Incompatibility (logic)

In the logic refers to a statement A as incompatible with a statement B if one of A and B contradiction follows . If A and B are not incompatible, they are called compatible or compatible .

Example: We consider the two sentences:

A: Skolem was Norwegian.

B: Skolem was Italian.

From A and B it follows first that “Skolem was Norwegian” (trivially, since this sentence is already contained in the premises ). In addition, “Skolem was not a Norwegian” follows. (We are assuming that no one can have two nationalities, so from “Skolem was Italian” it follows that he is not Norwegian.) From A and B follows a certain sentence and its negation, ie a contradiction. A and B are incompatible with this.

Logical laws of incompatibility

If A is incompatible with B, so is B with A.

This is clear, since a contradiction must follow from both together. So incompatibility is a symmetrical relation .

If two statements are incompatible, at least one of them must be wrong. However, both can be wrong.

It is true that everything that follows from true premises is also true. So if two incompatible statements were both true, then a contradiction would be true (since a contradiction follows from them). However, contradictions can never be true, so at least one of the statements must be false. However, both can also be wrong, as you can see in the following example: “Skolem was Swede” and “Skolem was Italian” are incompatible and yet both are wrong.

A is incompatible with B if and only if A does not follow-B

If A is incompatible with B, it follows from the assumption that A is true that B is false (this was just shown above). A then does not imply then -B. Conversely: if A does not follow-B, then A and B still follow non-B, but at the same time also trivially B, i.e. the contradiction B and non-B. The fact that “Skolem is Norwegian” and “Skolem is Italian” are incompatible means the same thing as that “Skolem is not Italian” follows from “Skolem is Norwegian”. This connection can therefore also be used to define the concept of incompatibility.

If A is incompatible with B and if A follows from C, then C is also incompatible with B.

If A is incompatible with B, then (as we have seen) it follows from A not-B. But if the following applies: From C follows A and: From A does not follow-B, then also applies: From C follows not-B. But C and B are incompatible with this. Since “Skolem is Norwegian” and “Skolem is Italian” are incompatible, a sentence that “Skolem is Norwegian” is also implied, e.g. B. “Skolem was born in Sandsvaer (a village in Norway)” is incompatible with “Skolem is Italian”.

If A is incompatible with B and C follows from A, then C is not always incompatible with B.

For example, “Skolem is Norwegian” follows “Skolem is European”. This sentence is obviously not incompatible with “Skolem is Italian”.

If A is incompatible with B and B with C, then A is not always incompatible with C.

In other words: the relation of incompatibility is not transitive . Example: B, “Skolem is Italian” is incompatible with both A “Skolem is Norwegian” and C “Skolem is Scandinavian”, but A and C are not incompatible.

If a statement is inconsistent , then it is incompatible with every statement and vice versa .

An inconsistent statement A is one from which a contradiction follows. That means, no matter which further premise B we accept, there will still be a contradiction, so A and B are inconsistent. Conversely: If a statement A is incompatible with any other statement, then in particular with itself. However, this already implies a contradiction from the set of premises, which consists of A alone, which means nothing other than that A is inconsistent. The inconsistent statement “Skolem is a Norwegian Swede” would therefore be incompatible with any statement.