# Who invented a way to measure the consistency of logical statements using cosine distance?

It seems as if the consistency of logical propositions is amenable to being measured using cosine distance. let $$\ell$$ be such a metric.

1. For any proposition $$P$$, it is the case that $$P$$ is consistent with itself, so $$\ell(P, P) = 1$$

2. For any proposition $$P$$, it is the case that $$P$$ is not consistent with its negation, so $$\ell(P, \neg P) = -1$$.

3. However, it is additionally the case that a proposition $$P$$ and its negation $$\neg P$$ can be consistent with another set of logical statements. Let $$P$$ be the parallel postulate of euclidean geometry, and let $$A_1, A_2, A_3, A_4$$ be the remaining four axioms. Then it is the case that $$\ell(P, A_i) = 0$$. That is, one statement can be independent from another.

Surely some smarter mathematician saw this and created a way to measure the cosine distance of logical propositions. Who is that mathematician, and where can I find their papers?

• But the parallel postulate is consistent with the remaining four axioms. Perhaps you intend $\ell(P,Q) = 1$ to hold precisely if $\Gamma \vdash P \leftrightarrow Q$ for some fixed $\Gamma$? May 31 at 3:02
• @Z.A.K. Thank you for pointing out spotty thinking. I am going to update the OP to better clarify what I meant. I don't know enough about sequent calculus to determine if what you wrote matches my intention, but it might. May 31 at 3:06