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.


Anonymous said...

It's good to see logicians exercising their right to arm bears.

Rafal Urbaniak said...

good one:)