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philosophy note

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James Hildebrand

THE DERIVATION SYSTEM SDIntroduction Chapter 1Accessibility and Closure Exercise 1 Chapter 2A and R rules Chapter 3 I Exercise 2 Chapter 4 IE rulesExercise 3 Chapter 5 vEExercise 4 Chapter 6 EExercise 5 Exercise 6Chapter 7 v rules Exercise 7Chapter 8rulesExercise 8 Chapter 9 Metatheoretical conclusionsExercise 9 Introduction The syntax for the language SL contains two kinds of rulesThere are formation rules which tell us how to string the vocabulary elements together to create those wellformed expressions that we call sentencesThose rules have just been studiedThere are also rules of a different sort the sentential derivation or SD rulesThey are the subject of this chapterThe two sets of rules are made for entirely different purposes and may not be mixed with one anotherWhereas the formation rules tell us how to build sentences out of vocabulary elements the derivation rules state conditions under which we can derive sentences from other sentencesor in some cases from nothing at allThe study of derivations brings us to the heart of what logic is traditionally considered to be the study of reasoningReasoning is drawing conclusionsWe will eventually use the derivation system SD to symbolize how we draw conclusions when we reason in EnglishThere are two parts to the study of reasoningThere is the study of the art of reasoningThis is the study of how to apply the rules of correct reasoning to derive whatever conclusions are in fact derivableFor us at this point this reduces to the study of how to apply the rules of SD to derive whatever sentences of SL are in fact derivableSecondly there is the study of what the rules of correct reasoning ought to beConsistency and derivability For now we will not worry about the second of these topicsI will not attempt to prove to you that the derivation rules I am about to go on to present to you are the right rules for logical reasoningIn fact some of them may strike you as arbitrary or even illogicalWe will not worry about thatFor now we will take the rules for grantedLater we will take a good deal of time and care to investigate whether the rules are correct and to prove exactly why they are For now we will place a very minimal constraint on an acceptable set of derivation rulesWe will say that a set of derivation rules is acceptable or as we will call it consistent if and only if there is at least one sentence that cannot be derived using that set of rulesThis is not saying a lotHowever it does put us in a position to draw a crucial distinction the distinction between what is derivable and hence rational or logical according that set of rules and what is not derivable according to that set of rulesA rule set that does not provide for the possibility of drawing this distinction is not going to be useful for purposes of logic since it cannot even begin to discriminate between logical and illogical inferences let alone do so in ways we consider to be the right onesThis is something to keep in mind when you struggle to find a way to derive a sentence and cant find a way to do itNot every sentence is derivableNor should it beI will not for the moment even go so far as to prove that the derivation rules I am going to give are consistentYou will however experience over the course of the derivation exercises in this set of notes and in Chapter 5 of the text that the rules of SD are in fact powerfully consistentThere are not a lot of sentences that they will allow you to derive and even when there is a sentence that is derivable it will be a challenge to figure out how to do so correctlyA rigorous proof of the consistency of the derivation system SD will also be given laterI close these introductory comments with some definitionsDerivation A derivation in SD is a numbered list of sentences of SL each of which is either an assumption or is derived from earlier sentences or complete earlier derivations in accord with one of the rules of SDDerivations accordingly consist of three columnsA left hand column is occupied by a list of numbers from 1 on numbering the successive lines of the derivation and allowing us to refer to those lines by their line numbersWe will find it practical to allow that these numbers need not run consecutivelysome can be skipped overthough they must run in ordergreater cannot come before lesserThe middle column contains a list of sentences that have been assumed or derivedThe right hand column contains a list of justificationsJustifications take the form of naming the rule that permits the sentence to be derived and the individual line numbers of the sentences or inclusive line numbers of the derivations from which it is derived 1 Assumption AB11A v B AE 2 A AE 16 A v B 2 vI 17 A v B11 R 18A 21617 E 19B 1 18 I 20A v B 19 vI 21A v B 11 R 22 A v B 113839 E An example of a derivation showing the three column structure line numbers are on the left justifications are on the right derived sentences with an apparatus of marker lines to be discussed later in the middle Note that line numbers need not run without omissionor interruption Derivations can go on indefinitelyYou just keep applying rules to the sentences and derivations you have so far generated to generate more sentencesHowever in the standard case proving that a set of sentences is contradictory is an exception you are given a certain sentence as a goal that you must work towardsIn most cases proving theorems is an exception you are also given some sentences called initial assumptions to begin withSometimes finding a way to arrive at the goal sentence by following the rules can be quite challenging and therefore quite rewarding when the effort meets with successAt other times the goal sentence may be underivableDerivability I have already remarked that the system of derivation rules I am going to go on to give you SD is consistent meaning that there is at least one sentence in fact a great number of sentences as it turns out that cannot be derived using the system of rulesThis fact establishes the fundamental concept associated with derivations using the rules of SD the concept of derivability in SDDerivability is a relation that obtains between a possibly empty set of sentences of SL and a further sentence of SLIn stating this definition I make use of a convention that is commonly employed in the textbook that of using the Greek letteras a symbol or metavariable to stand for an unspecified set of sentencesRather than writeIP Pusing the metavariables with which you are already familiarBut could write P123 costs me far fewer keystrokes particularly of the highly irritating bolditalicsubscripted typeThe longer notation is also unacceptably restrictive because there is no way to use it without either prejudicing the question of whether the set is infinite or finite or doubling the work by explicitly mentioning both possibilitiesTo proceed here is the definition of the relation of derivability in SDA sentence P of SL is derivable in SDfrom a possibly empty setof sentences of SL if and only if there is a derivation in SD that takes some of the sentences inas its initial assumptions and that yields P under the scope just of those assumptions by repeated applications of the rules of SDWhen P is derivable fromusing the rules of SD I will also say thatyields P in SDThe textbook additionally uses the single turnstileto represent derivabilityyieldingP means that the setyields P in SD or equivalently that P is derivable fromusing rules of SDA slash through the turnstile means the opposite Some further comments on the definition of derivability are in orderFirst the definition explicitly makes allowance for the possibility of deriving some sentences from nothingThis is the point of stipulating thatcould in some cases be the empty setSecond the definition makes allowance for the possibility of deriving P under the scope of just some of the sentences in When I say that P is obtained under the scope of the assumption of just some of the sentences inI mean that no additional assumptions are required in order to derive PWhen I say that the derivation takes some of the sentences inas its initial assumptions I do not mean to rule out the possibility that it might take all of them as initial assumptionsI just mean to open the possibility that it might take fewer than allAfter all just as arguments might sometimes contain irrelevant premises so sets might sometimes contain more sentences than you actually need to use to derive a goal sentenceAdmittedly if you focus on too few eg none you might not have enough to derive the thing you wantI warn you now that almost all the exercises will require you to use all the sentences in whatever set you are givenSo you will for most purposes want to start by listing them allI nonetheless bother to specify that you do not have to list them all at a significant risk of not being able to complete the exercise in order to cover the case where you may not need to list them all or the case where you simply cannot list them all as is the case wheneveris infinitely largeIn the latter case either P is derivable from a finite subset of the sentences inor P is not derivable at allAs it turns out this is not a
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