The Versatility of Universality inPrincipia Mathematica

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)
DOI 10.1080/01445340.2011.552323
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
Download options
PhilPapers Archive

Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 16,667
External links
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
Bernard Linsky (1999). Russell's Metaphysical Logic. Center for the Study of Language and Inf.

View all 19 references / Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles
Peter Milne (2008). Russell's Completeness Proof. History and Philosophy of Logic 29 (1):31-62.

Monthly downloads

Added to index


Total downloads

14 ( #186,222 of 1,726,249 )

Recent downloads (6 months)

1 ( #354,178 of 1,726,249 )

How can I increase my downloads?

My notes
Sign in to use this feature

Start a new thread
There  are no threads in this forum
Nothing in this forum yet.