Primitive Recursion

In an interesting thread titled “Recursive” on FOM last week there was a discussion on the history of primitive recursive functions. Of course, already Grassmann, Dedekind, and Peano gave primitive recursive definitions of individual functions such as addition and multiplication, and Skolem’s 1923 article

Independence of Goodstein’s Theorem from PA

I was asked in email about a good source about Goodstein sequences and the independence of Goodstein’s Theorem from Peano Arithmetic. The independence result is due to Kirby and Paris in a 1983 paper in the Proceedings of the London Mathematical Society (vol. 14), using the method of indicators. Georg Moser suggested the following paper by Cichon, which appeals to the characterization of provably recursive functions in PA only:

E. A. Cichon, A Short Proof of Two Recently Discovered Independence Results Using Recursion Theoretic Methods. Proceedings of the American Mathematical Society 87/4 (1983), 704-706. JSTOR

Cichon’s proof can also be found in Fairtlough and Wainer’s chapter on “Hierarchies of Provably Recursive Functions” in the Handbook of Proof Theory, S. Buss, ed. (Elsevier, 1998).

Philosophy of Language Texts?

I’m going to be teaching philosophy of language next term. It’s the first time–if you can believe that–we’re offering a course with that title. We used to have a course called “Analytical Philosophy”, which served that purpose, but it was also a history of analytic philosophy course. Anyway. I’d like to give my students a textbook, and was wondering if something new and good has shown up at the APA book exhibits in the last two years. Otherwise I’d probably use Ken Taylor’s Meaning and Truth.

SSHRC Grants in Philosophy for 2006

The Social Sciences and Humanities Research Council of Canada has posted a list of new Standard Research Grants for 2006. This year’s stats: 85 applications (2005: 96, 2004: 92), 32 grants, for a success rate of 37% (2005: 38%, 2004: 48%). This year, new scholars (≤ 5 years beyond PhD) had a 29% success rate (2005: 38%, 2004: 29%). Full stats here.

A list of successful proposals follows. I’ve included the dollar figure (in CAD), but these shouldn’t be taken as an indication of the quality of the project. The funding rate depends on the requirements of the project (travel, research support) and on the amount of graduate student funding, not just on the ranking of the proposal. I’ve certainly missed quite a few: I went by the titles in the full list (not broken down by subject area) and included grants that I guessed to be philosophy projects from the title and/or where I could ascertain that the applicant was in a philosophy department. Email me if you think I should include a grant not on here.

  1. Bartha, Paul , The University of British Columbia. Infinite Decision Theory. $31,500
  2. Campbell, Neil , Wilfrid Laurier University. Explanatory epiphenomenalism: at the crossroads of mental causation and consciousness. $57,106
  3. Davies, David A., McGill University. How making matters: provenance and the epistemology, ontology, and axiology of art. $49,469
  4. Duchesneau, François , Université de Montréal. Leibniz: système de la nature et organisation vitale. $78,948
  5. Joy, Morny M., University of Calgary. The confluence of head and heart: religion, ethics and the feminine in Hannah Arendt, Simone de Beauvoir and Edith Stein. $75,780
  6. Gauthier, Yvon , Université de Montréal. Logique arithmétique et philosophie de l’arithmétique. $49,542
  7. Griffin, Nicholas J., McMaster University. The collected letters of Bertrand Russell. $92,691
  8. Hacking, Ian , University of Toronto. Philosophical illustrations from the ultracold. $54,240
  9. Heath, Joseph M., University of Toronto. An adversarial approach to business ethics. $51,000
  10. Hudson, Robert G., University of Saskatchewan. The epistemology and metaphysics of dark matter research. $53,698
  11. Lin, Martin T., University of Toronto. Spinoza’s conatus doctrine. $38,621
  12. King, Peter , University of Toronto. Mediaeval souls and modern minds. $57,148
  13. Miller, Jon A., Queen’s University. Happiness in early modern philosophy. $48,423
  14. Norman, Wayne J., Université de Montréal. A skeptical business ethics. $41,381
  15. Moran, Brendan P., University of Calgary. Prose, myth, and time in late works of Walter Benjamin. $26,471
  16. Pickavé, Martin , University of Toronto. Medieval theories of the emotions (passions of the soul). $53,271
  17. Raffman, Diana , University of Toronto. Vagueness without paradox. $39,650
  18. Ripstein, Arthur S., University of Toronto. Authority and coercion: Kant’s doctrine of right. $49,813
  19. Russell, Paul , The University of British Columbia. The limits of free will. $37,502
  20. Schmitter, Amy M., University of Alberta, Representation in Early Modern philosophy: the 17th century. $62,540
  21. Seymour, Michel , Université de Montréal. Les droits collectifs linguistiques et le droit à l’autodétermination. $50,617
  22. Speaks, Jeffrey J., McGill University. The role of mental states in the philosophies of action and language. $54,506
  23. Sullivan, Arthur M., Memorial University of Newfoundland. The externalism/individualism debates. $55,093
  24. Sumner, Wayne L., University of Toronto. Matters of life and death. $41,503

Notions of Logical Independence

In Prague this past week, David Miller gave a talk in which (among many other interesting things) he distinguished two notions of logical independence. One he credits to Moore (the mathematician, not the philosopher) and Wittgenstein, and that’s the notion of independence at work when we say, e.g., that an axiom system is independent. A set Γ is independent if for each A ∈ Γ, Γ\A is consistent with ¬A. Moore’s notion of complete independence is a generalization of that, where we require that for each Δ ⊆ Γ, Γ\Δ is consistent with ¬Δ.

The other notion he credits to H. M. Sheffer, and that’s the notion of maximal independence: Γ is maximally independent if any two A, B ∈ Γ have no consequences in common, other than tautologies.

{p, q}, for instance, is (completely) independent in the first sense, but not maximally independent (p and q have the non-tautological consequence pq in common).

I think these are interesting concepts, and I should find out more about them. David makes use of them in comparing (false) theories in a 1974 paper. I hadn’t heard of Sheffer’s notion before; maybe that’s because the paper he defines it in is unpublished. But from David’s paper I see that Tarski uses it as well.

David Miller, 1974. On the comparison of false theories by their bases. The British Journal for the Philosophy of Science 25(2) 178–188.

Eliakim Hastings Moore, 1910. Introduction to a form of general analysis. The New Haven Mathematical Colloquium 1–150.

Henry Maurice Sheffer, 1921. The general theory of notational relativity. (Mimeograph)

Dear Matt

I am very sorry. There will be more logic blogging very soon, I promise. I’m off to Prague for the Vagueness and Uncertainty workshop, and if the Academy of Sciences also has internet access in the villa they’re putting me up in, I will liveblog it. Rosanna Keefe! Stewart Shapiro (who has a new book, which y’all should check out)! Peter Milne! Roy Sorensen! And my man Chris Fermüller.
(And UPDATE: Patrick Greenough! Sorry.)