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18. Boolian Algebra. First Lection

Charles S. Peirce Indiana University Press PDF

18

Boolian Algebra. First Lection c. 1890

Houghton Library

§1. INTRODUCTORY

The algebra of logic (which must be reckoned among man’s precious possessions for that it illuminates the tangled paths of thought) was given to the world in 1842; and George Boole is the name, an honoured one upon other accounts in the mathematical world, of the mortal upon whom this inspiration descended. Although there had been some previous attempts in the same direction, Boole’s idea by no means grew from what other men had conceived, but, as truly as any mental product may, sprang from the brain of genius, motherless. You shall be told, before we leave this subject, precisely what Boole’s original algebra was; it has, however, been improved and extended by the labors of other logicians, not in England alone, but also in France, in

Germany, and in our own borders; and it is to one of the modified systems which have so been produced that I shall first introduce you, and shall for the most part adhere. The whole apparatus of this algebra is somewhat extensive. You must not suppose that you are getting it all in the first, the second, or the third lection. But the subject-matter shall be so arranged that you may from the outset make some use of the notation described, and even apply it to the solution of problems.

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3. Six Lectures of Hints toward a Theory of the Universe

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3

Six Lectures of Hints toward a Theory of the Universe

Spring 1890

Houghton Library

Lecture I. Right reasoning in philosophy is only possible if grounded on a sound theory of logic.

Fruitful thinking and experimentation are only two branches of one process. They are essentially one. Thinking is experimentation; its results as startling, as inexplicable. Experimentation is thinking.

The law of the development of fruitful conceptions, made out from the history of science. A genuine development.

The nature of assurance. Induction & Hypothesis.

Lecture II. The ideas of philosophy must be drawn from logic, as Kant draws his categories. For so far as anything intelligible and reasonable can be found in the universe, so far the process of nature and the process of thought are at one.

What are the fundamental conceptions of logic? First, Second,

Third. Explanation and illustrations.

Chance, Law, and Continuity must be the great elements in the explanation of the universe.

Lecture III. Critical survey of mental development in the last three centuries, and the ideas of today.

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50. [Plan for a Scientific Dictionary]

Charles S. Peirce Indiana University Press PDF

50

[Plan for a Scientific Dictionary]

Winter–Spring 1892

Houghton Library

Plan for a scientific dictionary, to be called Summa Scientiæ; or,

Summary of Human Knowledge. To be contained in one volume of 1500 pages of 1000 words per page.

The articles, though elementary, to be masterly summaries valuable even to specialists. C. S. Peirce to be editor and to write about a third of the whole. The other writers to be young men, specialists who have not yet achieved great reputations, but found out and selected by the editor as having exceptional mental power and special competence. These men to conform to certain rules as to matter, arrangement, and style; and required to rewrite until they became trained in the kind of composition required.

Economy of space to be effected by every device that ingenuity and many years’ reflection upon this problem can suggest. Facts to be tabulated as far as possible. The style of writing to be extremely compact, yet scrupulously elegant. The ideas dominant in each branch of science to be emphatically indicated, and its leading principles distinctly stated.

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33. Algebra of the Copula [Version 2]

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33

Algebra of the Copula

[Version 2]

Spring 1891

Houghton Library

The copula of consequence, R, may be defined as follows: a. If b is true, then a R b is true. b. Either a or a R b is true. g. If a R b and a are true, b is true.

The letters a and b may be replaced by propositions, that replacing a proposition which is the antecedent of another being written in parenthesis. Thus, we shall have the forms following:

With 0 copula:

A.

With 1 copula:

A R B.

With 2 copulas: (A R B ) R C

A R B R C.

With 3 copulas: [(A R B ) R C] R D

(A R B R C ) R D

(A R B ) R C R D

A R (B R C ) R D

A R B R C R D.

With 4 copulas, there are 14 forms; with 5, 42; with 6, 132; etc. The last letter of a proposition is called its consequent; all those which are followed by copulas not under parentheses are called antecedents. In like manner, the propositions under parentheses have consequents and antecedents.

The copula may also be considered as defined by the following two propositions.

PROPOSITION I. If from a proposition, P, a proposition, Q, follows, we may write P R Q.

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32. Algebra of the Copula [Version 1]

Charles S. Peirce Indiana University Press PDF

32

Algebra of the Copula

[Version 1]

Spring 1891

Houghton Library

Logical quantity has but two values, t the true and f the false. But one operation is necessary, defined as follows: a. If b is true, a R b is true. b. Either a or a R b is true. g. If a and a R b are true, b is true.

This gives the following: fRtϭt

fRfϭt

tRtϭt

t R f ϭ f.

Any proposition written is supposed to be true. In writing propositions parentheses are employed to enclose compounds to be treated as single letters in combining them with letters or other such compounds. These may be called clauses. Parentheses ending clauses or propositions are omitted, and the clauses they would have included are not commonly regarded as such. The last letter of a proposition or clause is called its consequent. Its other immediate parts, letters or clauses, are called antecedents. Thus in the proposition a R [(b R c ) R d R e] R f the antecedents are a and (b R c ) R d R e, and the antecedents of the latter clause are (b R c ) and d.

ALGEBRAIC RULES

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