By Robert S. Boyer
Not like so much texts on common sense and arithmetic, this ebook is ready tips to end up theorems instead of facts of particular effects. We provide our solutions to such questions as: - whilst should still induction be used? - How does one invent a suitable induction argument? - while should still a definition be multiplied?
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As in our theory, it often forms part of the logical un derpinnings of richer theories. In addition, it offers a simple way of introducing certain important ideas such as soundness, completeness, and decision procedures. Because of its foundational role, discussions of the propositional calculus are usually carried on in the informal "metalanguage. " For example, a common definition of the value of the formula Mp Λ q" is that it is "true if both p and q have the value true, and false otherwise/' In this chapter we exercise the expressive power of our theory, and clarify certain aspects of it, by formalizing the semantics of a version of propositional calculus in our theory.
FLATTEN have b e e n introduced as in Chapter II. T h e first induction performed in the proof of the FLATTEN. MC . FLATTEN theorem of Chapter II is obtained by the following instance of our induction principle, p is the term ( EQUAL ( MC . LESSP and CDR. LESSP establish the two theorems required by condition (g). T h e base case and the induction steps produced by this application of the induction principle are those exhibited in Chapter II. We now prove that our induction principle is sound. Suppose we have in mind particular p , r , m, Xi, q t , hi, and s ifj satisfying condi tions (a) through (g) above, and suppose the base case and induction steps are theorems.
Xn) be an RM-minimal member of Dn such that for some two distinct values, Z l and Z2 say, both ((XI, . . , X n ) , Z l ) and « X I , . . , Xn),Z2) are members of FO. Let F l 50 / III. A PRECISE DEFINITION OF THE THEORY and Gl be partially correct functions that contributed ((XI, . . , X n ) , Z l ) a n d « X I , . . , X n ) , Z 2 ) t o F 0 . L e t F l ' and Gl ■ be the ex tensions of F l and G l . Both F l and Gl are defined upon all members of Dn RM-smaller than (XI, . . , Xn) because both are partially correct.
A Computational Logic by Robert S. Boyer