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By Robert S. Boyer

ISBN-10: 0121229505

ISBN-13: 9780121229504

Not like such a lot texts on good judgment and arithmetic, this publication is set the way to turn out theorems instead of facts of particular effects. We provide our solutions to such questions as: - while may still induction be used? - How does one invent a suitable induction argument? - while should still a definition be multiplied?

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T h e logical functions are defined with the following equations: Definition (NOT P) (IF P F T), Definition (AND P Q) ( I F P ( I F Q T F) F), C. WELL-FOUNDED RELATIONS / 31 Definition (OR P Q) (IF P T (IFQTF)), Definition (IMPLIES P Q) ( I F P ( I F Q T F) T). We adopt the notational convention of writing ( AND a b c ) for ( AND a ( AND b e ) ) , ( AND a b e d ) for ( AND a ( AND b ( AND c d ) ) ) , and so on. We make the same convention for OR. , we may refer to the "theo­ r e m " p , where p is a term).

Imagine that LESSP is well founded and that the axioms CAR. LESSP and CDR. LESSP have been added as in Chapter II. The defining equa­ tion for MC. FLATTEN is added to our theory by the following instantia­ tion of our principle of definition, f is the function symbol MC . FLATTEN (CDR X) ANS)) (CONS X ANS)); r is LESSP ; m is the function symbol COUNT 1, where ( COUNT 1 X Y ) is 46 / III. A PRECISE DEFINITION OF THE THEORY defined to be ( COUNT X) . FLATTEN (CDR X) ANS)) (C0UNT1 X A N S ) ) ) .

For example, the natural numbers can be thought of as shells: a number is either a blue 0 or a blue 1-tuple containing another blue object (namely, the predecessor of the tuple). Ordered pairs can be red 2-tuples containing arbitrary objects. The type consisting of lists of numbers can be either the green, empty list of numbers or else green 2-tuples (x, y ) such that x is a number (blue) and y is a list of numbers (green). The fact that ordered pairs and lists of numbers have different colors prevents an ordered pair consisting of a number and a list of numbers from being confused with a list of numbers.

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A Computational Logic by Robert S. Boyer

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