[CG] Re: Functional Logic • Inquiry and Analogy

Cf: Functional Logic • Inquiry and Analogy • 14 (Part 2) https://inquiryintoinquiry.com/2022/05/17/functional-logic-inquiry-and-anal… Inquiry and Analogy • Umpire Operators (Part 2) https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Ump_Abs All, The “absolute umpire operator”, also known as the “umpire measure”, is a higher order proposition Υ₁ : (B × B → B) → B defined by the equation Υ₁(f) = Υ(1, f). In this case the subscript 1 on the left and the argument 1 on the right both refer to the constant proposition 1 : B × B → B. In most settings where Υ₁ is applied to arguments it is safe to omit the subscript 1 since the number of arguments indicates which type of operator is meant. Thus, we have the following identities and equivalents. • Υf = Υ₁(f) = 1_B ⇔ ¬( 1 ¬( f )) = 1 ⇔ f = 1_{B × B → B} The umpire measure Υ₁ is defined at the level of boolean functions as mathematical objects but can also be understood in terms of the judgments it induces on the syntactic level. In that interpretation Υ₁ recognizes theorems of the propositional calculus over [u, v], giving a score of 1 to tautologies and a score of 0 to everything else, regarding all contingent statements as no better than falsehoods. Regards, Jon