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By V Sankrithi Krishnan

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6? arises from a tautology W by substitution. 13, there is a proof of W in L. In such a proof, make the same substitutions of wfs of K for statement letters as were used in obtaining & from W, and, for all statement letters in the proof which do not occur in W , substitute an arbitrary wf of K. Then the resulting sequence of wfs is a proof of &, and this proof uses PROOF. only Axiom Schemas (1)-(3) and MP. 62 QUANTIFICATION THEORY SEC. 4 SEC. 4 PROPERTIES OF FIRST-ORDER THEORIES Example. 1 in a proof will be indicated by writing "Tautology".

Of theories in the following way. J, is K. Assume J, defined, with n > 0. If it is not the case that tJn-then let J,,, be obtained from J, by adding as an additional axiom. On the other hand, if tJn-let J,+, = J,. Clearly, J,+, is an extension of J,, and J is an extension of all the Ji7s,including Jo = K. To show that J is consistent, it suffices to prove that all the J,'s are consistent, because a proof of a contradiction in J, involving as it does only a finite number of axioms, is also a proof of a contradiction in some J,.

X 2 I) ,. (x, y ) is xY. (x, y ) is xy, f:(x) is x + 1 , and a , is 0. (x,Y ) ,f:(a~))). (u, u, w) means u + u = w. & logically implies '3 if and only if & 3 '3 is logically valid. &) & and 93 are logically equivalent if and only if & = '3 is logically valid. (c) If & logically implies '3, and @ is true in a given interpretation, so is '3. (x, y ) means x 57 < v, A wf & is said to be logically valid if and only if & is true for every interpretation. & is said to be satisfiable if and only if there is an interpretation for which 8 is satisfied by at least one sequence in Z.

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