Set Theory and Logic

1807 Submissions

[27] viXra:1807.0514 [pdf] submitted on 2018-07-30 18:56:24

Confirmation that Every Straight Line Through a Point Inside a Circle Intersects the Circumference

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

"An interior point inside the circle and an exterior point outside the circle imply that every straight line intersecting both points intersects a point on the circle." This conjecture is comfirmed.
Category: Set Theory and Logic

[26] viXra:1807.0509 [pdf] submitted on 2018-07-29 06:56:29

Confirmation of Playfair's Axiom

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

Per "De Morgan that this proposition is logically equivalent to Playfair’s axiom. ... Let X be the set of pairs of distinct lines which meet and Y the set of distinct pairs of lines each of which is parallel to a single common line. If z represents a pair of distinct lines, then the statement, For all z, if z is in X then z is not in Y, is Playfair's axiom, and its logically equivalent contrapositive, For all z, if z is in Y then z is not in X, is Euclid I.30, the transitivity of parallelism." This is confirmed as tautologous.
Category: Set Theory and Logic

[25] viXra:1807.0499 [pdf] submitted on 2018-07-29 12:12:31

Confirmation of the Triangle Inequality

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

"[T]he triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side." We rewrite this as contradiction F implies tautology T using absolute values of distances: "If qr is not greater than pq with pr, then both qr is greater than pq and qr is greater than pr." Hence the triangle inequality is confirmed as tautologous. Remark: This exercise indirectly speaks to the fact that the vector space is not bivalent.
Category: Set Theory and Logic

[24] viXra:1807.0498 [pdf] submitted on 2018-07-29 12:41:53

Confirmation of a Trivial Vector Conjecture

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

"From the distance between two points, a third point always has the same distance to the other points." We rewrite this as: "If two points imply the necessity of a third point, then the respective distances are possibly the same." This is tautologous, hence confirming the conjecture. Remark: This exercise indirectly speaks to the fact that the vector space is not bivalent.
Category: Set Theory and Logic

[23] viXra:1807.0478 [pdf] submitted on 2018-07-28 16:50:58

Refutation of the Heisenberg Principle as a no-go Axiom, and Its Trivial Replacement Theorem

Authors: Colin James III
Comments: 2 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The Heisenberg uncertainty principle "states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa". This is refuted as not tautologous. In terms of the statistic of deviation, a trivial replacement theorem is: "If position is greater than momentum, then position and momentum are greater than position or momentum."
Category: Set Theory and Logic

[22] viXra:1807.0472 [pdf] submitted on 2018-07-27 15:42:55

Refutation of Relevance Logic Via Routely and Meyer

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The conjecture "A → B is true at a world a if and only if for all worlds b and c such that Rabc (R is the accessibility relation) either A is false at b or B is true at c" is not tautologous.
Category: Set Theory and Logic

[21] viXra:1807.0452 [pdf] submitted on 2018-07-26 23:41:32

Implication Combinations Derived from (P>q)>r

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

12 tautologous equations are named the general forms of (p>q)>r by implication on Meth8/VŁ4. (They are not listed here because the abstract block mungs line spacing.)
Category: Set Theory and Logic

[20] viXra:1807.0440 [pdf] submitted on 2018-07-27 10:42:14

Shortest Refutation of Prenex Normal Form

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

The implication rule for removing quantifiers from the antecedent of (∃xϕ)→ψ as equivalent to ∀x(ϕ→ψ), and the implication rule for removing quantifiers from the consequent of ϕ→(∃xψ) as equivalent to ∃x(ϕ→ψ), are not tautologous and hence refute the prenex normal form.
Category: Set Theory and Logic

[19] viXra:1807.0431 [pdf] submitted on 2018-07-25 18:48:36

Trivial Proofs

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

For all p, p+p=p (1.1). #p>((p+p)=p) ; TTTT TTTT TTTT TTTT (1.2). For one p, p+p=p (2.1). %p>((p+p)=p) ; TTTT TTTT TTTT TTTT (2.2). Axiom of associativity (3.1). ((p+q)+r)=(p+(q+r)) ; TTTT TTTT TTTT TTTT (3.2).
Category: Set Theory and Logic

[18] viXra:1807.0420 [pdf] replaced on 2018-07-28 20:36:49

Refutation of the Proscriptive Principle (Conceptivistic Logic) and Containment Logic

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

Three equations seminal to the proscriptive principle (conceptivistic logic) and containment logic are not tautologous, hence refuting the systems.
Category: Set Theory and Logic

[17] viXra:1807.0418 [pdf] replaced on 2018-09-18 12:32:44

Visualizing the Escape Paths of Quaternion Fractals

Authors: Shawn Halayka
Comments: 8 Pages.

The escape paths of the points in some quaternion fractal sets are visualized using OpenGL. C++ source code is provided.
Category: Set Theory and Logic

[16] viXra:1807.0417 [pdf] submitted on 2018-07-24 22:55:24

A Dual Identity Based Symbolic Understanding of the Gödel’s Incompleteness Theorems, P-NP Problem, Zeno’s Paradox and Continuum Hypothesis

Authors: Arun Uday
Comments: 29 Pages.

A semantic analysis of formal systems is undertaken, wherein the duality of their symbolic definition based on the “State of Doing” and “State of Being” is brought out. We demonstrate that when these states are defined in a way that opposes each other, it leads to contradictions. This results in the incompleteness of formal systems as captured in the Gödel’s theorems. We then proceed to resolve the P-NP problem, which we show to be a manifestation of Gödel’s theorem itself. We then discuss the Zeno’s paradox and relate it to the same aforementioned duality, but as pertaining to discrete and continuous spaces. We prove an important theorem regarding representations of irrational numbers in continuous space. We extend the result to touch upon the Continuum Hypothesis and present a new symbolic conceptualization of space, which can address both discrete and continuous requirements. We term this new mathematical framework as “hybrid space”.
Category: Set Theory and Logic

[15] viXra:1807.0414 [pdf] replaced on 2018-07-31 12:17:13

Modal Logic Model Checker Challenge, Unanswered

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com .

We offer to use the Meth8/VŁ4 modal logic model checker to evaluate difficult or unsolved logic problems because we are simply running out of artifacts to reduce. Since 2015, we evaluated 287 artifacts in 1479 assertions, to confirm 325 as tautology (22%) and 1154 as not (78%). See: ersatz-systems.com/RA.Meth8.refut.valid.abstract.pdf The checker is available with a wrapper for unique CPU-installation and meter for 10,000 accesses per renewable license. Asymmetric key encryption is not used, so the checker is fully exportable outside USA. Academic interest expressed in 2017-18 was by: an Oxford professor wanting it free for his students so he could use it; an unsupervised post-doc reviewer for a Polish logic magazine, so as to share; and a supervised Disqus troll seeking proofs for her classwork. POC: Colin James III, Colorado Springs, info@cec-services dot com, USA 719.210.9534.
Category: Set Theory and Logic

[14] viXra:1807.0410 [pdf] submitted on 2018-07-23 22:31:40

Refutation of Stit Logic (Sees to it That)

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

From four papers in the relevant literature for 2017-2018, 13 equations are not tautologous. For example, an equation claimed for necessitation as p>[]p is TNTN TNTN TNTN TNTN (not all T proof values). Hence Stit logic is refuted
Category: Set Theory and Logic

[13] viXra:1807.0392 [pdf] submitted on 2018-07-24 11:54:04

Refutation of Levi-Identity and Agm Postulates

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The Levi-identity as rendered is not tautologous, as a basis for the subsequent AGM expressions. Four of the eight AGM equations are not tautologous, hence refuting them as a postulate system for fictional logic.
Category: Set Theory and Logic

[12] viXra:1807.0380 [pdf] submitted on 2018-07-22 08:28:58

Refutation of Connexive Logic Based on Wansing's Nightmare

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Connexive logic turns on Wansing's nightmare of ∼(A→B)↔(A→∼B), or the falsity-weakend ∼(A→B)↔(∼A→B), both of which are not tautologous.
Category: Set Theory and Logic

[11] viXra:1807.0373 [pdf] submitted on 2018-07-22 17:06:11

Refutation of Aristotle's and Boethius' Theses

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to this author's email address: info@ersatz-systems dot com

Aristotle's Theses and Boethius' Theses are below. ~(~A> A)=(A=A) ; TNCF TNCF TNCF TNCF (AT) . ~( A>~A)=(A=A) ; FCNT FCNT FCNT FCNT (AT)'. (A> B)>~(A>~B) ; FCNT FCNT FCNT FCNT (BT) . (A>~B)>~(A> B) ; FCNT FCNT FCNT FCNT (BT)'. Eqs. AT, AT', BT, and BT' are not tautologous, hence refuting those theses of Aristotle and Boethius. Because ~AT = AT' = BT = BT', what follows is that using conditionals to justify connexive logic makes Wansing's nightmare worse.
Category: Set Theory and Logic

[10] viXra:1807.0365 [pdf] submitted on 2018-07-21 09:51:40

Refutation of the Improved Adams Hypothesis of Conditional Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 2 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The seven equations as rendered are not tautologous. This means the probability axioms of Popper and McGee and the improved hypothesis of Adams are refuted.
Category: Set Theory and Logic

[9] viXra:1807.0357 [pdf] submitted on 2018-07-21 23:55:29

Refutation of Conditional Events in Quantum Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The following conjectures are refuted: Definition of simultaneous verifiability; Definition of simultaneous falsifiability; Corollary on simultaneous verifiability and falsifiability; and Theorem on uniqueness of relative negation. The concluding conjecture to map superposition, while tautologous, as (((A∩B)∪(C∩D))|(B∪D))⊃((A|B)∪(C|D)) is replaced with this non-tautologous equation: (((A∩B)∪(C∩D))|(B∩D))⊃((A|B)∩(C|D)).
Category: Set Theory and Logic

[8] viXra:1807.0300 [pdf] submitted on 2018-07-17 15:04:21

Refutation of the Postulate of Contradiction Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The conjecture that a postulate of contradiction for superposition implies contradiction to map quantum logic is refuted.
Category: Set Theory and Logic

[7] viXra:1807.0294 [pdf] submitted on 2018-07-18 00:57:44

Refutation that Classical Logic is a Completion of Quantum Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 3 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

The distributivity of classical conjunction and disjunction does not fail. The conclusion to translate quantum negation as not necessity is not tautologous. This refutes quantum logic as a fragment of classical logic (or vice versa, as others write).
Category: Set Theory and Logic

[6] viXra:1807.0288 [pdf] submitted on 2018-07-16 21:41:03

Refutation of Quantum Logic as Tautologous Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 5 Pages. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

We evaluate quantum logic as described from two sources with different readership levels. We find quantum logic is not tautologous.
Category: Set Theory and Logic

[5] viXra:1807.0279 [pdf] submitted on 2018-07-15 15:01:34

Refutation of Whewell's Axiom of Causality Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

"Everything that becomes or changes must do so owing to some cause; for nothing can come to be without a cause." — Plato in Timaeus William Whewell [1794-1866] refined this as the concept of causality to depend on three axioms: nothing takes place without a cause; the magnitude of an effect is proportional to the magnitude of its cause; and to every action there is an equal and opposed reaction. The second axiom is tautologous, but the others are not, hence refuting the conjecture. From a metaphysical view, the axiom of causality is a bar to miracle because first cause is always assumed. This is overcome with rewriting the conjecture as "The necessity of effect implies the possibility of cause or no cause".
Category: Set Theory and Logic

[4] viXra:1807.0231 [pdf] submitted on 2018-07-12 23:29:05

If at Least One Question Implies Any Answer, Then Any Question Implies at Least One Answer. Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Is it true that any question implies at least one answer? No. Is it true that at least one question implies any answer? No. Is it true that if at least one question implies any answer, then any question implies at least one answer? Yes.
Category: Set Theory and Logic

[3] viXra:1807.0186 [pdf] submitted on 2018-07-09 22:53:57

Meth8/VŁ4 Self-Proves in One Variable for Validity, Consistency, Completeness, and Soundness Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

We test Meth8/VŁ4 using itself in one variable for the four qualities of a perfect logic system as validity, consistency, completeness, and soundness. The equations are tautologous. This means Meth8/VŁ4 proves itself, and in one variable. Remark: This also serves as the contra-example to the incompleteness theorem of Gödel which states a logic system cannot prove itself (and certainly not in one variable).
Category: Set Theory and Logic

[2] viXra:1807.0157 [pdf] submitted on 2018-07-09 06:47:49

Shortest Refutation of Gödel's Completeness Theorem Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

"By Gödel's completeness result, the formula (∀x.R(x,x))→(∀x∃y.R(x,y)) holds in all structures, and hence must have a natural deduction proof." We prove the formula is not tautologous, meaning it does not hold in all structures and serves as a contra-example. Hence Gödel's completeness theorem is refuted.
Category: Set Theory and Logic

[1] viXra:1807.0051 [pdf] submitted on 2018-07-02 23:08:38

Definition of Nothing in Mathematical Logic Copyright © 2018 by Colin James III All Rights Reserved.

Authors: Colin James III
Comments: 1 Page. Copyright © 2018 by Colin James III All rights reserved. Note that comments on Disqus are not forwarded or read, so respond to author's email address: info@cec-services dot com.

Nothing is defined as "not necessarily a thing". This leads to how to collect not everything as nothing in multiple variables into a larger nothing variable, implying a set of nothing as a null set. We write this as nothing in p and nothing in q and nothing in r are all greater than nothing in s: ((~#p&~#q)&~#r)>~#s TTTT TTTT CTTT TTTT. As rendered, this is not tautologous, although nearly so with one deviant C contingency (falsity) value. Hence a collection of nothing does not imply anything outside itself. By extension, the null set is not logically feasible and cannot exist: a collection must contain something even though it is nothing.
Category: Set Theory and Logic