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Showing posts from May, 2009

Adaptive logics in Cracow

Tomorrow, I'm off to Cracow, to give two 90 minutes long workshops on adaptive logics. I think I'll talk about various ways paraconsistent logics can be obtained, the way they generate inconsistency-adaptive logics, the standard format of adaptive logics and some other examples of adaptive logics, not related to paraconsistency.

Anyway, if you're in Cracow and speak Polish, feel free to pop in (more details here). Please do remember what happens to those who ask tricky questions, though. ;)

"Numbers" by M. & G. Fittings online

I've just noticed that a very nice book about number theory by Melvin Fitting and Greer Fitting is freely available online here. Here's a bit from the introduction.
A preface is supposed to explain why you should read the book. Like most prefaces, this one will make more sense after the fact. Nevertheless, here goes.
Most mathematicians believe (rather strongly) that numbers behave in certain well-defined ways. This belief can not be justified by personal experience. No mathematician has `seen' more than a finite, probably small, collection of numbers. Instead mathematicians justify their beliefs by giving proofs. In practice, this means that certain facts about numbers are accepted as `obvious', and used in carefully reasoned arguments for the correctness of other facts that are less obvious, or possibly not obvious at all. Since mathematicians generally are concerned to establish the nonobvious, little thought is customarily given as to why the `obvious' facts are …

Philosophy Talk

Philosophy Talk is a radio show, their past programs are available online - well, you can listen online without paying, at least to those that I've tried, but it seems you have to pay to download. ;)

Their list of topics sounds pretty cool. It includes:
Beliefs gone wildCapital punishmentThe Copyright warsDifferent cultures, different selvesAnimal mindsBodies for saleDigital selvesMorality of foodWe've been framed: how language shapes politicsVarietes of loveAthletic beautyPhilosophy of wineImmigrationMarriage and monogamyIf truth is so valuable, why is there so much BS?The erotic vs. the pornographic and many others.

Jerzy Perzanowski has passed away

During the night from 16th to 17th of May, Jerzy Perzanowski (1943-2009), a fine Polish logician and a legend of Polish logic has passed away. Brief information about his work can be found here.

Prof. Perzanowski concerned himself with applications of logic (esp. modal logics) in formal ontology. Three parts of his Locative Ontology are available here, here, and here. His essay Towards Combination Metaphysics is available here.

He also gave some thought to paraconsistent logics (he was a friend of Diderik Batens, and cooperated with the Ghent Centre for Logic and Philosophy of Science). His 50 years of parainconsistent logics is available here and his Parainconsistency, or inconsistency tamed, investigated and exploited can be found here.

Prof. Perzanowski was well known for his slightly idiosyncratic and yet very clear terminology and engaging manner of discussion (I recall a discussion we had in Warsaw in September about Godel-style ontological proofs, and man, he was difficult to argu…

König's paradox and the modal view of plural quantification

In the last chapter of this thing, I defended the view according to which plural quantification:
For some a, .....(where a can be singular, empty, or general) can (roughly speaking) be read nominalistically as:
It is possible to introduce a (singular, empty or general) name-token, such that...
One of the primafacie reasons to reject the substitutional interpretation of plural quantification was that we run out of tokens (finite sequences over a finite alphabet), if the domain is large enough.

My solution was to distinguish between different possible worlds where possible tokens are introduced, so that (assume we believe in Real Numbers):
For every real number, it is possible that it has a name.comes out true, whereas:
It is possible that every real number has a name.comes out false. So the basic idea is that even if in every possible world, there are only countably many names, the union of names in all accessible possible worlds doesn't have to be countable.

Now, I've been thinking …

PhilPapers Editorship

The editors-in-chief (David Bourget and David Chalmers) kindly offered me (well, after I applied) the editorship of the following sections of PhilPapers (they fall under Ontology of Mathematics):
Mathematical FictionalismMathematical NominalismMathematical PlatonismMathematical StructuralismMathematical Neo-FregeanismIndeterminacy in MathematicsIndispensability Arguments in MathematicsNumbersand I gladly accepted. My main motivation is, this will force me to actually spend some time checking out new stuff, and to read all those interesting papers lying around that my evil procrastinating twin would never read otherwise.

If you feel like helping out and moving stuff down the branches on the categorization tree, please do so! Also, feel free to drop me a line if you know of something available online that's not listed there!


Representing consistency

This is a response to Richard's remark on the previous post:
This has an unfortunate property that RCon doesn't have. If T is inconsistent, then, PA |- Con'(T). So this doesn't really have much claim to being called a consistency statement, does it? Since it was too long to post in one piece in the comments, here it is.

Well, yeah, I didn't say it's the most useful or intuitive one. I just said it's simple. The main question is whether the formula strongly represents in T the consistency of T, no?

So if you check out Mostowski's "Thirty years of Foundational Studes" his way of defining this is this. He first assumes that T is consistent (and extends PRA). Then he says:
"We shall say that a formula F with one free variable is a weak description of a set X of integers if for any integer n the formula F(\bar n) is provable in T just in case n is an element of X. If F is a weak description of X and ~F is a weak description of the complement of X,…

A strikingly simple consistency statement

Suppose T is a consistent formalization of arithmetic containing PRA (Primitive Recursive Arithmetic). Use some standard arithmetical encoding of formulas. Goedel's well-known second underivability theorem says that Con(T), the standardly constructed consistency statement for T, is not derivable in T.

There are, however, formulas equivalent to Con(T), which are derivable in T (although, the derivability conditions aren't satisfied). Usually, Rosser's or Feferman's examples are quoted; these are a bit complicated and involve reference to an ordering of formulas or proofs or axioms.

Here is another, strikingly simple non-standard consistency statement which is provable in T (due to Mostowski).

Let Pr express T's provability relation (it's the standard derivability predicate, nothing kinky is going on there). Define:
Pr'(x,y) ⇔ Pr(x,y)& ¬ Pr(x, ⌈0=S0⌉)
As T contains PRA, T proves 0≠S0. Since T is also consistent, T doesn't prove 0=S0. So ¬ Pr(x, ⌈0=S0⌉) is…

Peer instruction in philosophy

Here is a website devoted to something that looks like an interesting lecturing strategy. The basic idea is that every fifteen minutes or so, the lecture is interrupted and students are asked a quiz question. Then, those who got the aswers right are supposed to convince their neighbors who did not get the right answer about it. The method (i) provides the lecturer with insight into how many of the students understand the material, and (ii) makes students think harder. Here is a paper where the effectiveness of this method is studied. I might try it out some time.

Similarity, bots, uhm ...varia

I'm writing a paper on modeling various kinds of similarity relations with relational models (these are modified Bugajski models, [JPL 1983 vol. 12] for similarity), so that those of Williamson's constraints on four-place similarity relations [NDJFL 1988, vol . 29] that I find convincing are satisfied.

Also, contra Bugajski, who argued that a set of properties generating a similarity relation has to contain vague properties if the resulting structure is to be non-trivial, I rather argue that even with sharp properties we get fairly intuitive and yet quite non-trivial structures, if we assume that our concepts are more like dynamic frames (a fairly new theory of concepts uses this idea and does seem to have some empirical support, see Barsalou's stuff).

Anyway, I was looking for a good similarity jokes to use as examples, found two I like:

Whats the difference between a fish and a mountain bike?
Both can swim, except for the mountain bike.

How does a shotgun with a broken firing…

Lesniewski and Frege's way out

Recall that Frege, when faced with Russell's paradox, replaced the left-to-right direction of his Basic Law V:
{x:F(x)}={x:G(x)} → ∀x[F(x)↔G(x)]with something like:
{x:F(x)}={x:F(x)}→∀y[[y≠{x:F(x)}]→ F(y)↔G(y)]This move came to be referred to as Frege's way out (I think this name dates back to Quine's 1955 Mind paper, but that's just a guess).

Right now, I'm reading Giaquinto'sThe Search for Certainty. It's a very clear and accessible account of main debates surrounding foundations of mathematics pretty much from Dedekind and Cantor to Gödel. I really like the book, it's fun to read, the account is clear and succinct, and the author provides well-defended critical assessment of the views he discusses.

One remark. In footnote 22 (to ch. 3, part 2) Giaquinto's remarks that Lesniewski has already shown that Frege's way out still leads to contradiction.

This isn't exactly right. Strictly speaking, what Lesniewski has shown is rather that Frege's …