Showing papers in "Studies in logic and the foundations of mathematics in 1989"

Geometry of Interaction 1: Interpretation of System F

Abstract: Publisher Summary This chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism (that leads to static modification) and subjectivism (that leads to syntactical abuses, in other terms, bureaucracy). The new approach initiated in this chapter rests on the use of a specific C*-algebra Λ* that has the distinguished property of bearing a (non associative) inner tensor product. The chapter describes that a representative class of algorithms can be modelized by means of standard mathematics.

Trends in Logic

Abstract: Publisher Summary This chapter describes the dynamics of model theory and some threats of set theory It is often risky to predict movements in logic It is noted that something remains of forcing and reduced products in categorical model theory, which provides universe and semantics more relevant than the set-theoretic ones A clear tendency is to focus on definability rather than decidability or structure theory Most of the decidability results identify definable sets, and relations, specifically equivalence relations A major contemporary theme is the topological structure of definable sets Kreisel had stressed the gigantic difference in importance between algebraic topology and set-theoretic topology The latter is ubiquitous in routine arguments and formulations, but the former is effective in advancing mathematical understanding Applied model theory is often concerned with number theory or analytic function theory

Semantic Parallels in Natural Language and Computation

Abstract: Publisher Summary This chapter describes two major themes: (1) techniques for local strengthening of logical inference via minimization of models and (2) the more general dynamics of progressive handling of information in interpretation and argument. The chapter provides a coherent pattern behind some recent developments in these areas and discusses their value as affecting logic in general. The chapter also provides a mathematical analysis of the minimization operator on classes of models while also investigating several special systems in which minimal models play a central role. The chapter develops an analogy with earlier work in the philosophy of science on so-called “Ramsey eliminability” of theoretical terms in scientific theories. A technical connection is found between the general inferential properties of circumscription and more traditional conditional logic. It considers possible reductions of circumscriptive inference to standard first-order logic, establishing a high complexity for the question just when this is possible. The chapter reviews a number of results on dynamical semantics and several reductions of proposed dynamic systems to standard first-order logic. The latter system provides a promising tool for investigating dynamic modes of handling propositions.

Degrees of Functions with no Fixed Points

Abstract: Publisher Summary The fixed-point form of Kleene's recursion theorem asserts that for every recursive function f : ω→ ω, there exists an e ∈ ω with W e = W fe . This chapter discusses the results about the Turing degrees of functions that are fixed-point free (FPF) in the sense that We≠ W fe for all e ∈ ω. This is easily seen to coincide with the class of degrees of all functions, g, which are diagonally nonrecursive (DNR). The chapter reviews the studies on the degrees of functions whose nonrecursiveness is ensured by the diagonal method. The degrees of the recursively bounded DNR functions coincide with the degrees of the effectively immune, nonhyperimmune sets.

Comparative Logics and Abelian l-Groups

Abstract: Publisher Summary It is noted that comparative logic requires, that the propositions should in any case be true or false and this clearly forced the totality of the order in the group-theoretical framework. Going beyond the use of totally ordered abelian groups, so to encompass the lattice-ordered and, more generally, the partially ordered groups has proved very successful. Following this, the setting up of a general frame becomes possible, in which, beside the old comparative logic, other interesting logics also find a quite satisfactory assessment. This chapter focuses at the presentation of this frame however only for the propositional case. The chapter describes basic syntactical notions and facts, basic semantical notions and facts, some other stronger logics, and comparative logic.

On the use of Diagonally Nonrecursive Functions

Abstract: Publisher Summary This chapter describes the functions that diagonalize all partial recursive functions possess important properties and the study of these functions gives a new view on some results in recursion theory as well as new methods to prove them. The chapter explains a function f , which is called a “diagonally nonrecursive function” (a DNR function) and the function g is called a fixed-point free function (an FPF function). A degree containing an FPF function is called an “FPF degree.” DNR functions differ from FPF functions, FPF degrees coincide with degrees of DNR functions. It is more convenient to work with the class of DNR functions instead of the class of FPF functions. A degree containing a 0-1 valued DNR function is called a “PA degree.” The chapter reviews some known facts about FPF degrees and PA degrees. From a global point of view, the class of FPF degrees has (Lebesgue) measure 1, while the class of PA degrees has measure 0.

On the Existence of end Extensions of Models of Bounded Induction

Abstract: Publisher Summary This chapter focuses on the existence of end extensions of models of bounded induction. From the results of Friedman, it follows that any nonstandard countable model of ∑ n induction (I∑ n ) for n > 1 is isomorphic to a proper initial segment of itself, and hence, has a proper end extension to a model of I∑ n. This was later extended to the case n = 1. For n = 0, this result is false because in proposition 1, if M and K are models of IΔ 0 , and K is a proper end extension of M (M ⊂ K), then M must also satisfy ∑ 1 collection (B∑ 1 ). However, there are models of IΔ 0 that do not satisfy B∑ 1 and, hence, do not have proper end extensions to models of IΔ 0 . This then raises the question of finding necessary and sufficient conditions on a countable model M of IΔ 0 for M to have a proper end extension to a model of IΔ 0.

Intuitionistic formal spaces and their neighbourhood

Abstract: Publisher Summary The chapter provides simple and complete propositional semantics for Girard's linear logic and for usual intuitionistic logic Pretopology is obtained from formal topology by isolating and disregarding those conditions which correspond to the structural rules of weakening and contraction Pretopologies are the weakest possible structures in which an operation corresponding to implication is definable They generally provide a sound and complete semantics for a ground logic, commonly called minimal linear The completeness of intuitionistic logics, with or without either structural rule, is considered an easy corollary It is noted that by restricting to pretopologies in which every open, or saturated, subset is regular, one obtains exactly Girard's phase semantics, and hence completeness for the classical system, and also an extension to the linear case of the usual double negation interpretation It is also possible to treat exponentials, as an essential feature of Girard's linear logic in the pretopology framework

The C*-Algebras of Three-Valued Logic

Abstract: Summary We show that Grothendieck's functor K o maps 3-subhomogeneous AF C * -algebras with Hausdorff structure space one-one onto countable Lindenbaum algebras of 3-valued logic. Whereas in the interpretation of Birkhoff and von Neumann propositions arc identified with projections and form an uncountable nondistributive lattice, in our interpretation propositions are unitary equivalence classes of projections, and form a countable MV 3 algebra of Chang and Grigolia, alias a 3-valued Lukasiewicz algebra in the sense of Moisil and Monteiro, that is, a Kleene algebra equipped with an operation ∇ obeying x * ∧ ∇ x = x * ∧ x and x * ∧ ∇ x = 1.

On the Nature of a Social Order

Abstract: Publisher Summary This chapter focuses on the nature of a social order. The chapter discusses three categories: control, influence, and normative regulation. Control is a matter of what an agent does in relation to another agent; influence a matter of what an agent can do in relation to another; and normative regulation a matter of what an agent shall or may do in relation to another. Control and influence may be “combined”. Control may be defined in relation to an agent with respect to his influence positions and, conversely, his influence in relation to an agent with respect to his control positions. Studies shows that work in these areas are important for the understanding of the foundations of social science. There is a close structural affinity between act positions and normative positions.

The Justification of Negation as Failure

Abstract: Publisher Summary Prolog is a logic programming language, which is used to answer queries on the basis of information provided by the programmer. For the most part, the logic employed by Prolog is standard. But, it uses a highly unorthodox rule for establishing negative facts. This rule, the rule of negation as failure, allows denying a statement on the grounds that a certain attempt to prove it has failed. The rule is not classically valid and the question arises as to how it is to be justified. There are three different kinds of justification that have been proposed in the literature. The first is to re-interpret negation to mean something like unprovability. The second is to assume that the program is complete with respect to truths; all truths are derivable. The third is to suppose that the program is complete with respect to conditions; all sufficient conditions for the application of the predicates have been specified.

Probability in Dynamical Systems

Abstract: Publisher Summary Probability theory is applied mathematics. Probabilities are numbers between 0 and 1 that are additive in a special way. The probability that one of two alternative events occurs can be computed as a sum of the probabilities of the alternatives, minus the probability that the alternatives occur simultaneously. Interpretations of probability are broadly divided into epistemic and objective types. In the first type, probabilities are degrees of belief. The most common objectivist view, on the other hand, says that probabilities are the same as limits of relative frequencies. They are estimated from data according to a well-established statistical methodology. Time averages are some kind of continuous counterparts to limits of relative frequencies. By discretizing the phase space and by considering the occurrence of events at discrete intervals of time, one gets directly limits of relative frequencies from limits of time averages. The discretization of a stationary or ergodic dynamical system does not merely lead into limits of relative frequencies.

The Logical Ideas of N.A. Vasiliev and Modern Logic

Abstract: Publisher Summary At present, the logical ideas of N.A. Vasiliev are attracting the attention of many logicians. He is often considered the forerunner of many-valued logics, intuitionistic logic, and para-consistent logic. He was one of the first to proclaim and construct nonclassical, non-Aristotelean logics. The type of nonclassical logics proposed by Vasiliev is original, not coinciding with many-valued, intuitionistic, or para-consistent logics. Vasiliev's basic idea is the distinction between two levels in logic. The first or external level is connected with epistemological commitments. It is the logic of truth and falsity. Vasiliev calls it the metalogic. According to him, metalogic does not vary, it is absolute. The second level in logic depends on ontological (empirical) commitments in relation to cognizable objects. This part of logic can be varied.

Problems of Admissibility and Substitution, Logical Equations and Restricted Theories of free Algebras

Abstract: Publisher Summary The necessity of simplifying derivations in formal systems has led to the study of the class of all rules of inference such that the use of these rules in derivations does not change the set of provable formulas. This class has been called “the class of admissible rules of inference.” Investigations of the class of admissible rules have, for the most part, dealt with the intuitionistic propositional calculus H of Heyting. The substitution problem for propositional logics λ consists in the recognition, given an arbitrary formula A(xi, pj), whether there exist formulas Bi such that A(Bi, pj) ∈ A. The chapter describes the solution of the above stated problems. The chapter adopts an algebraic approach, using properties of free algebra. A solution is obtained for H by a reduction to the analogous problems for the systems S4 and Grz.

Non-Monotonic Reasoning by Axiomatic Extensions

Abstract: Publisher Summary Non-monotonic reasoning is the modern name for a variety of scientific activities that are characterized by the idea that the traditional deductive approach to inference systems is too narrow for many applications and that new formalisms are required that make arrangements for default reasoning, common sense reasoning, and autoepistemic reasoning. The recent interest in this field is caused by questions in artificial intelligence (AI) and computer science, which has led to a series of ad hoc methods and isolated case studies. This chapter reviews connections between the sets of formulae and individual formulae. The chapter discusses the forms of non-monotonic reasoning that are induced by default operators. In addition, closed-world assumption (CWA) is also discussed. Advantages of the CWA are its clear methodological conception and its efficiency for elementary database. Disadvantages are the restricted range of applicability and its complicated proof procedure.

Categoricity and Permutation Groups

Abstract: Publisher Summary This chapter focuses on the interaction of model theory and permutation groups. The chapter explains the model theory versus the group theory. The topology on automorphism group and recovering the structure from the topological group are discussed in the chapter. The chapter provides many theorems and examples to better understand these theories.

Rationality and Social Norms

Abstract: Publisher Summary This chapter discusses rationality and social norms. Rationality is defined as a relation between action, belief, desire, and evidence. An observed action is rational if it is the best means to realize the agent's desire, given his beliefs about relevant factual matter. The beliefs of the agent are themselves subject to a rationality constraint; they must be well grounded in the evidence available to the agent. The amount of evidence must also be scrutinized from the point of view of rationality or optimality. The canons of rationality tell people what to do if they want to achieve a given end. Social norms tell people what to do or not to do, either unconditionally or conditionally upon other people's behavior. Rationality is consequentialist; it is concerned with outcomes. Social norms are nonconsequentialist; they are concerned directly with actions, for their own sake.

Logic and Pragmatic Truth

Abstract: Publisher Summary This chapter reviews the field of pragmatic truth. There are three main conceptions of pragmatic truth, to wit: those of Peirce, of James, and of Dewey. This chapter focuses on Peirce's stance. A hypothesis is pragmatically true when it does not have consequences that contradict a primary statement—the better the hypothesis, the more it predicts. A proposition is pragmatically true when things happen as if it were true. For some contingent propositions, basic or decidable, truth and pragmatic truth do coincide. In addition, a basic statement must be such that its truth or falsehood can be settled. There are numerous situations, in the field of the empirical sciences, in which the concept of pragmatic truth can find applications. A statement is quasi-true or quasi-false only in relation to a given domain of knowledge and within fixed limits of applicability.

Type Theory and Explicit Mathematics

Abstract: Publisher Summary This chapter explains explicit mathematics as a framework for a unifying approach to the kind of type theories and lambda calculi that are used in present day computer science. The chapter describes some general considerations about monomorphic and polymorphic type structures and presents the basic ideas and definitions of this approach. The chapter also explains the conceptual interplay between type theories and explicit mathematics. In computer science, types and typed languages are useful to structure the data, prevent forbidden operations, and support the correctness of programs.

Unidimensional Theories: An introduction to geometric stability theory

Abstract: Publisher Summary This chapter introduces the stability theory from the geometric point of view. The unidimensional theories are those theories, which involve only one geometry. Most of the recent achievements in this area are reflected in the chapter. The best understood corner and the basis for more general conjectures is the subclass of totally categorical structures.

The rediscovery of time

Abstract: Publisher Summary This chapter discusses the role of probability and irreversibility in the conception of the universe. Different approaches to this question are well exemplified by the three excerpts from Einstein (1916), Kuznetzov (1987), Lucretius (∼ −60 BC). The analogy between Einstein and Lucretius is that the precise time of elementary processes is determined by chance. The basic problem is the conflictual situation between the static description proposed by classical physics, based on deterministic and time-reversible laws, and the world as it is known now, which includes probability and irreversibility as basic elements. According to the standard model, the entropy of the universe would remain constant, while its temperature decreases with the adiabatic expansion of the universe. the classical view expresses a dualistic structure: the phenomenological level corresponds indeed to irreversible and stochastic laws, while at the fundamental level, classical or quantum, there are time-reversible, deterministic laws.

Automorphisms of the Lattice of Recursively Enumerable Sets and Hyperhypersimple Sets

Abstract: Publisher Summary This chapter discusses the automorphisms of the lattice of recursively enumerable (r.e.) sets and hyperhypersimple (hh-simple) sets. While Maass proved a sufficient criterion for hh-simple sets to be automorphic, in Herrman, those properties of these sets has been analyzed; and it can be concluded when such sets are not automorphic even if their r.e. superset structures are isomorphic. The Lachlan's construction of hh-simple sets is universal from the point of view of their lattice position. The automorphism analysis of the hh-simple sets is an extensive topic for itself. There are still many open problems and questions. The main problem is to find a necessary and sufficient condition for two hh-simple sets to be automorphic. The isomorphism type of the family P*I(A) for the hh-simple set A could be such a condition.

The Descriptive Set Theory of σ - Ideals of Compact Sets

Abstract: We present here a survey of the definability theory of σ-ideals of closed sets in compact metric spaces and its structural implications We also include proofs of some recent results in this area

Some Aspects of Impredicativity: Notes on Weyl's philosophy of Mathematics and on todays Type Theory

Abstract: Publisher Summary This chapter describes some aspects of impredicativity. The chapter explains logic in mathematics and in computer science, objectivity and independence of formalism, predicative and nonpredicative, and the rock and the sand. Logic is viewed in computer science and in mathematics. Mathematical logic provides a foundation and a justification for all or parts of mathematics as an established discipline. Hilbert's strong stand towards the independence of mathematics is fascinating and clearly summarizes the basic perspective of modern mathematics. Predicative classification cannot be disordered by the introduction of new elements and the nonpredicative classifications in which the introduction of new elements necessitates constant modification.

How Natural is Natural Language

Abstract: Publisher Summary This chapter examines whether human language is natural or conventional. According to the modular view of mind, the mind is not one integrated whole, but a structure composed of various more or less autonomous components. Almost all systems of a certain degree of complexity are modular in this sense. Likewise, the language faculty is not a unitary system, but something composed of relatively independent subsystems. One of the most fascinating aspects of our language system is that it has the property of discrete infinity. Other systems of animal communication of infinite range lack this property of discrete infinity. The capacity to handle discrete infinities in connection with conceptual content is unique to humans. The chapter illustrates that the link between the computational structures found in human language and the conceptual systems is not intrinsic, but accidental. This means that the crucial link that makes human language so powerful is not a matter of biological necessity, but of human culture.

An Introduction to Extenders and Core Models for Extender Sequences

Abstract: This article surveys the theory of core models for non-overlapping coherent sequence of extenders. These models of set theory arise canonically when one wants to construct inner models for strong cardinals. Strong cardinals are defined in terms of elementary embeddings of V, and we define extenders as a way of coding elementary embeddings. The natural inner model for a strong cardinal is of the form L[E].

Solving Equations in Lambda-Calculus

Abstract: Part 1 reviews principal solvable and unsolvable equations and equation systems encountered in λ-calculus and Combinatory Logic Particular emphasis is given to the solvability of discriminability, separability and X-separability problems All these problems are special cases of (and are improved by) the predicate “F is X-weakly separable”, whose decidability is proved in part 2 The more general X-weak separability problem still remains open

Evolutionary Altruism and Psychological Egoism

Abstract: Publisher Summary This chapter focuses on evolutionary altruism and the thesis of psychological egoism Evolutionary altruism is a historical concept If a trait is an example of evolutionary altruism, this implies something about how it could have come into existence When groups compete against groups, psychological altruism can evolve as a group adaptation The trait exists because it is advantageous to the group, even though it is disadvantageous to the individuals possessing it The chapter examines the ways in which evolutionary altruism and psychological altruism are related Altruism can evolve by group selection It is essential that groups vary with respect to their local frequencies of altruism The chapter also analyzes the difference between psychological egoism and psychological altruism It assumes a distinction between self-directed and other-directed preferences The self-directed involves preferences about what happens to one's self; the other-directed concerns preferences about what happens to others This distinction is not entirely unproblematic because many preferences appear to be inherently relational

Unbounded Filters on ω

Abstract: Publisher Summary This chapter reviews unbounded filters on w = {0, 1, 2 . . .}. Filters will be non-principal and the Frechet filter F = {a ⊆ w: w–a is finite} will be a subset of filters. The study of these objects has given answers to many interesting problems in descriptive set theory and in the theory of forcing. The chapter describes the connection between filters on w and sets of real numbers, which are not Lebesgue messurable, do not have the Baire property, are not Ramsey and Kτ-regular. These properties involve a different characterization of the real line. In the case of the Lebesgue measure and the Baire property, the set of reals will be the set 2 w , the set of all w-sequences of 0's and 1's. The Lebesgue measure for 2w is obtained by taking the product measure over the equidistributed probability for {0, l} = 2. The topology for 2w is achieved by taking the product topology for the discrete space 2.

A Dilworth Decomposition Theorem for λ Suslin Quasi-Orderings of R

Abstract: Publisher Summary This chapter focuses on Dilworth decomposition theorem for λ-suslin quasi-orderings of ℝ. The chapter shows that if ≤ is a Borel quasi-ordering of ℝ with no perfect set of incomparable elements, then (1) ℝ = ∪n∈ω Xn, where each Xn is ≤ −linearly ordered and Borel and (2) there is a strictly ordered preserving map F : (ℝ,≤)→ (2α,