Showing papers on "Formal system published in 1971"

Examples of formal semantics

Abstract: A technique of formal definition, based on relations between "attributes" associated with nonterminal symbols in a context-free grammar, is illustrated by several applications to simple, yet typical, problems. First we define the basic properties of lambda expressions, involving substitution and renaming of bound variables. Then a simple programming language is defined using several different points of view. The emphasis is on "declarative" rather than "imperative" or "algorithmic" forms of definition.

An example of the passage from the concrete to the manipulation of formal systems

Abstract: Everybody knows that mathematics is an abstract subject. It follows that most people who have studied the problem of learning such an abstract subject, would agree that some passage from the concrete to the abstract must be mapped. By concrete, we mean usually, our immediate contact with the real world. We come into contact with objects and events and we re-act to them. This is the concrete level at which all organisms behave until they are able to organize their re-actions to events into re-actions to sets of events. This is the first stage towards abstraction — when the organism begins to classify. It is of interest to try and investigate the details of the abstraction process, not only from concrete experiences to classification, but to the learning of extremely complex abstract systems such as exist in mathematics and which more and more people, including children, are called upon to learn. The difficulties, in the way of such studies, are great, one of the main difficulties being that mathematicians on the whole are not interested in learning, and psychologists on the whole do not know enough mathematics to be able to formulate the problem in a way in which a possible solution might be sought. At the Sherbrooke Psychomathematics Center, for the past few years, we have been studying the abstraction process as it proceeds from the concrete to the final stage of wielding a mathematical formal system. We have rather few laboratory results of this aspect of our work as yet but we have enough classroom evidence that we can postulate certain regularities that seem to occur and certain pre-requisites that appear to have to be satisfied before certain stages of learning can successfully be undertaken.

Formal Message Theory

Abstract: Formal communication is the sort of communication used in large organizations; the unit of formal communication is the formal message. Typically, a formal message indicates that it comes from a certain person (in a certain status, at a certain time), to another person (in a certain status, at a certain time), with an aim or point connected with previous messages; it has a body, and perhaps also an interpretation intention that includes a special glossary or other aids to interpreting the body.

Incompleteness of a formal system for infinitary finite-quantifier formulas

Abstract: In [1], various formal proof systems for infinitary formulas were defined.2 3 The formal proof system 13 .(Z; Q2) is the result of extending the basic predicate calculus 93 . by adding a collection Z of axiom schemes and a collection Q of rules of inference. Let Taut be the collection of all infinitary propositional tautologies, considered as axiom schemes. Let QI consist of all the quantificational rules of independent choices. We will show, in ?2 (see Theorem 2.1), that 3 ..(Taut; 0) is not complete for L., (i.e., infinitary finite-quantifier) sentences; that is, we will exhibit an L.. sentence + such that is true in all models, but -o is not provable in W3. (Taut; 0). (The unprovability is shown by a weak forcing version of Boolean general models.) This answers a question of Karp in [1, 12.1(i)]. In ?4, we will show that our cb is "93.. complete for L.. sentences." By Theorem 11.5.1 of [1], 93..(Taut; El,) is complete for L,,,, sentences. Since we will show that 93 .,(Taut; 0) is not, there is a rule of independent choice which is not a derived rule of inference in 93,3(Taut; 0). At the end of ?2, we will give a specific instance of a rule of choice not derivable in 93x,(Taut; 0). In ?3, we will show that each rule of independent or dependent choice is a derived rule of inference in the formal system 930.. This will be a corollary of the following fact: if 13 is a S1 notion of proof such that only valid formulas are provable, and if L.. formula 0 is provable in 93, then 0 is provable in 93... We will assume the informal axioms of Zermelo-Fraenkel set theory with the axiom of choice. In ??3, 4 we will require both knowledge of the (bottom level of the) hierarchy in Levy's [3] and the ability to relate this hierarchy to (as in [2]) proof systems and to satisfaction of L.. formulas by Boolean models. For these sections, we will also use the fact (mentioned in [1, p. 141], with different notation) that an La, formula is consistent in 93. iff it is satisfiable in some Boolean model.

A programmed second‐year course on the psychology of learning part 2: content

Abstract: The aim of this course was to enable the students to write operational definitions of concepts used in the psychology of learning, and to relate them in a formal system. . Accordingly, after the preliminary stages (reported in the first article) students participated in a number of classroom demonstrations of concepts in action. This transition from formal lectures was followed by a series of individual, self‐paced exercises. In these, continual stress was placed on operational definitions, as the scope of the material was widened to include strategies of learning, thinking, information theory and perception. Theories of learning were introduced, and finally all aspects of the course were brought together within a framework based on a model of the individual.

Some Methodological Aspects of a System Theory

Abstract: The purpose of this lecture is two-fold. In the first part the role of the mathe­matical notion of a functional is emphasized and it is shown to be the underlying notion in straightforward practice in more or less formal systems constructions. In particular it is argued that in many cases the simple conception of a system as a set of input-output relations is inadequate.