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There is some excellent news, posted by Richard Zach. (Although, I imagine most of my readers will have seen this on his blog. I think this information should be spread as much as possible. It is a Very Good Thing.) The Association of Symbolic Logic has decided to release several out of print logic books for free through Project Euclid. These include books from the Perspectives in Logic series and the Lectures Notes in Logic series. They are available in chapter-by-chapter .pdf. Be careful about downloading a lot at once or your IP will get banned. As Richard says,

This includes classics like

- Shoenfield’s Recursion Theory,
- Lindström’s Aspects of Incompleteness in the LNL,
- Sack’s Higher Recursion Theory,
- Hájek and Pudlák’s Metamathematics of First-order Arithmetic,
- Shelah’s Proper and Improper Forcing,
- Barwise’s Admissible Sets and Structures, and
- Barwise and Feferman’s Model-theoretic Logics in the PiML.

It is an excellent selection of books. The Barwise volumes, in particular, are gems that are nigh impossible to obtain elsewhere. (HT: Richard Zach)

This is an attempt to think through some topics from the philosophy of math seminar I’m attending.

If you are in the Pittsburgh area, you’ve probably already heard about this. The weekend of April 3-4, Pitt will be hosting a conference in honor of Nuel Belnap, called “Logics of Consequence.” The speaker line-up consists mostly of Belnap’s students, and it is quite impressive. A selection: Mike Dunn, Phil Kremer, and Alasdair Urquhart. There are also others coming for discussion. I, for one, am quite excited.

A topic that recurs in the interviews with Quine is his views on analyticity and how his views have changed since “Two Dogmas”.

There was a lot in the Search on Russell, so I will continue my notes mainly on Russell. Read the rest of this entry »

A large part of the Search for Mathematical Roots focuses on Russell and the development of Principia (PM, hereafter). I found these chapters, which roughly comprise the latter half of the book, to be quite helpful since I’m less familiar with Russell than Frege and Wittgenstein and the chapters do a good job of explaining the influence of PM. In an interesting bit of trivia, Grattan-Guinness says that its name is a nod, not to Newton’s book, but to Moore’s Principia Ethica, Moore being a huge influence on Russell’s philosophical development. Grattan-Guinness makes Russell out to be heavily influenced by Peano. In the historical narrative, Russell’s interests seem to change along the same lines that Peano’s do. An exception to this is that Russell maintains that math is a part of logic whereas Peano thinks they merely overlap. Peano even wrote a paper on “the” in which he gave the same principles for its meaning that Russell did in “On Denoting.” According to Grattan-Guinness, Russell seems to have been familiar with that paper, at least reading it once, but seems to have forgotten about it.

In my previous post on Search for Mathematical Roots, I mentioned that one of the things that Frege complained about, according to Grattan-Guinness, was the overloading of symbols. I want to expand on that briefly in this post. I said that the same thing was done in proof theory. It seems that this is not completely accurate. The situation, from what I can tell from the Search for Mathematical Roots, was that often logicians, especially in the algebraic tradition, would use one symbol to designate many different things. For example, some would use ‘1’ to designate the universe of discourse as well as the truth-value true. The equals sign ‘=’ would do double duty as identity among elements as well as equivalence between propositions. It would also get used to indicate a definition. This would not be so bad, but often there would not be any clear way to tell what sense a sign would be used in, sometimes appearing in one sentence in multiple ways. Things get hairier when quantifiers are added, since the variables quantified over would sometimes be propositions and sometimes not. Read the rest of this entry »

In Hodges’s model theory book, there is a proof that sort of surprised me. The proof is for lemma 9.1.5 and runs as follows.

A Skolem theory is axiomatized by a set of ∀_{1} sentences and modulo the theory, every formula is equivalent to a quantifier-free formula. Now quote Lemma 9.1.3 and Theorem 9.1.1.

What surprised me is the last sentence. It is more like a recipe, telling you what to do. Some other proofs in the book have this character to a small degree, but this one stood out a little. The proof is fine, but, unlike many other proofs, this one seems to encourage the reader to do the proof as well. Is this sort of “more active” style of writing proofs common? I don’t think I’ve come across it much at all in the things I read. I’d expect the last line of the proof to go: “The result then follows from Lemma 9.1.3 and Theorem 9.1.1.”

Reading the Revision Theory of Truth has given me an idea that I’m trying to work out. This post is a sketch of a rough version of it. The idea is that circular definitions motivate the need for a sharper conception of content on an inferentialist picture, possibly other pictures of content too. It might have a connection to harmony as well, although that thread sort of drops out. The conclusion is somewhat programmatic, owing to the holidays hitting. Read the rest of this entry »

Does anyone know of any proof systems in which some but not all contents have negations? I’m looking for examples for a developing project.

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