Set Theory and Logic

1909 Submissions

[23] viXra:1909.0634 [pdf] submitted on 2019-09-30 13:31:47

Refutation of Possible/certain Functional Dependencies for Incomplete Data

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 three equations from a continuing example. The last equation is supposed to be contradictory which is not tautologous. This refutes the seminal example, disallows the counter example as proffered, and refutes the conjecture of allowing incomplete information for possible/certain functional dependencies. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[22] viXra:1909.0595 [pdf] submitted on 2019-09-27 11:17:41

Refutation of the Unified Language for Linear and Relational Algebra (Lara)

Authors: Colin James III
Comments: 2 Pages. .© Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate two proofs of Th. 3 of the Lara language as not tautologous. By extension the Lara language as a unification of linear and relational algebra is also refuted. The conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[21] viXra:1909.0569 [pdf] submitted on 2019-09-26 21:59:53

Refutation of Graph Neural Reasoning

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

Definitions of three embedded rules are not tautologous. This refutes the conjecture of proving unsatisfiability by applying neural reasoning to graphs, and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[20] viXra:1909.0568 [pdf] submitted on 2019-09-26 23:36:34

Refutation of the Weisfeiler-Lehman Theorem

Authors: Colin James III
Comments: 1 Page. .© Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the join equation of Weisfeiler-Lehman method as not tautologous, refuting the conjectured theorem and forming a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[19] viXra:1909.0563 [pdf] submitted on 2019-09-25 06:36:10

Refutation of Second Incompleteness Theorem and Well-Foundedness of Jump Hierarchies

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 Gödel’s second incompleteness theorem (equated to well-founded Kripke-frames and the least element principle) and modal Löb’s theorem (equated to induction). Both are not tautologous and converse. Hence to imply well-foundedness, such as to jump hierarchies, is also not tautologous. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[18] viXra:1909.0554 [pdf] submitted on 2019-09-25 10:42:28

Refutation of Hilbert Logic System HG

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 Axiom 1 (A→(B→A)) of the Hilbert system HG as not tautologous. This refutes it and those conjectures to follow: Gödel algebras; Heyting algebras with the prelinearity law; intuitionistic Kripke models; Gödel LG; and modal multi-valued Gödel logic. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[17] viXra:1909.0531 [pdf] submitted on 2019-09-24 08:00:29

A Flaw in Separation Axioms

Authors: Hannes Hutzelmeyer
Comments: 18 Pages.

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 geometry, number theory, recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers etc. . The conventional treatment of axiomatic set theory (ZFC) is replaced by the abstract calcule sigma so that certain shortcomings can be avoided by the use of Funcish-Mencish language hierarchy: - precise talking about formula strings necessitates a formalized metalanguage - talking about open arities, general tuples, open dimensions of spaces, finite systems of open cardinality and so on necessitates a formalized metalanguage. 'dot dot dot … ' just will not do - the Axiom of Infinity is generalized in order to allow for certain other infinite sets besides the natural number representation according to von Neumann (i.e. general recursion) - the Axioms of Separation is modified as it seems more convenient - there are only enumerably many properties that can be constructed from formula strings, as these are finite strings of characters from a finite alphabet; this should be kept in mind in connection with the Axioms of Replacement - a new look at Cantor's continuum hypothesis in abstract axiomatic set theory leads to the question of so-called basis-incompleteness versus proof-incompleteness - the Axioms of Separation seem to have a flaw; there is a caveat for axiomatic set theory.
Category: Set Theory and Logic

[16] viXra:1909.0522 [pdf] submitted on 2019-09-24 13:33:15

Refutation of Tarski's Quantifier Elimination (Qe) as a Formal Method Technique

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 evaluate the first order polynomial case of Tarski’s quantifier elimination (QE) for recursive induction using definitions of the crucial theorem. QE is not tautologous. This also refutes the conjecture of QE to enhance safety assurance in deep neural networks. These conjectures form non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[15] viXra:1909.0494 [pdf] submitted on 2019-09-23 22:43:34

Refutation of Lindström Theorems for Fragments of First-Order Logic

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 evaluate three equations for the example property as not tautologous. This refutes virtually all conjectures, such as Lindström theorems for the six upper fragments of logic in Fig. 1, and the two proof approaches claimed as clever. These form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[14] viXra:1909.0470 [pdf] submitted on 2019-09-23 06:56:53

Refutation of Ceers (Computably Enumerable Equivalence Relations)

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 definitions from two papers of computably enumerable equivalence relations (ceers) as not equivalent, not tautologous, and not computably reducible. This refutes ceers as a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[13] viXra:1909.0469 [pdf] submitted on 2019-09-23 07:00:20

Refutation of Examples of Monotonicity

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 antecedents and consequents of four disjunctions of monotonicity in the system Natural Logic. The conjecture that these claim different interpretations for disjunction is refuted. The claims form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[12] viXra:1909.0431 [pdf] submitted on 2019-09-20 19:11:30

Refutation of Interpretability Logic, and Vaught and Adjunctive Set Theory

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 axioms and inference rules of interpretability logics ILM and TOL with none tautologous, making doubtful the claimed completeness. We then turn to Vaught and adjunctive set theory with neither tautologous. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[11] viXra:1909.0411 [pdf] submitted on 2019-09-19 13:22:46

Refutation of a Modal Logic for Supervised Learning

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 ten conjectures which are not tautologous, and with four as contradictory. This refutes the approach and models of modal logic for supervised learning as a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[10] viXra:1909.0369 [pdf] submitted on 2019-09-17 14:33:15

Resolution in Classical Modal Logic of a Security Barrier Model

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 evaluate three states of a security barrier example. For the security barrier as equivalent to the road width, the two cases are staying on the road, or staying on the road or going off of the road. Staying on the road is not a theorem. However staying on the road or going off road is a theorem. For the security barrier as larger than the road width, then staying on the road is effectively enforced as a theorem. This evaluation concludes that casting a problem to substructural epistemic resource logic is an unneeded effort for model resolution. The resolution in classical model logic of a security barrier model relegates the authors’ conjectures to non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[9] viXra:1909.0365 [pdf] submitted on 2019-09-17 21:57:03

Refutation of Axiomatizing Logics of Fuzzy Preferences Using Graded Modalities

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 the fuzzy preference relation of the assumed ∧-transitivity theorem as not tautologous, relegating it, along with the subsequent minimal modal logics of a finite residuated lattice and the Bulldozed method, as non tautologous fragments of the universal logic VŁ4.
Category: Set Theory and Logic

[8] viXra:1909.0335 [pdf] submitted on 2019-09-17 01:14:44

Refutation of Modal Operators on Rings of Continuous Functions

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 five expressions for a definition (2), two remarks (2), and a lemma proof (1). None is tautologous. This refutes the titled conjecture which drags in the Hausdorff and Stone topologies, Kripke frames, and dualities of Gelfand-Naimark-Stone and Esakia-Goldblatt. These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[7] viXra:1909.0284 [pdf] submitted on 2019-09-13 19:42:35

Refutation of Predicate Transformer Semantics for Hybrid Systems

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 steps of the approach to verify hybrid systems in the style of dynamic logic. Top tier input assumes modal Kleene algebras which are not bivalent. Middle tier processing invokes binary relations (which we do not test). Bottom tier output produces Lipschitz continuous vector fields as verification of hybrid store dynamics, which are not bivalent. This refutes the Isabelle framework for hybrid systems verification, and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[6] viXra:1909.0256 [pdf] submitted on 2019-09-11 11:50:18

Square of Opposition as Meth8 Corrected

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 modern revision of the square of opposition is not tautologous and forms a non tautologous fragment of the universal logic VŁ4. Consequently we redefine the square so that it is validated as tautologous. Instead of definientia using implication for universal terms or conjunction for existential terms, we adopt the equivalent connective for all terms. The modal modifiers necessity and possibility map quantifiers as applying to the entire terms rather than to the antecedent within the terms.
Category: Set Theory and Logic

[5] viXra:1909.0240 [pdf] submitted on 2019-09-10 16:35:12

Refutation of Bitstring and Question-Answer Semantics

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 evaluate bitstring semantics and its follow-on by partition. Its ordered set of exhaustive predicates is not bivalent but a probabilistic vector space. Its calculus of relations is not tautologous. Hence its broader framework of question-answer semantics (QAS) is not tautologous. The conjecture of “generalizing the Aristotelian square within one common gathering” is denied. What is affirmed is the Meth8 corrected, modern, revised square of opposition is a square, to mean the following conjectures are probabilistic vector spaces: collapsed number line of opposition; non-standard quadrilateral of oppositions; and colored square of oppositions. Bitstring semantics and the extended QAS form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[4] viXra:1909.0237 [pdf] submitted on 2019-09-10 20:36:44

Refutation of Cook-Reckhow Definition

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 Cook-Reckhow definition as represented for conjectures of a polynomial equation and integer linear inequalities with both as not tautologous. Hence we do not evaluate the conjecture of a relational form on which the paper subsequently relies. These results form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[3] viXra:1909.0081 [pdf] submitted on 2019-09-04 19:09:54

Refutation of Automorphism and Borel Dynamics

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 the definition of automorphism as not tautologous, hence refuting conjectures of Borel dynamics, and forming non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[2] viXra:1909.0079 [pdf] submitted on 2019-09-04 23:26:42

Refutation of Skolem Witness and Counter-Example Functions

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 the Skolem witness function and counter-example function. The functions are not tautologous and not counter-exemplary. This refutes the conjecture of a witness function as opposite a counter-example function and forms a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic

[1] viXra:1909.0058 [pdf] submitted on 2019-09-03 23:11:03

Refutation of the Russell-Prawitz Embedding

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 the Russell-Prawitz embedding as not tautologous. Hence atomization of universal instantiation does not follow (nor does proof reduction, weakening of dinaturality conversion, or strict simulation). These conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Set Theory and Logic