[39] **viXra:1905.0610 [pdf]**
*submitted on 2019-05-31 10:08:06*

**Authors:** Colin James III

**Comments:** 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Abstract: We evaluate the eight defining equations of the Spencer-Brown system. None is tautologous. This refutes the subsequent primary arithmetic renamed as BF calculus. We previously refuted the Dunn-Belnap 4-valued bilattice as not bivalent and thus non tautologous, so to draw in refinements and extensions by others and apply BF to it compounds the mistakes. Further producing a square root operation on negative 1 is also not tautologous. Spencer-Brown and BF systems subsequently form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[38] **viXra:1905.0598 [pdf]**
*submitted on 2019-05-30 11:26:45*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the Schaefer theorem for the P, NP problem by two examples for Graph-SAT(Ψ ). Neither example is tautologous; while claimed to be different, they result in the same truth table values. (The injection of NP-intermediate does not describe our result.) This refutes NP-complete (and P, NP, NP-hard). We also evaluate the P, NP problem as based on P≤NP with the same result. Therefore P, NP, NP-complete, NP-hard, NP-intermediate form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[37] **viXra:1905.0590 [pdf]**
*submitted on 2019-05-31 03:08:42*

**Authors:** Hannes Hutzelmeyer

**Comments:** 10 Pages.

In connection with his so-called incompleteness theorem Gödel discovered the beta-function. The beta-function theorem is important for the representation of recursive functions in the concrete calcule ALPHA of Robinson arithmetic. The other features are composition and minimization of primitive recursive functions. Recursive functions are no part of Robinson arithmetic, but they are representable by certain formulae.
The author has developed an approach to logics that comprises, but goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and so on.
The concrete calcule LAMBDA of a natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. They include e.g. Ackermann's function and similar constructions. All recursive functions (that are obtained by minimization too) can be represented in LAMBDA . As long as there is no proof that all processive functions are minimitive recursive (recursive but not primitive recursive) one has the problem of representing them in concrete calcule ALPHA of Robinson arithmetic. As long as the challenge of such a proof is not met there is the conjecture that there are calculative functions that are not representable in Robinson arithmetic.
An abstract calcule alphakappa of Robinson-Crusoe arithmetic shows that there exists an even weaker adequate arithmetic than Robinson's.

**Category:** Set Theory and Logic

[36] **viXra:1905.0586 [pdf]**
*submitted on 2019-05-29 08:13:43*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate six seminal equations, none of which is tautologous. (The author mistakenly labels the Löb axiom as a “fact” as proved by another author.) Therefore modal logics bounding the circumference of transitive frames is refuted and becomes another non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[35] **viXra:1905.0583 [pdf]**
*submitted on 2019-05-29 09:28:18*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In NIP theory, the Keisler measure as φ(x)↦μ(φ(x) ∩X)/μ(X)) is not tautologous, relegating it to a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[34] **viXra:1905.0577 [pdf]**
*submitted on 2019-05-29 11:24:49*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Church’s thesis (CT) is not tautologous as an essential consistency property to fulfill the requirement of the intensional level of a constructive foundation proposed of the minimalist foundation (MF) for constructive mathematics. Therefore, this relegates CT and MF to a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[33] **viXra:1905.0554 [pdf]**
*submitted on 2019-05-28 10:54:57*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

“In descriptive set theory and mathematical logic, Lusin's separation theorem states that if A and B are disjoint analytic subsets of Polish space, then there is a Borel set C in the space such that A ⊆ C and B ∩ C = ∅.” We evaluate two renditions of that equation, both non tautologous, refuting it. Therefore, the separation theorem of Lusin forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[32] **viXra:1905.0547 [pdf]**
*submitted on 2019-05-28 14:26:43*

**Authors:** Colin James III

**Comments:** 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The separation theorem of Gobbay takes eight basic cases and four cases for disjunction. None is tautologous. In fact, three groups of the basic cases share unique truth table result values, and one group of the disjunctive cases shares the same truth table result values. This refutes the theorem and adds it as another non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[31] **viXra:1905.0524 [pdf]**
*submitted on 2019-05-27 21:13:09*

**Authors:** Colin James III

**Comments:** 5 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

From the 11 equations tested, we refute 13 artifacts:
1. a condition for "an existential quantifier ∃ … on a Boolean algebra";
2. "a quantifier ∃ as closure operator on B, for which every open element is closed";
3. the interior operator on abstract topological Boolean algebra;
4. the kernel of a homomorphism from a Heyting algebra into another as a filter;
5. deductive systems and filters as equivalent;
6. the atomic definition of p ≤ ∃p in Halmos algebra;
7. a ‘concrete’ Rauszer Boolean algebra;
8. two conditions for the definition of a filter (and Heyting algebra using the filter);
9. a De Morgan algebra as a Kleene algebra;
10. equivalences of symmetrical Heyting algebras;
11. equivalences in Heyting algebras;
12. intuitionistic implication of intuitionistic logic; and
13. a theorem and a proposition of Nelson algebras.
As a result, the following seven areas are non tautologous fragments of the universal logic VŁ4:
1. Topological Boolean algebra;
2. Heyting algebra;
3. Intuitionistic logic;
4. Halmos algebra;
5. Rauszer algebra;
6. Kleene algebra; and
7. Nelson algebra.

**Category:** Set Theory and Logic

[30] **viXra:1905.0519 [pdf]**
*submitted on 2019-05-28 03:49:06*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

In Presburger arithmetic, Axiom 2 as x+1 = y+1 → x=y is not tautologous. Therefore Presburger arithmetic is a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[29] **viXra:1905.0482 [pdf]**
*submitted on 2019-05-25 06:15:15*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate Moroianu’s and the Tarski-Bourbaki fixed point theorem and axiom of choice (AC). Two versions of the theorem and then seven theorems and corollary which follow are also not tautologous. Therefore these conjectures form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[28] **viXra:1905.0477 [pdf]**
*submitted on 2019-05-23 08:12:13*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

A fundamental proposition for bounded and Σ1 formulas in PA is not tautologous. While the author states that the informal notes are full of errors, this fundamental mistake causes the entire section about Rosser’s form of Gödel’s theorems to collapse. Therefore the proposition is a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[27] **viXra:1905.0476 [pdf]**
*submitted on 2019-05-23 08:13:29*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Definitions for ◻H and ¬ ◊H are supposed to be equivalent for a classical mapping of agnostic hypothesis tests. While each definition reduces to a theorem in the conjecture, they are not tautologous. This refutes that agnostic hypothesis tests are proved to be logically consistent. Hence the characterization of credal modalities in agnostic hypothesis tests cannot be mapped to the hexagon of oppositions to explain the logical relations between these modalities. Therefore the 11 definitions tested form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[26] **viXra:1905.0475 [pdf]**
*submitted on 2019-05-23 08:14:35*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The seminal formula of C causes E iff (~C □→ ~E) is not tautologous, that is, it is not a theorem, from which the conjecture is derived. Hence reversing the counterfactual analysis of causation is refuted. Therefore the conjecture forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[25] **viXra:1905.0469 [pdf]**
*submitted on 2019-05-23 17:41:36*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The equations for condition/decision duality are not tautologous, hence refuting what follows as internal logic of extensive restriction categories. These conjectures form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[24] **viXra:1905.0440 [pdf]**
*submitted on 2019-05-22 12:02:07*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Strawson’s presupposition and Russell’s entailment are of the same form, equivalent, and hence not different. These conjectures form a tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[23] **viXra:1905.0437 [pdf]**
*submitted on 2019-05-22 18:19:33*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluated 12 equations for the assertions with none tautologous. Therefore this conjecture is a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[22] **viXra:1905.0434 [pdf]**
*submitted on 2019-05-22 21:44:39*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate two example equations as not tautologous, thereby refuting the rooted hypersequent calculus for modal propositional logic S5. The sequent calculus forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[21] **viXra:1905.0358 [pdf]**
*submitted on 2019-05-20 01:17:30*

**Authors:** Bornali Paul

**Comments:** 9 Pages.

Reasoning carried out in ordinary language, can not avoid using non-referring names if occasion arises. Semantics of classical logic does not fit well for dealing with sentences with non-referring names of the language. The principle of bivalence does not allow any third truth-value, it does not allow truth-value gap also. The outcome is an ad hoc stipulation that no names should be referentless. The aim of this paper is to evaluate how far free logic with supervaluational semantics is appropriate for dealing with the problems of non-referring names used in sentences of ordinary language, at the cost of validity of some of the classical logical theses/ principles.

**Category:** Set Theory and Logic

[20] **viXra:1905.0310 [pdf]**
*replaced on 2019-05-19 22:47:28*

**Authors:** Colin James III

**Comments:** 3 Pages.

Of 13 equations evaluated, five are tautologous and eight are non tautologous. This refutes Tarski’s geometric axioms and betweenness which form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[19] **viXra:1905.0276 [pdf]**
*submitted on 2019-05-17 09:44:55*

**Authors:** Colin James III

**Comments:** 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Of seven examples for counterpart theory, none is tautologous. In fact, a translation is not tautologous in the counterpart model or in QMT, but rather shares the same truth table result. Two definitions of intensionality are also not tautologous and hence refuted. Therefore counterpart theory forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[18] **viXra:1905.0275 [pdf]**
*submitted on 2019-05-17 10:56:36*

**Authors:** Hannes Hutzelmeyer

**Comments:** 17 Pages. The details given here relate to document viXra: 1905.0221

In addition to the publication 'The Snark, a counterexample for Church's thesis?' examples and details are offered in the form of two appendices C6 and C7 that allow for better understanding of the general method and the particular problem related to Church's thesis.
The author has developed an approach to logics that comprises, but also goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers etc.
The most usual approach to calculative (effectively calculable) functions is done by register machines or similar storage-based computers like the Abacus or Turing machines. Another usual approach to computable functions is to start with primitive recursive functions. However, one has to find a way to put this into a form that does not rely on a pre-knowledge about functions and higher logic. The concrete calcule LAMBDA of decimal pinitive arithmetic allows for such an access. It is based on a machine that is completely different from the storage-based machines: the PINITOR does not use storages but rather many microprocessors, one for each appearance of a command in the code of primitive recursive functior. the codes are decimal numbers, called pinons , where only the characters 0 1 2 8 9 appear. There are four kind of commands only: 0 nullification, 1 succession, 2 straight recursion and 8 composition. The PINITOR is a calculator which means that there is no halting problem. Computers have halting problems, per defintion calculators do not.
Appendix C6 gives the programming of the codes of most of the usual primitive function and goes even farther, e.g. it introduces generator technique that allows for the straight-forward calculation of so-called processive function, that are not primitive recursive. The most famous examples of Ackermann and other hyperexponential functions are programmed.
Appendix C7 turns to the Boojum-function and the Snark-function that have been introduced as calculative functions in the above publication in connection with Church's thesis. A list is provided that gives the lowest values for these functions and gives some more insight into these functions.

**Category:** Set Theory and Logic

[17] **viXra:1905.0254 [pdf]**
*submitted on 2019-05-16 08:59:09*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The seminal theorem of the dissertation states a cardinal is greatly inaccessible if and only if it is Mahlo. Three non trivial equations of the proof are not tautologous, thereby refuting theorems derived therefrom. These conjectures form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[16] **viXra:1905.0231 [pdf]**
*submitted on 2019-05-15 08:39:24*

**Authors:** Ricardo Alvira

**Comments:** 19 Pages.

In recent times we have witnessed proliferation of indicators and models for measuring sustainability. This reveals the lack of common and shared scientific paradigm/common framework from which to confront the issue of quantitatively assessing the sustainability of our society. With the aim of moving forward the definition of such common framework, in this article we explain an easy formal methodology for designing urban sustainability indicators based on Fuzzy Logic / Fuzzy Sets Theory. The interest of this methodology is threefold: Firstly, formal procedures enable easier testing, a most fundamental issue forgotten in many current proposals of sustainability indicators. Secondly, a formal procedure is easily understandable and can become a common language allowing shared use of the indicators and facilitating their continuous improvement. And thirdly, fuzzy logic is widely used in computing and artificial intelligence, thus facilitating the progressive automation of our sustainability monitoring models. To help understand the procedure, the design of two indicators is reviewed.

**Category:** Set Theory and Logic

[15] **viXra:1905.0223 [pdf]**
*submitted on 2019-05-15 18:33:17*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The definition of the Jaccard index is not tautologous, hence refuting it with derivations and forming a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[14] **viXra:1905.0221 [pdf]**
*submitted on 2019-05-16 03:15:48*

**Authors:** Hannes Hutzelmeyer

**Comments:** 7 Pages.

In 1936 Alonzo Church put forward his thesis that recursive functions comprise all effectively calculative functions. Whereas recursive functions are precisely defined, effectively calculative functions cannot be defined with a rigor that is requested by mathematicians. There has been a considerable amount of talking about the plausibility of Church's thesis, however, this is not relevant for a strict mathematical analysis. The only way to end the discussion is obtained by a counterexample.
The author has developed an approach to logics that comprises, but goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and so on.
The concrete calcule LAMBDA of natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. All recursive functions can be represented in LAMBDA too. The unary Snark-function is defined by a diagonalization procedure such that it can be calculated in a finite number of steps. However, this calculative function transcends the compinitive functions and presumably the recursive functions. The defenders of Church's thesis are challenged to show that the Snark-function is recursive. Another challenge asks for an example of a recursive function that cannot be expressed as a compinitive function, i.e. without minimization.

**Category:** Set Theory and Logic

[13] **viXra:1905.0215 [pdf]**
*submitted on 2019-05-14 07:54:57*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate the nine axioms for intutionistic Zermelo-Fraenkel set theory (IZF). None is tautologous. This refutes IZF and its use in blended models and denies De Jongh’s classical theorem and similar results for constructive ZF (CZF). These segments form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[12] **viXra:1905.0195 [pdf]**
*submitted on 2019-05-13 12:10:48*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

We evaluate four definitions for reverse mathematics (3) and nets (1). None is tautologous.
This refutes reverse mathematics and nets. Therefore these definitions form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[11] **viXra:1905.0172 [pdf]**
*submitted on 2019-05-12 23:34:42*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

A definition with variant to establish a sublanguage in support of the Barwise compactness theorem is not tautologous. By extension the theorem is also refuted. These conjectures form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[10] **viXra:1905.0160 [pdf]**
*submitted on 2019-05-11 19:27:34*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

Logic VŁ4 is defined as a bivalent classical logic that maps quantifiers to modalities as a tautology making VŁ4 complete. Paraconsistent, non bivalent, vector logics are defined as non tautologous fragments of VŁ4 as a universal logic.

**Category:** Set Theory and Logic

[9] **viXra:1905.0153 [pdf]**
*submitted on 2019-05-10 18:37:20*

**Authors:** Analytic Birb

**Comments:** 2 Pages.

The apparent contradiction of the status of women as both thots and queens, as expressed by the sentences ‘if she breathes, she’s a thot’ and ‘all women are queens’ is examined. The problem is formalized using the law of the excluded middle and a counterexample to the assertion is provided. The implications of the lack of a paradox are discussed.

**Category:** Set Theory and Logic

[8] **viXra:1905.0116 [pdf]**
*submitted on 2019-05-07 15:52:46*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The antecedent and consequent of the thought experiment of Pascal’s wager are not tautologous. However, to determine gain by one wager or the other is tautologous. This refutes the conjecture of Pascal’s wager as ultimately not allowing reason to determine faith. In other words, the “existence of God is possible to prove by human reason”. What follows furthermore is that the existence of God is more profitable from this thought experiment. Therefore the conjecture forms a tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[7] **viXra:1905.0097 [pdf]**
*submitted on 2019-05-06 21:21:49*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate a definition of the visibility graph as not tautologous to deny a fast algorithm for forecasting time series. Hence the conjecture of a forecasting algorithm is denied. This forms a non tautologous fragment of the universal logic VŁ4. However, we resuscitate the conjecture using the Kanban cell neuron network (KCNN), a linear step-wise function, for the desired conjecture without injected data.

**Category:** Set Theory and Logic

[6] **viXra:1905.0063 [pdf]**
*submitted on 2019-05-04 23:23:21*

**Authors:** Colin James III

**Comments:** 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate eight equations for provability logic (GL) and the derived polymodal logic of Japaridze (GLB, GLP). None is tautologous, hence refuting provability logic. These form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[5] **viXra:1905.0062 [pdf]**
*submitted on 2019-05-04 23:24:52*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate three equations as examples of non Sahlqvist formulas. None is tautologous. What follows is that Fine’s theorem and monotonic modal logic are refuted. Therefore those conjectures form a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[4] **viXra:1905.0051 [pdf]**
*submitted on 2019-05-03 10:27:21*

**Authors:** Colin James III

**Comments:** Pages.

A meta-rule for structural induction in the prover assistant Isabelle/HOL is not tautologous. This refutes the assistant and denies it can effect a cross-fertilization of computer science and metaphysics. Therefore Isabelle/HOL is a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[3] **viXra:1905.0038 [pdf]**
*submitted on 2019-05-02 13:39:14*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

A first order language of sets is proposed, but the first example A⊂B iff (x∈A then x∈B) is not tautologous. This refutes the conjecture of model theory = universal algebra + mathematical logic, which forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[2] **viXra:1905.0015 [pdf]**
*submitted on 2019-05-01 11:33:18*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate two versions of Arrow’s impossibility theorem with disjunctive or conjunctive results. Both as rendered are not tautologous. This means Arrow’s framework is refuted, hence coloring the conjecture of Arrow’s theorem before pivotal voters or dictators can be derived. Therefore Arrow’s impossibility theorem forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[1] **viXra:1905.0011 [pdf]**
*submitted on 2019-05-01 17:44:36*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate four equations which are not tautologous, but in fact produce the equivalent logic table values result. This means that the stated problem of applying singletons to the powerset is equivalent to proving singletons are atoms and that every subset satisfying a singleton is also an atom. Hence, overlap algebras do not constructively prove complete Boolean algebras. Therefore that conjecture for intuitionistic logic forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic