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By N. Dequoy

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Let 'll~ (U, N, V) be this model: is closed un- -44- 0 = {Q, 1,2,3 j , Ni - {{0, i], (0, 1, 2], {0, 1, 3}, i0, 2, 3 j , (0, 1, 2, 3}], for - 0, 1, 2, 3 ; i f { 0 , 1] , i f n - 0 , V (n) = P [{0, 2}, if n > 0 . Then (^GPq , ^ D ( P However, each — > P^ ), and J^DP^ q , which violates 1 is closed under supersets and hence satis­ fies (r) above, and therefore If. is a model for ER. 3 THEOREM. P ro o f. Let Neither K nor R is_ derivable in EK'. 1L «* 0 .

This is our basic convention; others will be intro­ duced later. First a remark on completeness proofs, which consti­ tute the bulk of this essay. They always consist of two parts, the consistency part (sometimes also called the soundness part) and the completeness part proper. The consistency part consists in proving that all theorems of the logic under examination are valid in the class of frames with respect to which determination is to be proved. 4). For the most part such checks are very simple to carry out, and we shall rarely reproduce them -40- here.

Stop! The answer is YES. STOP 2. Stop! The answer is NO. Here it is of course important to note that for any finite model one can check in a finite number of steps whether it is a model for a certain axiomaticable logic. Tnis is so because it is enough to check the modal axioms of any - 26 - axiomatization of the logic, and they are the instances of only finitely many schemata. Tnis ends the proof. The term "finite model property" was evidently coined by R. Harrop, who did not use the term "model" in exactly the same sense as we do.

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