On 12 Mar 2014, at 21:51, LizR wrote:
On 13 March 2014 04:33, Bruno Marchal <[email protected]> wrote:
Hello Terren,
On 12 Mar 2014, at 04:34, Terren Suydam wrote:
Hi Bruno,
Thanks, that helps. Can you expand a bit on <>t? Unfortunately I
haven't had the time to follow the modal logic threads, so please
forgive me but I don't understand how you could represent reality
with <>t.
Shortly, "<>A" most "general" meaning is that the proposition A is
possible.
Modal logician uses the word "world" in a very general sense, it can
mean "situation", "state", and actually it can mean anything.
To argue for example that it is possible that a dog is dangerous,
would consist in showing a situation, or a world, or a reality in
which a dog is dangerous.
so you can read "<>A", as "A is possible", or possible(A), with the
idea that this means that there is a reality in which A is true.
Reality is not represented by "<>A", it is more "the existence of a
reality verifying a proposition".
In particular, <>t, which is "t is possible", where t is the
constant true, or "1=1" in arithmetic, simply means that there is a
reality.
You mean <>t asserts there is a reality in which the relevant
proposition is true (e.g. one in which the dog is dangerous) ?
Exactly. (Although possible(dog is dangerous) is more <>(dog-is-
dangerous) than <>t, which is more like possible(dog is dog).
That's Kripke semantics: <>A is true in alpha IF THERE IS A "REALITY"
beta verifying A.
so
<>t is true in alpha if there is a reality beta verifying t.
Now, Kripke semantics extends classical propositional logic, and t is
verified in all worlds. So, if alpha verifies <>t (if <>t is true in
alpha), then <>t means simply that there is some world beta accessible
(given that t is true in all world).
<>t = "truth is possible" = "I am consistent" = "there is a reality
out there" = "I am connected to a reality" ="truth is accessible".
Note that this well captured by modal logic, but also by important
theorem for first order theories. In particular Gödel completeness
theorem, which can put in this way: a theory is consistent if and only
the theory has a model.
Gödel completeness (two equivalent versions):
- provable(p) (in a theory) entails p is true in all models of the
theory.
- consistent(p) (in a theory) entails there is at least one model in
which p is verified (true).
Bruno
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