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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?

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  • $\begingroup$ 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 $\Gamma$? $\endgroup$
    – Z. A. K.
    May 31 at 3:02
  • $\begingroup$ @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. $\endgroup$
    – HolyKnowing
    May 31 at 3:06

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