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Unformatted text preview: 25 LECTURE TWENTY-SIX 106 25 Lecture Twenty-Six 25.1 Some Difficulties with the Tree Rules I want to give a few examples of trees, in the first place some examples of trees which close, for instance, ones showing that an argument is valid, and after that some examples of trees which remain open, and from which we can reconstruct an interpretation making the initial sentences true, thus showing them consistent. Since time is short, and I know youve already had some practice with trees in your conferences, I will do some of them with all the rules in them, including the rules for identity. But before we start, a word about predicate trees in general. In the sentential case, we can show (we didnt, but we could) that every tree that starts with a finite set of sentences of SL terminates eventually with either a closed tree or a tree which includes a completed open path. This is one sense, then, in which the rules for SL trees are foolproof. In the predicate logic case, though, this helpful property no longer obtains, for a couple of different reasons. In the first place, there are some sets of statements of predicate logic whose trees simply will never terminate. For instance, consider the tree: ( x )( y ) Fxy ( y ) Fay check Fab ( y ) Fby check Fbc . . . Clearly each application of universal instantiation to the first statement will produce a new existential statement, so ( y ) Fay , ( y ) Fby , ( y ) Fcy , and so on. But each applica- tion of existential instantiation will require the introduction of a new name, producing Fab , Fbc , Fcd , . . . . But each time a new name is introduced, we have to go back and use the universal to instantiate for this new name, producing a new existential. Clearly, we are in a infinite loop, a loop that is not going to terminate. Note that this is not our fault; the rules dictate this. Secondly, the rules for classical propositional logic required no particular ingenuity on the part of the person constructing the trees. Sometimes ingenuity might lead one to a short, efficient tree associated with a set, while applying the rules in a different order would have led to a more complex tree for the same set, but in the end both the ingenious and the foolish will be led to the same result (i.e., either to a closed tree or to a completed tree with an open branch) as long as they apply the rules correctly. This is another sense in which the rules for sentential trees are foolproof. However, in the predicate logic case it is possible to apply rules correctly but, owing to a lack of ingenuity, to fail to obtain...
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