Re: Non-Implication

Re: Kirsti Määttänen
At: http://permalink.gmane.org/gmane.science.philosophy.peirce/8798

JA: This is what makes sense in terms of adding information
     to a term in order to make it more precise, which is the
     sense of “implication” that Peirce uses when he equates it
     with “information”.

KM: What about the senses of “implication” when
     Peirce DOES NOT equate if with “information”?

I thought of 3 or 4 things that Peirce might have meant by “non-implication”.
I tried to pick the one that best fit the context of what Peirce was saying
in that quotation, and also fit Tom's context of “Diagrammatic Thinking”,
at least, as I thought he might be alluding to Peirce's graphical logics.
If I had not started my reply back then from my phone in a coffee shop,
I might have gone straight to CP or CE or NEM for full enlightenment,
but as it was I had to wing it on my own e-lightenment.

Backing up a bit, here are some of the possibilities that come to mind:

Let P be a predicate term or a proposition.

In propositional logic, implications have the form P => Q.
In Peirce's alpha graphs, implications have the form (P (Q)),
where parenthetical enclosures in a line of text can be used
to simulate circles around expressions in existential graphs.

Briefly, “(P)” means “not P”, while “P Q” means “P and Q”,
so “P (Q)” means “P and not Q”, in other words, “P without Q”,
so “(P (Q))” has the very intuitive reading “not P without Q”.

To avoid confusion in mixed contexts, I'll use square brackets
for the ordinary use of parentheses everywhere below.

In quantificational logic, we have the related form [For all x] [Px => Qx].
We draw the same Venn diagrams for both of these forms, merely reading them
in slightly different ways.

1. One thing Peirce might have meant by a “non-implication” is
    the negation of an implication.  For the implication (P (Q))
    the corresponding non-implication would be ((P (Q))), which
    reduces to P (Q).  In order to understand what a proposition
    like that tells us about P, which seemed to be the point of
    the exercise, it is best to shift up to the next level.

2. Taking an implication to be a quantified proposition of the form
    [For all x] [Px => Qx], the corresponding negated implication is
    a quantified proposition of the form [For some x] [Px and not Qx].
    That is a way of saying that the original implication has counter-
    examples, so we might understand a “non-implication” as being any
    one of these counter-examples or the information that they exist.

3. Moving up to a level where the predicates themselves are not fixed,
    a “non-implication” of P might be any predicate whatsoever that is
    not implied by P.  Visualized in a Venn diagram, that would be any
    region whatever that does not wholly contain the region where P is
    true.  That is one possibility for what Peirce meant, but it gives
    us very little information that might be said to “determine” P.

4. The last resort is the one to which I resorted before.  We take an
    implication to have the form (P (Q)) and we take the corresponding
    non-implication to have the form (P ((Q))), which reduces to (P Q).
    That is a statement that P and Q are mutually exclusive predicates,
    and so we might understand a “non-implication” of P to be any such
    predicate Q whose extension lies wholly outside the extension of P.
    That is in fact a reasonable way to bound the extension of P, very
    much on a par with bounding P by giving a predicate Q whose region
    wholly contains it.

At any rate, that is why I went with the last resort as my first guess.

JA: Removing the double negation, we get (A B),
     which can be read as “Not both A and B”,
     or “A and B are mutually exclusive”.

KM: Do you equate “Not both A and B” with “A and B are mutually exclusive”?

KM: In asking this I have in mind the rule of the excluded middle (and its destinies).

Yes, that brings in Peirce's whole logic of information,
including the consideration of generality and vagueness.

But enough for a Monday morning ...

Regards,

Jon

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