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 <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?
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, which was my objection to writing
<>t. In such a formula, t can only be regarded as shorthand for some tautology. 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."
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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