To explain more comprehensively for anyone who does not understand.

The letters 'A' and 'B' are propositional-variables. The relation 'A ? B' can be read as 'if A is true then B is true'. The turnstile represents a valid inference, so 'A |= B' can be read as 'there is no consistent assignment of truth-values where "A" is true and "B" is false'. Finally, the comma is used to seperate premises and conclusions where there is more than one. For example, consider the following argument.

A, A ? B |= B

Here we have the two premises 'A' and 'A ? B', from which we validly infer the conclusion 'B'. This simple deduction employs the rule known as *modus ponens*, which states: if the antecedent of a conditional is implied by any of the premises then its consequent can be validly inferred. In this case, 'A' is implied by 'A', which is the antecedent of the conditional 'A ? B', and so according to *modus ponens* the consequent can be valdily derived, which is the conclusion 'B'. With this argument in mind it should be clear that the following argument is also valid.

A, A ? B |= A, B

In fact, this argument is the same as the previous one but with the addition of 'A' as a conclusion, and since 'A' is implied by the premise 'A' the argument is valid. However, the next argument I wish to present is somewhat less intuitive.

B |= A ? B

This argument states that the conditional 'A ? B' can be validly derived from the premise 'B'. Now some people might look askew at that argument, but it is actually valid. To understand why we need do no more than reconsider the definition of the turnstile symbol: the argument 'B |= A ? B' can be read as 'there is no consistent assignment of truth-values where "B" is true and "A ? B" is false'. In other words, if 'A' is true then 'B' is true, but since 'B' is true anyway then it must be true that *if* 'A' is true then 'B' is true. From this we can obtain the following valid argument.

A, B |= A, A ? B

It should be clear that this argument is valid, since we have simply added 'A' into the premises and derived 'A' as a conclusion, while changing nothing else. Now this argument, combined with our previous result, can reveal our problem.

A, A ? B |= A, B

A, B |= A, A ? B

The above arguments are identical except for the fact that the premises and conclusions have been swapped. In other words, the premises imply the conclusions *and* the conclusions imply the premises. This relation can be captured by rewriting the above argument using a double turnstile.

A, A ? B =||= A, B

This means that everything which can be derived from the premises can also be derived from the conclusion, and vice versa. In other words, the premises 'A, A ? B' and 'A, B' are semantically equivalent i.e. their logical content is exactly the same. Therefore, 'A, A ? B' and 'A, B' are synonymous, and so any instance of one can be swapped with the other and retain the semantic interpretation of the argument. In consequence, the original argument can be rewritten from.

A, A ? B |= B

To.

A, B |= B

The final form presented here would surely do nothing to convince anyone that 'B' is true, and the addition of 'A' as a premise is entirely superfluous. The problem which arises for anyone who wants to argue *for* a conclusion is that the results of this procedure can be replicated for any valid or invalid argument.