Dear Organlearners,
I do not like to reply to my own contributions. But sometimes I do
have to make corrections. I am sorry for the following error. It
happened because it was late, I was tired and my concentration failed.
>Compare the length of Law of the Excluded Middle (LEM) of
>Classical Logic, namely
>p=>p == NOT(pANDp)
>with
>(p|(q|r))|((t|((t|t)t|((s|q)|((p|s)|(p|s)))
The LEM is indeed in the "outfix" notation
p=>p
or in the "infix" notation
IMPLY(p,p).
But it corrresponds (the "==") to
NOT(pAND(NOTp))
and not to
NOT(pANDp)
as I have indicated above.
The following truth table shows why:
p q p=>q NOTp p&(NOTp) NOT(p&(NOTp))
1 T T T F F T
2 *T *F *F *F *F *T
3 *F *T *T *T *F *T
4 F F T T F T
Rows (2) and (3) fall out because p and q have to be the same. We get
column (4) directly from column (1) by negation. We get column (5) by
using columns (1) and (4) in the function AND(p,q). We then negate
column (4) to get column (5).
Most systems of Classical logic do not use LEM as directly as I did
above. In the system of Rosser the LEM is somewhat disguised by
conjuncting a second antecedent to p which implies itself:
(pANDq)=>p
Here is its truth table:
p q pANDq (pANDq)=>p
1 T T T T
2 T F F T
3 F T F T
4 F F F T
We get column (4) by applying IMPLY(p,q) to columns (3) and (1). Note
that this "compound proposition" has the value T in all four its rows.
In other words, it is T for all values T (true) and F (false) of all
propositions p and q. This property of having only the value T for all
possible combinations of truth values in its constituent propositions
is what makes a compound proposition an axiom or theorem.
The form
(pANDq)=>p
makes LEM more open for effective connections (fruitfulness) than
p=>p
because q provides a handle (reactive centre) for connection.
The second error I made is in the axiom of the Nicod system, namely
>(p|(q|r))|((t|((t|t)t|((s|q)|((p|s)|(p|s)))
^
It should have been a ")" rather than a "t", namely
(p|(q|r))|((t|(t|t))|((s|q)|((p|s)|(p|s)))
In the next expression I have have symbolised the "scope" of the
various strokes by creating spaces around them when their scope
increases.
p | q|r | t | t|t | s|q | p|s
| p|s
This allows us to get rid of the brackets so that we can focus on the
deeper pattern between the "elements".
This issue of "scope" is seldom explained to students in mathematics.
Yet they are very important with respect to emergences. Every
conceivable mathematical operator (+, -, x, &, |, etc) have a scope.
In an expression like
((p|q)|(r|s))
the first and third stroke has the same scope, namely 1 level deep.
But the second stroke has a higher scope, namely 2 levels deep.
Therefore in
(p|(q|r))
the scope of the first stroke is two levels deep (we take the greatest
of 1 to the left or 2 to the right)whereas the scope of the second
stroke is 1 level deep. It means that in the expanded/spaced version
of the Nicod axiom the stroke with greatest space around it has the
greatest scope.
Best wishes
--At de Lange <amdelange@gold.up.ac.za> Snailmail: A M de Lange Gold Fields Computer Centre Faculty of Science - University of Pretoria Pretoria 0001 - Rep of South Africa
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