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We will now study some special kinds of non-standard quantifiers.

We will now study some special kinds of non-standard quantifiers. Definition 4 . Let (x), (x) be two fixed formulae of a language L n such that x is the

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We will now study some special kinds of non-standard quantifiers.

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  1. We will now study some special kinds of non-standard quantifiers. Definition 4. Let (x), (x) be two fixed formulae of a language Ln such that x is the only free variable in both of them and they don’t have common predicates. Let M and N be two models. Then we have the following two four-fold tables: We define: N is associational better than M if a2a1, d2d1, c1c2, b1b2. Moreover, a binary quantifier  is associational if, for all formulae (x) and (x), all models M, N: if vM((x)(x)) = TRUE, N associational better than M, thenvN((x)(x)) = TRUE. Obviously, the quantifier ofsimple association is associational: this follows by the fact that, under the given circumstances, a2d2a1d1>b1c1b2c2. Churchquantifier of implication is associational, too. Indeed, given a model M such that vM((x) =>C(x)) = TRUE, the corresponding four-fold table has a form Thus, any model N that is associational better than M has a form Quantifiers of founded p-implication are associational: if a2a1 n, b1b2, then a2b1a1b2 , therefore a2a1+ a2b1 a2a1+ a1b2 and finally, Thus, vN((x) =>C(x)) = TRUE. I (Today called : Basic implication)

  2. Definition 5. Let (x) , (x) be two fixed formulae of a language Ln such that x is the only free variable in both of them and they don’t have common predicates. Let M and N be two models. Then we have the following two four-fold tables: We define: N is implicational better than M if a2a1, b1b2. Moreover, a binary quantifier  is implicational if, for all formulae (x) and (x), all models M, N: if vM((x)(x)) = TRUE, N implicational better than M, thenvN((x)(x)) = TRUE. Churchquantifier of implication is implicational, quantifiers of founded p-implication are implicational [proof: by a similar argument that they are associational]. However, quantifier ofsimple association is NOT implicational: consider the following counter example: Clearly, N is implicational better than M, and a1d1c1b1 thus, vM((x)~(x)) = TRUE. However, a2d2<c2b2, thus vN((x)~(x)) = FALSE. Therefore ~ is not implicational. Lemma. Let  be an implicational quantifier. Then  is associational. Proof. Let  be implicational and vM((x)(x)) = TRUE If N is associational better than M, then N is clearly also implicational better than M, so vN((x)(x)) = TRUE. Therefore  is associational, too. 

  3. Theorem 2. Let (x), (x), (x) be formulae, and let  be an implicational quantifier. Then (x)  (x) (x)  [(x)(x)] Proof. Let M be a model such that vM((x)  (x)) = TRUE and is a sound rule of inference. In the proof we used an obvious fact: for all implicational quantifiers  , if vM((x)  (x)) = TRUE and We realise that a1 = #{x | vM((x)) = vM((x)) = TRUE}  #{x | vM((x)(x)) = vM((x))=TRUE} = a2and b1 = #{x | vM((x)) = vM((x)) = TRUE }  #{x | vM((x)  (x)) = vM((x)) = TRUE} = #{x | vM(((x)(x))) = vM((x)) = TRUE }. Thus, we have Then, for any other formulae *(x), *(x) such that we have vM(*(x)  *(x)) = TRUE, too. We will use this fact in the next Theorem, too. Since is implicational we conclude that vM((x)  [(x)(x)]) = TRUE, too. 

  4. Theorem 3. Let (x), (x), (x) be formulae, and let  be an implicational quantifier. Then [(x) (x)]  (x) (x)  [(x)(x)] Proof. Let M be a model such that vM( ([(x) (x)]  (x)) = TRUE and is a sound rule of inference. Theorem 4. Let (x) and (x) be formulae, and let ~ be the simple association quantifier. Then (x) ~ (x) (x) ~  (x) We realise that a1 = #{x | vM((x)(x))) = vM((x)) = TRUE}  #{x | vM((x)(x))) = vM((x)) = TRUE} = a2and b1 = #{x | vM((x)(x))) = vM((x)) = TRUE} = #{x | vM((x)) = vM((x) (x)))= TRUE } = #{x | vM((x)) = vM(((x)(x)))= TRUE } = b1. Thus, we have in the model M SYM NEG are sound rules of inference. Exercises 13. Prove Theorem 4. 14. Prove that Theorem 4 does not hold for founded p-implication quantifiers. and (x) ~ (x) (x) ~ (x) Since is implicational we conclude that vM((x)  [(x)(x)]) = TRUE, too. 

  5. We have introduced deduction rules (i.e. sound rules of inference) mainly to minimise the amount tautologies, called hypothesis i.e. outputs in practical GUHA data mining tasks. For example, Theorem 1 says that if  is an implicational quantifier and (x)  (x) is true in a given model M, so is (x)  [(x)(x)] true. Thus, we do not have to print (x)  [(x)(x)] as a data mining result. Next we will study some other useful deduction rules. Consider elementary conjunctions EC and elementary disjunctions ED, i.e. open formulae of a form P1(x) ... kPk(x) and P1(x)... kPk(x), where i:s are either ‘’ or empty sign. For example, P1(x)P5(x) and P1(x)P3(x)P5(x) are EC’s P2(x)P3(x)P4(x) and P2(x)P4(x) are ED’s. Denote EC’s or T by symbols , 2, 3,… (maybe empty) and denote ED’s or  by symbols  2 3… (maybe empty). Definition 6. An elementary association is a sentence of the form , where  is a quantifier and ,  are disjoint, i.e. have no common predicates. Let  and 22 be elementary associations. We say that  results from 22 by specification if either  and 22 are identical or there is an ED 0 disjoint from 1 such that 2 and 01 are logically equivalent(i.e. have always the same truth value) and  is logically equivalent to 20. [We say also:  despecifies to 22] Example. P1(x)P3(x)P5(x)  P2(x)P4(x)results from P1(x)P5(x)  P2(x)P3(x)P4(x) by specification [indeed, 0 = P3(x)]

  6. Moreover, we say that results from 22 by reduction [or  dereduces to 22] if  is 2 and 1 is a subdisjunction of 2 Example.[P1(x)P5(x)]  [P2(x) P3(x)P4(x)]results from [P1(x)P5(x)]  [P2(x)P3(x)P4(x)P6(x)P7(x)]by reduction [indeed, 2 = P6(x)P7(x)]. We introduce the despecifying-dereduction rules (SpRd-rules); they are of the form  22 where  results from 22 by successive reduction and specification, i.e. there is a ED 3 (a sub-ED of 2 ) such that 11 despecifies to 23 and 23 dereduces to 22. Example. [P1P3P5]  [P2P4] despecifies to [P1P5]  [P2P3P4] and [P1P5]  [P2P3P4] dereduces to [P1P5]  [P2P3P4P6P7] Thus, we have an SpRd-rule [P1P3P5]  [P2P4] [P1P5]  [P2P3P4P6P7] Theorem 5. For any implicational quantifier , SpRd-rules are sound rules of inference. Proof. In a same manner than Theorem 4 and Theorem 5.ž Remark. Theorem 5 can be reformulated in the following way: whenever  22 is a SpRd-rule, then ( )(22 ) [i.e. is a tautology].

  7. Theorem 5. SpRd-rules are transitive, that is, if and22 then11 22 33 33 Proof. The result is obvious as soon as we realise that the order of despecification and dereduction can be reverted, i.e. ( 2 ) ( 2 ) (2 ) dereduces to despecifies to despecifies to dereduces to    (  2 )   (2 2 ) We introduce two more types of quantifiers: p- equivalence quantifiers, where 0 < p  1. (today: Basic equivalence) For any model M, v(x ((x) p (x))) = TRUE iff (a+d)  p(a+b+c+d), in particular, in a model M such that p- equivalence quantifiers, also called -double quantifiers, where 0 < p  1. (Basic double implication) For any model M, v(x ((x) p(x))) = TRUE iff a  p(a+b+c), in particular, in a model M such that I I b+c > 0, v(x ((x) p (x))) = FALSE d > 0, v(x ((x) p (x))) = FALSE Exercises. Prove that 15. p- equivalence quantifiers and 16. p - equivalence quantifiers are associational

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