical Antinomies with That of Tarski. Journal of Symbolic Logic 41 (4):747-760.
William Demopoulos & Peter Clark (2005). The Logicism of Frege, Dedekind, and Russell. In Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. 129--165.
Nicholas Griffin (1980). Russell on the Nature of Logic (1903–1913). Synthese 45 (1):117 - 188.
Citations of this work BETA
No citations found.
Similar books and articles
Fairouz Kamareddine & Twan Laan (2001). A Correspondence Between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems. Journal of Logic, Language and Information 10 (3):375-402.
M. Randall Holmes, Automated Type-Checking for the Ramiﬁed Theory of Types of the Principia Mathematica of Russell and Whitehead.
M. Randall Holmes, Polymorphic Type Checking for the Type Theory of the Principia Mathematica of Russell and Whitehead.
Twan Laan & Rob Nederpelt (1996). A Modern Elaboration of the Ramified Theory of Types. Studia Logica 57 (2-3):243 - 278.
Kevin C. Klement (2010). The Functions of Russell's No Class Theory. Review of Symbolic Logic 3 (4):633-664.
Gregory Landini (1991). A New Interpretation of Russell's Multiple-Relation Theory of Judgment. History and Philosophy of Logic 12 (1):37-69.
Gregory Landini (1987). Russell's Substitutional Theory of Classes and Relations. History and Philosophy of Logic 8 (2):171-200.
Bernhard Weiss (1994). On Russell's Arguments for Restricting Modes of Specification and Domains of Quantification. History and Philosophy of Logic 15 (2):173-188.
Gregory Landini (2005). Quantification Theory in *8 ofPrincipia Mathematicaand the Empty Domain. History and Philosophy of Logic 26 (1):47-59.
Fairouz Kamareddine, Twan Laan & Rob Nederpelt (2002). Types in Logic and Mathematics Before 1940. Bulletin of Symbolic Logic 8 (2):185-245.
Gregory Landini (2000). Quantification Theory in *9 of Principia Mathematica. History and Philosophy of Logic 21 (1):57-77.
Kevin C. Klement (forthcoming). PM's Circumflex, Syntax and Philosophy of Types. In Bernard Linsky & Nicholas Griffin (eds.), Principia Mathematica at 100. Cambridge.
Peter Milne (2008). Russell's Completeness Proof. History and Philosophy of Logic 29 (1):31-62.
Kevin C. Klement (2003). Russell's 1903 - 1905 Anticipation of the Lambda Calculus. History and Philosophy of Logic 24 (1):15-37.
Gregory Landini (1998). Russell's Hidden Substitutional Theory. Oxford University Press.
Added to index2011-08-04
Total downloads12 ( #133,216 of 1,099,746 )
Recent downloads (6 months)3 ( #126,683 of 1,099,746 )
How can I increase my downloads?