By Robert S. Boyer
In contrast to so much texts on good judgment and arithmetic, this booklet is set how one can turn out theorems instead of evidence of particular effects. We supply our solutions to such questions as: - whilst may still induction be used? - How does one invent a suitable induction argument? - whilst should still a definition be improved?
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T r n , default values dvlf . . , d v n , and well-founded relation wfn, where: (a) c o n s t is a new function symbol of n arguments, (btm is a new function symbol of no arguments, if a bottom object is sup plied), r , acx, . . , a c n are new function symbols of one argu ment, wfn is a new function symbol of two arguments, and all the above function symbols are distinct; (b) each t r j is a term that mentions no symbol as a variable be- 38 / III. A PRECISE DEFINITION OF THE THEORY sides Xi and mentions no symbol as a function symbol besides I F , TRUE, FALSE, previously introduced shell recognizers, and r ; and (c) if no bottom object is supplied, the dVi are bottom objects of previously introduced shells, and for each i , (IMPLIES (EQUAL Xi dVi) t Γΐ) is a theorem; if a bottom object is supplied, each dVi is either ( b t m ) or a bottom object of some previously introduced shell, and for each i , (IMPLIES (AND (EQUAL Xi dVi) (r (btm))) is a theorem, means to extend the theory by doing the following (using T for ( r ( btm ) ) and F for all terms of the form ( EQUAL x ( btm ) ) if no bot tom object is supplied): (1) assume the following axioms: (OR (EQUAL (r X) T) (EQUAL (r X) F)), (r (const XI ...
Xn), then (G XI . . Xn) = body[F0 ' ] = b o d y [ G ' ] . If ( Y l , . . , Yn) ^ (XI, . . , Xn), then ( G XI . . Xn) = ( FO XI . . Xn ) = body[F0 ' ] = body[G ' ] . D. That concludes the proof that the definition principle is sound. No constructivist would be pleased by the foregoing justification of recur- J. LEXICOGRAPHIC RELATIONS / 51 sive definition because of its freewheeling, set-theoretic character. The truth is that induction and inductive definition are more basic than the truths of high-powered set theory, and it is slightly odd to jus tify a fundamental concept such as inductive definition with set the ory.
H. ORDERED PAIRS We axiomatize ordered pairs as follows: Shell Definition Add the shell CONS of two arguments with recognizer LISTP, accessors CAR and CDR, default values "NIL" and " N I L " , and well-founded relation CAR. CDRP . We sometimes think of ordered pairs as sequences, binary trees, or terms. For example, (CONS 1 (CONS 2 (CONS 3 "NIL"))) may be thought of as the sequence 1, 2 , 3 . ( CONS ( CONS 1 2 ) 3 ) may be thought of as the binary tree: Finally, (CONS "PLUS" (CONS "X" (CONS 3 " N I L " ) ) ) .
A Computational Logic by Robert S. Boyer