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.

The cylindric algebras of three-valued logic

1998 ◽  
Vol 63 (4) ◽  
pp. 1201-1217
Author(s):  
Norman Feldman
Keyword(s):  
Effective Procedure ◽  
Cylindric Algebras ◽  
Truth Values ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Truth Value ◽  
The One ◽  
Definition Of

In this paper we consider the three-valued logic used by Kleene [6] in the theory of partial recursive functions. This logic has three truth values: true (T), false (F), and undefined (U). One interpretation of U is as follows: Suppose we have two partially recursive predicates P(x) and Q(x) and we want to know the truth value of P(x) ∧ Q(x) for a particular x0. If x0 is in the domain of definition of both P and Q, then P(x0) ∧ Q(x0) is true if both P(x0) and Q(x0) are true, and false otherwise. But what if x0 is not in the domain of definition of P, but is in the domain of definition of Q? There are several choices, but the one chosen by Kleene is that if Q(X0) is false, then P(x0) ∧ Q(x0) is also false and if Q(X0) is true, then P(x0) ∧ Q(X0) is undefined.What arises is the question about knowledge of whether or not x0 is in the domain of definition of P. Is there an effective procedure to determine this? If not, then we can interpret U as being unknown. If there is an effective procedure, then our decision for the truth value for P(x) ∧ Q(x) is based on the knowledge that is not in the domain of definition of P. In this case, U can be interpreted as undefined. In either case, we base our truth value of P(x) ∧ Q(x) on the truth value of Q(X0).

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 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.

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).

scholarly journals On Equivalent Truth-Tables of Many-Valued Logics

1954 ◽  
Vol 10 (2) ◽  
pp. 56-61 ◽  
Author(s):  
J. Kalicki
Keyword(s):  
Nineteenth Century ◽  
Truth Table ◽  
Table Method ◽  
Image Position ◽  
Truth Tables

Many-valued or non-Aristotelian calculi of propositions (logics) were originally introduced by generalisation of the truth-table method. It was known by the end of the nineteenth century that ordinary “binary” formulae of the calculus of propositions, such ascould be verified directly by means of the truth-table:although the terminology and symbolism used were different.

Structure Information From Electron Diffraction Intensities

1990 ◽  
Vol 48 (2) ◽  
pp. 516-517
Author(s):  
J. Gjønnes ◽  
N. Bøe ◽  
K. Gjønnes
Keyword(s):  
Computational Effort ◽  
Unit Cell ◽  
Small Unit ◽  
Structure Information ◽  
Dispersion Surface ◽  
Integrated Intensity ◽  
The One ◽  
Image Position ◽  
Convergent Beam ◽  
Beam Patterns

Structure information of high precision can be extracted from intentsity details in convergent beam patterns like the one reproduced in Fig 1. From low order reflections for small unit cell crystals,bonding charges, ionicities and atomic parameters can be derived, (Zuo, Spence and O’Keefe, 1988; Zuo, Spence and Høier 1989; Gjønnes, Matsuhata and Taftø, 1989) , but extension to larger unit cell ma seem difficult. The disks must then be reduced in order to avoid overlap calculations will become more complex and intensity features often less distinct Several avenues may be then explored: increased computational effort in order to handle the necessary many-parameter dynamical calculations; use of zone axis intensities at symmetry positions within the CBED disks, as in Figure 2 measurement of integrated intensity across K-line segments. In the last case measurable quantities which are well defined also from a theoretical viewpoint can be related to a two-beam like expression for the intensity profile:With as an effective Fourier potential equated to a gap at the dispersion surface, this intensity can be integrated across the line, with kinematical and dynamical limits proportional to and at low and high thickness respctively (Blackman, 1939).

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

On functional Nörlund methods

1970 ◽  
Vol 67 (1) ◽  
pp. 47-60 ◽  
Author(s):  
B. Choudhary
Keyword(s):  
The Other ◽  
Integral Transformations ◽  
Other Hand ◽  
Nörlund Means ◽  
The One ◽  
Image Position

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a functionregular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

scholarly journals Tensile Strength of NaCl Ice

1962 ◽  
Vol 4 (31) ◽  
pp. 25-52 ◽  
Author(s):  
W. F. Weeks
Keyword(s):  
Tensile Strength ◽  
Sea Ice ◽  
Fresh Water ◽  
Eutectic Point ◽  
Water Ice ◽  
Average Value ◽  
Freshwater Ice ◽  
The One ◽  
Image Position ◽  
Bodies Of Water

AbstractTo resolve some of the factors causing strength variation in natural sea ice, fresh water and five different NaCl–H2O solutions were frozen in a tank designed to simulate the one-dimensional cooling of natural bodies of water. The resulting ice was structurally similar to lake and sea ice. The salinity of the salt ice varied from 1‰ to 22‰. Tables of brine volumes and densities were computed for these salinities in the temperature range 0° to −35° C. The ring-tensile strength σ of fresh-water ice was found to be essentially temperature independent from −10° to −30°C., with an average value of 29.6±8.5 kg./cm.2at −10° C. The strength of salt ice at temperatures above the eutectic point (–21.2° C.) significantly decreases with brine volumev;. The σ–axis intercept of this line is comparable to the a values determined for fresh ice indicating that there is little, if any, difference in stress concentration between sea and lake ice as a result of the presence of brine pockets. The strength of ice containing NaCl.2H2O is slightly less than the strength of freshwater ice and is independent of the volume of solid salt and the ice temperature. No evidence was found for the existence of either phase or geometric hysteresis in NaCl ice. The strength of ice at sub-eutectic temperatures, however, is decreased appreciably if the ice has been subjected to temperatures above the eutectic point; this is the result of the redistribution of brine during the warm-temperature period. Short-term cooling produces an appreciable (20 per cent) decrease in strength, in fresh-water and NaCl.2H2O ice. The present results are compared with tests on natural sea ice and it is suggested that the strength of freshwater ice is a limit which is approached but not exceeded by cold sea ice and that the reinforcement of brine pockets by Na2SO4.10H2O is either lacking or much less than previously assumed.

Proof that there are infinitely many modalities in Lewis's system S2

1940 ◽  
Vol 5 (3) ◽  
pp. 110-112 ◽  
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Finite Number ◽  
Infinite Matrix ◽  
Image Position

In this note I show, by means of an infinite matrix M, that the number of irreducible modalities in Lewis's system S2 is infinite. The result is of some interest in view of the fact that Parry has recently shown that there are but a finite number of modalities in the system S2 (which is the next stronger system than S2 discussed by Lewis).I begin by introducing a function θ which is defined over the class of sets of signed integers, and which assumes sets of signed integers as values. If A is any set of signed integers, then θ(A) is the set of all signed integers whose immediate predecessors are in A; i.e., , so that n ϵ θ(A) is true if and only if n − 1 ϵ A is true.Thus, for example, θ({−10, −1, 0, 3, 14}) = {−9, 0, 1, 4, 15}. In particular we notice that θ(V) = V and θ(Λ) = Λ, where V is the set of all signed integers, and Λ is the empty set of signed integers.It is clear that, if A and B are sets of signed integers, then θ(A+B) = θ(A)+θ(B).It is also easily proved that, for any set A of signed integers we have . For, if n is any signed integer, then

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