Set Theory and Logic

1711 Submissions

[14] viXra:1711.0378 [pdf] submitted on 2017-11-20 08:04:42

Refutation of Chaitin's Theorem of Incompleteness © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages.

We evaluate Chaitin's incompleteness theorem of 1974. Martin Davis described it as “a dramatic extension of Gödel’s incompleteness theorem” in 1978. We find the approach of the conjecture is moot, refute Chaitin's theorem of incompleteness, and remark that Chaitin's constant is suspicious.
Category: Set Theory and Logic

[13] viXra:1711.0364 [pdf] submitted on 2017-11-19 10:38:24

The Shortest Refutation of Gödel's Theorem of Incompleteness © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © Copyright 2017 by Colin James III All rights reserved.

Our example in the positive and contra example show the shortest known definitive refutation for Gödel's incompleteness theorem.
Category: Set Theory and Logic

[12] viXra:1711.0357 [pdf] submitted on 2017-11-18 08:01:42

Note on Grilliot's Trick and Effective Implication © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We evaluate two of the simpler equations for tautology which relate to Grilliot's trick and the standard extensionality trick via effective implication. Eqs., as rendered, are not tautologous. According to variant sysem VŁ4, this means Grilliot's trick, effective implication, and the subsequent non standard extensionality trick are not bivalent, but rather are a vector space.
Category: Set Theory and Logic

[11] viXra:1711.0323 [pdf] submitted on 2017-11-15 14:07:02

Logic not Tautologous in Neutrosophic Sets © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

We test a theorem and two properties from above. Eqs. 1.1, 5.2, and 6.1 should be tautologous, but are not.
Category: Set Theory and Logic

[10] viXra:1711.0320 [pdf] submitted on 2017-11-15 18:23:23

Law of Self-Equilibrium: not Law; not Paradox © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

The law of self-equilibrium sometimes uses this example: Too much work produces sickness; sickness produces less work; therefore, too much work implies less work. We rewrite the sentence to replace the connective verb with "causes" for better meaning and also include a modal operator for clarity: Too much work causes possible sickness; sickness causes less work; therefore, too much work causes less work. ... the law of self-equilibrium is not tautologous, and hence not a theorem and not a paradox.
Category: Set Theory and Logic

[9] viXra:1711.0318 [pdf] submitted on 2017-11-14 17:23:56

Lothar Collatz Conjecture is Tautologous © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

Given a positive integer n, if it is even, calculate n/2, otherwise if it is odd then calculate 3n+1; repeat this process with the resulting value. (1.0) We assume the apparatus of the Meth8 modal logic model checker implementing variant system VŁ4. Meth8 allows to mix four logical values with four analytical values. The designated proof value is T. (p>(p@p))>(((p=(%p>#p))>(p\(%p<#p)))+((p=(%p<#p))>((p&(p=p))+(%p>#p)))) ; TTTT TTTT TTTT TTTT (1.1) Eq. 1.1 shows the conjecture is tautologous.
Category: Set Theory and Logic

[8] viXra:1711.0312 [pdf] submitted on 2017-11-15 06:04:19

Re: Deducibility Theorems in Boolean Logic Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, Usa e-Mail: Smarand@unm.edu Http://vixra.org/abs/1003.0171

Authors: Colin James III
Comments: 1 Page.

As presumably a basis for neutrosophic logic these mistakes were found: Theorems 1 and 2 are not tautologous.
Category: Set Theory and Logic

[7] viXra:1711.0290 [pdf] submitted on 2017-11-12 12:05:19

Rationale of Rendering Quantifiers as Modal Operators © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 14 Pages. © 2016 by Colin James III All rights reserved.

The rationale for rendering quantifiers as modal operators in Meth8 has arguments from satisfiability and reproducability of invalidating and validating syllogisms. The Square of Opposition (original) produced four combinations for each corner A, I, E, O for 4 ^ 4 = 256 syllogisms. Medieval scholars determined 24 of the 256 syllogisms were valid deductions. Of those, 9 were made valid but only after additional known assumptions were applied as fix ups. Meth8 Tautologous none of the 24 syllogisms before fix ups. Meth8 also discovered correct additional assumptions to render the other 15 syllogisms Tautologous. We use Meth8 to replicate the 24 valid syllogisms derived from the original Square of Opposition. In the process we make three recent advances. 1. A third assumption is needed to fix up Modus Cesare EAE-2 2. The third assumption cannot be removed from Modus Camestros AEO-2 (as in other syllogisms with known third assumptions); and 3. No third assumptions are required for the other 22 syllogisms.
Category: Set Theory and Logic

[6] viXra:1711.0289 [pdf] submitted on 2017-11-12 12:09:05

Square of Opposition Meth8 Corrected © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. © 2016 by Colin James III All rights reserved.

The modern revision of the square of opposition is not validated as tautologous by the Meth8 logic model checker, as based on system variant VŁ4. Consequently we redefine the square so that it is validated as true my Meth8. 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. We note the validating connectives for the edges on the square are: \ Nand for the Contraries and Contradictories; > Imply for the Subalterns; and + Or for the Subcontraries.
Category: Set Theory and Logic

[5] viXra:1711.0288 [pdf] submitted on 2017-11-12 12:11:20

Square of Opposition Modern Revised: not Validated as Tautologous © 2016 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. © 2016 by Colin James III All rights reserved.

The modern, revised square of opposition is not validated as tautologous by the Meth8 logic checker in five models for all expressions. This leads us to consider that any logic system based on the square of opposition is spurious. What follows then is that a first order predicate logic based on the square of opposition is now suspicious.
Category: Set Theory and Logic

[4] viXra:1711.0278 [pdf] submitted on 2017-11-11 07:04:37

Evaluation of Computer Assisted Proofs on Gödel Incompleteness Theorems

Authors: Colin James III
Comments: 3 Pages. © Copyright 2017 by Colin James III All rights reserved.

© Copyright 2017 by Colin James III All rights reserved. The Gödel incompleteness theorem states in effect that sequences of logic symbols can be assigned to strings of natural numbers, but because it is assumed there are more natural number than sequences of logic symbols, the symbols are incomplete as a self-referencing mechanism to describe repeatedly themselves as yet more numbers. (The Gödel completeness theorem states in effect that the sequences of logic symbols may be consistent to form a logic system that is sufficiently complete enough to prove theorems as tautology.) The arguments ultimately turn on the mapping of sequences of symbols into strings of natural numbers. The arguments also assume a function to map numbers as a domain into symbols as an image in a co-domain, also collectively named a range. The incompleteness theorem is that such a function exists and operates where all of its domain is larger than the smaller image existing within all of the co-domain. The completeness theorem is that all of the image, existing solely for its purposes, is self-sufficient unto the existence of itself. We are interested in mapping the image in the co-domain as sequences of logic symbols back into the originating domain which is named a preimage. Attempting such an inverse function is not allowed by the one-way definition of a function to an image as dictated by the incompleteness theorem. We show this invertive approach or reverse process is not tautologous by evaluating the misuse of the application of quantified operators in the one-way functional mapping in the first place. Our experimental results in mapping are the definitive evidence that constructionist logic and subsequently constructivistic logics are not complete (deny the completeness theorem) and hence deny the incompleteness theorem. The most important evidence is that those logics can exist only by ignoring the law of excluded middle (LEM) that "p or not p is a tautology", (p+~p)=(p=p). We argue that any system denying the LEM is not tautologous and hence unworkable.
Category: Set Theory and Logic

[3] viXra:1711.0271 [pdf] submitted on 2017-11-10 16:06:05

Rule of Necessitation: True, But not Tautologous © Copyright 2017 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages.

1. The axiom or rule of necessitation N states that if p is a theorem, then necessarily p is a theorem: If ⊢ p then ⊢ ◻p. We show this is non-contingent (a truth), but not tautologous (a proof). We evaluate axioms (in bold) of N, K, T, 4, B, D, 5 to derive systems (in italics) of K, M, T, S4, S5, D. We conclude that N the axiom or rule of necessitation is not tautologous Because system M as derived and rendered is not tautologous, system G-M also not tautologous. What follows is that systems derived from using M are tainted, regardless of the tautological status of the result so masking the defect, such as systems S4, B, and S5. We also find that Gentzen-sequent proof is suspicious, perhaps due to its non bi-valent lattice basis in a vector space.
Category: Set Theory and Logic

[2] viXra:1711.0263 [pdf] submitted on 2017-11-10 10:13:03

Recent Advances in Refutations and Validations Using Meth8 Modal Logic Model Checker © Colin James III 2016-2017 All Rights Reserved.

Authors: Colin James III
Comments: 189 Pages.

© Colin James III 2016-2017 All rights reserved. In applied and theoretical mathematics, assertions are categorized in alphabetical order as: axiom; conjecture; definition, entry; equation; expression; formula; functor; hypothesis; inequality; metatheorem; paradox; problem; proof; schema; system; theorem; and thesis. We evaluate 130 objects for 519 assertions to validate 156 as tautology and 363 as not (70%). We use Meth8 that is a modal logic checker in five models. The semantic content or predicate basis of some expressions on their face does not disqualify them from evaluation by Meth8 in classical modal logic. However, the rules of classical logic, as based on the corrected Square of Opposition by Meth8, apply to virtually any logic system. Consequently some numerical equations are mapped to classical logic as Meth8 scripts. The rationale for mapping quantifiers as modal operators is in the Appendix based on satisfiability and reproducability of validation of syllogisms. A table lists what was tested with separated results. The names are numbered in alphabetical order. Test results are Invalidated as Not Validated Tautology (nvt) or Validated as Tautology (vt). For a paradox, invalidated means it is not validated as true, that is, it is not a paradox or contradiction. The experimental tests used variables for 4 propositions, 4 theorems, and 11 propositions. The size of truth tables are respectively for 16-, 256-, and 2048- truth values. One formula of Popper in 250-characters processed in 125-steps instantly due to recent advances in look up table indexing. The Meth8 modal theorem prover implements the logic system variant VŁ4 which corrects the quaternary Ł4 of Łukasiewicz. There are two sets of truth values on the 2-tuple {00, 10, 01, 11} as respectively and . The model checker contains recent advances in parsing technology and is U.S. Patent Pending. The mapping of formulas into Meth8 script was performed by hand, checked, and tested for accuracy of intent. (A semi-automation of that process is underway.) The Meth8 script uses literals and connectives in one-character. Propositions are p-z, and theorems are A-B. The connectives for are <&; +; >; =). The negated connectives for are <\; -; <; @>. The operators for are <~; %; #>. Some expressions are adopted for clarity such as: (p=p) for true; (p@p) for false; and (xy). Def Axiom Sym Name Meaning 11 p=p T Tautology proof 00 p@p F Contradiction absurdum 01 %p>#p N Non-contingency truth 10 %p<#p C Contingency falsity The designated proof value is T tautology. Note the meaning of (%p>#p): a possibility of p implies the necessity of p; and some p implies all p. In other words, if a possibility of p then the necessity of p; and if some p then all p. This shows equivalence and interchangeability of respective modal operators and quantified operators, as proved in Appendix. For Meth8 an immediate further application to "validate as tautologous" is mapping sentences of natural language into logical formulas. The approach identifies parts of speech as nouns, verbs, and modifiers. These translate into logical symbols for literals, connectives, and operators. For example: the conjunction "and" becomes the connective "&"; and the modifier articles "the" and "a" become the modal box # and lozenge %. Expressions for consecutive sentences are linked by the imply connective to build paragraphs to form requirements documents.
Category: Set Theory and Logic

[1] viXra:1711.0113 [pdf] submitted on 2017-11-03 07:20:31

Tautology Problem and Two Dimensional Formulas

Authors: Deniz Uyar
Comments: 38 Pages.

Finding whether a boolean formula is a tautology or not in a feasible time is an important problem of computer science. Many algorithms have been developed to solve this problem but none of them is a polynomial time algorithm. Our aim is to develop an algorithm that achieve this in polynomial time. In this article, we convert boolean functions to some graph forms in polynomial time. They are called two dimensional formulas and similar to AND-OR graphs except arches on them are bidirectional. Then these graphs are investigated to find properties which can be used to differentiate tautological formulas from non tautological ones.
Category: Set Theory and Logic