On 3/13/2014 8:33 AM, Bruno Marchal wrote:
On 12 Mar 2014, at 21:51, LizR wrote:
On 13 March 2014 04:33, Bruno Marchal <marc...@ulb.ac.be
<mailto:marc...@ulb.ac.be>> 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.
This is a point that confuses me in trying your exercises (which I'm attempting to do
without reading your exchanges with Liz). There you refer to a formula being "respected"
when it is true in all worlds for all valuations. But does "all valuations" of a formula
A include f when A=p->p? Are we to assume that "t" is a formula in all worlds and it's
value is always t? And then is f also a formula in every world?
Brent
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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