Download PDF by Robert S. Boyer: A Computational Logic

By Robert S. Boyer

ISBN-10: 0121229505

ISBN-13: 9780121229504

In contrast to such a lot texts on common sense and arithmetic, this ebook is ready tips to turn out theorems instead of evidence of particular effects. We provide our solutions to such questions as: - while should still induction be used? - How does one invent a suitable induction argument? - while should still a definition be multiplied?

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The term s u b t e r m i is a variable. The proof in this case is im­ mediate. Case 2. T h e term s u b t e r m i has function symbol I F . T h e n s u b t e r m i is ( I F s u b t e r m a s u b t e r m b s u b t e r m c ) for some a , b , and c all less than i . Hence we have previously proved *2 (IMPLIES testsa *3 (IMPLIES testsb (EQUAL s u b t e r m a [ F l ' ] s u b t e r m a [ G l f ]) ) , (EQUAL s u b t e r m b [ F l ' ] s u b t e r m b [ G l ' ]) ) , and *4 (IMPLIES testsc (EQUAL s u b t e r m c [ F l ' ] s u b t e r m c [ G l ' ]) ) .

If we can show that G is partially cor­ rect a contradiction will arise because then G would be a subset of FO by the definition of FO . The domain of G is an RM-closed subset of Dn because it was formed by adding to an RM-closed subset of Dn an RMminimal element of Dn not in that subset. Let Gf be the extension of G. We need to show that for every n-tuple (XI, . . , Xn) in the domain of G that (GX1 . . Xn) = b o d y [ G ' ] . For every (XI, . . , Xn) in the domain of G, we may apply the lemma for G, G ' , FO, FO ' , and (XI, .

For example, if f de­ notes the function symbol G, and t denotes the term (ADDI Y) , then ( f t X) denotes the term (G (ADDI Y) X) . When we are speaking in naive set theory we use both upperand lowercase words as variables ranging over numbers, sets, func­ tions, etc. Context will make clear the range of these variables. " Whenever we add a new shell or function defini­ tion, we insist that certain function symbols not have been men­ tioned in any previous axiom. We call a function symbol new until an axiom mentioning the function symbol has been added.

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A Computational Logic by Robert S. Boyer

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