From Peirce to Skolem: A Neglected Chapter in the History of Logic
Elsevier, 22 de nov. 2000 - 480 pàgines
This book is an account of the important influence on the development of mathematical logic of Charles S. Peirce and his student O.H. Mitchell, through the work of Ernst Schröder, Leopold Löwenheim, and Thoralf Skolem. As far as we know, this book is the first work delineating this line of influence on modern mathematical logic.
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a-chain addition Algebra der Logik algebra of relatives argument axioms binary relatives Boole Boole's Boolean algebra calculus of relatives coefficients condition countable Dedekind's defined definition denote disjunction disjunctive normal form domain elimination equal equation equivalent example existential exists expression factor finite first-order logic follows formal formulas Frege given gives identity calculus individual induction inference infinite interpretation introduced latter lattice least upper bound Lecture logic of relatives Löwenheim mapping rule mathematical means Mitchell Mitchell's multiplication negation normal form notation notion operations pairs of elements paper partial order Peirce Peirce's possible predicate logic premises prenex problem proof propositional calculus propositional logic proved quantifiers relation relative product represent respect result root rules of inference Russell satisfied scheme Schröder Schroeder second-order logic set theory simply Skolem functions solution statement stipulations subsumption symbols true truth functions universe of discourse valid vanish variables write x-image
Pàgina 113 - It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science.
Pàgina 90 - THESE papers, the work of my students, have been so instructive to me, that I have asked and obtained permission to publish them in one volume. Two of them, the contributions of Miss Ladd (now Mrs. Fabian Franklin) and of Mr. Mitchell, present new developments of the logical algebra of Boole. Miss Ladd's article may serve, for those who are unacquainted with Boole's " Laws of Thought," as an introduction to the most wonderful and fecund discovery of modern logic.
Pàgina 113 - The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation ; namely, deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts.
Pàgina 91 - Miss Ladd and Mr. Mitchell also use two signs expressive of simple relations involving existence and non-existence; but in their choice of these relations they diverge both from McColl and me and from one another.
Pàgina 61 - Thus, to say that all men are mortal is the same as to say that if any man possesses any character whatever then a mortal possesses that character. To say, 'if A, then B' is obviously the same as to say that from A, B follows, logically or extralogically. By thus identifying the relation expressed by the copula with that of illation, we identify the proposition with the inference, and the term with the proposition.
Pàgina 56 - A habit of inference may be formulated in a proposition which shall state that every proposition c, related in a given general way to any true proposition p, is true. Such a proposition is called the leading principle of the class of inferences whose validity it implies. When the inference is first drawn, the leading principle is not present to the mind, but the habit it formulates is active in such a way that, upon contemplating the believed premiss, by a sort of perception the conclusion is judged...
Pàgina 128 - I call a sign which stands for something merely because it resembles it, an icon. Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness...
Pàgina 139 - Now, to say that a lot of objects is finite, is the same as to say that if we pass through the class from one to another we shall necessarily come round to one of those individuals already passed ; that is, if every one of the lot is in any one-to-one relation to one of the lot, then to every one of the lot some one is in this same relation.
Pàgina 133 - IV. — Second^intentional Logic. Let us now consider the logic of terms taken in collective senses. Our notation, so far as we have developed it, does not show us even how to express that two indices, i and j, denote one and the same thing. We may adopt a special token of second intention, say 1, to express identity, and may write ly.
Pàgina 432 - Primitive propositions. Some propositions must be assumed without proof, since all inference proceeds from propositions previously asserted. These, as far as they concern the functions of propositions mentioned above, will be found stated in *1, where the formal and continuous exposition of the subject commences. Such propositions will be called "primitive propositions.