By Wolfgang Bibel (auth.)

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**Extra resources for Automated Theorem Proving**

**Example text**

Nc"Sc,t,Rc,t}) 'a\t( 1 , ,Nc) U 'wt( 1 , ,Sc, t) U ''Wt( 1 , Rc, t) {} U { {} } U {} Similarly, and (I. 1) Ei for i 2,3,4, E the = {{}} = whole T • formula are true in this model which may be easily checked by the reader. Further there is no other model with this property differing on the four propositional variables in question only. l). I t is unique if we disregard any other facts or propositional variables which have not been mentioned in this description. If a natural statement consists of substatements combined by any of our boolean connectives then the knowledge of the truth values for the substatements naturally determine a truth value for the whole statement.

T -L-J L ~ 1M MJ L t K ~L IK 1M MJ Figure 1. A connection deduction for the matrix E As before B2 is divided into the set B20 of al1 paths containing {M,I M} and the set B2, B20 of the remaining ones. 4 third clause containing no literals without a dot. For this reason the method may now turn to any set which still has to be processed in this way. In the present example only B 1 is left over which is settled in the same way as B2 just before. At this point all paths in E have been shown to be complementary.

In order to show the converse, assume that F is complementary. In particular, this says that -p CJ {L c } and -p CJ { 1 L C } is complementary each containing a connection want to show that p {M,lM} and {N,lN}, respectively. We is complementary, which is trivial if one of these two connections is contained in is not the case then thus {L r ,1 Lr} 1;; p and {M,lM} p = {Lc,lLr} -p . 4). 6 (think of any statements in natural language for Land K). In this case one says that K subsmnes LAK. e. LAK+K is a valid formula.