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Notions of Logical Independence

Submitted by Richard Zach on Sun, 09/10/2006 - 2:49pm

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 ? is independent if for each A ? ?, ?\A is consistent with ¬A. Moore's notion of complete independence is a generalization of that, where we require that for each ? ? ?, ?\? is consistent with ¬?.

The other notion he credits to H. M. Sheffer, and that's the notion of maximal independence: ? is maximally independent if any two A, B ? ? 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 p ? q 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 I hadn't heard of Sheffer's notion before; maybe that's because the paper he defines it in is unpublished.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

Submitted by Richard Zach on Mon, 09/04/2006 - 12:50am

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.)

Grue Forever!

Submitted by Richard Zach on Mon, 08/07/2006 - 11:11am

The Austrian newspaper Der Standard, of all things, reminded me of another centenary: Nelson Goodman would have turned 100 today.

PhD Student Position in Logic at Bristol

Submitted by Richard Zach on Thu, 08/03/2006 - 8:07am

PhD Student Position in Logic and Cognitive Science, Department of
Philosophy, University of Bristol (UK):

A newly established research group headed by Hannes Leitgeb at the Department of Philosophy, University of Bristol, has an open position for a PhD student. The group, which will be part of an international EUROCORES Collaborative Research Project on Metacognition, will be funded by the AHRC and the European Science Foundation (as should be confirmed officially in the first half of August 2006). The Bristol group will be dealing with:

Logical Constraints on Conditionals and Introspection in Systems of Belief Revision and Non-Monotonic Reasoning.

The doctoral student will have an undergraduate background in philosophy, logic, and cognitive science, and will be expected to work on a suitable PhD thesis topic within the project. The group will be able to fund three years of tuition fees and maintenance (funds will also be available for computer equipment, conference fees, and travel costs).

There is no official deadline, but since the student's doctoral studies have to be taken up in the beginning of October 2006, we strongly recommend applications for this position to be submitted as early as possible.

Applications (including a CV and references) should be sent electronically to

Informal enquiries may be directed to:

Dots as Brackets in Formulas

Submitted by Richard Zach on Wed, 07/19/2006 - 7:49am

Ever tried reading logical texts from the 20s or before (e.g., C. I. Lewis's Symbolic Logic)? Confused by the absence of parentheses and all the dots and colons? Here's Carnap's explanation of the notation (from Abriss der Logistik):

4 c. The Dot Rules

The dot symbols (. : :. :: etc.) replace the bracketing of propositions. The dot signs fall into three distinct levels, depending on whether they occur

  1. between two propositions in a conjunction,
  2. after an operator (x), (? x), [(?x)(?x)],
  3. after |-, before and after the sign ?, ? ?, |, =Df.

Dot rules for reading: The scope of a dot symbol (for 1, to the left and to the right, for 2 to the right, for 3 to left or right, depending) extends either to the end of the proposition or to a dot symbol with more dots or to a symbol of the same or a higher level with the same number of dots.

Dot rules for writing: If the scope of a dot sybol is to extend beyond that of another, it must, if it is of a higher level than the latter, contain at least as many dots, and otherwise more dots.


p ? . q . r means p ? (q . r)
|- : (p, q) : p ? q . ? . q ? p " |- {(p, q) . [(p ? q) ? (q ? p)]}
p : ? : q . ? . q ? p " p ? [q ? (q ? p)]
(x) . ?x . ? . p ? q " [(x) . ?x] ? (p ? q)
(x) : ?x . ? . p ? q " (x) . [?x ? (p ? q)]
(x) : ?x ? p . ? q " (x) . [(?x ? p) ? q]
(x) : ?x ? p : ? q " [(x) . (?x ? p)] ? q

Dartmouth AI Conference 50 Years Ago

Submitted by Richard Zach on Wed, 07/19/2006 - 7:41am

50 years ago this summer, McCarthy, Minsky, Rochester, and Shannon organized a summer conference at Dartmouth which turned out to be a milestone in Artificial Intelligence research. For the logically minded, this item in the funding proposal to the Rockefeller Foundation is perhaps most interesting:

4. Theory of the Size of a Calculation

If we are given a well-defined problem (one for which it is possible to test mechanically whether or not a proposed answer is a valid answer) one way of solving it is to try all possible answers in order. This method is inefficient, and to exclude it one must have some criterion for efficiency of calculation. Some consideration will show that to get a measure of the efficiency of a calculation it is necessary to have on hand a method of measuring the complexity of calculating devices which in turn can be done if one has a theory of the complexity of functions. Some partial results on this problem have been obtained by Shannon, and also by McCarthy.

Computability in Swansea

Submitted by Richard Zach on Mon, 07/10/2006 - 2:25pm

Last week I had the pleasure of attending the Computability in Europe conference in the lovely seaside town of Swansea, Wales. Lots of interesting talks on all kinds of aspects of computation, including a number of talks on the (limits of) hypercomputation, a tutorial on proof complexity by Sam Buss, and special sessions on Gödel's legacy for computability theory with talks by Arnon Avron, John Dawson, Andrew Hodges, and Wilfried Sieg. I won't talk about the details--the abstracts are up on the website, the slides for many of the talks should be up very soon, and the proceedings volumes (invited, contributed [big PDF!] talks) are online as well.

Oh, and this guy stopped by to visit the building across from the conference site:

HRH The Prince of Wales

I also had an enjoyable bus ride back to Heathrow with Bob Meyer, talking about relevant logic and the epsilon calculus.

A Sequitur of Logicians?

Submitted by Richard Zach on Thu, 06/29/2006 - 12:46pm

Reading over my previous post, I was wondering if a group of scholars can be referred to as a "gaggle". So I did some research (i.e., I asked google) and happened upon this wiki page, which lists the appropriate collective noun for logicians as "sequitur", and attributes this to Bertrand Russell. Anyone got a reference for this? (It doesn't list a special collective noun for scholars, though--maybe "a school of scholars" would be appropriate?)

UPDATE: The wikipedia page for Quine has him saying it, but maybe he got it from Russell?

Online Collaboration for Scholars Follow-up

Submitted by Richard Zach on Thu, 06/29/2006 - 6:34am

I mentioned it in comments on the other post already, but I thought of a few solutions to my question: how do you keep a gaggle of scholars jointly working on a publication project organized? They are outlined in a wiki page. Comments welcome (you can edit the wiki, of course, or write something on the talk page).

Conference Blogging

Submitted by Richard Zach on Thu, 06/29/2006 - 6:28am

Greg is reporting from the Realism/Anti-Realism Workshop in Nancy, and Yarden from the CMU Summer School in Logic. Stay tuned for reports from Computability in Europe next week from yours truly.

It's Alan's Birthday!

Submitted by Richard Zach on Fri, 06/23/2006 - 12:58pm

Friendster reminded me that today is Alan Turing's birthday.

Online Multi-Author Collaboration?

Submitted by Richard Zach on Thu, 06/22/2006 - 12:59pm

If you've ever coauthored a paper, you know what a hassle it is to send versions back and forth, keep track of changes, avoid conflicts when you're simultaneously changing things, and so on. Now software engineers have the same problems when working on large development projects and have developed very sophisticated Revision Control Systems. I've used the most well-known of these, CVS, for collaborating on a couple of papers. But CVS is not exactly easy to use, and for a larger project with more people who all use different systems (Linux, Mac, Windows) and who aren't as geeky as I, CVS is not a good choice. So: I'm looking for an easy way to collaborate on editing text documents. It should work cross-platform, it should have version control (keeping track of changes, ways of telling what parts of a document were edited by whom and when, reverting to previous versions), it should be easy to use (graphical user interface), it should be easy to install (preferably it should be completely distributed with the client programs doing all the work). It doesn't have to have fancy security, support for branching and merging projects, include a build environment, and all the other stuff that's important to programmers. I also don't want to have to set up a server or persuade my university's IT people to give accounts to my collaborators. Anyone have any ideas?

Schlick Killed 70 Years Ago Today

Submitted by Richard Zach on Thu, 06/22/2006 - 12:50pm

On June 22, 1936, Moritz Schlick, influential philosopher and center of the Vienna Circle of logical empiricists, was gunned down on the steps of the University of Vienna. The Institute Vienna Circle is holding a small memorial event at the University of Vienna today, which will include the presentation of the first two volumes of Schlick's Collected Works.

Carnap, Epistemology of Math in Berlin

Submitted by Richard Zach on Wed, 06/21/2006 - 2:05pm

There will be two interesting workshops following GAP 6 in Berlin (September 14 & 15): one on Carnap, and one with the promising title "Towards a New Epistemology of Mathematics".

History of Logic at HOPOS

Submitted by Richard Zach on Thu, 06/15/2006 - 5:13pm

I'm at HOPOS, which is loads of fun. All my history of analytic/history of logic buddies are here. But more to the point:

Paolo Mancosu just gave the most amazing talk about the debate within the Vienna Circle about Tarski's theory of truth, in particular, the opposition that Neurath had voiced against it from the mid 1930s onward. I had always thought that the Vienna Circle wholeheartedly accepted Tarski's theory, and that Tarski's paper made truth an "acceptable" notion (from a logical empiricist standpoint). What Paolo showed, using correspondence between Carnap, Neurath, Tarski and others from 1935 and later, was that there was a debate raging over it. Neurath was opposed not, as one might suppose, because of the set theoretic metalanguage in which Tarski's theory was couched, but because he feared that people would use Tarski's theory in areas where it wasn't applicable, i.e., in non-formalized languages, and that it would lead people to return to "metaphysics". (I guess, that has actually happened decades later in the whole thing about deflationary conceptions of truth!) Moreover, we now know why no-one is aware about the tensions within the Vienna Circle around the theory of truth: at the Congrès Descartes in Paris in 1937, there was a private meeting with Carnap, Tarski, Neurath, Naess, Lutman-Kokoszynska, Hempel, and others where the proponents and opponents of Tarski's theory put forward their arguments. It was agreed (or, decreed by Carnap) that noone was to bring up the differences within the Circle regarding this issue in print.

Paolo's talk was followed by Johannes Hafner on the origins of model theory in Hilbert and Tarski. Johannes' main point was that whereas Tarski's notion of interpretation of an axiom system was a lot clearer about the syntax of the language (precise recursive definition of syntax in meta-language), Hilbert's notion of interpretation was a lot closer to the modern one--Tarski's notion of truth in the 1935 paper did not allow for varying domains. It wasn't until the 1950s in Tarski's and Vaught's work on model theory that the modern notion of truth in a structure emerged.

SciBlog Launch

Submitted by Richard Zach on Wed, 06/07/2006 - 7:53pm

SciBlog, a blog project of a bunch of science writers in Vienna, goes live tomorrow with an event at Depot. If you're in Vienna, come on by.

New Version of Peter Smith's Gödel Book

Submitted by Richard Zach on Wed, 06/07/2006 - 5:28pm

Peter has posted a new version of the first 22 chapters of his Gödel Book.

Philosophia Mathematica Issue on Gödel

Submitted by Richard Zach on Thu, 06/01/2006 - 9:39pm

The new issue of Philosophia Mathematica is devoted to Gödel. There are essays by Sol Feferman, Peter Koellner, Wilfried Sieg, Bill Tait, Rick Tieszen, and Mark van Atten, as well as a review of Torkel Franzén's Gödel's Theorem: An Incomplete Guide to its Use and Abuse.

I'm Back

Submitted by Richard Zach on Thu, 06/01/2006 - 5:11pm

After a week in Singapore and a week in Melbourne, I'm back in cold and rainy Vienna, nursing a cold and trying to finish two papers. Thanks to John and Belle, who I got to hang out with in Singapore, and to Allen, Conrad, Graham, Greg, Kate, Maren, Tama, and Zach for making it a thoroughly enjoyable visit to Melbourne.