By PH. D. Raymond J McCall

One of the issues integrated are induction and deduction, basic apprehension, phrases, judgment, propositions, and hypothetical and express syllogisms.

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Thus we define quo(x,y) to be (µ'z ~y)(xz ~y). As mentioned earlier, this definition is not the same as the previous definition of quo when x· is 0, but this does not matter, as we shall never be interested in this case. Finally, we define rem(x ,y) to bey . :. quo(x ,y), which is also primitive recursive. (V) The predicates 'composite' and 'prime' are primitive recursive. By definition xis composite iff it is the product of two integers greater than 1. Since an integer greater than 1 is y + 2 for some yeN, we have x composite iff 3y3z((y + 2)(z + 2) = x).

This is illustrated in the examples in the next section. 3 EXAMPLES USING BOUNDED MINIMISATION (I) The square root function, defined by sqrn = y if y 2 ~ n but (y + 1) 2 > n, is primitive recursive. 2. But we can now simply observe that sqrn = (µ,'y ~ n)(y 2 ~ n). (II) The functions Kand L (which are the components of 1- 1) are primitive recursive. By definition Kr is the unique m for which there is n with J(m,n) = r. Thus Kr= µ,m3n(J(m,n) = r). This is not enough to show K primitive recursive, since we have unbounded minimisation and quantification.

Note that (by induction on r) each fn and hence f itself, will then be total. (i) The set of all primitive recursive functions is a primitive recursively closed set. (ii) Any primitive recursively closed set contains every primitive recursive function. 1 Proof (i) If/ is one of the initial functions then the sequence with one term/ shows that/ is primitive recursive. Let f be obtained by primitive recursion from the primitive recursive function 8 and h. Then there are defining sequences 81> ...