Fwd: Re: Lattice of theories
John,
I bet that the principle of the lattice of theories that is
under disussion here has been greatly clarified recently.
Isn't it just good to make it crystal clear? Here is just
an example from theory to practice.
In the lattice of theories, there are only tautologies in
Top that are known a priori. This reminds of D.M.Armstrong's
principle of a posteriori realism. According to it,
universals can be discovered only a posteriori, by total
science. That is why the theories in Top are not universals,
and have no causal powers upon anything.
In the successors of Top are all theories; those theories
that really tell the truth about the actual states of
affairs (aka. universals), and those that do not. This is
one subset of the lattice of all theories, where a,b,c denote
theories:
T
|
a
/ \
a&b a&c
\ /
a&b&c
There are all the tautologies in the Top, and in the
successors of Top are some example theories. Bot is
the conjunction of all the theories. Let us suppose
that b is theory about roundness, c is theory about
angulatedness, and a is theory about being between
-100 and +100 celsius. Above, bottom would not
instantiate any particulars.
We don't want useless categories of theories nor
tautologies in our applications, and therefore it is
intentional to use this with practical applications:
a
/ \
a&b a&c
This sort of semantics is used generally with standard
W3C Web ontologies for example.
Avril
----- Forwarded message from "John F. Sowa" <sowa@bestweb.net> -----
Avril,
The top is a theory that has zero axioms, but it has every
theorem that can be proved from zero axioms -- and there are
infinitely many of them:
p -> p. (p & q) -> p. (p & q) -> q. (p & ~p) -> q....
This is just elementary logic. There is nothing profound
about it.
>
> T
> |
> a
> / \
> a&b a&c
> \ /
> a&b&c
>
>There are no theories in the Top, and in the bottom
>there are all the theories. Bot is the conjunction
>of all the theories.
The first clause is false, and the second is true.
> The case is closed.
Most definitely. It should never have been opened.
John