Re: question regrading tautology and valid formula

On Mon, 29 Oct 2007 13:12:11 -0000, translogi
<wilemien@xxxxxxxxxxxxxx> wrote:

On Oct 29, 12:19 pm, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx>
wrote:
On Sun, 28 Oct 2007 16:42:58 -0000, translogi





<wilem...@xxxxxxxxxxxxxx> wrote:
On Oct 28, 1:23 pm, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:
On Sat, 27 Oct 2007 17:08:34 -0000, translogi

<wilem...@xxxxxxxxxxxxxx> wrote:
On Oct 27, 4:03 pm, Min JIANG <mail.minji...@xxxxxxxxx> wrote:
Hi, all,

I'm studying modal logic, but one problem puzzles me. Can someone help
me to understand the differentia between tautology and valid formula
in modal logic? or they are the same?

TIA

Which book are you studying?
(different authers have different opinions about what a tautology and
a valid formula is

some posibilities (going from broad to narrow)

(very broad)
A valid formula is a well formed formula. It is not just a line of
(modal) logical symbols

(a bit less broad)
A valid formula is a formula that is satisfiable ( It can be true but
does not have to be true in every situation)

Can you give a reference where either of those is the definition?

A valid formula is a formula that is (probably true) biut not proven /
provable to be so.

Huh? Can you give a reference where the definition of "valid formula"
has the property that a formula can fail to be valid one day but
become valid the next day?

Or is a valid formula a formula that is a theorem
.

I know at least one author (Smullyan, but i don't know if he wrote
about modal logic) who has the opinion that
a tautology must be a propositional statement.
p -> p is a tautology

(Ax) ( Fx -> Fx) is not a tautology (but it is a theorem)

In the last situation the main connective is (Ax) and that is not a
propositional connective so it is not a tautology.

That's close to the standard definition. A tautology in
first-order logic must be a _substitution instance_ of
a tautology in propositional logic. For example,
AxP(x) -> AxP(x) is a tautology, according to the
definition that I _bet_ is what Smullyan actually
gives, even though it's not a formula of propositional
logic.

Maybe the auther of your book has one of these ideas in mind.
(Or maybe other ideas there are more posibilities)
Like Humpty Dumpty (Trough the looking-glass, Lewis Carroll) says, It
means just what I choose it to mean -- neither more nor less.

Not all metalogical terms in logic are not as rigourous defined as
logic itself is.

Good luck hope this helps,
Otherwise please state to wich book you are refering or a quote or
so.
(if it is not a such widly known book)

Or ask your lecturer.
He will find it a smart question

It shows that you have thought about it.

************************

David C. Ullrich

Addition

A valid formula is a formula that is (probably true) biut not proven /
provable to be so.

Huh? Can you give a reference where the definition of "valid formula"
has the property that a formula can fail to be valid one day but
become valid the next day?
<<<
No no definition

I meant more like the famous Con (T) or so that is true (valid?)
But cannot be proven to be so.
(I find it famous not infamous)

Assuming we're thinking of the same "famous Con(T)":
Yes, it's true. Who says it's _valid_? Your post
iis the first I've heard of this.

But my first intension was to show a lot of possibilities so that he
could think about it.

Uh, right. Why didn't you include the possibility that "valid"
means "purple", as long as you're including possibilities that
you know are not the actual definition?

And it is interesting to notice that Smullyan and Smith do disagree
about this...

Really? I don't see where.

************************

David C. Ullrich- Hide quoted text -

- Show quoted text -

According to Smullyan only propositional statementschema's can be
tautologies.
he even goes so far in Ax(Px) -> ~Ex(~Px) is not a tautology because
it is in the form A -> B what is not a tautology.

according to Smith
(from his lecture notes pag 17)

J is a modal tautology iff for any modal valuation Val(i) over
any i, Val(i) (J, a) = T.
so his definition is much broader.

Which is why he's careful to call this a "modal tautology"
instead of just a tautology.

Smullyan is not talking about modal logic, is he? Assuming
not: Then the two notions of tautology _agree_ when
restricted to propositional logic.

[](P->P)
Would be a tautology in Smith's sense. (because it is always true )
But it would not be a tautology in Smullyan's sense. (it is not a
propositional statementschema )

True not equal to valid?
You are right, but they are close related ideas.

Yes. Just as 4 and 5 are closely related ideas.
In spite of the fact that 4 and 5 are closely related
ideas you shouldn't tell people that 2 + 2 = 5 when
they're asking for help with their arithmetic.

I know in argumentation
Valid does not mean sound.
and things like that.

So what?


Maybe Smullyans tautology has more to do with valid than with true.

Maybe.

(it is a valid inference step)












************************

David C. Ullrich
.

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