By Robert S. Boyer, J. Strother Moore
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It is possible to prove the consistency of the theory resulting from the addition of a finite number of shells by exhibiting a model. ” Note that merely because we add a finite number of shells we are not assured that every object in the world is in one of our shell classes. That is, we do not have an axiom that says: for any x, x is either T, or x is F, or x satisfies one of the shell recognizers. Indeed, this is an intended feature of the shell principle; we desire that any extension produced by adding shells can be further extended by additional shells without giving rise to inconsistency.
Formally, we define the function APPEND so that (APPEND X Y) is the concatenation of X and Y: Definition (APPEND X Y) = (IF (LISTP X) (CONS (CAR X) (APPEND (CDR X) Y)) Y). APPEND is a particularly simple recursive function. , the) recursive call. Later in the book we will introduce more interesting recursive functions – functions for which a measure as obvious as the size of one argument will not suffice to justify their definition. By the axioms of equality, we can replace any instance of (APPEND X Y) with the corresponding instance of the righthand side of the definition above.
Xn of variables and some function m such that (m x1 . . xn ) is getting r-smaller. ” Instead of case splitting on q, we consider k+1 cases, of which one is a base case and the remaining k are induction steps. We permit each of the k induction steps to have several induction hypotheses. 4. , hk are positive integers; and (g) for 1≤i≤k and 1≤j≤ hi , si,j is a substitution and it is a theorem that: (IMPLIES qi (r (m x1 ... xn )/si,j (m x1 ... xn ))). Then p is a theorem if (IMPLIES (AND (NOT q1 ) ...
A Computational Logic (ACM monograph series) by Robert S. Boyer, J. Strother Moore