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3.3-5 answers.pdf

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James Hildebrand

Common mistakes Most answers to the metatheory questions in 3335 were quite well done with only minor infelicities in the presentation of generally sound argumentsTwo infelicities both easily repaired stood out1Confusing sentences with setsThis mistake leads to strictly meaningless assertions likeset is trueorsentence is consistent Only sentences can be true or false on a truth value assignment or truthfunctionally true false or indeterminateSets cannot beRather than say that a set is true on a truth value assignment or tf true etc you have to say that all the sentences in the set have this propertySimilarly only sets can be tf consistent or inconsistentSentences cannot beIf you want to talk about the sentences contained in tf consistent sets you cannot say that they are tf consistentYou must say that they are true on at least one truth value assignmentIf you want to talk about the sentences contained in tf inconsistent sets you cannot say they are tf inconsistentYou must say that on every tva at least one of them is false2Relativizing tf equivalence and entailment to truth value assignmentsTf equivalence and tf entailment are allornothing conceptsEither all truth value assignments are a certain way or you cant claim that there is tf equivalence or entailmentIf you are part way through your argument and have only established that some truth value assignments are the way they need to be you cannot draw a preliminary conclusion that there is tf equivalence or tf entailment on those truth value assignmentsIt makes no sense to talk of a pair of sentences being tf equivalent on a truth value assignment or to talk of a set entailing a sentence on a truth value assignment because it takes more than oneIn fact it takes all of themInstead of saying that sentences are equivalent on some truth value assignments restrict yourself to saying that the sentences have the same truth value on those truth value assignmentsOnly draw the conclusion that they are tf equivalent after you have shown that this so for all tvasSimilarly for equivalenceJust say that on all the tvas considered so far either at least one set member is false or the entailed sentence is trueWait until all tvas have been shown to be this way before drawing the conclusion that there is entailmentThough most submissions were significantly improved over the previous assignment I noted two persistent problems shame on those of you who are continuing to do this1Just proving the if part when you are asked to prove an if and only if statement or alternatively just proving the only if partNote 355bTo prove that A if and only if B requires proving first that if you assume that A is the case then B has to follow and second that if you assume that B is the case then A has to followFar too many people are still doing only half the jobSome do one half others do the other half2Giving long arguments to explain why things do not always have to be a certain way rather than just giving a counterexampleWhen things have to be a
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