Note on truth-tables

1950 ◽  
Vol 15 (3) ◽  
pp. 174-181 ◽  
Author(s):  
Jan Kalicki
Keyword(s):  
Finite Number ◽  
Truth Table ◽  
Truth Value ◽  
Truth Tables

In this paper a method will be described which is intended to exhibit some relationships between sets of tautologies determined by truth-tables. This method is an attempt to form an algebra of truth-tables. The results sketched below are restricted to sets of tautologies determined by truth-tables with a finite number of elements and involving a single binary connective Δ. However, most of the results can be easily extended to the case of Tarski's logical matrix and even to a more general case.We denote by S() the set of all tautologies (-tautologies) according to a given truth-table . Let describe a binary connective Δ. Then Δ()(x, y) stands for the truth-value of ΔPQ, when P has the truth-value x and Q has the truth-value y. If no ambiguity may arise we write Δ(x, y) or Δ() for Δ()(x, y).

A test for the existence of tautologies according to many-valued truth-tables

1950 ◽  
Vol 15 (3) ◽  
pp. 182-184 ◽  
Author(s):  
Jan Kalicki
Keyword(s):  
Finite Number ◽  
Truth Table ◽  
Effective Procedure ◽  
Single Variable ◽  
Truth Values ◽  
Contraction Process ◽  
The One ◽  
Image Position ◽  
Truth Tables

Theorem. There is an effective procedure to decide whether the set of tautologies determined by a given truth-table with a finite number of elements is empty or not.Proof. Let W(P) be a w.f.f. with a single variable P and n a given n-valued truth-table with elements (values)Substitute 1, 2, 3, …, n in succession for P. By the usual contraction process let W(P) assume the truth-values w1, w2, w3, …, wn respectively. The sequencewill be called the value sequence of W(P).Value sequences consisting of designated elements of exclusively will be called designated; others will be called undesignated.All the W(P)'s will be classified in the following way:(a) to the first class CL1 of W(P)'s there belongs the one element P,(b) to the (t + 1)th class CLt + 1 belong all the w.f.f. which can be built up by means of one generating connective from constituent w.f.f. of which one is an element of CLt and all the others (if any) are elements of CLn ≤ t.For example, if N and C are the connectives described by a truth-table etc.Let ∣CLn∣ stand for the set of value sequences of the elements of CLn.

A test for the equality of truth-tables

1952 ◽  
Vol 17 (3) ◽  
pp. 161-163 ◽  
Author(s):  
Jan Kalicki
Keyword(s):  
Finite Number ◽  
General Method ◽  
Truth Tables

In this paper the symbolism, definitions and results of my two papers, namely A test for the existence of tautologies according to many-valued truth-tables (hereafter referred to as ET) and Note on truth-tables (hereafter referred to as TT) will be presupposed.The problem whether two arbitrary truth-tables with finite number of elements are equal or not was reduced in TT to the question of equality of two identical tables in which different elements have been designated. However, no general method for testing the equality of truth-tables was given there, although some cases were discussed for which such a method is available. In the present paper we shall describe a general method using the considerations of ET and TT.

Graphing True-False Statements

1969 ◽  
Vol 62 (7) ◽  
pp. 553-556
Author(s):  
Margaret Wiscamb
Keyword(s):  
Truth Table ◽  
Graphic Representation ◽  
Truth Values ◽  
Truth Tables ◽  
True False ◽  
Tedious Process ◽  
Three Components

In elementary logic the construction of truth tables, while not difficult, can be a long and tedious process. In this article I would like to present a simple graphic representation of the truth values of compound statements involving two or three components. The graph gives all the information found in a truth table and pictures the statement as an easily recognizable pattern. By using this graphing procedure, the simplifying of statements is shortened considerably. In fact, for statements involving only two components, with a little practice it can usually be done by inspection. Proving that a statement is a tautology becomes almost trivial.

scholarly journals Design of Combinational Logic Circuit Prober

2019 ◽  
Vol 24 (4) ◽  
pp. 317-325
Author(s):  
Mohanad Abdulhamid ◽  
Okoth Masimba
Keyword(s):  
Logic Circuit ◽  
Truth Table ◽  
The Other ◽  
Combinational Logic ◽  
Oscilloscope Screen ◽  
Sinusoidal Signals ◽  
Combinational Logic Circuit ◽  
Truth Tables

Abstract The objective of this paper is to design and implement a logic circuit prober to display truth tables of a three input combinational logic circuit. The truth table is to be as “1” and “0” on an ordinary 60 MHz oscilloscope. This paper meets this objective by using Lissajous Patterns to plot a “0” or a “1” on the oscilloscope screen. To plot a “0” on the oscilloscope screen, two sinusoidal signals in quadrature are supplied to the two inputs of the oscilloscope with the scope set to X-Y mode. To plot a “1” on the oscilloscope, only the signal to the Y input is allowed to reach the oscilloscope screen. To display all the 32 patterns required to obtain a three input truth table, two staircase waveforms are employed. The staircase waveforms, one eight-step and the other four-step, are added to the two sinusoidal signals to shift the patterns along the X and Y directions to produce all the 32 patterns.

A Truth Table on the Island of Truthtellers and Liars

2001 ◽  
Vol 94 (9) ◽  
pp. 730-732
Author(s):  
Christopher Baltus
Keyword(s):  
Truth Table ◽  
Critical Reasoning ◽  
Experience Teaching ◽  
Truth Tables

My experience teaching truth tables left me looking for problems that exercise students' critical reasoning. This article grew from efforts to find such problems.

Post, Emil Leon (1897–1954)

2018 ◽  
Author(s):  
Michael Scanlan
Keyword(s):  
Finite Number ◽  
Propositional Logic ◽  
Decision Procedures ◽  
Truth Table ◽  
Formal Language Theory ◽  
Theory Of Computation ◽  
Symbol Manipulation ◽  
Canonical Systems ◽  
Finite Set ◽  
And Mathematics

Emil Post was a pioneer in the theory of computation, which investigates the solution of problems by algorithmic methods. An algorithmic method is a finite set of precisely defined elementary directions for solving a problem in a finite number of steps. More specifically, Post was interested in the existence of algorithmic decision procedures that eventually give a yes or no answer to a problem. For instance, in his dissertation, Post introduced the truth-table method for deciding whether or not a formula of propositional logic is a tautology. Post developed a notion of ‘canonical systems’ which was intended to encompass any algorithmic procedure for symbol manipulation. Using this notion, Post partially anticipated, in unpublished work, the results of Gödel, Church and Turing in the 1930s. This showed that many problems in logic and mathematics are algorithmically unsolvable. Post’s ideas influenced later research in logic, computer theory, formal language theory and other areas.

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

2020 ◽  
Vol 13 ◽  
Author(s):  
Maher Nabulsi ◽  
Nesreen Hamad ◽  
Sokyna Alqatawneh
Keyword(s):  
Digital Circuits ◽  
Truth Table ◽  
New Method ◽  
Alternative Methods ◽  
Discrete Mathematics ◽  
New Approach ◽  
Logical Proposition ◽  
The Given ◽  
First Time ◽  
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.

scholarly journals Truth-Graph Method: a Handy Method Different from that of Leśniewski’s

2015 ◽  
Vol 42 (1) ◽  
pp. 79-111
Author(s):  
Lei Ma
Keyword(s):  
Visual Representation ◽  
The Other ◽  
Truth Conditions ◽  
Truth Values ◽  
Graph Method ◽  
Convenient Tool ◽  
Other Hand ◽  
Truth Value ◽  
The One ◽  
Truth Tables

Abstract The paper presents a method of truth-graph by truth-tables. On the one hand, the truth-graph constituted by truth value coordinate and circumference displays a more visual representation of the different combinations of truth-values for the simple or complex propositions. Truth-graphs make sure that you don’t miss any of these combinations. On the other hand, they provide a more convenient tool to discern the validity of a complex proposition made up by simple compositions. The algorithm involving in setting up all the truth conditions is proposed to distinguish easily among tautologous, contradictory and consistent expressions. Furthermore, the paper discusses a certain connection between the truth graphs and the symbols for propositional connectives proposed by Stanisław Leśniewski.

On the structures inside truth-table degrees

2001 ◽  
Vol 66 (2) ◽  
pp. 731-770 ◽  
Author(s):  
Frank Stephan
Keyword(s):  
Finite Number ◽  
Open Problem ◽  
Affirmative Answer ◽  
Truth Table ◽  
Turing Degrees ◽  
Positive Degree

AbstractThe following theorems on the structure inside nonrecursive truth-table degrees are established: Dëgtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside every truth-table degree. The latter implies an affirmative answer to the following question of Jockusch: does every truth-table degree contain an infinite antichain of many-one degrees? Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmune-free truth-table degrees) which consist only of 2-subjective sets and therefore do not contain any objective set. Furthermore, a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enumerable semirecursive sets, one of coenumerable semirecursive sets and one of sets, which are neither enumerable nor coenumerable nor semirecursive. So Jockusch's result that there are at least three positive degrees inside a truth-table degree is optimal. The number of positive degrees inside a truth-table degree can also be some other odd integer as for example nineteen, but it is never an even finite number.

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