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).

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.

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.

operators and alternating sentences in arithmetic

1980 ◽  
Vol 45 (1) ◽  
pp. 144-154 ◽  
Author(s):  
Larry Manevitz ◽  
Jonathan Stavi
Keyword(s):  
Standard Model ◽  
Natural Number ◽  
Combinatorial Problem ◽  
The Other ◽  
Predicate Calculus ◽  
Completeness Proof ◽  
The Standard Model ◽  
Truth Value ◽  
The One ◽  
Definition Of

Determining the truth value of self-referential sentences is an interesting and often tricky problem. The Gödel sentence, asserting its own unprovability in P (Peano arithmetic), is clearly true in N(the standard model of P), and Löb showed that a sentence asserting its own provability in P is also true in N (see Smorynski [Sm, 4.1.1]). The problem is more difficult, and still unsolved, for sentences of the kind constructed by Kreisel [K1], which assert their own falsity in some model N* of P whose complete diagram is arithmetically defined. Such a sentence χ has the property that N ⊨ iff N* ⊭ χ (note that ¬χ has the same property).We show in §1 that the truth value in N of such a sentence χ, after a certain normalization that breaks the symmetry between it and its negation, is determined by the parity of a natural number, called the rank of N, for the particular construction of N* used. The rank is the number of times the construction can be iterated starting from N and is finite for all the usual constructions. We also show that modifications of, e.g., Henkin's construction (in his completeness proof of predicate calculus) allow arbitrary finite values for the rank of N. Thus, on the one hand the truth value of χ in N, for a given “nice” construction of N*, is independent of the particular (normalized) choice of χ, and we shall see that χ is unique up to (provable) equivalence in P. On the other hand, the truth value in question is sensitive to minor changes in the definition of N* and its determination seems to be largely a combinatorial problem.

Definability via enumerations

1989 ◽  
Vol 54 (2) ◽  
pp. 428-440 ◽  
Author(s):  
Ivan N. Soskov
Keyword(s):  
The Other ◽  
Partial Structure ◽  
Recursive Functions ◽  
Other Hand ◽  
Partial Recursive Functions ◽  
Priority Method ◽  
Basic Functions ◽  
Definition Of ◽  
Definable Functions ◽  
Finite Domains

The notion of ∀-recursiviness, introduced by Lacombe [1], is intended to describe the effectively definable functions and predicates in abstract structures with equality and denumerable domains. The fact that on every such structure ∀-recursiviness and search computability are equivalent is proved by Moschovakis in [2].The definition of search computability [3] does not require the presence of the equality among the basic predicates of the structure. There exist abstract structures where the equality is not search-computable and even not semicomputable. On the other hand, in some structures the equality is not an “effective” predicate. Consider, for example, a structure whose domain consists of all partial recursive functions.A notion of relative computability in abstract structures with denumerable domains, which we shall call here ∀-admissibility, was introduced by D. Skordev in 1977. The notion of ∀-admissibility is a generalization of Lacombe's ∀-recursiviness and does not require the presence of the equality among the basic predicates. In 1977 Skordev conjectured that, in every partial structure with denumerable domain, ∀-admissibility and search computability are equivalent.Since 1977 some attempts have been made to establish Skordev's conjecture. It is proved in [4] for structures with total basic functions and without basic predicates, and in [5] for structures with finite domains. The proofs in [4] and [5] make use of the priority method and are very complicated.

ON TRUTH-FUNCTIONALITY

2010 ◽  
Vol 3 (4) ◽  
pp. 628-632
Author(s):  
DANIEL J. HILL ◽  
STEPHEN K. McLEOD
Keyword(s):  
Alternative Definition ◽  
Functional Operator ◽  
Truth Values ◽  
Traditional Definition ◽  
Truth Value ◽  
Definition Of

Benjamin Schnieder has argued that several traditional definitions of truth-functionality fail to capture a central intuition informal characterizations of the notion often capture. The intuition is that the truth-value of a sentence that employs a truth-functional operator depends upon the truth-values of the sentences upon which the operator operates. Schnieder proposes an alternative definition of truth-functionality that is designed to accommodate this intuition. We argue that one traditional definition of ‘truth-functionality’ is immune from the counterexamples that Schnieder proposes and is preferable to Schnieder’s alternative.

Diagonals and -maximal sets

1994 ◽  
Vol 59 (1) ◽  
pp. 60-72 ◽  
Author(s):  
Eberhard Herrmann ◽  
Martin Kummer
Keyword(s):  
Important Class ◽  
Computable Numbering ◽  
Halting Problem ◽  
Recursive Functions ◽  
Elementary Lattice ◽  
Partial Recursive Functions ◽  
Definition Of ◽  
Simple Sets

In analogy to the definition of the halting problem K, an r.e. set A is called a diagonal iff there is a computable numbering ψ of the class of all partial recursive functions such that A = {i ∈ ω: ψi(i)↓} (in that case we say that A is the diagonal of ψ). This notion has been introduced in [10]. It captures all r.e. sets that can be constructed by diagonalization.It was shown that any nonrecursive r.e. T-degree contains a diagonal, that for any diagonal A there is an r.e. nonrecursive nondiagonal B ≤TA, and that there are r.e. degrees a such that any r.e. set from a is a diagonal.In §2 of the present paper we show that the property “A is a diagonal” is elementary lattice theoretic (e.l.t.). This result complements and generalizes previous results of Harrington and Lachlan, respectively. Harrington (see [16, XV. 1]) proved that the property of being the diagonal of some Gödelnumbering (i.e., of being creative) is e.l.t., and Lachlan [12] proved that the property of being a simple diagonal (i.e., of being simple and not hh-simple [10]) is e.l.t.In §§3 and 4 we study the position of diagonals and nondiagonals inside the lattice of r.e. sets. We concentrate on an important class of nondiagonals, generalizing the maximal and hemimaximal sets: the -maximal sets. Using the results from [6] we are able to classify the -maximal sets that can be obtained as halfs of splittings of hh-simple sets.

Understanding functional illiteracy from a policy, adult education, and cognition point of view: Towards a joint referent framework

2019 ◽  
Vol 30 (2) ◽  
pp. 109-122
Author(s):  
Aleksandar Bulajić ◽  
Miomir Despotović ◽  
Thomas Lachmann
Keyword(s):  
Adult Education ◽  
Operational Definition ◽  
Literacy Skills ◽  
Point Of View ◽  
Industrialized Countries ◽  
Functional Illiteracy ◽  
Scale Levels ◽  
Components Of Reading ◽  
The One ◽  
Definition Of

Abstract. The article discusses the emergence of a functional literacy construct and the rediscovery of illiteracy in industrialized countries during the second half of the 20th century. It offers a short explanation of how the construct evolved over time. In addition, it explores how functional (il)literacy is conceived differently by research discourses of cognitive and neural studies, on the one hand, and by prescriptive and normative international policy documents and adult education, on the other hand. Furthermore, it analyses how literacy skills surveys such as the Level One Study (leo.) or the PIAAC may help to bridge the gap between cognitive and more practical and educational approaches to literacy, the goal being to place the functional illiteracy (FI) construct within its existing scale levels. It also sheds more light on the way in which FI can be perceived in terms of different cognitive processes and underlying components of reading. By building on the previous work of other authors and previous definitions, the article brings together different views of FI and offers a perspective for a needed operational definition of the concept, which would be an appropriate reference point for future educational, political, and scientific utilization.

Preliminaries

2017 ◽  
Author(s):  
David J. Lobina
Keyword(s):  
Data Structures ◽  
Production Systems ◽  
Specific Data ◽  
Recursive Functions ◽  
Recursive Solution ◽  
Partial Recursive Functions ◽  
Recursive Structures ◽  
Recursive Data Structures ◽  
Recursive Data ◽  
The Relationship

Recursion, or the capacity of ‘self-reference’, has played a central role within mathematical approaches to understanding the nature of computation, from the general recursive functions of Alonzo Church to the partial recursive functions of Stephen C. Kleene and the production systems of Emil Post. Recursion has also played a significant role in the analysis and running of certain computational processes within computer science (viz., those with self-calls and deferred operations). Yet the relationship between the mathematical and computer versions of recursion is subtle and intricate. A recursively specified algorithm, for example, may well proceed iteratively if time and space constraints permit; but the nature of specific data structures—viz., recursive data structures—will also return a recursive solution as the most optimal process. In other words, the correspondence between recursive structures and recursive processes is not automatic; it needs to be demonstrated on a case-by-case basis.

Democracy and Political Culture

2019 ◽  
Author(s):  
Ross McKibbin
Keyword(s):  
Political Culture ◽  
Case Studies ◽  
Social System ◽  
Political System ◽  
Democratic Politics ◽  
Social Structures ◽  
The Political ◽  
The Subject ◽  
The One ◽  
Definition Of

This book is an examination of Britain as a democratic society; what it means to describe it as such; and how we can attempt such an examination. The book does this via a number of ‘case-studies’ which approach the subject in different ways: J.M. Keynes and his analysis of British social structures; the political career of Harold Nicolson and his understanding of democratic politics; the novels of A.J. Cronin, especially The Citadel, and what they tell us about the definition of democracy in the interwar years. The book also investigates the evolution of the British party political system until the present day and attempts to suggest why it has become so apparently unstable. There are also two chapters on sport as representative of the British social system as a whole as well as the ways in which the British influenced the sporting systems of other countries. The book has a marked comparative theme, including one chapter which compares British and Australian political cultures and which shows British democracy in a somewhat different light from the one usually shone on it. The concluding chapter brings together the overall argument.

An LQ Approach to Active Control of Vibrations in Helicopters

1996 ◽  
Vol 118 (3) ◽  
pp. 482-488 ◽  
Author(s):  
Sergio Bittanti ◽  
Fabrizio Lorito ◽  
Silvia Strada
Keyword(s):  
Optimal Control ◽  
Active Control ◽  
Linear Quadratic ◽  
Control Effort ◽  
Vibration Effect ◽  
Suitable Definition ◽  
Lq Optimal Control ◽  
The One ◽  
Definition Of ◽  
And Control

In this paper, Linear Quadratic (LQ) optimal control concepts are applied for the active control of vibrations in helicopters. The study is based on an identified dynamic model of the rotor. The vibration effect is captured by suitably augmenting the state vector of the rotor model. Then, Kalman filtering concepts can be used to obtain a real-time estimate of the vibration, which is then fed back to form a suitable compensation signal. This design rationale is derived here starting from a rigorous problem position in an optimal control context. Among other things, this calls for a suitable definition of the performance index, of nonstandard type. The application of these ideas to a test helicopter, by means of computer simulations, shows good performances both in terms of disturbance rejection effectiveness and control effort limitation. The performance of the obtained controller is compared with the one achievable by the so called Higher Harmonic Control (HHC) approach, well known within the helicopter community.

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