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N10SDSound - Soundness of SD.pdf

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James Hildebrand

THE SOUNDNESS AND CONSISTENCY OF SD AND THE EXTENSIONALITY OF THE SENTENCES OF SLUp to now we have treated the derivation system SD as an arbitrarily constructed system of rules to be used to derive sentences from other sentencesIn fact the rules were not arbitrarily constructedThey were designed to serve as a means for demonstrating the truthfunctional validity of argumentsEach of the straight derivation rules was designed with the thought in mind that if the sentences being appealed to as justifications for the rule application are true the sentence derived in accord with that rule must be true as wellAnd the derivation system as a whole with its restrictions on accessibility and closure was designed to ensure that whatever sentences can be derived from a set of assumptions ie occur under the scope just of those assumptions must be truthfunctionally entailed by those assumptionsIt is not immediately obvious that this goal is in fact achieved by the rules of SDStudents who begin working with derivations under the disadvantage of already knowing something about the semantics of sentences of SL will often be suspicious about some of the rulesSo a proof that SD really is soundthat if a sentence is derivable just under the scope of a set of assumptions using the rules of SD then it is truth functionally entailed by that set of assumptionsis mandatedSoundnessIf a sentence P of SL is derivable in SD from a setof sentences of SL thentruthfunctionally entails PWe would also like to be assured that SD is consistent that you cannot in fact appearances notwithstanding derive just anything using the rules of SDConsistencyThere is at least one sentence that is not a theorem of SDAs well as being assured that SD is sound we would like to be assured that SD is soundProving this last point requires that we establish an important prior point concerning the sentences of SL that they are extensionalThe soundness of SDThe strategy behind the soundness proofThe soundness proofProof of the basis clauseProof of the inductive stepCase one the assumption ruleCase two the reiteration ruleCase three ICases four and five E and ICases six and seven vE and ICases eight to twelve the straight rulesCorollary resultsTheorems of SD are tf true sentences of SLSentences that are interderivable in SD are tf equivalentSets that are contradictory in SD are tf inconsistentThe consistency of SDThe soundness of SDAppendix Proof of the extensionality of the sentences of SLProofs of Metatheorems STRATEGY FOR PROVING THE SOUNDNESS OF SDIf SD consisted just of the straight rules E E E I vI and the straight versions of vE E I and I originally given in the SD notes then proving its soundness would be easyEach of the straight rules is truth preservingEach rule specifies up to three sentences that you need to have in order to derive a further sentence and it is easy to prove that for each rule if the sentences you need to have are all true on a truth value assignmentthen the sentence the rule allows you to derive must be true onas well Consider for example the straight version of the vE ruleIt specifies that from P v Q PR and QR you can derive RAnd as it turns out if the three sentences the rule requires you to have are true on a tvathen there is no way that the sentence it allows you to derive R can be false on To see why suppose P v Q PR and QR are all true on a tva Since P v QT either PT or QTBut if PT then since PRT RTAnd if QT then since QRT RTSo either way RT The same case can be made for each of the straight rules It follows that if you start off a straight derivation in SD with sentences that are all true on a truth value assignmentthen this truth must get carried down to the all the sentences derived from them using the straight rules to all the sentences derived from those sentences using the straight rules and so on down to the sentence on the last line of the derivation supposing the derivation has remained straightie no subderivations have been introducedSo by mathematical induction if all the initial assumptions of a straight derivation in SD are true on a tvathen the last line of that derivation must be true onas wellSo the set of initial assumptions truth functionally entails the sentence on the last line because there is no way the set of initial assumptions could be true on a truth value assignment and the sentence on the last line be false on that assignmentSo it follows that if a sentence P of SL is derivable using the straight rules of SD from a setof sentences of SL thentruthfunctionally entails PSo the system of straight rules of SD is soundUnfortunately the system of straight rules is not also completeWhile we can trust that any sentence derived using the straight rules must be tf entailed by the set of initial assumptions not all sentences that are truth functionally entailed by sets can be derived from those sets using just the straight rulesMost notably no sentence that is tf entailed by the empty set no tf true sentence is derivable using the straight rulesEach of the straight rules requires you to have some sentences to start with so none can be applied when there are no initial assumptionsSD becomes complete when we add the subderivation rules to itBut the addition of these rules significantly complicates our effort to prove that SD is soundWe can no longer argue that if the sentences higher up in a derivation are true on a tvathen the sentences lower down must be true on This is because the subderivation rules allows us to assume sentences that bear no deductive relation to the sentences higher up on the tree and so cannot be shown to be true if those sentences are true
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