I have been bored tonight, so here is an explanation of a quite surprising fact: there exists one, and only one, contradictory hypothesis.

Entailment

The turnstile symbol '⊢' represents entailment. If we assume that A is a set of statements and B is a set of statements, then '⊢' can be defined in the following way.

1. If 'A ⊢ B', then there is no intepretation of A, where A is true and B is false.

The word 'interpretation' here means any assignment of truth-values to every statement in some set A. In other words, 'A ⊢ B' means that if A is true then B is true, or B is a logical consequence of A, or B is a valid derivation from A, where 'valid' means truth-preserving. Here is another use of the turnstile.

2. If '⊢ B', then there is no interpretation where B is false.

To say '⊢ B' is simply to say that B is a tautology i.e. B exemplifies a logical form that cannot be false. The definition is precisely the same as before, but we just delete the reference to A, since here the set of premises is empty. The turnstile can be used again to express the following.

3. If 'A ⊢', then there is no intepretation where A is true.

Here again we leave one side of the entailment empty, and this time we express a contradiction. That is, 'A ⊢' says that there is no intepretation of every statement in A, where A is true. In other words, A is an inconsistent set of premises.

Logical Content

The concept of logical content can be defined in the following way.

4. The logical content for a set of statements A is the set which contains every logical consequence of A.

The logical content of any set of statements is infinite. For example, if '⊢ B' then any tautology can be derived from the empty set of premises. The empty set is a member of every set, therefore anything which is a consequence of the empty set of premises is a consequence of every set of premises. In short, the set of every tautology, of which there are infinitely many, is a subset of every set of premises.

5. If 'A ⊢ B', then the logical content of B is a subset of the logical content of A.

If 'A ⊢ B', then there is no interpretation of A, where A is true and B is false. That is, if A is true then B must also be true. It should be clear that the logical content of A i.e. the set of every logical consequence of A, has B as one of its members, otherwise A would not entail B. However, if B is a member of the logical content of A, then every member of the logical content of B is also a member of A. Therefore, the logical content of B is a subset of the logical content of A.

6. If 'A ⊢ B' and 'B ⊢ A' then A and B have the same logical content.

Following on from the above. It should be clear that there is only one subset of the logical content of A which entails A, namely, the entire logical content of A. Therefore, if 'A ⊢ B' and 'B ⊢ A', then A is equal to B.

Hypotheses

I wish to define the notion of a hypothesis in the following way.

7. If H is a hypothesis then H is a set of statements.

8. If 'H ⊢ B', then B is a member of the logical content of hypothesis H.

9. If hypothesis H1 and hypothesis H2 have the same logical content, then H1 and H2 are the same hypothesis.

The Contradiction

From a contradiction anything follows. This can be seen from the definition of entailment provided earlier.

10. If 'A ⊢' then 'A ⊢ B'.

That is, if there is no interpretation in which A is true, then there is surely no interpretation in which A is true and B is false. Therefore, B is a consequence of A where A is an inconsistent set of premises, whatever the logical content of B. This same point can be expressed in a formal proof, raplacing the metalogical variables of A, B, C, ... with the sentential variables of P, Q, R, ...

P → ~P, P ⊢ Q

Premise-----1. P → ~P

Premise-----2. P

Assumption--3. ~Q

MP (1, 2)----4. ~P

I& (2, 3)----5. P & ~Q

&E (2, 3)----6. P

AD (3, 6)----7. ~Q → P

MT (4, 7)----8. ~~Q

DNE (7)-----9. Q

The consequence of this argument is that anything, any statement whatever, is entailed by a contradiction.

The Contradictory Hypothesis

It follows from the definition of a 'hypothesis', and the fact that anything follows from a contradiction, that there is one, and only one, contradictory hypothesis. If anything follows from a contradiction, then the logical content of an inconsistent set of premises contains every other hypothesis as a subset.

Let us suppose that there exist two contradictory hypothesis, H1 and H2. If they are indeed different hypotheses then their logical content is not equal, which means that H1 or H2 must contain a member which the other does not. If we suppose that H2 contains a member B which is not a member of H1 then we run into a problem, since we can immediately show that B does indeed follow from H1 using the proof above, since anything does because it is a contradiction.

Therefore, the logical contents of H1 and H2 are equal--they are the same hypothesis.