[19] **viXra:1903.0510 [pdf]**
*submitted on 2019-03-28 19:36:20*

**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 a constructive proof of Craig’s interpolation theorem by way of Maehara’s technique. Six equations are not tautologous, and serve as antecedents for respective conclusions of two- or four-sequents. Hence, the concluding consequents in any state of proof value will always return a tautology. This means the technique of Maehara does not produce a constructive proof of Craig's interpolation theory as applied to sequent logic for interpolation of non-normal logics. Therefore the approach forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[18] **viXra:1903.0494 [pdf]**
*submitted on 2019-03-27 11:41:43*

**Authors:** Colin James III

**Comments:** 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.)

In see-to-it-that logic (stit logic), three deontic examples are presented of Horty's coin betting puzzle with two agents. The form of the examples is tautologous. However, a profitability analysis by contrasting outcome for the agents shows none is tautologous. The example for the agent initiating the state of the coin as more profitable than the other agent is more closely aligned to tautology and hence the more profitable strategic outcome. What follows is that stit logic is a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[17] **viXra:1903.0472 [pdf]**
*submitted on 2019-03-26 15:29:13*

**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.)

From the example of applying dyatic semantics to paraconsistent logic C1, four axioms are not tautologous, hence refuting that approach. Therefore paraconsistent logics are non tautologous fragments of the universal logic VŁ4.

**Category:** Set Theory and Logic

[16] **viXra:1903.0471 [pdf]**
*submitted on 2019-03-26 15:31:55*

**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. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We evaluate the Gödel quantification scheme of ∃*∀n∃*φ as tautologous. When it is extended to decidable and undecidable prefix-classes, none is tautologous. This refutes the Gödel class with identity as undecidable, to mean it is in fact solvable as not tautologous. Therefore the prefix-classes are non tautologous fragments of the universal logic VŁ4.

**Category:** Set Theory and Logic

[15] **viXra:1903.0461 [pdf]**
*submitted on 2019-03-27 05:43:26*

**Authors:** Thomas Limberg

**Comments:** 9 Pages. Language: German.

Like the title already tells, it is my goal to create a new foundation of mathematics.
In this mathematical article we define the term "mathematical system" and introduce a simple mathematical system called System 1.1, in which a proof sketch of a proof of the existence of the universal set and the empty set within the system is made.

**Category:** Set Theory and Logic

[14] **viXra:1903.0442 [pdf]**
*submitted on 2019-03-24 06:19:51*

**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.)

From the sections on OTM-realizabilty and intuitionistic provability and axioms and systems of constructive set theories, we evaluate an inference rule and two propositions. None is tautologous. The refutes OTM on set theory in Hilbert space for intuitionistic logic. Therefore that approach produces non tautologous fragments of the universal logic VŁ4.

**Category:** Set Theory and Logic

[13] **viXra:1903.0410 [pdf]**
*replaced on 2019-03-23 07:50:39*

**Authors:** Colin James III

**Comments:** 2 Pages.

Two definitions equations from Eqs. 4.4.1.2 and 4.4.4.2 as rendered are not tautologous, hence denying the monotone functor on KHaus. What follows is that the use of coalgebra to manufacture a geometric modal logic is refuted. Therefore the conjecture is a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[12] **viXra:1903.0406 [pdf]**
*submitted on 2019-03-22 14:56:52*

**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 six equations for conjectures in five subsections of origins, bases of objects, axiomatic conditions, adjunctions, finite domains, and join-approximable relations. None is tautologous, hence refuting the domain theory of Dana Scott. Therefore, Scott's domain theory is a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[11] **viXra:1903.0335 [pdf]**
*submitted on 2019-03-18 17:46:27*

**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.)

From the summary of standard and mutual induction and coinduction, we evaluated four formulas with non tautologous and hence refutations. Therefore these are non tautologous fragments of the universal logic VŁ4.

**Category:** Set Theory and Logic

[10] **viXra:1903.0302 [pdf]**
*submitted on 2019-03-15 10:50:26*

**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.)

From the elementary constructive set theory (ECST) of intuitionistic logic, we evaluate six axioms of equality for system CZF. None is tautologous. This refutes those axioms in set theory and by extension denies intuitionistic logic.

**Category:** Set Theory and Logic

[9] **viXra:1903.0289 [pdf]**
*submitted on 2019-03-14 07:25:10*

**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.)

For the relation of "a point is incident to a straight line", we find that proposition is not tautologous. This denies the conjectured approach of a constructive mapping Euclidean geometry into a Hilbert space and also refutes the Coq proof assistant as a bivalent tool. The conjecture and Coq are therefore non tautologous fragments of the universal logic VŁ4.

**Category:** Set Theory and Logic

[8] **viXra:1903.0237 [pdf]**
*submitted on 2019-03-12 15:08:14*

**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 examine a sentential logic description, as based on set theory, in support of Suzko's theorem that only two truth values are required as a universal logic. Under syntactic notions, we evaluate three definitions (monotonicity, transivity, permeability) out of six definitions (trivial are substitution-invariance, reflexivity, combined consequence relation). Monotonicity and transivity are not tautologous. Right-to-left permeability is not tautologous. What follows is that a Malinowski
extension of mixed-consequence by relaxation of the two values for three logical values is spurious, especially due to the fact that Suzko's theorem is a conjecture based on the assumption of set theory. What also follows is that compositionality as based on Suzko-Scott reductions are not bivalent and exact, but rather a vector space and probabilistic. Our results point further to the equations analyzed as being non tautologous fragments of the universal logic VL4.

**Category:** Set Theory and Logic

[7] **viXra:1903.0230 [pdf]**
*submitted on 2019-03-12 23:23:19*

**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.)

From the section on Płonka sums, we evaluate an equation derived thereform. It is not tautologous, hence coloring subsequent assertions in the conjecture. This means the non tautologous conjecture is a fragment of the the universal logic VŁ4.

**Category:** Set Theory and Logic

[6] **viXra:1903.0229 [pdf]**
*submitted on 2019-03-12 23:27:58*

**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.)

From the section partitions of frames, local finiteness, and the finite model property, we evaluate that definition. Because it is not tautologous, subsequent equations in the conjecture are denied. This means it is a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[5] **viXra:1903.0206 [pdf]**
*replaced on 2019-03-13 08:53:41*

**Authors:** Colin James III

**Comments:** 20 Pages.

This paper demonstrates why logic system VŁ4 is a universal logic composed of any refutation as a non-tautologous fragment. Recent advances are a definitive answer to criticism of logic Ł4, modal equations for lines and angles of the Square of Opposition, confirmation of the 24-syllogisms by updating Modus Cesare and Camestros, and proving that respective quantified and modal operators are equivalent. The parser Meth8 implements VŁ4 as the modal logic model checker Meth8/VŁ4. Over 430 assertions are tested in 2300 assertions with a refutation rate of 81%.

**Category:** Set Theory and Logic

[4] **viXra:1903.0205 [pdf]**
*submitted on 2019-03-11 21:36:16*

**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.)

From the introduction, we evaluate EA elementary arithmetic for induction formulas which are not tautologous. This further refutes the reflection property upon which subsequent assertions are based. These formulas constitute a non tautologous fragment of the universal logic VŁ4.

**Category:** Set Theory and Logic

[3] **viXra:1903.0146 [pdf]**
*submitted on 2019-03-08 09:52:22*

**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 canonically logical compatibility relations (CM) and complements (CRS), each in three sets of definitions. None is tautologous, so we avoid the subsequent ten relations. This refutes the "the possibility of a notion of compatibility that allows either for glutty or gappy reasoning". (By extension, paraconsistent logic is rendered untenable.) Therefore the bivalent standard notion of formal theory in logic is confirmed as allowing both assertion and denial as equally valid. In fact, this refutation further disallows injection of a bilateralist approach on many dimensions. This also indirectly reiterates that set theory is not bivalent, and hence derivations therefrom, such as the instant relations, are non tautologous fragments of the universal logic VŁ4.

**Category:** Set Theory and Logic

[2] **viXra:1903.0052 [pdf]**
*submitted on 2019-03-03 10:15:49*

**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.)

From the background definitions in the ordered Banach space, we evaluate equations to produce the term named positive generating cone. It is not tautologous, hence refuting the model.

**Category:** Set Theory and Logic

[1] **viXra:1903.0049 [pdf]**
*replaced on 2019-05-16 22:30:10*

**Authors:** Ron Ragusa

**Comments:** 6 Pages.

The interval sieve algorithm partitions a closed interval of real numbers [ri, rj] where ri < rj to create a complete list, L, of numbers in the interval. We will prove that the list L is complete, and lastly derive the bijective function f : ℕ ↔ [r1, r2].

**Category:** Set Theory and Logic