Another puzzle on logic and obligation

There are no ideal reasoners

Jacob W. Archambault
May 24 · 2 min read

Yesterday, I showed that axiom commonly called ‘ought implies can’ (OIC), if applied to logical reasoning itself, implies that a good part of what are considered logically valid deductions in logic various logics, including the standard formalization of deontic logic itself, would have to be rejected. Today, I’ll discuss a different principle — that obligation implies contingency — and draw another counterintuitive conclusion from it.

If you’re not sold on the axiom, I offer the following argument. For any obligation, the weight of that obligation is only given in experience via the possibility of its absence: in other words, one can only feel obliged to bring about something if it is possible for that thing not to be. Consequently, if there were obligation to bring about what is necessary, it could never be given in experience. While an obligation does not be require its being recognized as such, it at least presupposes the possibility that it be recognized. No such possibility exists in the case of necessities. Therefore, it must be that

(O~□) OA → ¬□A,

which is equivalent to

(O♢~) OA → ♢¬A

Given (OIC), this gives us

(OI𝒞) OA → (♢A ∧ ♢¬A)

We can then define a contingency operator 𝒞 such that

(𝒞) 𝒞A ≝ ♢A ∧ ♢¬A

This allows us to rewrite OI𝒞 as follows:

(OI𝒞) OA → 𝒞A

Now, apply this reflexively, as before, to the case of logical reasoning itself. The implication is that for any claim A and agent x, if x is obliged to infer A, x‘s actually inferring A must itself be a contingent act.

Now there are two ways to understand this statement, depending on whether we understand reasoning to include in its domain reasoning that can be engaged in, but isn’t, or instead to refer to an act of reasoning actually engaged in. On the less controversial reading, Ought-implies-contingency means that x could simply not engage in a given reasoning process: perhaps our logician has chosen instead to spend the afternoon bingeing on Taylor Swift songs. But on the stronger reading, it means that x, while engaging, could fail to get it. The latter, if correct, would imply something about the kinds of beings to whom logic actually applies, viz. that logic doesn’t apply to ideal reasoners. Consequently, even leaving aside the problem of circular definition, a correct set of inferences or axioms would not be able to be defined by appealing to the kinds of inferences ideal reasoners would make.


Originally published at http://jacobarchambault.com on May 24, 2019.

Jacob W. Archambault

Written by

Philosopher, logician, programmer.