An Investigation of the Laws of Thought: On which are Founded the Mathematical Theories of Logic and Probabilities
Walton and Maberly, 1854 - Logic, Symbolic and mathematical - 424 pages
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a₁ Algebra application arbitrary assume babilities c₁ C₂ cause chapter cloth coefficients combination compound event conception conclusion condition connexion consists constituents deduce definition deliberative assembly denoted determine DIONYSIUS LARDNER disjunctive proposition elements elimination equa equal events whose probabilities events x evil exist expression factor final logical functions X1 given Hence hypothesis indefinite independent individual interpretation involved language laws of thought logical equation logical function logical symbols logical value method minor limits nature necessary objects observed occur operations p₁ P₂ particular premises present principle problem Prop question reasoning reduced reference relation represent respectively result second member secondary propositions simple events sition solution stxy supposed syllogism t₁ t₂ theory of probabilities things tion treatise true truth universe V₁ whence X is true X₁ X₂ καὶ
Page 296 - From the Invention of Printing to the present time ; being Brief Notices of a large Number of Works drawn up from actual inspection.
Page 38 - We may, in fact, lay aside the logical interpretation of the symbols in the given equation ; convert them into quantitative symbols, susceptible only of the values 0 and 1 ; perform upon them, as such, all the requisite processes of solution ; and, finally, restore to them their logical interpretation.
Page 4 - ... to predict new combinations of them. They are in all cases, and in the strictest sense of the term, probable conclusions, approaching, indeed, ever and ever nearer to certainty, as they receive more and more of the confirmation of experience. But of the character of probability, in the strict and proper sense of that term, they are never wholly divested.
Page 297 - HODGSON -MYTHOLOGY FOR LATIN VERSIFICATION. A brief Sketch of the Fables of the Ancients, prepared to be rendered into Latin Verse for Schools.
Page 122 - tis evident that something always was. Otherwise, the things that now are must have been produced out of nothing, absolutely and without cause — which is a plain contradiction in terms. For to say a thing is produced, and yet that there is no cause at all of that production, is to say that something is effected, when it is effected by nothing — * First Principles, pp. 1 6, 17. f Vide Appendix ; Note A. that is, at the same time when it is not effected at all.
Page 28 - white things," and y for "sheep," let xy stand for "white sheep;" and in like manner, if z stand for "horned things," and x and y retain their previous interpretations, let zxy represent "horned white sheep," ie that collection of things to which the name "sheep," and the descriptions "white" and "horned
Page 8 - As the conclusion must express a relation among the whole or among a part of the elements involved in the premises, it is requisite that we should possess the means of eliminating those elements which we desire not to appear in the conclusion, and of determining the whole amount of relation implied by the premises among the elements which we wish to retain. Those elements which do not present themselves in the conclusion are, in the language of the common Logic, called middle terms; and the species...
Page 296 - Tables of Logarithms Common and Trigonometrical to Five Places. Under the Superintendence of the Society for the Diffusion of Useful Knowledge. Fcap. 8vo.
Page 23 - ... 1st. Literal symbols, as x, y, &c., representing things as subjects of our conceptions. 2nd. Signs of operation, as +, — , X, standing for those operations of the mind by which the conceptions of things are combined or resolved so as to form new conceptions involving the same elements. 3rd. The sign of identity, ==. And these symbols of Logic are in their use subject to definite laws, partly agreeing with and partly differing from the laws of the corresponding symbols in the science of Algebra.
Page 37 - The employment of the uninterpretable symbol \/ — i in the intermediate processes of trigonometry furnishes an illustration of what has been said. I apprehend that there is no mode of explaining that application which does not covertly assume the very principle in question. But that principle, though not, as I conceive, warranted by formal reasoning based upon other grounds, seems to deserve a place among those axiomatic truths which constitute in some sense the foundation of general knowledge,...