Wednesday, May 27, 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. ;)

Postdoc position in History and Philosophy of Logic

There's a postdoc position in Amsterdam, related to history of logic, Bolzano, Tarski, Lesniewski etc. More details here.

Tuesday, May 26, 2009

"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 correct.
Now, it is an observation as old as Aristotle that one can not provesomething from nothing. One must always begin with some body of `obvious' facts and proceed from there. In practice, most mathematicians contentedly place hundreds of facts in this `obvious' category in order to get on with their proper business of discovery and verification of the non-obvious.
But at least once in a mathematician's career, it is good to take a sharp look at the status of the `obvious' facts; and it is probably best to do it early, and get it over with. As we remarked above, it is not possible to do away with all assumptions, even in mathematics. But, one of the great achievements of 19th and early 20th century mathematics was the careful and precise limitation of exactly what a working mathematician must `accept on faith'. That is, it was discovered what can constitute an irreducible minimum of `obvious' facts.
It is the purpose of this book to present such an irreducible minimum, and show how most commonly assumed facts about numbers follow directly. Nonetheless, this book is a bit of a fraud, because like all mathematicians we still assume that some obvious facts are more obvious than others. This is a book about the number systems, so for our purposes we assume as `sufficiently obvious' a variety of pre-numerical facts. Specifically, we assume, without being too explicit about the matter, several principles about the behaviour of sets or collections. Now this material too has been subjected to a similar treatment, also around the turn of the century. Today one can find basic set theory developed from a small number of axioms in many books on elementary set theory. But our book is long enough already, so we elected to omit this material here. For us the issue is: given a variety of `obvious facts' from set theory, what elementary properties of numbers must one accept in order to logically derive the entire basic framework of mathematics.

Friday, May 22, 2009

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 wild
  • Capital punishment
  • The Copyright wars
  • Different cultures, different selves
  • Animal minds
  • Bodies for sale
  • Digital selves
  • Morality of food
  • We've been framed: how language shapes politics
  • Varietes of love
  • Athletic beauty
  • Philosophy of wine
  • Immigration
  • Marriage and monogamy
  • If truth is so valuable, why is there so much BS?
  • The erotic vs. the pornographic
and many others.

Wednesday, May 20, 2009

An amazing bit of poetry

Here you can find an interesting poem by Geoffrey K. Pullum, Scooping the Loop Snooper, treating about the halting problem and its undecidability.


Monday, May 18, 2009

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 argue with).

He also played a political role as one of the leading figures of the Krakow section of Solidarność.

He will be missed.

Friday, May 15, 2009

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 prima facie 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 about König's paradox (this is G. Priest's formulation from his essay Paraconsistency and Dialetheism, in Handbook of the History of Logic, vol 8):
There is an uncountable infinitude of ordinal numbers, but there is only a countable number of descriptions in English. Hence, there are many more ordinal numbers than can have names. In particular to turn the screw, since the ordinal numbers are well-ordered, there is a least ordinal number that has no description. But we have just described it.
Now, in the framework I like to think in (the modal framework mentioned above), the problem doesn't seem to arise (even given the assumption that ordinal numbers actually exist, which I'm not inclined to accept but that's a different story). Why?

Well, even if in any possible situation the number of English descriptions is countable, it doesn't mean that there is an unnameable ordinal number. So say, we are at certain time t. There is a set of actually existing English tokens which describe ordinals. Then at t+1 we produce the token:
The least ordinal number that has no description.
Now, as I see it, there are two different things that can be meant here.
  1. The least number for which no description actually existed at t.
  2. The least number for which no English description can be introduced, i.e. for which there is no possible description.
If 1. is meant, then no problem arises, because the newly introduced description exists at t+1 but not at t. If 2. is meant, then the description introduced doesn't pick an ordinal, because there is no ordinal for which no English description can be introduced, and empty sets don't have least members. Basically, from the fact that necessarily, the number of English descriptions is countable, it doesn't follow that there are ordinals that can't be described.

Of course, they can't be simultaneously described in one possible world. Say one points this out and tries the description:
The least ordinal that can't be possibly described in a simultaneous description of (some) ordinals.
But here, again, from the uncountability of ordinals it doesn't follow that there is such a number, because there are many ways we can go about naming things, and an ordinal not named in one scenario might be named in another one.




Thursday, May 14, 2009

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 Fictionalism
  • Mathematical Nominalism
  • Mathematical Platonism
  • Mathematical Structuralism
  • Mathematical Neo-Fregeanism
  • Indeterminacy in Mathematics
  • Indispensability Arguments in Mathematics
  • Numbers
and 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!


Wednesday, May 13, 2009

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, then we call F a strong description of X in T."
Right, so what we care about in this setting is whether Pr' strongly represents in T provability in T. And indeed, it does.

A slightly different approach is not to assume the consistency of T, and to say that Pr' captures (in T) provability (in T) iff both:
If [n] (the object whose godel number is n) is a proof of [m], then T proves Pr'(n,m), and

if [n] is not a proof of [m], then T proves ~Pr'(n,m).
If T is consistent, then everything works out just fine for Pr', and if T is inconsistent, then vacuously also both conditions are satisfied, right?

(also, correct me if I'm wrong, but if T is PA, then Con'(PA) captures PA's consistency, and there's nothing wrong about PA proving Con'(PA) if PA is inconsistent.)

Now, if you call a consistency claim of T any formula that in T strongly represents T's consistency, Mostowski's formula is a consistency claim.

This all only shows that strong representation requirement isn't as hard to satisfy as one might initially think. And of course, there are other requirements you might put on formulas to count as "real" consistency claims.



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 true of all numbers x. So Pr(x,y)& ¬ Pr(x, ⌈0=S0⌉) is true of exactly the same numbers as Pr(x,y). So Pr and Pr' have the same extension.

Now, construct the consistency statement Con'(T) as follows:
Con'(T) ⇔ ∀x ¬ Pr'(x,⌈0=S0⌉)
This boils down to:
Con'(T) ∀x ¬[Pr(x,⌈0=S0⌉)& ¬ Pr(x,⌈0=S0⌉)]
which says that nothing is a number of a T-derivation that proves both 0=Sn and its negation. Clearly, proving Con'(T) in T doesn't require much effort.


Monday, May 11, 2009

Troubles with Elsevier

Following Brian Weatherson I just allow myself to spread the word that Elsevier has been publishing fake journals for money (see here). Interesting reactions here.

Friday, May 8, 2009

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.

Lectures online, Arche

Arche research centre is posting lecture multimedia online. So far, they have Crispin Wrigh, Graham Priest and Ole Hjortland. Also, the format is pretty cool (it's sound AND slides).

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 pin resemble a government worker?
It won't work and you can't fire it.
More importantly, I stumbled upon an article about work being done by Julia Taylor and Larry Mazlack to get bots understand jokes based on puns. The task is quite non-trivial, considering the vast computational complexity of bacground knowlege searches etc.

Of course, this doesn't mean a great breakthrough is to be expected anytime soon (or anytime at all), but still, finding similarities between words and employing them efficiently in jokes isn't really that easy. The way the bot works is supposed essentialy to be this:
The program then checks to see if the message is consistent with what would make sense. If it doesn’t, the bot searches to see if the word sounds similar to a word that would fit. If this is the case, the bot flags it as humor.
Three things come to my mind: (1) defining what you mean by consistency with what would make sense and finding a way to check it might be a serious problem. (2) It's not sure whether (a) the bot is supplied with a list of similarities between words, or (b) can search its dictionary and identify words that sound similar (case (b), of course, sounds a bit more interesting). (3) It would be even more interesting if the bot could crack new jokes.

Another interesting thing I found following a link from Wikipedia (I was looking for OpenCyc to play around with, but didn't know what it was called). Wiki, while writing about Cyc mentions a fairly new application of it:
The comprehensive Terrorism Knowledge Base is an application of cyc in development that will try to ultimately contain all relevant knowledge about terrorist groups, their members, leaders, ideology, founders, sponsors, affiliations, facilities, locations, finances, capabilities, intentions, behaviors, tactics, and full descriptions of specific terrorist events. The knowledge is stored as statements in mathematical logic, suitable for computer understanding and reasoning.
So, in fact, there are moments where logic comes handy. Good to know. A slightly more serious report about how this database works is available here.

Monday, May 4, 2009

Many thanks, Greg!

Some time ago, I won a slightly geeky quiz by Greg Restall by providing a false answer (well, interestingly, a false answer was needed). Today I received the prize, it's a copy of The Law of Non-Contradiction. Now I have two copies on my desk (one is from the library, though). So, many thanks, Greg!

Sunday, May 3, 2009

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's The 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 way out leads to contradiction with three additional assumptions (well, after translation from Lesniewski's language):
  1. ∀F[∃xF(x)→∃y(y={z:F(z)})]
  2. ∀F∀x,y[x={z:F(z)} & y={u:F(u)} → x=y]
  3. ∃x,y,z[x≠ y & x ≠ z & y≠z]
The first says that if something is F, then the class of F's exists. The second says that the class operator is a function (this is especially non-trivial in Lesniewski's systems), so that when you have a concept F, 'the extension of F' will name at most one object. (In a sense, assumption 2 corresponds to an instance of the right-to-left direction of Basic Law V). The third says that there are at least three distinct objects.

For an excruciatingly boring but a fairly detailed account of these (and related) things, see this paper. If you want to see a streamlined version of Lesniewski's proof on the blog, let me know in a comment and I might come up with a short version.