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 problems pertaining to the real number line. So, a conditional can be proven whose antecedent states the existence of inacessible cardinals and whose consequent says something about Lebesgue measurability. Which, in a sense, is quite exciting, because it shows that certain abstract questions that few people seem to care about have certain implications for stuff that every real number theorist should be interested in.

Reinhard Muskens (Tilburg), on the other hand, discussed type theory and introduced intensional models for type theory. They are meant to serve as a formal model of intensional contexts (so that, e.g., you can play around and introduce an explicit knowledge operator and yet avoid logical omniscience). The axiom of extensionality fails, but you still get classical reasonings as far as the usual operators go. Muskens also discussed a way one can construct possible worlds out of these intensional models (they’d be properties of propositions).

One thing we slightly disagreed about was the behavior of proper names. Muskens says (in his abstract):

When you look at arithmetic, it turns out that you can model it not only in infinite domains but also in finite but potentially infinite models (roughly, instead of the standard model, you take a set of its finite initial segments, making sure that all needed segments are in the domain). Quite surprisingly, even though this doesn’t seem to make much of a difference, in finite model theories of multiplication or division are undecidable, even though they are decidable in finite models. The weakest known arithmetic for which undecidability in finite models is proven is arithmetic of coprimality. Here, Marek’s work kicks in. He shows that even though the finite arithmetic of coprimality is quite poor, you can still code semantics in it, and obtain the undefinability of truth theorem for that system.

Jean Paul van Bendegem delivered a tutorial on many uses of paraconsistent logics. Basically, this was a very neat and quite accessible introduction to adaptive logics and their many uses. It was pretty informal and was fairly easy to follow (although, this might be an overstatement - I’ve seen the systems before).

Karl-Georg Niebergall (Berlin) gave an entertaining talk about Gödel’s incompleteness theorems. Given the talk title, I was afraid this might go in the direction of murky waters of philosophical interpretations of the theorems. I’m glad I was wrong. Karl-Georg, actually, discussed various versions of proofs of the first theorem and compared the types of soundness(/adequacy) needed for the proof. Given that, he briefly explained how we get the second theorem. This brought him to an explanation of the role of Rosser’s provability predicate (well, if you use Rosser’s notion, the soundness requirement is weaker). Now, the second theorem says that consistency of arithmetic is not provable within it. What’s less know is that even though this indeed is the case if you use the classical provability predicate, if you use Rosser’s predicate, things get funky, because you can prove within arithmetic that it is consistent in this sense. This brings us to a dilemma: either the modified definition expresses the consistency and consistency is provable (in which case the original formula doesn’t express consistency), or the modified provable formula fails to express consistency. The answer hinges on how you understand “expressing”. Karl-Georg discussed quite a few prima facie plausible definitions of what it is for a formula to express consistency, and showed why none of this definitions can be accepted.

Martin Mose Bentzen (Amsterdam) talked about game-theoretic formalizations of situations where situation participants would be better off if they trusted each other. Martin introduced neat formal tools to represent some of the examples he discussed intuitively, and discussed how those situations are to be assessed (interestingly, it doesn’t seem to be the case that we always would be better off if we trusted each other!).

Christian Straβer developed an adaptive logic for deontic dilemmas allowing for factual detachment. First, he explained why usually dyadic deontic operators are preferred to monadic ones (with monadic approach it is hard to avoid unrestricted applications of Strengthening the Antecedent, also, it is hard to model the Chisholm paradox or the Gentle-Murderer paradox). Second, he discussed one of the key difficulties for the dyadic approach. Roughly, A and the commitment under A to do B should imply the actual obligation to do B. But in standard dyadic deontic logics this fails (or: its acceptance either brings Strengthening the Antecedent back, or collides with certain intuitive natural language examples). To deal with this issues, Christian “adaptivized” Lou Bogle’s CDPM logics, so that the resulting (non-monotonic) theory handles restricted uses of factual detachment and strengthening the antecedent without allowing problematic applications.

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 problems pertaining to the real number line. So, a conditional can be proven whose antecedent states the existence of inacessible cardinals and whose consequent says something about Lebesgue measurability. Which, in a sense, is quite exciting, because it shows that certain abstract questions that few people seem to care about have certain implications for stuff that every real number theorist should be interested in.

Reinhard Muskens (Tilburg), on the other hand, discussed type theory and introduced intensional models for type theory. They are meant to serve as a formal model of intensional contexts (so that, e.g., you can play around and introduce an explicit knowledge operator and yet avoid logical omniscience). The axiom of extensionality fails, but you still get classical reasonings as far as the usual operators go. Muskens also discussed a way one can construct possible worlds out of these intensional models (they’d be properties of propositions).

One thing we slightly disagreed about was the behavior of proper names. Muskens says (in his abstract):

The semantics to a large degree implements Frege’s idea that all signs that have a referent, also come with a sense.Now, if I understood him correctly, Muskens would like to treat proper names as having senses: those senses would be just Quine-style predicatized proper names. So, for example, the sense of `Aristotle’ would be `the unique object that Aristotelizes’ or something like that. This, to my mind, has certain drawbacks. Most importantly, this doesn’t seem to be a fulfillment of the descriptivists’ program: for on the descriptive view, senses of names are purely qualitative, whereas Quineazing names this way doesn’t give you a qualitative specification of the reference of a name. But, I guess, the discussion should go deeper: why would we prefer purely qualitative descriptions? Why does the procedure seem kinda fishy, especially in the light of Kripke-style modal arguments against descriptivism? Anyway….

Marek Czarnecki (Warsaw) gave a talk on Semantics coded by coprimality in finite models. Marek has recently won one of the Polish Society for Logic and Philosophy of Science awards in the competition for best MSc theses in logic, and this talk was about his results.

When you look at arithmetic, it turns out that you can model it not only in infinite domains but also in finite but potentially infinite models (roughly, instead of the standard model, you take a set of its finite initial segments, making sure that all needed segments are in the domain). Quite surprisingly, even though this doesn’t seem to make much of a difference, in finite model theories of multiplication or division are undecidable, even though they are decidable in finite models. The weakest known arithmetic for which undecidability in finite models is proven is arithmetic of coprimality. Here, Marek’s work kicks in. He shows that even though the finite arithmetic of coprimality is quite poor, you can still code semantics in it, and obtain the undefinability of truth theorem for that system.

Jean Paul van Bendegem delivered a tutorial on many uses of paraconsistent logics. Basically, this was a very neat and quite accessible introduction to adaptive logics and their many uses. It was pretty informal and was fairly easy to follow (although, this might be an overstatement - I’ve seen the systems before).

Karl-Georg Niebergall (Berlin) gave an entertaining talk about Gödel’s incompleteness theorems. Given the talk title, I was afraid this might go in the direction of murky waters of philosophical interpretations of the theorems. I’m glad I was wrong. Karl-Georg, actually, discussed various versions of proofs of the first theorem and compared the types of soundness(/adequacy) needed for the proof. Given that, he briefly explained how we get the second theorem. This brought him to an explanation of the role of Rosser’s provability predicate (well, if you use Rosser’s notion, the soundness requirement is weaker). Now, the second theorem says that consistency of arithmetic is not provable within it. What’s less know is that even though this indeed is the case if you use the classical provability predicate, if you use Rosser’s predicate, things get funky, because you can prove within arithmetic that it is consistent in this sense. This brings us to a dilemma: either the modified definition expresses the consistency and consistency is provable (in which case the original formula doesn’t express consistency), or the modified provable formula fails to express consistency. The answer hinges on how you understand “expressing”. Karl-Georg discussed quite a few prima facie plausible definitions of what it is for a formula to express consistency, and showed why none of this definitions can be accepted.

Martin Mose Bentzen (Amsterdam) talked about game-theoretic formalizations of situations where situation participants would be better off if they trusted each other. Martin introduced neat formal tools to represent some of the examples he discussed intuitively, and discussed how those situations are to be assessed (interestingly, it doesn’t seem to be the case that we always would be better off if we trusted each other!).

Christian Straβer developed an adaptive logic for deontic dilemmas allowing for factual detachment. First, he explained why usually dyadic deontic operators are preferred to monadic ones (with monadic approach it is hard to avoid unrestricted applications of Strengthening the Antecedent, also, it is hard to model the Chisholm paradox or the Gentle-Murderer paradox). Second, he discussed one of the key difficulties for the dyadic approach. Roughly, A and the commitment under A to do B should imply the actual obligation to do B. But in standard dyadic deontic logics this fails (or: its acceptance either brings Strengthening the Antecedent back, or collides with certain intuitive natural language examples). To deal with this issues, Christian “adaptivized” Lou Bogle’s CDPM logics, so that the resulting (non-monotonic) theory handles restricted uses of factual detachment and strengthening the antecedent without allowing problematic applications.

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