Re: truth/falsity of sentences in first-order logic

On 27 Feb 2006 17:18:15 -0800, "Charlie-Boo" <shymathguy@xxxxxxxxx>
wrote:


H. J. Sander Bruggink wrote:
ali_tofigh@xxxxxxxxxxx wrote:

Then this is not what "true" means in (formal) logic. Truth is
always w.r.t. a specific model.

Truth has nothing to do with models.

Huh?

Models Are Backwards. Models are
just a way to construct a sentence,

Huh?

but after it is created (whether by
using a model or not), you still have a sentence that is either true or
false according to the meaning of the logical symbols,

Huh?

and functions
and relations, used, which has nothing to do with the model (if one
was) used. The same sentence can be created using different models,

Huh?

but the truth value is the same!

Huh???

When people sometimes comment that truth and provability are defined so
differently, they are typically referring to models. However, the
definition of truth does not actually involve models.

Huh?

Why don't you learn some logic, and then come back and post
a corrected version of all this?

Systems which
force you to use variables for functions and relations when you are
writing the axioms are only postponing (interfering with) your creation
of the actual sentences that have a truth value.

Considering different interpretations can much better be done after the
specific sentence (with no function or relation variables) is defined.

C-B

A sentence is *valid* if it is
true in all models.

[snip example]

Now, if a sentence is false, then the negated sentence must be true.

This is not true. If a sentence is not valid ("false"), then
there is at least one model (A) in which it is false, but there
may also be a model (B) in which it is true. In that case, the
negated sentence is true in (A), but false in (B), so it's still
not valid (not "true").

On the other hand, there are sentences which are false in all
models. Then, of course, the negation is true in all models, and
thus valid.

So, to continue with the same example, ~((ForAll x)(Fx OR Gx) -->
(ForAll x)Fx) must be true (tilde = NOT). Is this sentence true in all
possible interpretations? In sentential logic ~(p --> q) means p & ~q.
If we were to apply this to our negated sentence we would get (ForAll
x)(Fx OR Gx) & ~(ForAll x)Fx. This last sentence does not seem to be
true in all possible interpretations.

Indeed! So you found a counter example yourself.

groente
-- Sander


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

David C. Ullrich
.

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