Ed and Patrick, (01)
I agree that it is essential to distinguish the notation of
logic from the proof theory and the model theory. I was
trying to be brief and omitted a lot of the qualifications. (02)
My main point was to make it clear that logic can be expressed
in any precisely defined declarative notation. And that includes
precisely specified subsets of natural languages, diagrams of
various kinds, etc. The definition I prefer is the one I
quoted from Peirce: (03)
"the formal science of the truth of representations" (04)
EB> If you look at the UML standard (and discard the noise),
> you will find that there is a defined reasoning process,
> but it is almost entirely based on subsumption -- there is
> no general support for modus ponens, for example, until
> you add OCL. (05)
My recommendation is to replace that standard with a translation
of each UML diagram type to Common Logic. That automatically
brings in the model theory, etc. (06)
EB> And (most?) natural languages support many reasoning processes,
> because they have if/then structures and type/instance notions,
> and so on. (07)
All major ones do. There are many still unresolved questions
about some of the languages with very small numbers of speakers
and insufficiently recorded and analyzed corpora. The Pirahã
of Brazil are one of the most intriguing and frustrating tribes: (08)
http://www.spiegel.de/international/spiegel/0,1518,414291,00.html
Brazil's Pirahã Tribe: Living without Numbers or Time (09)
Without logic, it's hard to define numbers, and the Pirahã can't
count -- they can't even be taught how to count, even when they
realize that they're routinely being cheated. When they get a
vague feeling that they've been cheated, they make up for it
by staging a midnight raid. (010)
Re Heraclitus, Plato, Aristotle, and John the Evangelist: My main
point is that the connection between logic and language is very old.
I should have left out John, but I couldn't resist countering the
idea that I was making the definition of logic too broad by showing
that the logos is even broader. In most of my discussions I just
cite Aristotle. (011)
PD> ... is there some universal notion of "precise" and "truth" that
> I have missed along the way? I suspect both of those are as hotly
> debated as many of the other terms that have been discussed in this
> forum. (012)
There are no serious debates on the standards of precision for
a notation. With computer languages, the standards are decided
instantly. Can you write a formal grammar for the notation?
If you do and the computer compiles it, it's precise. If you
can't, go back to your cubicle until you get it right. (013)
The standards for "truth" in logic are Tarski's model theory
or some variation. See the following paper (014)
http://www.jfsowa.com/logic/tarski.htm
The Semantic Conception of Truth (015)
There is no question that Tarski's approach captures part of what
is meant by truth in ordinary language. But even Tarski admitted
that there are aspects of meaning in natural languages that his
approach cannot handle. The debates are about how much is left out. (016)
PD> Rather difficult to say whether the placement in Hell was a
> true or false assertion. (017)
There is a difference between making a precise assertion, and
determining whether it is true or false. (018)
As I said, a notation that can be used in logic must be (a) precise
and (b) able to make statements that can be judged true or false.
Determining whether a particular statement is true or false,
however, is a totally different matter. (019)
You don't even have to go to hell to find hard questions.
For example, how many bacteria are in the room you are sitting in?
And there are infinitely many true statements about ordinary
arithmetic for which there is no way to prove that they are true. (020)
John (021)
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