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Buffalo State College

Philosophy

1002

James Hildebrand

Spring

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PROOF OF THE SOUNDNESS CONSISTENCY AND COMPLETENESS OF THE TREE METHOD Ts and Fs do not appear on trees unless of course they are there as sentence letters rather than as truth value assignmentsIt is therefore a serious question whether the results obtained on the truth table will be reflected by trees in all casesIf the tree method tells us that a set is tf inconsistent by producing a closed tree for that set will that set always be tf inconsistent or will the tree method falsely identify some tf inconsistent sets as tf consistent by producing completed open branches under the sentences in those setsAnd if a set is tf inconsistent will the tree method always detect that fact by producing a closed tree for that setThese two questions concern the soundness and the completeness of the tree method respectively SOUNDNESSIf the tree method tells us that a set is tf inconsistent then that set really is tf inconsistent COMPLETENESSIf a set is tf inconsistent the tree method will tell us that it isNote that the two questions are distinctThe tree test might not pick out all the tf inconsistent sets even though any sets that it does pick out are tf inconsistentIn that case it would be sound but not completeOr it might not only pick out all the tf inconsistent sets but also some tf consistent onesIn that case it would be complete but not soundSo both claims need to be proven separatelyPROOF OF THE SOUNDNESS OF THE TREE METHOD Here is a brief proof of the soundness of the tree method Brief soundness proofSupposeis a tf consistent set of sentences of SLThis means that there is at least one truth value assignment call iton which all the sentences inare trueThe initial stage of the tree forplaces all and only the sentences inon the top of that treeConsequently the initial tree consists exclusively of sentences that are true on Now note that the decomposition rules were designed in such a way as to carry truth down along at least one branchThat is if the sentence you are decomposing is true on a truth value assignment at least one of the branches you generate when you decompose that sentence must consist of sentences that are also all true on that same truth value assignmentSince i after the initial stage the tree only grows as a consequence of the application of decomposition rules to sentences already on the tree ii every time you apply a decomposition rule you will end up with at least one branch consisting of sentences that are all true on any truth value assignment on which the sentence you are decomposing is true and iii all trees must be completed after a finite number of rule applications the completed tree formust have at least one branch consisting of sentences that are all true onthe truth value assignment on which all the set members on the top of the tree are trueBut a completed branch that has sentences on it that are all true on one and the same truth value assignment cannot contain an atomic sentence and the negation of that atomic sentence since no truth value assignment can make both

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