Hume's guillotine

November 9, 2023

On the ride home from the office today, I remembered a statement I had once heard: “One cannot prove an ‘ought’ statement from an ‘is’ statement”. A rigorous answer as to why this statement was true was not as straightforward as I had originally thought, as to give a precise “proof” of the principle, one must establish what an ought statement, an is statement, and a proof even are. I will attempt to do this concisely here, although I am not an expert on philosophical matters. I am including this as a blog post instead of a note, as metamathematics falls somewhat outside the realm of topics I had hoped to post as notes.

First, one must define the relevant components of the statement, starting with the near self-explanatory “is” statement. To use the word “is” is to imply an equivalence relationship. That is, in any logical framework “is” is a reflexive (a is a), symmetric (a is b if and only if b is a), and transitive (a is b and b is c implies a is c) relation. Note here that when used with adjectives, one should think of the adjective as a set assignment to keep these ideas logically coherent. Next, we introduce the idea of proof: in order to prove some statement given some premises, the premises must be given in the context of a logical framework (equivalently, a set of axioms) where statements have truth values. Then, to prove something, one would build a sequence of true statements given the premises, axioms, and some logical operators (e.g. “if”, “then”, etc), that imply the truth of the desired conclusion within the logical framework. This logical framework is precisely what gives validity to the “is” statements, as within this framework, an “is” statement is necessarily axiomatically true, provably false, provably true, or not provable given the axioms. It is within this fourth category in which many ought statements lie.

An ought statement relates to the way things should be, and is distinct from an is statement in that a valid ought statement implicitly relates an “is” statement in a given framework to a logical framework in which the statement is true, regardless of the truth value in the current framework. In this sense, an “ought” statement is also an equivalence relation, as it is an “is” statement in another logical system. In a categorical sense, “ought” is more like a functor between logical systems with shared objects but potentially different truth values (True, False, neither True nor False), and an is statement is a morphism within a logical framework. Then, a trivial “ought” statement would map “is” statements to themselves, essentially stating that things should be how they are. Then, assuming all logical frameworks discussed are consistent, a non-trivial “ought” statement would be used to debate the validity of axioms or to introduce new axioms in a logical framework. Examples of such non-trivial “ought” statements would be the proposition that the Axiom of Choice should (or should not) be included in Zermelo-Frankel set theory, or that relativistic considerations should be (or need not be) considered in, say, determining the velocity of a charged particle.

It should now be obvious, even trivial, why an “ought” statement cannot in general prove an “is” statement (or vice-versa): one cannot use one logical framework to prove a statement in another logical framework without knowledge of the validity of statements in both logical frameworks a priori. That is, one can not use reason to prove something to someone else unless relevant axioms are true for all parties involved. In particular, one can never prove statements in a logical system using statements that are neither true or false in that system, which relates to the original problem posed by Hume if one assumes that any ethical proof requires an axiom that exists beyond reason. It is for this reason that lies and propaganda are so effective in controlling people or sewing division among a populus, as everyone has their own logical framework determining how they see the world. If these frameworks can be effectively molded, no amount of evidence or reason will change a person’s mind on anything. Clearly, nothing is more valuable than a good education, and no one should be more valued than a good educator. Unfortunately, good educators will not be valued as much as they should until people start agreeing on what “good” is, which would require most educators and parents to be good themselves (based on the definition of “good” from my logical framework, of course, which was formulated independent of reason).

I probably have more pressing things I should be thinking about. Even my category theory paper is more pressing!

-TJC

References: Algebra chapter 0 by Aluffi, Electrodynamics by Kovetz, and my old notes from a course on Formal Methods of Philosophy at JHU. If I remember the exact Youtube video I watched on this subject I will paste the link here, but the general concept is “Hume's guillotine”, from a google search. I am not well versed regarding scholarship on the idea, these are just my own independent thoughts.