Jon,
The way I learned it, (formal) implication is not the /assertion/ but
the /validity/ of the (material) conditional, so it's a difference
between 1st-order and 2nd-order logic, a difference that Peirce
recognized in some form. If the schemata involving "p" and "q" are
considered to expose all relevant logical structure (as usually in
propositional logic), then a claim like "p formally implies q" is false.
On the other hand, a proposition /à la/ "if p then q" (or "p materially
implies q") is contingent, neither automatically true nor automatically
false. I agree that you can see it as the same relationship on two
different levels. That seems the natural way to look at it.
Another kind of implication is expressed by rewriting a proposition like
"Ax(Gx-->Hx)" as "G=>H". In other words "All G is H" gets expressed "G
implies H". In first-order logic, at least, it actually comes down to a
material conditional compound of two terms in a universal proposition.
If in addition to logical rules one has postulated or generally granted
other rules, say scientific or mathematical rules, then these lead to
scientific or mathematical implications, the associated conditionals
being true by the scientific or mathematical rules, not just
contingently on a case-by-case basis. Anyway, all these kinds of
implication do seem like the same thing in various forms.
It's not clear to me how any of this figures into the
concept-vs.-judgment question. The only connection that I've been able
to make out in my haze is that when we say something like "p formally
implies p", we're thinking of the proposition p as if it were a concept
rather than a judgment; our concern is limited to validity. If we say
'p, ergo p' or, in a kindred sense, "p proves p," we're thinking of p as
a judgment, and our concern includes soundness as well as validity.
Best, Ben
On 5/11/2012 2:25 PM, Jon Awbrey wrote:
Ben,
Just to give a prototypical example, one of the ways that the distinction
between concepts and judgments worked its way through analytic philosophy
and into the logic textbooks that I knew in the 60s was in the
distinction
between a "conditional" ( → or -> ) and an "implication" ( ⇒ or => ).
The
first was conceived as a function (from a pair of truth values to a
single
truth value) and the second was conceived as a relation (between two
truth
values). The relationship between them was Just So Storied by saying
that
asserting the conditional or judging it to be true gave you the
implication.
I think it took me a decade or more to clear my head of the dogmatic
slumbers
that this sort of doctrine laid on my mind, mostly because the
investiture of
two distinct symbols for what is really one and the same notion viewed
in two
different ways so obscured the natural unity of the function and the
relation.
Cf. http://mywikibiz.com/Logical_implication
Regards,
Jon
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