It is important to be able to distinguish between truth and facts. Facts can be relatively easy to establish, by experience, or by observation. Determining underlying truths is often much more difficult.

It was a well established observable fact, for centuries, that the Sun went round the Earth. But it was not true. Further indirect observations later showed that the Earth must go round the Sun. Another fact; but wherein lay the underlying truth? The force of gravity was an explanation, but it was less of a truth than a mechanism. How was it that this particular law held sway?

Eventually space-time concepts showed that gravity was better regarded not as a force, but simply as 4-dimensional geometric curvature property. This is much nearer a truth: a fundamental property which is universal, a characteristic of existence itself. Nevertheless, scientists are still puzzling how gravitons fit into their space-time models and whether the universe is "flat". Time will tell, perhaps.

The road to truth is rarely easy, but that does not mean we should not even try to find a path. The work of Gödel seems to imply that in certain circumstances there exist truths which can be known, but which cannot be proved by any sort of calculation. In everyday life, however, the search for falsehoods can be just as effective as the search for truth.

A basic difficulty in the search for truth of any kind is that it is possible to draw true conclusions from false premises. For example, it may be conceivable to believe that the only reason some toadstools are poisonous is because policemen spend their off-duty time injecting them with toxins. Yet the fact that some toadstools are poisonous does not prove that any policeman has this esoteric hobby. This may seem obvious. Nevertheless, it is common for people to accept that their fundamental beliefs are true simply because they lead to conclusions which match the facts observed or derived from their own experience. Experience can only give added weight to initial beliefs. Confirmation, however reassuring, is not always proof.

Although it may be hard to prove true a statement, or belief, directly from experience alone, it may be easy to prove it false. This is because a true statement can not lead logically to a false conclusion. Hence the power of a counterexample. That the exception proves the rule, is well known, (in this context, remember that "proves" means "tests").

Returning to our fable of the toadstools: all our believer has to do is to grow a poisonous variety of toadstool under controlled conditions so that no policeman (nor anyone else) is allowed near it. The believer then eats the toadstool and makes a death-bed conversion!

This hypothetical example demonstrates another powerful logical technique. It is possible to prove a statement true indirectly, by showing that its negation is false. If assuming the opposite of a belief or proposition validly leads to a false conclusion then the original belief must be true.

Even so, both these approaches (seeking counterexamples, or examining negative hypotheses) sooner or later usually need an objective test against some criterion involving observation. In mathematics, this test can sometimes be avoided by arguing in a circular fashion in such a way as to create a paradox. Bertrand Russell's paradox about sets of sets was a landmark of its time. More readily described is the proof that the square root of two (ie 2^1/2, where ^ means "to the power of") cannot be represented by a proper fraction.

Let us assume that it could. Let this fraction be a/b, where "a" and "b" are therefore both whole numbers with no common factor (i.e. all the cancelling possible has been done). Thus:

Hence "a" and "b" have a common factor 2, in direct contradiction of our hypothesis. This paradox shows that the square root of two cannot be represented by any fraction, or ratio of whole numbers. It is therefore called an irrational number. Expressed in decimal form it would go on for ever: 1.141421356........

A more light-hearted example is the proof that all positive numbers are "interesting". The word "interesting" no doubt has many subjective interpretations, so let us suppose that some numbers are not interesting. Of these numbers there must be a smallest one. Is not that number therefore interesting?? Paradox created!

In everyday life this kind of proof is hard to develop, but there is another way. It is possible to prove a statement false, and hence its negation true, without any reference to experience at all.

The method goes like this: from the initial belief, statement, or proposition, find two (or more) distinct validly implied conclusions, then test these conclusions for mutual compatibility. If any pair are incompatible then the original belief is false. For those interested, the proof that this method works is given at the end of this chapter.

Note that it does not matter whether the conclusions themselves are true or false, nor whether they fit observable facts. The twin truths that:

a. the conclusions are validly implied by the hypothesis;
b. they are mutually contradictory; give the method its power.

This method is rare in pure mathematics, but has been valuable, for example, in quantum physics in wave-versus-particle debates. In everyday life, however, one is constantly meeting examples of disputes between people who claim to agree on a common premise, yet draw opposing conclusions. They often fall into the trap of still believing their premise to be true, simply because they both believe it.[Footnote 1]

If, over the years, a particular belief consistently leads to conflicting conclusions, then it should be at least re-examined, if not discarded. This has been the long-term way with Science in its broadest sense. When conflicts arise, sooner or later (too often later some may feel), hypotheses are reviewed by the scientific community, on the grounds that:

"Everything that is not what it seems, and not what it reasonably should be, must have significance." {Ellis Peters}

Eventually, better hypotheses emerge from which conclusions do converge. Until the next contradiction appears, therefore, these new hypotheses are accepted as truths, or at least meta-truths. I would claim that the search for truth, of any kind, scientific or otherwise, should be such a convergent evolutionary paradox-investigating process. I would go further. It is no coincidence that the Universe itself on the grand scale follows simple rules, the same everywhere. If the existence of our Universe is true, then ultimate truth is universal.

The above fundamental logical result, that inconsistency of conclusions proves that initial hypotheses are suspect, is largely ignored by the great religions of the world. From the same hypothesis "monotheism", Jews, Christians, and Moslems, for example, come to so many incompatible conclusions, both within their respective creeds and across them, that the only logical conclusion which can be drawn is that their common hypothesis must be false. The expression of religious truth is characterised by divergence of views, not convergence.

Consider just two aspects of these three major religions:

It is irrelevant whether these conclusions, believed by millions of members of the various faiths to be validly deduced from their common belief in the very same God, are true or false. The power of the logical theorem is that their mere conflicting is enough to make their common premise invalid. If there is one universal God, there can logically only be one religion. That there patently is not. "One Church, one faith, one Lord" the hymn says. Indeed?!

There is a fascinating book by Arthur Koestler, called "The 13th Tribe". He tells of the ruler of a small Caucasian State which was threatened with invasion by either or both of the early Christian Empire, or the newly emerging Islamic Empire. In their missionary zeal they wished to "convert" this pagan kingdom. The ruler therefore called emissaries from both faiths to discuss their respective religions. In the course of the discussions both the Christians and the Moslems agreed that they both worshipped the God of the Jews. The ruler therefore decided that his whole nation would convert to Judaism. He was then left in peace!

I have every sympathy with this ruler. I am sure he took a very wise, (Solomon-like?) decision. He maintained the independence of his people. As Neil Kinnock once said: "You must be here to make it better when you go." Martyrdom is no way to live! Logically, however, he could be said to have drawn the wrong conclusion. From the same premise the Christians and Moslems disagreed. Their premise should therefore be more suspect than their conclusions.

I realise, of course, that an alternative way of regarding the conflicting religions of the world is to say that their hypothesis is correct: i.e. a God exists, but that only one of the sects of one of the religions is correct, and all others are mistaken in some way. This is logically acceptable if, but only if, one assumes that only one set of people, a particular sect of "true believers", is capable of deriving valid conclusions from the basic monotheistic belief.

There are two difficulties with this defence of monotheism. The first is that there is no way of testing which sect has the truth, they all could well claim that honour. In 1997 there were reportedly over 500 sects in the UK alone. To which island of faith in the sea of conflict should an honest sailor steer?

Secondly, over the thousands of years, and millions of human thinkers, it is beyond reasonable acceptance that so many people have been incapable of drawing valid conclusions (remember, their correctness is not important, merely their validity as conclusions from a premise). In all other fields the human race has amply demonstrated its capacity for deduction or reason.

Over so much time, there can be little wrong with the reasoning powers of religious men, from eminent theologians, to the man or woman in the street. It is their hypothesis which must be flawed. When everyone in the parade is out of step with each other, they are all marching to the wrong tune.

This result, extended by analogy, across all religions, conclusively proves that no Gods exist of any kind that mankind has so far considered. Believers in deities past or present have often dodged the challenge of atheists by saying such things as "You cannot prove there is no God, and we believe. So there!" I agree! I do not claim to have proved there is no God. I have merely pointed out the existence of a useful theorem in logic. It is believers themselves, collectively, who have proved, (yes actually proved, beyond reasonable doubt, by sheer weight of numbers of conflicts) that there is no God. Multi-church, multi-faith, no Lord!

"Oh, yes there is", battling pantomime-like with "Oh, no there isn't", is no way to resolve conflicts. It is for believers to demonstrate there just might be a God by deriving a common creed from their hypothesis. Regrettably, they usually do not even try. Each faction instead relying on something they call faith. I note that a Chambers dictionary defines faith as "belief in the statement of another". It is time people began thinking for themselves. It is said that faith can move mountains. Maybe, but it cannot reveal the truth.

I remember the first ever pun I was taught at school. Supposedly, two people walking down a narrow street observed two others arguing vigorously across the road from upstairs windows. One observer remarked to the other: "They will never agree. They are arguing from different premises!".

The theist religions of the world are all arguing from the same premise: a personalised creator, but still disagreeing vigorously. All these "wrongs" show that none of them is arguing from the "right" premises.

Let me hasten to remind you of an earlier point, however. It is perfectly possible to derive valid and correct conclusions from a false premise. We should not, therefore, reject all of what the multifarious religions do conclude: respect for one's fellows, concern for the less fortunate, doing one's best, etc.. What concerns me is that when they have to refer back to first principles, to consider a novel situation, they are starting from a false assumption, and hence may equally validly draw a false conclusion as a correct one.

This is a particularly ironic position for Christians, who make such a point of their house being built on a rock. Peter's rock is built of quicksand. If there is no God, Jesus Christ was not the Son of God. The son of Man he may be, in that many of the ideas attributed to him were derived from the cumulative experience of a particular section of mankind (the Jews), but there was, and is as yet, no identifiable divine source of wisdom to tap. That is probably the most important truth of all. The sooner the whole world realises it, and takes it into account, the better for all life on earth. Collectively we must accept responsibility for ourselves, and our fellows, not pass the buck to some supernatural deity, nor to blind fate. We do not need a Decade of Evangelism, we need a century of Aptious Thinking.

Footnote 1 "In formal logic, a contradiction is the signal of defeat; but in the evolution of real knowledge it marks the first step in progress towards a victory." [Alfred North Whithead]

PROOF:

First let us consider all possibilities involving logical implication between two statements, an hypothesis "A", say, and a valid conclusion "B", say, independently of the truth, or otherwise, of either. For example, if A is "The moon is made of green cheese" and B is "The moon is green", then B is a valid result of A. It is possible, therefore for one statement to be a valid conclusion from another, even when both are false.

In the preceding chapter we have already shown that it is possible to deduce a true statement from a false one (the policemen and the toadstools), so it is also possible for A to be false, but B to be true. On the other hand if A really is true, and B is a valid conclusion, then B must be true. You cannot validly draw a false conclusion from a true hypothesis. This latter assertion is fundamental to all logic.

We can simplify these observations by constructing what is called a Truth Table for the relationship: A implies B (A=>B):

We shall use this table to prove the theorem.

The situation we wish to address is where there are 3 statements, A, B, and C, say, where A is the hypothesis, B and C are valid conclusions, and A implies B, and A implies C. There is an additional constraint, however, that B and C contradict each other. Putting it another equivalent way: B implies the opposite, or negation, of C.

Now study the larger truth table below. It lists all combinations involving these relationships.

For convenience only the initial letter of the table entries are displayed: F for False; T for True; P for Possible, and I for Impossible.

Each of the 3 columns to the right of the table is derived from the corresponding pair of the 4 columns on the left, by applying our original 4-value truth table. For example, consider the relationship B implies not C. The first pair of relevant values in the left set of four columns are B is F(alse), but not C is T(rue). Looking up this combination in the smaller truth table we see that it is possible. The next pair, under these headings, are F/F, also possible, and so on for all eight rows and all three right-hand columns.

It is important to appreciate that this table does cover all combinations. There are no others. Hence, from this exhaustive list, it is clear that the only ways all three of the implications on the right could possibly be valid are covered by the top three rows. For all of these rows, whatever the actual truth of B or C, A is seen to be FALSE. QED

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