Inductive definitions and proofs

2018 ◽  
Author(s):  
Vann McGee
Keyword(s):  
Negative Information ◽  
Predicate Calculus ◽  
Inductive Definition ◽  
Positive Information ◽  
Inductive Definitions ◽  
Closure Conditions ◽  
Definition Of ◽  
Special Case

An inductive definition of a predicate R characterizes the Rs as the smallest class which satisfies a basis clause of the form (β(x)→Rx), telling us that certain things satisfy R, together with one or more closure clauses of the form (Φ(x,R)→Rx), which tell us that, if certain other things satisfy R, x satisfies R as well. ’Smallest’ here means that the class of Rs is included in every other class which satisfies the basis and closure clauses. Inductive definitions are useful because of inductive proofs. To show that every R has property P, show that the class of Rs that have P satisfies the basis and closure clauses. The closure clauses tell us that if certain things satisfy R, x satisfies R as well. Thus satisfaction of the condition Φ(x,R) should be ensured by positive information to the effect that certain things satisfy R, and not also require negative information that certain things fail to satisfy R. In other words, the condition Φ(x,R) should be monotone, so that, if R ⊆S and Φ(x,R), then Φ(x,S); otherwise, we would have no assurance of the existence of a smallest class satisfying the basis and closure conditions. While inductive definitions can take many forms, they have been studied most usefully in the special case in which the basis and closure clauses are formulated within the predicate calculus. Initiated by Yiannis Moschovakis, the study of such definitions has yielded an especially rich and elegant theory.

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

Monotone inductive definitions over the continuum

1976 ◽  
Vol 41 (1) ◽  
pp. 188-198 ◽  
Author(s):  
Douglas Cenzer
Keyword(s):  
Mathematical Logic ◽  
Recursion Theory ◽  
Axiom System ◽  
Baire Space ◽  
Natural Numbers ◽  
Inductive Definition ◽  
Inductive Definitions ◽  
Definition Of ◽  
The Continuum

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

Infinitary formulas preserved under unions of models

1972 ◽  
Vol 37 (3) ◽  
pp. 449-465 ◽  
Author(s):  
Bienvenido F. Nebres
Keyword(s):  
Predicate Calculus ◽  
Compactness Theorem ◽  
Mathematical Research ◽  
First Order ◽  
Syntactic Form ◽  
Preservation Theorems ◽  
First Results ◽  
Closure Conditions ◽  
Special Case ◽  
Order Predicate Calculus

So-called “preservation theorems” relate the (possible) syntactic form of the axioms of a theory to certain closure conditions on its class of models. Such results are well known for the first-order predicate calculus, Lω, ω, and there are various expositions; e.g., Keisler [14], [15]. For the language , the first results were the theorems of Lopez-Escobar on sentences preserved under homomorphic images and of Malitz on formulas preserved under substructures. More recently, Feferman added a result on formulas preserved under (or persistent for) ∈-extensions. Some of these theorems will be considered in subsequent sections. A more thorough treatment may be found in Makkai [17]. The main new preservation result obtained here characterizes the sentences preserved under ω-unions. This notion and the statement of the theorem will be explained shortly.It is a familiar experience in mathematical research that concepts which are equivalent in a special case diverge in general. In the case at hand, one must expect to consider different possible statements for , which generalize a known result for Lω, ω. Moreover, diverse proofs may yield the same result in the special case, not all of which can be extended to the general case. Again, since the compactness theorem fails for , one cannot expect to extend the arguments from Lω, ω which use this in an essential way.

An intuitionistic proof of Tychonoff's theorem

1992 ◽  
Vol 57 (1) ◽  
pp. 28-32 ◽  
Author(s):  
Thierry Coquand
Keyword(s):  
Set Theory ◽  
Free Space ◽  
Direct Proof ◽  
Axiom Of Choice ◽  
Inductive Definition ◽  
Compact Spaces ◽  
Inductive Definitions ◽  
Constructive Set Theory ◽  
Definition Of ◽  
The Axiom Of Choice

Tychonoff's theorem states that a product of compact spaces is compact. In [3], P. Johnstone presents a proof of Tychonoff's theorem in a “localic” framework. The surprise is that the point-free formulation of Tychonoff's theorem is provable without the axiom of choice, whereas in the usual formulation it is equivalent to the axiom of choice (see Kelley [5]).The proof given in [3], however, is classical and seems to use the replacement axiom of Zermelo-Fraenkel. The aim of this paper is to present what we believe to be a more direct proof, which is intuitionistic and can be proved using as primitive only the notion of inductive definition, as it is for instance presented in Martin-Löf [6]. One main point of the paper is to show that the theory of locales can be developed rather naturally in the framework of inductive definitions. We think that our arguments can be presented in the constructive set theory of Aczel [2].The paper is organized as follows. In §1 we show the argument in the case of a product of two spaces. This proof has a direct generalisation to the case of a product over a set with a decidable equality.§1. Product of two spaces. We first recall a possible definition of a point-free space (see Johnstone [3] or Vickers [10]). It is a poset (X, ≤), together with a meet operation ab, written multiplicatively, and, for each a ∈ X, a set Cov(a) of subsets of {x ∈ X ∣ x ≤ a}. We ask that if M ∈ Cov(a) and b ≤ a, then {bs ∈ s ∈ M} ∈ Cov(b). This property of Cov will be called the axiom of covering. The elements of Cov(u) are called basic covers of u ∈ X.

Does Performing Other Audit Tasks Affect Going-Concern Judgments?

1999 ◽  
Vol 74 (4) ◽  
pp. 493-508 ◽  
Author(s):  
Stephen E. Rau ◽  
Donald V. Moser
Keyword(s):  
Information Processing ◽  
Review Process ◽  
Negative Information ◽  
The Other ◽  
Positive Information ◽  
Going Concern ◽  
Staff Members

This study examines whether personally performing other audit tasks can bias supervising seniors' going-concern judgments. During an audit, the senior performs some audit tasks him/herself and delegates other tasks to staff members. When personally performing an audit task, the senior would focus on the evidence related to that task. We predict that such evidence will have greater influence on the senior's subsequent going-concern judgment. The results of our experiment are consistent with our predictions. When provided with an identical set of information, seniors who performed another audit task for which the underlying facts of the case reflected positively (negatively) on the company's viability, subsequently made going-concern judgments that were relatively more positive (negative). Our results also demonstrate that the well-documented tendency of auditors to attend more to negative information does not always dominate auditors' information processing. Subjects who performed the task for which the underlying facts reflected positively on the company's viability directed their attention to such positive information and, consequently, both their memory and judgments were more positive than those of subjects in the other conditions. Recent findings indicating that biases in seniors' going-concern judgments may not be fully offset in the review process are discussed along with other potential implications of our results.

scholarly journals Bad World: The Negativity Bias in International Politics

2019 ◽  
Vol 43 (3) ◽  
pp. 96-140 ◽  
Author(s):  
Dominic D.P. Johnson ◽  
Dominic Tierney
Keyword(s):  
International Relations ◽  
Fundamental Principle ◽  
Human Mind ◽  
Negative Information ◽  
Theory And Practice ◽  
Negativity Bias ◽  
Positive Information ◽  
Psychological Phenomena ◽  
Wide Range ◽  
New Research

A major puzzle in international relations is why states privilege negative over positive information. States tend to inflate threats, exhibit loss aversion, and learn more from failures than from successes. Rationalist accounts fail to explain this phenomenon, because systematically overweighting bad over good may in fact undermine state interests. New research in psychology, however, offers an explanation. The “negativity bias” has emerged as a fundamental principle of the human mind, in which people's response to positive and negative information is asymmetric. Negative factors have greater effects than positive factors across a wide range of psychological phenomena, including cognition, motivation, emotion, information processing, decision-making, learning, and memory. Put simply, bad is stronger than good. Scholars have long pointed to the role of positive biases, such as overconfidence, in causing war, but negative biases are actually more pervasive and may represent a core explanation for patterns of conflict. Positive and negative dispositions apply in different contexts. People privilege negative information about the external environment and other actors, but positive information about themselves. The coexistence of biases can increase the potential for conflict. Decisionmakers simultaneously exaggerate the severity of threats and exhibit overconfidence about their capacity to deal with them. Overall, the negativity bias is a potent force in human judgment and decisionmaking, with important implications for international relations theory and practice.

μ-definable sets of integers

1993 ◽  
Vol 58 (1) ◽  
pp. 291-313 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Fixed Point ◽  
Free Variable ◽  
Order Logic ◽  
First Order Logic ◽  
Inductive Definition ◽  
Natural Class ◽  
First Order ◽  
Inductive Definitions ◽  
Definable Sets ◽  
The Universe

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

scholarly journals The Structure of n Harmonic Points and Generalization of Desargues’ Theorems

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1018
Author(s):  
Xhevdet Thaqi ◽  
Ekrem Aljimi
Keyword(s):  
Fourth Harmonic ◽  
New Findings ◽  
Rank 2 ◽  
Definition Of ◽  
Special Case

: In this paper, we consider the relation of more than four harmonic points in a line. For this purpose, starting from the dependence of the harmonic points, Desargues’ theorems, and perspectivity, we note that it is necessary to conduct a generalization of the Desargues’ theorems for projective complete n-points, which are used to implement the definition of the generalization of harmonic points. We present new findings regarding the uniquely determined and constructed sets of H-points and their structure. The well-known fourth harmonic points represent the special case (n = 4) of the sets of H-points of rank 2, which is indicated by P42.

A proof of the cut-elimination theorem in simple type theory

1973 ◽  
Vol 38 (2) ◽  
pp. 215-226
Author(s):  
Satoko Titani
Keyword(s):  
Boolean Algebra ◽  
Type Theory ◽  
Arbitrary Number ◽  
Formal System ◽  
Cut Elimination ◽  
Simple Type ◽  
Inductive Definition ◽  
Simple Type Theory ◽  
Maximal Ideals ◽  
Definition Of

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

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