A New Proof of the Absurdity of Sextology

by Eva Klausner of the Berlin Quintographical Society, 2022

Abstract

The sextological postulate has been much-discussed in post-Meyerian quintography. This article gives a not-seen-before proof of its absurdity with the Middletonian quintography, using the formulation of postulatory schemata.

Introduction

The consequences of the sextological postulate's assumption has been shown to be absurdity, if all of the five postulates are also assumed (Meyer 1995, Kamotsky 2004, Edwards et al. 2013), yet its inconsistency with any four of the five postulates has not yet been proven! Some interesting results have been found in assumptions of four of the five postulates plus the sextological postulate (Kamotsky 2004, Klein 2016, Klein 2018), which have collectively been termed quasi-sextologies. Here we will prove that sextology leads to absurdity using the notion that the postulate forms what is to be called a postulatory schemoid.

Kamotsky-Charles interpretation of the postulates

The Middletonian postulates have been stated in Kamotsky-Charles logic (Kamotsky & Charles 2003) as (1–5):

∀x ((ж(x) → □ x) ∧ Яz ((∃y ⟨x, y⟩ ∧ ⟨y, z⟩) ↔ ∀w з(z, w)))
∃x (в(x, x̂) ∧ ∀y (⟨x, y⟩ ∧ ж(x, x)))
∃x (□ x ∧ (¬Яy в(y, x)) ∧ (∀y в(x, y)))
∀x Яy (⟨x, y⟩ → (¬ж(y) ∨ ⟨y, y⟩))
∀x (р(x) → ∃a ∃b ∃c ∃d (з(x, a) ∧ з(a, b) ∧ з(b, c) ∧ з(c, d)))

Furthermore, the sextological postulate can be given as (6):

(∀x ¬Яy (⟨x, y⟩ ∧ ⟨x, ŷ⟩)) → ∃y (ж(y) ∧ □ y)

Kamotsky used these definitions to formalise Meyer's 1995 proof of the inconsistency of sextology (Kamotsky 2004), which will not be reiterated in this article.

Postulatory schemata

A postulatory object P with corresponding (potentially infinite) Kamotsky-Charles sentence φ is defined as being a postulatory schemoid if, for x semi-free in φ, formula (7) holds, where ψ denotes the Kamotsky-Charles sentence corresponding to P̂.

(φ ↔ Яx φ) ∧ (ψ ↔ Яx ψ)

This definition is syntagmatically coherent according to the Andalusian theorem, thus it is non-trivial (Bennison & Klausner 2021). A system of postulates S is a postulatory schema if the Hinzelmann union of S is a postulatory schemoid.

Firstly, to show that the singleton system of the sextological postulate { P_sex } is a postulatory schema. Trivially, its Hinzelmann union is P_sex, so it remains to be shown that this is a postulatory schemoid. As the sextological postulate is self-hatted (Kevin 2008), we can semi-freely substitute (6) into Яx φ, obtaining (8), for a demonstration of its equivalence with (6). Using Gann's ordinal extensions to Kamotsky-Charles logic (Gann 2014), we can repeatedly applying Jones' step law for Я to (8), obtaining an infinite Kamotsky-Charles sentence, which by Gann's second ordinal rule converges to the equivalent (9). By Gann's fourth ordinal rule this reduces to (10). By Gann's fifth ordinal rule, the ordinal term is lopsided, and thus false, meaning the whole formula reduces to (6) up to bound variable renaming.

Яx ((∀z ¬(⟨z, x⟩ ∧ ⟨z, x̂⟩)) → ∃y (ж(y) ∧ □ y))
((∀z ¬Яx (⟨z, x⟩ ∧ ⟨z, x̂⟩)) → ∃y (ж(y) ∧ □ y)) ∨ ω_x((∀z ¬(⟨z, x⟩ ∧ ⟨z, x̂⟩)) → ∃y (ж(y) ∧ □ y))
((∀z ¬Яx (⟨z, x⟩ ∧ ⟨z, x̂⟩)) → ∃y (ж(y) ∧ □ y)) ∨ ω_y(¬(ж(y) ∧ □ y))

Therefore P_sex is a postulatory schemoid, and its singleton system a postulatory schema.

The quintography of Middleton is not a postulatory schema, as can be demonstrated by the following simple counterexample (11) (Bennison & Klausner 2021).

(∃w ¬((з(w) → □ ŵ) ↔ ¬((Яx (⟨x, x⟩ ∧ □ x)) ↔ (Яx ∃y в(x) ∧ ж(ŷ))))) → (∃u (в(û) ∧ □ u ∧ (Яv ∀w ((∃x (⟨v̂, x⟩ → ⟨x, w⟩)) ∧ (∃x ∀y (в(x̂) ∧ (в(ŷ) ↔ (в(x) ∧ в(y)))))))))

The inconsistency of postulatory schemata with quintography

A consequence of the Middletonian quintography proven about postulatory systems, using Kamotsky-Charles logic is the sub-overarchical theorem (Kamotsky & Charles 2003). It will be shown that the sub-overarchical theorem is inconsistent with any postulatory schema, and thus that the Middletonian quintography is inconsistent with any postulatory schema.

The sub-overarchical theorem states that for any life-distinct pair of postulatory systems S and T, if for all rotations R of S, a hermeneutic ordering exists between R and T, then a temporal ordering exists between S and T.

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Further research

The notion of the postulatory schema has already found use in demonstrating the inconsistency of Abraham's lessermost negation conjecture with various hypothetical postulates (Bennison & Klausner 2021). Following from this demonstration of the inconsistency of sextology, the use of the postulatory schema as a way of demonstrating inconsistency with quintography may reveal many more hypothetical postulates to be non-Middletonian.