IN DEFENSE OF THE SELF-REFERENCING QUANTIFIER Sx. APPROXIMATION OF SELF-REFERENTIAL SENTENCES BY DYNAMIC SYSTEMS

Arguments in defense of introducing the self-referencing quantifier Sx and its approximation on dynamical systems are consistently presented. The case of classical logic is described in detail. Generated 3-truth tables that match Priest’s tables (Priest 1979). In the process of constructing 4-truth tables, two more truth values were revealed that did not coincide with the original ones. Therefore, the closed tables turned out to be 6-digit.

Logical Principles in Ternary Mathematics

Introduction/Background:Our new research called “Logical Principles in Ternary Mathematics“ is an attempt to establish connection between logical and mathematical principles governing Ternary Mathematics and address issues that appeared earlier while making truth tables for “Ternary addition” and “Ternary Multiplication” presented by the same author in “Ternary Mathematics Principles Truth Tables and Logical Operators 3 D Placement of Logical Elements Extensions of Boolean Algebra” publication.The title “Logical Principles in Ternary Mathematics“ is not randomly chosen To be able to set up relations between elements in the given discipline one usually employs the basic principle of meaning-form and function In the same way we propose a logical triangle “Component”,”Vector”,”Decimal” to prove fundamental principle governing “Ternary Mathematics” presented in the given research. Aims/Objectives: The aim of the article is to set up connection between mathematical and logical rules governing Ternary Mathematics The main postulates of the Ternary Mathematics can be demonstrated by the abstract scheme or a triangle the vertices of which are “Component”,”Vector”,”Decimal” We use a triangle diagram to prove the functionality of the chosen principle. The three components are each connected with other two and transition is possible from one to another without changing the shape of a diagram and the principle applied. Methodology: The most difficult part is to “translate” Algebra and Numeric Analysis into Mathematical Logic and vice versa Traditional methods of logic fail to do this transition therefore a new functional approach is chosen. Results and Conclusion: As a result of this functional approach a new Ternary addition Truth Table is made The new Ternary Truth Table consists of the 3 literals (Т, ₸,F) Truth Negative, False and the last column of the table is the logical sum of the two. For example: Т+T=T Unlike the old table it presents a sum of two numbers in a vector form and therefore makes it possible to use it in mathematics as well as in logic.

Supporting Information "Quantum logic gates based on DNAtronics, RNAtronics and Proteintronics"

This Supporting Information includes the extended description of the superposition state of the asymmetric double-well system in vacuum system and in solution, truth tables for the residue pairs and their corresponding quantum logic gates, and figures for the double well potential energy surfaces and transmission spectra of the residue pairs. Corresponding Authors Email: [email protected] and [email protected]

Proof Systems for 3-valued Logics Based on Gödel’s Implication

The logic $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ was introduced in Robles and Mendéz (2014, Logic Journal of the IGPL, 22, 515–538) as a paraconsistent logic which is based on Gödel’s 3-valued matrix, except that Kleene–Łukasiewicz’s negation is added to the language and is used as the main negation connective. We show that $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ is exactly the intersection of $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ and $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$, the two truth-preserving 3-valued logics which are based on the same truth tables. (In $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ the set ${\cal D}$ of designated elements is $\{1\}$, while in $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ ${\cal D}=\{1,0.5\}$.) We then construct a Hilbert-type system which has (MP) for $\to $ as its sole rule of inference, and is strongly sound and complete for $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$. Then we show how, by adding one axiom (in the case of $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$) or one new rule of inference (in the case of $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$), we get strongly sound and complete systems for $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ and $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$. Finally, we provide quasi-canonical Gentzen-type systems which are sound and complete for those logics and show that they are all analytic, by proving the cut-elimination theorem for them.

Truth Tables Without Truth Values: On 4.27 and 4.42 of Wittgenstein’s Tractatus

In 4.27 and 4.42 of his Tractatus Wittgenstein introduces quite complicated formulas, which are equivalent to $$2^n$$ 2 n and $$2^{2^{n}}$$ 2 2 n . This paper shows, however, that the formulas Wittgenstein presents fit particularly well with the way he thinks about truth values, logical connectives, tautologies, and contradictions. Furthermore, it will be shown how Wittgenstein could have avoided truth values even more radically. In this way it is demonstrated that the reference to truth values can indeed be substituted by talking of existing and non-existing facts.

Ternary Mathematics Principles Truth Tables and Logical Operators 3 D Placement of Logical Elements Extensions of Boolean Algebra

The approach presented in this article aims at transition between two systems of counting binary and ternary I propose to use ternary math principle in coding the signal Instead of using duos of numbers 01 I propose to use triplets (1,0-1) and make a transition from binary to ternary so that the binary code is converted to ternary and vice versa That same principle can be used for building microcircuits where logical elements are placed in a 3 D space instead of a layer.

A New Approach for Simplification of Logical Propositions with Two Propositional Variables Using Truth Tables

Background: Propositions simplification is a classic topic in discrete mathematics that is applied in different areas of science such as programs development and digital circuits design. Investigating alternative methods would assist in presenting different approaches that can be used to obtain better results. This paper proposes a new method to simplify any logical proposition with two propositional variables without using the logical equivalences. Methods: This method is based on constructing a truth table for the given proposition, and applying one of the following two concepts: the sum of Minterms or the product of Maxterms which has not been used previously in discrete mathematics, along with five new rules that are introduced for the first time in this work. Results: The proposed approach was applied to some examples, where its correctness was verified by applying the logical equivalences method. Applying the two methods showed that the logical equivalences method cannot give the simplest form easily; especially if the proposition cannot be simplified, and it cannot assist in determining whether the obtained solution represent the simplest form of this proposition or not. Conclusion: In comparison with the logical equivalences method, the results of all the tested propositions show that our method is outperforming the current used method, as it provides the simplest form of logical propositions in fewer steps, and it overcomes the limitations of logical equivalences method. Originality/value: This paper fulfils an identified need to provide a new method to simplify any logical proposition with two propositional variables.

Addressing engineering threshold concepts in an African university of technology

Learning new skills isn’t only for the benefit of passing the exam but being able to apply those skills in a productive way. One cannot learn before one understands which is why student understanding is a priority for facilitators. This becomes especially important in threshold concepts where a student will be unable to progress to the next stage before the threshold concept is mastered, but facilitators do not focus on pedagogy as they rely on the support of instructional designers. This explorative paper looks at student perceptions of their understanding of the threshold concept in electrical engineering, logic gates, after completing a lesson designed using the proposed ten-step activity plan. The activity plan is derived from the learning theories of Gagne, Biggs, Vygotsky and Gibson. A sample of 18 students completed an online survey that focused on their acquiring of skills relating to logic gates, truth tables and Boolean algebra. Results showed a positive experience with 88.89% of participants indicating that they left the lesson with a good understanding of the threshold concept. This ten-step activity plan can assist to close the gap between instructional designer and facilitator to design threshold concept lessons based on sound learning theory.