Re: PC(1): An introductory formal logic


David C. Ullrich wrote:
On 7 Jul 2006 09:20:22 -0700, "George Dance" <georgedance04@xxxxxxxx>
wrote:

David C. Ullrich wrote:
On 6 Jul 2006 06:33:14 -0700, "George Dance" <georgedance04@xxxxxxxx>
wrote:

David C. Ullrich wrote:
On 5 Jul 2006 19:10:25 -0700, "George Dance" <georgedance04@xxxxxxxx>
wrote:

Frederick Williams wrote:
George Dance wrote:

In PC(1), there is no distinction between semantic and syntactic: the
truth table is the proof method, and the semantic interpretations are
the lines (m, m et al) of the truth table. It is only when the student
learns PC(1) and PC(2), and goes on from there to a formal deductive
system - where what is provable in the system may be different from
what's valid by the truth table, that the semantic/syntactic
distinction even comes up. So that is precisely the point to introduce
it; when it makes some sense to the students, not at the beginning
where it means nothing.

Then why do you insist that the symbol should be |- when
what you're really talking about is |= ?

|- isn't an operator of the system; it's a symbol used to talk about
the system.

Huh? Yes, we all know that - what does that have to do with the
question of whether it should be |- or |=.

For one thing, because PC(1) has nothing to do with what's PC-valid
(which is what |= is standardly used to mean in PC). All that PC(1)
is concerned with is the mechanics, or syntactics, of a formal system:
with what's proveable in it. What is proved are the "valid inferences"
as defined; the way to prove those is to construct a truth table. One
proves that an inference [A |- B] is valid by constructing the truth
tables for A and B and demonstrating that, on those lines where A takes
the value T, B also takes the value T.

After the schoolchildren know how to do that, it's time to give them
the axioms and move on into basic PC(2) (PC without the conditional) -
which will be their first sound and complete system. At that point
it's a good idea to tell them about soundness and completeness, and the
idea of |- (proveability in the system) and |= (PC-validity) and
that's when I think they should be taught it. Not on their first day
of school.

Which symbol one uses, in this case, makes absolutely no
difference; there is no difference between a proof using a truth table
test, and PC-validity. I already told you that.

Uh, yes, you told me that. In actual logic there is a huge
difference

Only if one assumes that PC(1) is not an 'actual logic' - which
assumption I'll reject, as it looks like nothing but a cheap shot from
an obfuscator.

- if there were no difference then the Soundness
and Completeness Theorem, to the effect that |- and |= are
actually equivalent, would have no significance.

The Soundness and Completeness Theorems are only significant when one
is discussing (possibly) sound and complete systems. Since you already
know that PC(1) is not complete (as it was designed to not be
complete), I would conclude that you're just trying to raise as many
red herrings as you can, and not trying to make any serious point here.


In fact in your system there is literally no difference, but
only because you've _defined_ |- to mean what everyone else
would call |=. If for whatever reason you're _going_ to insist
on teaching kids about these things, the idea that you
should use the wrong symbol, given that there's no compelling
reason not to use the right symbol, seems like a bad idea.

I mean if you were writing a book on arithmetic there's
also no reason you could not define "-" to mean what most
people call "+", and then teach them that 2 - 2 = 4. But
it doesn't seem like a good idea.

But, since you think it's quite important, I'll do something for you:
If I write a book or article on PC(1), I'll use |=, and be sure to
thank you for your help in the acknowledgements.

Fabulous. Make certain to let us know when it's published.

Which is the point of PC(1) as well.


.

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