[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
--
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
More information about the FOM
mailing list