On Logic (1970-1979)

"In general, one might define a complex of semantic components connected by logical constants as a concept. The dictionary of a language is then a system of concepts in which a phonological form and certain syntactic and morphological characteristics are assigned to each concept. This system of concepts is structured by several types of relations. It is supplemented, furthermore, by redundancy or implicational rules […] representing general properties of the whole system of concepts. […] At least a relevant part of these general rules is not bound to particular languages, but represents presumably universal structures of natural languages. They are not learned, but are rather a part of the human ability to acquire an arbitrary natural language." (Manfred Bierwisch, "Semantics", 1970)

"Engineers, as a rule are not and do not pretend to be philosophers in the sense of building up consistent systems of thought following logically from certain premises. If anything, they pride themselves on being hard-headed practical men concerned only with facts, disdaining mere speculation or opinion. In practice, however, engineers do make many assumptions about the nature of the universe, of man, and of society." (Edwin T Layton Jr., "The Revolt of the Engineers", 1971)

"Physics is not a finished logical system. Rather, at any moment it spans a great confusion of ideas, some that survive like folk epics from the heroic periods of the past, and others that arise like utopian novels from our dim premonitions of a future grand synthesis." (Steven Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity", 1972)

"The result of the implementation, the logical design, is traditionally shown as a series of block diagrams. These blocks represent in effect a series of statements, Actually, a direct presentation of these statements is more suitable and, although less familiar, more easily understood. The Harvard Mark IV was to large degree designed and described by such statements, as has been the case with several subsequent developments." (Gerrit Blaauw, "Computer Architecture", 1972)

"There are two subcategories of holist called irredundant holists and redundant holists. Students of both types image an entire system of facts or principles. Though an irredundant holist's image is rightly interconnected, it contains only relevant and essential constitents. In contrast, redundant holists entertain images that contain logically irrelevant or overspecific material, commonly derived from data used to 'enrich' the curriculum, and these students embed the salient facts and principles in a network of redundant items. Though logically irrelevant, the items in question are of great psychological importance to a 'redundant holist', since he uses them to access, retain and manipulate whatever he was originally required to learn." (Gordon Pask, "Learning Strategies and Individual Competence", 1972)

"There is little point in demanding minor concessions and relaxations of the abstract, timeless general equilibrium. The light it can throw on human affairs is throw by its most austere and formal version. We are not concerned to ask: How could it possibly work? The useful question is: What does its logical structure imply?" (George L S Shackle, "Epistemics and Economics", 1972)

"This parallel, between cybernetic explanation and the tactics of logical or mathematical proof, is of more than trivial interest. Outside of cybernetics, we look for explanation, but not for anything which would simulate logical proof. This simulation of proof is something new. We can say, however, with hindsight wisdom, that explanation by simulation of logical or mathematical proof was expectable. After all, the subject matter of cybernetics is not events and objects but the information 'carried' by events and objects. We consider the objects or events only as proposing facts, propositions, messages, percepts, and the like. The subject matter being propositional, it is expectable that explanation would simulate the logical." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)

"And yet, on looking into the history of science, one is overwhelmed by evidence that all too often there is no regular procedure, no logical system of discovery, no simple, continuous development. The process of discovery has been as varied as the temperament of the scientist." (Gerald Holton, "Thematic Origins of Scientific Thought: Kepler to Einstein", 1973)

"[Fuzzy logic is] a logic whose distinguishing features are (1) fuzzy truth-values expressed in linguistic terms, e. g., true, very true, more or less true, or somewhat true, false, nor very true and not very false, etc.; (2) imprecise truth tables; and (3) rules of inference whose validity is relative to a context rather than exact." (Lotfi A. Zadeh, "Fuzzy logic and approximate reasoning", 1975)

"An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true." (Ian Stewart, "Concepts of Modern Mathematics", 1975)

"A model […] is a story with a specified structure: to explain this catch phrase is to explain what a model is. The structure is given by the logical and mathematical form of a set of postulates, the assumptions of the model. The structure forms an uninterpreted system, in much the way the postulates of a pure geometry are now commonly regarded as doing. The theorems that follow from the postulates tell us things about the structure that may not be apparent from an examination of the postulates alone." (Allan Gibbard & Hal R. Varian, "Economic Models", The Journal of Philosophy, Vol. 75, No. 11, 1978)

"All mathematical problems are solved by reasoning within a deductive system in which basic laws of logic are embedded." (Martin Gardner, "Aha! Insight", 1978)

"Common to both logical positivism and transformational linguistics is their view of language-as-mathematics. Both focus on language as a system of primitive or elementary units which can be combined according to fixed rules. However useful this analogy may be in certain limited ways, it creates problems in understanding how the purely formal system of elements and rules relates to something other than itself. Both create dualistic systems which oppose formal linguistic competence to empirical components." (Stephen A Tyler, "The said and the unsaid: Mind, meaning, and culture", 1978)

"On the face of it there should be no disagreement about mathematical proof. Everybody looks enviously at the alleged unanimity of mathematicians; but in fact there is a considerable amount of controversy in mathematics. Pure mathematicians disown the proofs of applied mathematicians, while logicians in turn disavow those of pure mathematicians. Logicists disdain the proofs of formalists and some intuitionists dismiss with contempt the proofs of logicists and formalists." (Imre Lakatos,"Mathematics, Science and Epistemology" Vol. 2, 1978)

"[...] despite an objectivity about mathematical results that has no parallel in the world of art, the motivation and standards of creative mathematics are more like those of art than of science. Aesthetic judgments transcend both logic and applicability in the ranking of mathematical theorems: beauty and elegance have more to do with the value of a mathematical idea than does either strict truth or possible utility." (Lynn A Steen, "Mathematics Today: Twelve Informal Essays", Mathematics Today, 1978)

"[…] a body of practices widely regarded by outsiders as well organized, logical, and coherent, in fact consists of a disordered array of observations with which scientists struggle to produce order." (Bruno Latour & S Woolgar, Laboratory Life: The Social Construction of Scientific Facts, 1979)

"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements."  (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)