Chance or Intelligent Design ? (part 1)

By James Andrew Choury, serving with WorldVenture in North Brazil September, 2005

[Leia este artigo em português]

Logic and Mathematics. What have these disciplines to do with evidence of God’s existence?

The Law of the Non Contradiction is a fundamental law of logic. The law states that a proposition and its contradictory can never both be true. In symbols we write [ ~ (A & -A)] where the & symbol means that both A and ~A are true and the tilde (~) means “not”. This phrase is logically equivalent to (A v ~ A) which tells us that one or the other (A or ~A) must be true without exceptions (this by the way is called the Law of the Excluded Middle). In short, the Law of Non Contradiction tells us that a proposition and its contradictory are never both true and the Law of the Excluded Middle tells us that one of them must be true. A more classical way of describing contradictories is that one is the negation of the other and they cannot both be true and they cannot both be false. Logical conversation about any subject must observe these basic laws of deduction.

Examples of the Law of Non Contradiction/Law of the Excluded Middle: *

This table is rectangular or this table is not rectangular.
Two plus two is four or two plus two is not four.
The Bible was inspired by God or the Bible was not inspired by God.
Christ rose from the dead or Christ did not rise from the dead.
My name is James or my name is not James.

In order to more fully understand this important concept let’s imagine a typical closet. We will very likely find shoes, shirts, trousers, umbrellas, coats, hats, baseball bats, gloves, table games on the shelves and many other odds and ends. Let’s have the closet represent any given collection of things. We can go through the closet and sort out everything in it. We can divide everything into two categories, say shoes and not shoes. In another sorting we can do trousers and not trousers. Stuff to throw out and stuff not to throw out. In any given sorting everything in the closet will fall into one of two categories. In every case we will end up with two piles (one pile may not have anything in it and that, in mathematics, is called the empty set). That is what the Law of the Excluded Middle tells us. Nothing will be both a shoe and not a shoe. Every item will be either a shoe or not a shoe.

Of course we must first define our terms and eliminate any ambiguities. But, when all has been clarified, any rational conversation must recognize and respect the Laws of the Excluded Middle and Non Contradiction. Let’s always remember that it is impossible for a proposition and its contradictory to both be true. It is logically necessary that either a proposition or its contradictory be true, i.e., one of them must be true!

Still unconvinced? Try to disprove the Law. Think of any collection of things (in mathematics called a set) and try to come up with something within the set that is both X and not X where X is some unambiguous quality or quantity. Don’t forget that X must be defined unambiguously.

More examples:

The set of all animals can be sorted into mammals and non-mammals.
The set of all vehicles can be sorted into front wheel drive and not front wheel drive vehicles.
The set of all athletes can be sorted into Olympic Gold Medal winners and not Olympic Gold Medal winners.
The set called “furniture” can be sorted into chairs and non chairs.
The set of all beings can be sorted into the categories of divine and not divine.

Remember that a very important part of logical debate and conversation is clear definition. When things are clearly defined we will never find a case where a proposition and its contradictory are both true. We will always find that one of them must be true.

* Aristotle correctly codified these basic laws of deduction together with the Law of Identity and for well over a thousand years they were believed to be the fundamental “laws of thought”. With the development of modern symbolic logic one plainly sees that these three statements are logically equivalent and express the same truth in three different ways. Although they may not deserve the elevated position attributed to them by the ancients they are very definitely inescapable and true.

Note: Some objections have been raised asserting that the Law of the Excluded Middle makes everything either black or white. This is a misunderstanding of the Law. The Law of the Excluded middle actually makes everything either black or not black, round or not round, tall or not tall, etc. when, of course, these terms are well defined.

Exercises:

1. See if you can formulate the Law of Identity for yourself or do some research and find it formulated.
2. Come up with five more examples of the Law of the Excluded Middle.
3. Define the following items or concepts leaving as little ambiguity as possible:
A. a tree
B. a book
C. a box
D. imperialism
E. rectangular
4. Check your definition against that of a good dictionary. How did you do? How did the dictionary do?