Cf: Functional Logic • Inquiry and Analogy • 12 https://inquiryintoinquiry.com/2022/05/07/functional-logic-inquiry-and-anal… Interpretive Categories for Higher Order Propositions (n = 1) https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#HOPE_1… All, Referring to “Table 11. Higher Order Propositions (n = 1)” from the previous post: https://inquiryintoinquiry.files.wordpress.com/2022/05/higher-order-proposi… Table 12 presents a series of “interpretive categories” for the higher order propositions in Table 11. I’ll leave these for now to the reader’s contemplation and discuss them when we get two variables into the mix. The lower dimensional cases tend to exhibit “condensed” or “degenerate” structures and their full significance will become clearer once we get beyond the 1‑dimensional case. Table 12. Interpretive Categories for Higher Order Propositions (n = 1) https://inquiryintoinquiry.files.wordpress.com/2022/05/interpretive-categor… Regards, Jon

Cf: Functional Logic • Inquiry and Analogy • 14 https://inquiryintoinquiry.com/2022/05/17/functional-logic-inquiry-and-anal… Umpire Operators (Part 1 of 2) https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Umpire… All, [Note. Please follow the first link above for better math formatting.] The 2¹⁶ measures of type (B × B → B) → B present a formidable array of propositions about propositions about 2-dimensional universes of discourse. The early entries in their standard ordering define universes too amorphous to detain us for long on a first pass but as we turn toward the high end of the ordering we begin to recognize familiar structures worth examining from new angles. Instrumental to our study we define a couple of higher order operators, • Υ : (B × B → B)² → B and • Υ₁ : (B × B → B) → B, referred to as the relative and absolute “umpire operators”, respectively. If either operator is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established. Let X = ⟨u, v⟩ be a 2-dimensional boolean space, X ≅ B × B, generated by 2 boolean variables or logical features u and v. Given an ordered pair of propositions e, f : ⟨u, v⟩ → B as arguments, the relative umpire operator reports the value 1 if the first implies the second, otherwise it reports the value 0. • Υ(e, f) = 1 if and only if e ⇒ f Expressing it another way: • Υ(e, f) = 1 ⇔ ¬( e ¬( f )) = 1 In writing this, however, it is important to observe that the 1 appearing on the left side and the 1 appearing on the right side of the logical equivalence have different meanings. Filling in the details, we have the following. • Υ(e, f) = 1 ∈ B ⇔ ¬( e ¬( f )) = 1 : ⟨u, v⟩ → B Writing types as subscripts and using the fact that X = ⟨u, v⟩, it is possible to express this a little more succinctly as follows. • Υ(e, f) = 1_B ⇔ ¬( e ¬( f )) = 1_{X → B} Finally, it is often convenient to write the first argument as a subscript. Thus we have the following equation. • (Υ_e)(f) = Υ(e, f). Regards, Jon

Cf: Functional Logic • Inquiry and Analogy • 15 https://inquiryintoinquiry.com/2022/05/20/functional-logic-inquiry-and-anal… Inquiry and Analogy • Measure for Measure https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Measur… All, Let us define two families of measures, α_i, β_i : (B × B → B) → B for i = 0 to 15, by means of the following equations: • α_i f = Υ(f_i, f) = Υ(f_i ⇒ f), • β_i f = Υ(f, f_i) = Υ(f ⇒ f_i). Table 14 shows the value of each α_i on each of the 16 boolean functions f : B × B → B. In terms of the implication ordering on the 16 functions, α_i f = 1 says that f is “above or identical to” f_i in the implication lattice, that is, f ≥ f_i in the implication ordering. Table 14. Qualifiers of the Implication Ordering α_i f = Υ(f_i, f) https://inquiryintoinquiry.files.wordpress.com/2022/05/qualifiers-of-implic… Table 15 shows the value of each β_i on each of the 16 boolean functions f : B × B → B. In terms of the implication ordering on the 16 functions, β_i f = 1 says that f is “below or identical to” f_i in the implication lattice, that is, f ≤ f_i in the implication ordering. Table 15. Qualifiers of the Implication Ordering β_i f = Υ(f, f_i) https://inquiryintoinquiry.files.wordpress.com/2022/05/qualifiers-of-implic… Applied to a given proposition f, the qualifiers α_i and β_i tell whether f is above f_i or below f_i, respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those which occupy the limiting positions in the Tables. • α₀f = 1 iff f₀ ⇒ f iff 0 ⇒ f, hence α₀f = 1 for all f. • α₁₅f = 1 iff f₁₅ ⇒ f iff 1 ⇒ f, hence α₁₅f = 1 iff f = 1. • β₀f = 1 iff f ⇒ f₀ iff f ⇒ 0, hence β₀f = 1 iff f = 0. • β₁₅f = 1 iff f ⇒ f₁₅ iff f ⇒ 1, hence β₁₅f = 1 for all f. Expressed in terms of the propositional forms they value positively, α₀ = β₁₅ is a totally indiscriminate measure, accepting all propositions f : B × B → B, whereas α₁₅ and β₀ are measures which value the constant propositions 1 : B × B → B and 0 : B × B → B, respectively, above all others. Regards, Jon

Cf: Functional Logic • Inquiry and Analogy • 17 https://inquiryintoinquiry.com/2022/05/25/functional-logic-inquiry-and-anal… Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory (Part 1) https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#App_Qu… All, Our excursion into the expanding landscape of higher order propositions has come round to the point where we can begin to open up new perspectives on quantificational logic. Though it may be all the same from a purely formal point of view, it does serve intuition to adopt a slightly different interpretation for the two-valued space we take as the target of our basic indicator functions. In that spirit we declare a novel type of “existence-valued functions” f : Bⁿ → E where E = {-e, +e} = {empty, existent} is a pair of values indicating whether anything exists in the cells of the underlying universe of discourse. As usual, we won’t be too picky about the coding of those functions, reverting to binary codes whenever the intended interpretation is clear enough. With that interpretation in mind we observe the following correspondence between classical quantifications and higher order indicator functions. Table 17. Syllogistic Premisses as Higher Order Indicator Functions https://inquiryintoinquiry.com/syllogistic-premisses-as-higher-order-indica… Regards, Jon

Cf: Functional Logic • Inquiry and Analogy • 19 https://inquiryintoinquiry.com/2022/05/31/functional-logic-inquiry-and-anal… Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory (Part 3) https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#App_Qu… <QUOTE John Dewey> Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it. John Dewey • How We Think </QUOTE> All, Tables 19 and 20 present the same information as Table 18, sorting the rows in different orders to reveal other symmetries in the arrays. Table 18. Simple Qualifiers of Propositions (Version 1) https://inquiryintoinquiry.files.wordpress.com/2022/05/simple-qualifiers-of… Table 19. Simple Qualifiers of Propositions (Version 2) https://inquiryintoinquiry.files.wordpress.com/2022/05/simple-qualifiers-of… Table 20. Simple Qualifiers of Propositions (Version 3) https://inquiryintoinquiry.files.wordpress.com/2022/05/simple-qualifiers-of… Regards, Jon

Cf: Functional Logic • Inquiry and Analogy • 21 https://inquiryintoinquiry.com/2022/06/09/functional-logic-inquiry-and-anal… Inquiry and Analogy • Generalized Umpire Operators https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Gen_Um… All, To get a better handle on the space of higher order propositions and continue developing our functional approach to quantification theory, we’ll need a number of specialized tools. To begin, we define a higher order operator Υ, called the “umpire operator”, which takes 1, 2, or 3 propositions as arguments and returns a single truth value as the result. Operators with optional numbers of arguments are called “multigrade operators”, typically defined as unions over function types. Expressing Υ in that form gives the following formula. • Υ : ⋃_{ℓ = 1, 2, 3} ((B^k → B)^ℓ → B). In contexts of application, that is, where a multigrade operator is actually being applied to arguments, the number of arguments in the argument list tells which of the optional types is “operative”. In the case of Υ, the first and last arguments appear as indices, the one in the middle serving as the main argument while the other two arguments serve to modify the sense of the operation in question. Thus, we have the following forms. • Υₚ^r q = Υ(p, q, r) • Υₚ^r : (B^k → B) → B The operation Υₚ^r q = Υ(p, q, r) evaluates the proposition q on each model of the proposition p and combines the results according to the method indicated by the connective parameter r. In principle, the index r may specify any logical connective on as many as 2^k arguments but in practice we usually have a much simpler form of combination in mind, typically either products or sums. By convention, each of the accessory indices p, r is assigned a default value understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition 1 : B^k → B for the lower index p and the continued conjunction or continued product operation ∏ for the upper index r. Taking the upper default value gives license to the following readings. • Υₚ(q) = Υ(p, q) = Υ(p, q, ∏). • Υₚ = Υ(p, _, ∏) : (B^k → B) → B. This means Υₚ(q) = 1 if and only if q holds for all models of p. In propositional terms, this is tantamount to the assertion that p ⇒ q, or that ¬(p ¬(q)) = 1. Throwing in the lower default value permits the following abbreviations. • Υq = Υ(q) = Υ₁(q) = Υ(1, q, ∏). • Υ = Υ(1, _, ∏) : (B^k → B) → B. This means Υq = 1 if and only if q holds for the whole universe of discourse in question, that is, if and only q is the constantly true proposition 1 : B^k → B. The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter. Resources ========= Multigrade Operators ( https://oeis.org/wiki/Multigrade_operator ) Parametric Operators ( https://oeis.org/wiki/Parametric_operator ) Regards, Jon