The Algebra of Logic, by Louis Couturat potx

[...]... consequent is the thesis of the theorem It is often said that the hypothesis is the sucient condition of the thesis, and the thesis the necessary condition of the hypothesis; that is to say, it is sucient that the hypothesis be true for the thesis to be true; while it is necessary that the thesis be true for the hypothesis to be true also When a theorem and its reciprocal are true we say that its hypothesis... and 1 the modulus of multiplication On the other hand, the product of any term whatever by 0 is 0 and the sum of any term whatever with 1 is 1 These formulas justify the following interpretation of the two terms: C I.: The part common to any class whatever and to the null class is the null class; the sum of any class whatever and of the whole is the whole The sum of the null class and of any class whatever... summands of the development of 1 are what Boole called the constituents (of the universe of discourse) We may equally well call them, with Poretsky, 29 the minima of discourse, because they are the smallest classes into which the universe of discourse is divided with reference to the n given terms In the same way we shall call the factors of the development of 0 the maxima of discourse, because they are the. .. in the universe of discourse by means of the n given terms 0.22 Properties of the Constituents The constituents or minima of discourse possess two properties characteristic of contradictory terms (of which they are a generalization); they are mutually exclusive, i.e., the product of any two of them is 0; and they are collectively exhaustive, i.e., the sum of all exhausts the universe of discourse The. .. which the analogy of the relation < with the mathematical relation of inequality suggests, it may be said that the product of two classes is the greatest class contained in both, and the sum of two classes is the smallest class which contains both.18 Consequently the product of two classes is the part that is common to each (the class of their common elements) and the sum of two classes is the class of. .. calculations, to transpose the negative from one member of an equality to the other The law of double negation makes it possible to conclude the equality of two terms from the equality of their negatives (if a = b then a = b), and therefore to cancel the negation of both members of an equality From the characteristic formulas of negation together with the fundamental properties of 0 and 1, it results that... simplication because by means of them the premises of an argument may be simplied by deducing therefrom weaker propositions, either by deducing one of the factors from a product, or by deducing from a proposition a sum (alternative) of which it is a summand Formulas (2) and (4) are called the principle of composition, because by means of them two inclusions of the same antecedent or the same consequent... Huntington Sets of Independent Postulates for the Algebra of Logic (Transactions of the American Mathematical Society, Vol V, 1904) ix THE ALGEBRA OF LOGIC 1 0.1 Introduction The algebra of logic was founded by George Boole (18151864); it was developed and perfected by Ernst Schröder (18411902) The fundamental laws of this calculus were devised to express the principles of reasoning, the laws of thought... properties; their combined product is equal to 0, as we have seen; and the sum of any two of them is equal to 1, inasmuch as they dier in the sign of at least one of the terms which enter into them as summands For the sake of simplicity, we shall conne ourselves, with Boole and Schröder, to the study of the constituents or minima of discourse, i.e., the developments of 1 We shall leave to the reader the. .. ; the principle of excluded middle must be added which equally deserves the name of principle of contradiction This is why Mrs Ladd-Franklin proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the rst, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of