Wednesday, February 17, 2010
Wednesday Math, Vol. 109: Logic and Duality
Logic has been studied as a subject unto itself for about 2,500 years now. What our confused penguin friend here is trying to do is construct a syllogism, and he's making a common mistake. The study of mathematical logic made a major stride back in the 1850's when George Boole published The Laws Of Thought, and now there are many mathematical systems, including set theory, computer architecture as well as mathematical logic itself, that follow the rules known as Boolean algebra.
Instead of syllogisms being the basic building blocks, mathematical logic breaks down the system into smaller bits, and builds all logic connections by using the basic connectors AND, OR and NOT, and the logical states TRUE and FALSE. Much like the duality of trigonometry discussed last week, we have two way connections between objects, though unlike trig, the connections do not have an easy physical representation as to why they are duals.
The basics are:
AND is the dual of OR.
TRUE is the dual of FALSE.
NOT is the dual of NOT, which is a situation called self duality.
All of the axioms of logic have a dual axiom, by switching all ANDs for ORs and vice versa, and doing the same for any TRUE or FALSE that shows up.
commutative AND: a AND b = b AND a
commutative OR: a OR b = b OR a
Domination by truth: a OR TRUE = TRUE
Domination by falsity: a AND FALSE = FALSE
DeMorgan's Law #1: NOT(a AND b) = (NOT a) OR (NOT b)
DeMorgan's Law #2: NOT(a OR b) = (NOT a) AND (NOT b)
This is not to say that the study of logic is very simple, far from it. But duality in a system can be a very nice tool for cracking open some very difficult problems.