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Showing posts from February, 2009

PHDs in Logic – news from the trenches

The Ghent workshop in logic is over. Since my life has been quite hectic (I was moving in to a new apartment), I couldn’t make it to all talks. Since (around) half of the people at the workshop were pure mathematicians, some of the talks that I went to were either above my head or beyond what I’m interested in. Here are some comments on those talks that I went to and that I feel I can say something about.

The workshop hit off with a tutorial on inaccessible cardinals by Benedikt Löwe (Amsterdam). It was quite entertaining and really informative. The main gist was this. Take the hierarchy of ordinals. Play around with it to get a hierarchy of cardinals. It turns out, all succesor cardinals are regular and there are also non-regular limit cardinals. Now, ZFC cannot prove the existence of regular limit cardinals (also called inaccessible cardinals). Interestingly, however, we can relate the existence of something as weird and prima facie useless as inaccessible cardinals to certain proble…

A reference for Lindenbaum Theorem

If you've read FOM recently, you probably already know, if you don't, here is a fun fact about Lindenbaum Theorem.

Arnold Neumaier asked why exactly Lindenbaum is credited with Lindenbaum Theorem, and the answer (given by Panu Raatikainen) is that it was never published by Lindenbaum, but it is credited to Lindenbaum by Tarski in On fundamental concepts of metamathematics (1930), theorem 12 and footnote, p. 34 in the English translation of Logic, Semantics, Metamathematics.

Since I did some Polish logic and still was lame enough not to know this, I'm grateful to Panu for the reference.

What Hempel didn't say about Ayer and Church

I’ve started prepping Philosophy of Language class for the next academic year (yeah, I don’t have to teach till Oc!). As for the text, The Philosophy of Language (third edition), edited by Martinich is the classic.

Right now, I’m looking at Hempel’s Empiricist Criteria of Cognitive Significance: Problems and Changes. At some point, he mentions Ayer’s revised verifiability requirement and Church’s argument against it, without explaining either of them. Below, a minor addendum which covers these things.

On p. 29 (Martinich pagination) Hempel discusses Ayer’s formulation of the testability criterion. The first formulation from Language, Truth and Logic is pretty much this:
A sentence S has empirical import if from S in conjunction with suitable subsidiary hypotheses it is possible to derive observation sentences which are not derivable from the subsidiary hypotheses alone. Hempel then explains why this criterion is too wide: it allows empirical import to any sentence whatsoever. Take S to …