Set Theory and Logic

1807 Submissions

[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