Question

Turing and Wittgenstein seem to agree that if you run into a contradiction, then anything might happen. In other words, any logic that is inconsistent is also trivial in the sense that every sentence may be proved. Could there be a logic in which it is not possible to prove every sentence from a contradiction? What might such a logic look like? Explain your answer.

Answer 1

Turing says the value of a variable 'x' is less than or equal to 4

Wittgenstein says the value of x is greater than 4.

In this case, the contradiction can have one definite output, that is, either x is less than or equal to 4; or x is greater than 4

Now, lets look at another case:

Turing says the value of a variable 'x' is 4

Wittgenstein says the value of x is 7

This is a contradiction.

But in this case we still dont know the value of x. Therefore, this is an example when there is no proof, given a contradiction