By J. M. Bocheński (auth.)
The paintings of which this is often an English translation seemed initially in French as summary de logique mathematique. In 1954 Dr. Albert Menne introduced out a revised and just a little enlarged variation in German (Grund riss der Logistik, F. Schoningh, Paderborn). In making my translation i've got used either variations. For the main half i've got the unique French variation, due to the fact that i presumed there has been a few virtue in preserving the paintings as brief as attainable. notwithstanding, i've got incorporated the extra large old notes of Dr. Menne, his bibliography, and the 2 sections on modal good judgment and the syntactical different types (§ 25 and 27), that have been now not within the unique. i've got endeavored to right the typo graphical blunders that seemed within the unique variations and feature made a number of additions to the bibliography. In making the interpretation i've got profited greater than phrases can inform from the ever-generous aid of Fr. Bochenski whereas he used to be educating on the collage of Notre Dame in the course of 1955-56. OTTO chook Notre Dame, 1959 I normal rules § O. advent zero. 1. concept and heritage. Mathematical good judgment, often known as 'logistic', ·symbolic logic', the 'algebra of logic', and, extra lately, easily 'formal logic', is the set of logical theories elaborated through the final century simply by a man-made notation and a conscientiously deductive method.
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64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. CApAqrArAqp HISTORY: The axiomatization of the logic of sentences was undertaken by Frege and Peano and completed in PM, which employs five axioms. This number was reduced to four by Hilbert, to three by Lukasiewicz, to one by Nicod, and this one was notably shortened by Lukasiewicz and Sobocinski. LITERATURE: The system expounded in § 8 is that of Hilbert-Ackerman, but the method of deduction, which is not very rigorous in these authors, has been replaced by that of Lukasiewicz.
Y = z· ::::>. x = z. 15) Cf. § 22. 16. 'xIy' for: 'x = y'. 17. 'xly' for: 'x :f. y'. 18. (x, y): x = y' ::::>. (qJ) . qJx ::::> qJy. Explanation: If x and yare identical, y possesses all the predicates that x does. 2. 21. 'Description' for: 'a monadic matrix, preceeded by '1' (inverted iota) and a variable of the same shape as that in the matrix between parentheses'. 22. '(1X)(o/X)' for: 'the x such that o/x'. Description. Explanation: The description functor '(1X)' is like the quantifier in taking only a matrix for argument.
IIxyrpyx' for: 'IIxIIyrpxy'. 13. '(Ex, y)rp(x, y)' for: '(Ex) . (Ey)rp(x, y),. 'Ixyrpxy' for: 'IxIyrpxy'. 14. '(x)(Ey)rp(x, y)' for: '(x)· (Ey)rp(x, y),. 15. '(Ex)(y)rp(x, y)' for: '(Ex) . (y)rp(x, y),. 2. 21. (x, y)rp(x, y) . _ . 22. (Ex, y)rp(x, y) . - . 23. Rule: If the quantifiers of a sentence binding the arguments of the same individual functor are all universal or all existential, their order can be changed without changing the value of the sentence. 24. (Ex)(y)rp(x, y). :). (y)(Ex)rp(x, y) CIxIIyrpxyIIyIxrpxy Explanation: This law is only an implication, and not an equivalence, since its inverse: (x)(Ey)rp(x, y).
A Precis of Mathematical Logic by J. M. Bocheński (auth.)
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