On 13 Mar 2014, at 20:05, meekerdb wrote:
On 3/13/2014 11:43 AM, Bruno Marchal wrote:
On 13 Mar 2014, at 18:03, meekerdb wrote:
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> 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?
No, the valuations are defined only on the atomic p, q, r, ....
(in modal propositional logic).
Then the arbitrary formula get their value by the truth table, and
the modal formula get their value by the Kripke semantics, that is,
the truth values of the boxed an "diamonded" propositions depends
on the locally accessible worlds.
Then t and f cannot be treated as atomic propositions,
Why? Pi is constant, but still a (real) number. Why could we not have
constant proposition?
which was my objection to writing <>t. In such a formula, t can only
be regarded as shorthand for some tautology.
If you want. Any simple provable proposition would do.
So <>t doesn't mean "There is some reality" it means "There is some
tautology: a proposition that is t in virtue of the definition of
relations "&", "V", "~", etc."
<>t means, in Kripke semantics, that there is a world in which t is
true (and as t is true in any world, it does mean that there is a world.
Then when "<>A" is the diamond "consistency of A", it means that there
is a model verufying A, by Gödel's completeness theorem.
Bruno
Are we to assume that "t" is a formula in all worlds and it's
value is always t?
Yes. It is a boolean constant. You can suppress it and replaced it
by (p -> p), as this is true in all words (as this is true in the
worlds where p is true, and is true in the worlds where p is false).
And then is f also a formula in every world?
You can represent it by (p & ~p), or just ~t, and it is false in
every world.
The cul-de-sac worlds get close, as they verify []f.
Fortunately they don't verify []A -> A.
f is never met, in any world, but you can met []f, [][]f, [][]
[]f, ... G* proves ◊[]f, ◊[][]f,◊[][][]f, ... in the "G-
worlds".
You say (p & ~p) is false in every world, but f is never met in any
world. That seems contradictory. If p is a proposition in some
world, are we not always allowed to form (p & ~p), which will have
the value f for all valuations of p?
(Do everyone see a lozenge here: ◊ ?)
I see it.
Brent
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