Harish asked:

I would be extremely grateful, if you could define or place into words so that I could explain the
question below to my children: What is the concept of 'proof'?

The above question is difficult to explain.

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You're right about that!

It is a long time since I put my nose in a text book of symbolic logic, but as a first, rough pass it might
say something like this. A 'proof' in the formal sense is a series of strings of symbols, each of which
results from applying a particular rule to a previous string of symbols, or strings of symbols, such that
each string of symbols in the series is:

Either given as a 'premiss' or starting point,

Or derived from previous strings of symbols,

Or an assumption which will be 'discharged'. (For example, in a proof which takes the simple form,
'Suppose A. Then it follows that B. But not-B. Therefore not-A', the assumption A is discharged.)

The last string of symbols is the conclusion of the proof.

One vital piece of information is missing, however. And that is that each string of symbols is
interpreted as making, or standing for something that makes a statement, capable of truth or falsity. It
is perfectly possible to investigate formal aspects of proofs in symbolic logic while completely ignoring
this aspect. However, that is what makes the difference between a proofand a mere game with
meaningless symbols manipulated according to rules.

All I have said above is just a preliminary. For even when we distinguish between the purely formal
aspect of proof and its interpretation, we still have not said the most important thing about what
makes something a proof.

It is this. The aim of proof isrational persuasion.We seek to persuade someone to accept a
proposition, starting from an agreed basis. So that if you and I agree that A, and I can show that if you
accept A then you have to accept B, that amounts to a proof of B.

'Having to accept' in this context means bringing you recognize that it is impossible for A to be true
while B is false. So if you accept A, you have to accept B.

That still does not answer the question of what a proof looks like. What is this 'recognizing'? How is it
brought about?

Each step of the proof takes the the idea of 'having to accept' to the most basic level. For example, if I
have succeeded in getting you to accept the truth of C, and also succeeded in getting you to accept
that it is not the case that (C and D), then you have to seethat as a consequence of those two steps,
D cannot be true. It is simply self-evident that not-D.

In text books of symbolic logic, however, the idea of a correct step is defined in terms of a given set of
axioms and/or rules of inference. So, for example, in the case I have just given, one might be using a
system of logic where there is no rule of inference, 'From X and not-(X and Y) infer not-Y'. In that
case, in order to meet the requirements of a valid proof in the formal sense,additional steps have to
be inserted.

Now, there is an ancient rule called 'Reductio ad absurdum' which says that if you can derive a
contradiction from an assumption A, then that counts as a proof of not-A. We can use this rule to
'prove' that from premiss C, taken together with the premiss not-(C and D), entails not-D:

(1) Assume D

(2) From premiss C and the assumption D: (C & D)

(3) From premiss not-(C & D) and (2): (C & D) & not-(C & D).

(4) Contradiction! By Reductio ad absurdum, we can reject assumption (1). Hence, not-D.

This an example of what I referred to above as 'a provisional assumption which is discharged'. D is
assumed as the first step in this mini-proof, but it is not one of the 'premisses' or starting points of the
proof.

No system of logic completely coincides with our intuitive sense of what is 'self-evident'. That is why
logic text books so often seem to be stating the obvious, such as in the proof above. A formal 'proof'
is defined relative to a given system of formal logic. The idea of proof as such, however, does not
invoke any given set of rules or axioms. People argued logically before Aristotle invented the subject
called 'logic'.

Thus: A proof, in the informal sense, takes you from agreed premissesvia a series of self-evident
stepsto a conclusion which cannot be false, if the premisses are true.

Geoffrey Klempner