# [FOM] Counting models

Steve Stevenson steve at cs.clemson.edu
Mon Jul 2 13:35:10 EDT 2007

```Andy,

Just curious since you're in psychology. Is this work in reference to
mental models in Johnson-Laird theory?
--
steve
steve at cs.clemson.edu
also fatmarauder at gmail.com

“The surest way to become a pacifist is to join the infantry.” Bill
Mauldin

On Jul 1, 2007, at 4:32 AM, Andy Fugard wrote:

> 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
>
> --
> Andy Fugard, Postgraduate Research Student
> Psychology (Room F15), The University of Edinburgh,
>    7 George Square, Edinburgh EH8 9JZ, UK
> Mobile: +44 (0)78 123 87190   http://www.possibly.me.uk
>
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```