# [FOM] Counting models

Andy Fugard a.fugard at ed.ac.uk
Sun Jul 1 04:32:37 EDT 2007

```Dear all,

For various kinds of model I'm interested in how tricky it is to find
counter models for a given conjecture.  To begin playing with this, I
have enumerated first-order models of all 512 conjectures of
syllogistic form with the number of individuals from 1 to 5.

For instance for

forall x. A(x) => B(x)
forall x. B(x) => C(x)
----------------------
forall x. A(x) => C(x)

The number of models of the premises are:

individuals   1      2      3      4      5
models        4     16     64    256   1024

Presumably 2^(2n) in general.  (There are obviously no counter models
for the conclusion in the set of models of the premises.)

For

exists x. A(x) & ~B(x)
forall x. B(x) => C(x)
----------------------
forall x. C(x) => A(x)

the table looks like

individuals     1      2      3      4      5
models          2     20    152   1040   6752
countermodels   0      8     96    800   5760

where "countermodels" is how many of the models of the premises are
counter models of the conclusion.

My question: does anyone know of examples of work where these kinds
of things (not necessarily for syllogisms) are counted, e.g.
analytically?  I'm pretty sure for syllogisms it has been done for
Euler Circle type models (and as an aside, logically, I'm not quite
sure what those beasts are).

Best wishes,

Andy

--