Deep Meaning of
Reductio ad Absurdum
Armahedi Mahzar (c) 2011
Logic as a science for me is very hard to be learned, especially if it is explained verbally without formulas. Fortunately, George Boole simplified it for mathematically inclined mind who is thinking in algebraic formulas. It becomes very convenient for such person like me.
From Boole to Kauffman …
The Boolean algebra is not an ordinary algebra which is based on numerical arithmetic, but is based on logical truth arithmetic with 0 and 1 as the only values. As the operations we take conjuction, disjunction and negation as addition, multiplication and complementation respectively. Using them we calculate many logical inferences as easy as we do numeric algebra.
Later on, Charles Sanders Peirce made the algebra simpler, using the implication IF as the only operation and he developed a pictorial algebra of logic which is also developed by George Spencer-Brown
using NOR as the only operation later. Spencer-Brown gave two algebraic axioms as the basis of logical algebra. Spencer-Brown called his mathematics of primary arithmetic and primary algebra as Laws of Form which is prelogical.
Finally, Louis Kauffman using Box Notation simplified Brownian axiomatic formulation further by stating Huntington
tautology as the sole axiom of Laws of Form. For me it is the most elegant and economical formulation of modern logic. However, the Huntington axiom is seemed to me as a meaningless axiom.
From Huntington to Box Equation ...
Later on, I found out that it is not a meaningless formula at all. In fact, it is the algebraic formulation of the most ancient logical law which was used by pre-socratic Greek philosophers: reductio ad absurdum. It is becoming clearly so, if we take the Peircean NOT-AND interpretation for the Brownian forms rather than the Brownian NOT-OR interpretation.
In Boolean algebraic formulation the Huntington axiom is written as
(x’+ y’)’+ (x’+ y)’ = x
In Brownian cross notation it can be written as
-----+ ----+ -+ -+| -+ | x| y|| x| y| = x
In Kauffman Box notation it can be drawn as
+------++------+ |+-++-+||+-+ | ||x||y||||x| y | = x |+-++-+||+-+ | +------++------+
With NOT-OR interpretation this can be read as
NOT(OR(NOT(x),NOT(y))) OR NOT(OR(NOT(x),y)) = x.
In this Brownian interpretation,
a Box is read as NOT and
juxtaposition is read as OR.
Reductio ad Absurdum .…
However, if we use Peircean Interpretation which read
a Box as NOT and
juxtaposition as AND,
then the box diagram can be read as
NOT(AND(NOT(x),NOT(y))) AND NOT(AND(NOT(x),y)) = x.
This is equivalent with
(IF NOT(x) THEN y) AND (IF NOT(x) THEN NOT(y)) = x
since NOT(NOT(x))=x
In algebraic notation it can be written as
(x’->y) & (x’->y’) = x
or
x = (x’->y) & (x’->y’)
which can be read as
+----------------+ | x is TRUE | | IF AND ONLY IF | | x' IMPLY y & y'| +----------------+
or verbally as
+------------------------------------+ | Any proposition is TRUE | | IF AND ONLY IF | | its negation IMPLY a CONTRADICTION | +------------------------------------+
This verbal formulation is nothing but
the ancient REDUCTIO AD ABSURDUM principle.
So the verbal meaning of Huntington axiom
is nothing but the reductio ad absurdum.
Consequently it can be concluded that
the whole modern logic algebra is based
on the ancient traditional REDUCTIO AD ABSURDUM
principle as the sole axiom.
For me it is a beautiful and meaningful
deep fact of logic.
. Integralism 