Monday, February 28, 2011

Friday, February 25, 2011

Teddy bears, guns and modal logic

It's the time of the year when a new semester starts in Poland and I'm in Gdansk for a while (it's annoyingly and unusually cold, it feels like Calgary for some reason -  seems I haven't escaped after all. Damn you, global warming!). Anyway, one of the courses I'm teaching is non-classical logic and I'm using Graham Priest's awesome book. If you've ever taught modal logics, you probably observed that it's sometimes difficult to get the students to remember which normal modal logic is related to which properties of the accessibility relation. Here's a trick I invented last year, feel free to use it (just give credit where it's due).

First off, Priest uses Greek letters to denote the main properties of the accessibility relation: 

  •  \rho stands for reflexivity
  •  \sigma stands for symmetricity
  •  \tau stands ofr transitivity
  •  \eta stands for extendability

The main logics worth remembering in a basic course are T, D, B, S4 and S5:

- T is determined by the class of \rho-models
- D is determined by the class of \eta-models
- B is determined by the class of \rho\sigma-models
- S4 is determined by the class of \rho\tau-models
- S5 is determined by the class of \rho\sigma\tau-models

Here's a mnemotechnic to help people remember this.

First, you want people to remember the ordering of the logics:
T, D, B, S4, S5
instead, (make them) memorize:
TeDdy Bear with 45S
The coding here is quite obvious.

Next, you want people to remember the ordering:

\rho, \eta, (\rho \sigma), (\rho \tau), \(rho \sigma \tau)

instead,  (make them) memorize:
R stands for \rho, E stands for \eta, S stands for \sigma, T stands for \tau.

Thus, the matching of modal logics with properties of the accessibility relations is encoded by:

a Teddy bear with  45s rests.
Two problems:
  • You still have to remember that the first two logics are just \rho and \eta, and that the remaning one involve \rho
  • You have to remember to repeat "st", because it stands for "first \sigma, then \tau, then \sigma and \tau together".
Now, get people to imagine a teddy bear with a gun (preferably with a 45).

One option is to use this (source):

Another is to use this:

Or you can use this:

(I was also thinking of taking a picture of a teddy bear with a 45, I have everything I need apart from a teddy bear and a 45.)

Of course, the key sentence is not the best English phrase (I think there are at least some fragments of Yeats' poetry which trump its genius; not so sure about J. Conrad though), and there are some weak points (it doesn't extend easily to non-normal modal logics, you have to remember at least two extra assumptions I mentioned, and so on). So, if you have a better mnemotechnic, please share.

Wednesday, February 16, 2011

Reichenbach on the philosophical insignificance of G's second incompleteness theorem

Nowadays, one standard answer to various exaggerated claims about Gödel's second incompleteness theorem (the unprovability of consistency) is that even if an interesting mathematical theory could prove its own consistency, this wouldn't help us much because inconsistent theories also prove their own consistency, (so we would still have no idea whether the theory is consistent). (For instance, I think this sort of remarks, without further references, can be found in Franzen's and Smith's books, but I don't have them handy now).
In a somewhat uninspired moment of mine, I wondered why this rather straightforward observation didn't get through to the wider philosophical audience earlier and when it was formulated. The earliest mention I run into so far (although, it's not like I spent days browsing stuff systematically) is in Reichenbach's 1948 unpublished lecture notes (which, by now, have been published in 1978). The fragment comes from the first volume:  
This would mean that the proof of consistency of the language L could be given within L. A simple analysis shows that this would not improve the situation, since in this case our proof of consistency of L would be of value only if we were sure that L is consistent. In case L were not consistent, we could also deduce the statement of the consistency of L, with the qualification that then the negation of the statement were deducible too. Thus if the consistency of L were deduced within L, this fact would not prove the consistency of L.
So, one reason might be that the remark occurred in lecture notes which were unpublished for a while (by the way, if you know of Reichenbach's point being published earlier, I'd appreciate a reference), and even when they were published, it was in a two-volume set of Reichenbach's papers, which very few non-specialist would buy (or grab from a library shelf, for that matter).
But I don't think this is the whole story. The point is simple enough and certainly came to minds of many bright people who gave the issue proper consideration. The reason why this is often omitted by (hack?) philosophers is not that they are too dumb to get this point (well, at least some of them aren't). Rather it is that wishful thinking sometimes prevents people from considering objections to their view. The perspective of reaching strong philosophical conclusions by relying on some fancy mathematical apparatus, which (they think) would lend splendor to the philosophical claims themselves, is quite tempting.
Another reason is that Reichenbach's point applies to cases where one wants to claim that mathematics is somehow epistemologically flawed and that there is no mathematical certainty because mathematics cannot prove its own consistency. (... and if real mathematical knowledge is unattainable, then no knowledge is, science is useless, blah blah, blah...). It doesn't help much when one wants to suggest that since we, humans, can establish consistency claims, and mathematical "devices" cannot, our minds are not computing machines. In this argument, it seems that no deep epistemological considerations of the former sort are needed.
Incidentally, it seems that despite the fact that I've seen some philosophers using both strategies in one passage (or breath) both stances are incompatible. Either you say that mathematics is unreliable and then you cannot rely on it in an argument against computational theories of mind, or you rely on mathematics, give an argument in philosophy of mind, but cannot simultaneously deny the reliability of mathematics. (By the way, (1) even if you do the latter, you admit that mathematics can be done reliably by humans and the fact that a mathematical theory cannot prove its consistency has no bearing on whether humans can use it sensibly, (2) the sketched argument against computational theories of mind also doesn't work, but that's a whole different story).