Beginning Logic-Free Books

Beginning Logic
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in brackets refers either to a line of a proof or to a sentence or. formula so numbered earlier in the same section context will always. determine which,The Propositional Calculus I, My thanks are due to Father Ivo Thomas o p who read Chapter. 1 and to Professor James Thomson who read the whole book for I THE NATURE OF LOGIC. helpful comments and the correction of many errors I am greatly It is not easy and perhaps not even useful to explain briefly what. indebted to my wife and to Miss Susan Liddiard for typing assistance logic is Like most subjects it comprises many different kinds of. and to Mr Bruce Marshall for help with proof reading and indexing problem and has no exact boundaries at one end it shades oft into. lowe a lot to discussions with colleagues about the best way to mathematics at another into philosophy The best way to find out. formulate logical rules in particular to Professor Patrick Suppes what logic is is to do some None the less a few very general. and Michael Dummett whose idea it was in 1957 that I should remarks about the subject may help to set the stage for the rest of. write this book But my greatest debt is to the many students in this book. Oxford Texas and elsewhere who forced me by their questions and Logic s main concern is with the soundness and unsoundness of. complaints to write more clearly about the matters involved The arguments and it attempts to make as precise as possible the. many faults of exposition that remain of course are mine conditions under which an argument from whatever field of study. I should like to dedicate this book to Arthur Prior without whose is acceptable But this statement needs some elucidation we need. encouragement and enthusiasm I would never have entered logic to say first what an argument is second what we understand by. and to the memory of my father who I hope would have enjoyed it soundness third how we can make precise the conditions for sound. argumentation and fourth how these conditions can be independent. E J L of the field from which the argument is drawn Let us take these. points in turn, Claremont California Typically an argument consists of certain statements or pro. March 1965 positions called its premisses from which a certain other statement. or proposition called its conclusion is claimed to follow We mark. in English the claim that the conclusion follows from the premisses. by using such words as so and therefore between premisses and. conclusion Instead of saying that conclusions do or do not follow. from premisses logicians sometimes say that premisses do or do not. entail conclusions When an argument is used seriously by someone. and not for example just cited as an illustration that person is. asserting the premisses to be true and also asserting the conclusion. to be true on the strength of the premisses Thls is what we mean. by drawing that conclusion from those premisses, Logicians are concerned with whether a conclusion does or does. not follow from the given premisses If it does then the argument. in question is said to be sound otherwise unsound Often the. The Propositional Calculus J The Nature of Logic, terms valid and invalid are used in place of sound and What techniques does the logician use to make precise the. unsound The question of the soundness or unsoundness of conditions for sound argumentation The bulk of this book is in. arguments must be carefully distinguished from the question of the a way a detailed answer to this question but for the moment we. truth or fal sity of the propositions whether premisses or conclu may say that his most useful device is the adoption of a special. sion in the argument For example a true conclusion can be symbolism a logical notation for the use of which exact rules can. soundly drawn from false premisses or a mixture of true and false be given Because of this feature the subject is sometimes called. premisses thus in the argument symbolic logic It is someti mes also called mathematical logic. partly because the rigour achieved is similar to that already belonging. I Napoleon was German all Germans are Europeans, to mathematics and partly because contemporary logicians have.
therefore Napoleon was European, been especially interested in arguments drawn from the field of. we find a true conclusion soundly drawn from premisses the first mathematics In order to understand the importance of symbolism. of which is fa lse and the second true Again a false conclusion in logic we should remind ourselves of the analogous importance. can be soundly drawn from false premjsses or a mixture of true of special mathematical symbols. and fal se prem isses thus in the argument, Consider the following elementary alscbraic eq uation. 2 Napoleon was German aU Germans are Asiatics 4 xt yl x y x y. therefore Napoleon was Asiat ic, and imagine how difficult it would be t o express this proposition in. a fal se conclusion is soundly drawn from two false premisses On. ordinary English without the use of variables x Y t brackets. the other hand an argument is not necessarily sound just beeause. and the minus and plus signs Perhaps the best we could achieve. premisses and conclusion are true thus in the argument. 3 Napoleon was French all Frenchman are Europeans 5 The result of subtracting the square of one number from. therefore Hitler was Austrian the square of a second gives the same number as is. all the proposit ions arc true but no one would say that the con obtained by adding the two numbers subtracting the. clusion followed from the premisses first from the second and then mult iplying the results of. these wo calculations, The basic connection between the soundness or unsoundness of. an argument and the truth or fal sity of the constituent propositions Comparing 4 with 5 we see that 4 has at least three advantages. is the follow ing an argument cannot be sound if its premisses are over 5 as an expression for the same proposition It is briefer. all true and its conclusion false A necessary condition of sound It is c1earer at least once the mathematical symbols are under. reasoning is that from truths only truths follow This condition is stood And it is more exact The same advantages brevity clarity. of course not sufficient for soundness as we see from 3 where we and exactness are obtained for logic by the use of special logical. have true premisses and a true conclusion but not a sound argument symbols. But for an argument to be sound it must at least be the case that Equation 4 holds true for any pair of numbers x and y Hence. if all the premisses are true then so is the conclusion Now the if we choose x to be 15 and y to be 7 we have as a consequence. logician is primarily interested in conditions for soundness rather of 4. than the actual truth or fal sity o f premisses and conclusion but 6 15 7 15 7 15 7. he may be secondarily interested in truth and falsity because of. this connection between them and soundness If we now compare 6 with 4 we can see that 6 is obtained. The ProposlJional Calculus I Conditionals and Negation. from 4 simply by putting 15 in place of x and 7 in place affirms that therefore the object m does not have the property G. of y In this way we can check that 6 does indeed follow from We may state the common pattern of 7 and 8 as follows. 4 simply by a glance to see that we have made the right sub. 9 m has F nothing with F has 0, stitutions for the variables But if 6 had been expressed in ordinary.
therefore m does not have G, English as 4 was in 5 it would have been far harder to see. whether it was soundly concluded from 5 Mathematical symbols Once the common logical form has been ext racted as in 9 a new. make both the doing and the checking of mathematical calculations feature of it comes to light Whatever objec t m is picked out. far easier Similarly logical symbols are humanly indispensable if whatever properties F and G are chosen to be the pattern 9 will. we are to argue correctly and check the soundness of arguments be valid 9 as it stands is a pattern of a valid argument For. efficiently example take m to be Jenkins F and 0 to be the properties respect. If in the sequel it seems irritating that a special notation for ively of being a bachelor and being married then 9 becomes. logical work bas to be learned the reader shouJd remember that he. 10 Jenkins is a bachelor no bachelors are married, is only mastering for argumentation what he masters for calculation. therefore Jenkins is not married,when he learns the correct use of and so on This. device which logic has copied from mathematics is the logician s which like 7 and 8 is a sound argument Yet 9 is not tied to. most powerful tool for checking the soundness and unsoundness of any particular subject matter whether it be ornithology chemistry. arguments or the law the spedal terminology migrant molecular. Our final question in this section is how the conditions for valid bachelor has disappeared in favour of schematic letters F. argument can be studied independenlly of the fields from which G m. arguments are drawn if this could not be done there wouJd be no Form can thus be studied independently of subject matter and. separate study called logic A simple example will suffice for the it is mainly in virtue of their form as it turns out rather than their. moment If we compare the two arguments subject matter that arguments are valid or invalid Hence it is the. forms of argument rather than actual arguments themselves that. 7 Tweety is a robin no robins are migrants,logic investigates. therefore Tweety is not a migrant, To sum up the contents of this section we may define logic as.
the study by symbolic means of the exact conditions under which. 8 Oxygen is an element no elements are molecular, patterns of argument are valid or invalid it being understood thai. therefore oxygen is not molecular, validity and invalidity are to be carefully distinguished from the. both of which are sound one drawn from ornithology the other related notions of truth and falsity But this account is provisional. from chemistry it is hard to escape the feeling that they have in the sense that it will be better understood in the light of what is. something in common This something is called by logicians their to follow. logical form and we shall have more to say about it later For the. moment let us try to analyse out in a preliminary way this common. form The first premiss of both 7 and 8 affirms that a certain 2 CONDITIONA LS AND NEGA TIO N. particular thing call it m Tweety in 7 oxygen in 8 has a certain When we analyse the logical form of arguments as we did in the. property call it F being a robin in 7 being an element in 8 last s tion to obtain 9 from 7 and 8 words which relate to. The second premiss of 7 and 8 affirms that nothing with this specific subject matters disappear but other words remain this. property F has a certain other property call it G being a migrant residual vocabulary constitutes the words in which the logician is. in 7 being mol ular in 8 And the conclusion of 7 and 8 primarily interested for it is on their properties that validity hinges. The Propositional Calculus 1 Conditionals and Negation. Of particular importance in this vocabulary are the words if two propositions then we shall write the proposition that if P then. then and either or and not This Qas, chapter and the next are in fact devoted to a systematic study of P Q. the exact rules for their proper deployment in arguing We have Again let P be any proposition then we shall write the proposition. no single grammatical term for these words in ordinary speech but that it is not the case that P as. in logic they may be called sentenceJormmg operators on sentences. I shall try to explain why they merit this formidable title Po. In arguments as we have already seen propositions occur an The proposition P Q will be called a conditional proposition or. argument is a certain complex of propositions among which we simply a conditional with the proposition P as its antecedent and the. may distinguish premisses and conclusion Propositions are proposition Q as its consequent For example the antecedent of the. expressed in natural languages in sentences However not all proposition that if it is raining then it is snowing is the proposition. sentences express propositions some are used to ask questions such that it is raining and its consequent is the propoSition that it is. as Where is Jack others to give orders such as Open the snowing The proposition P will be called the negation of P For. door Where it is desirable t distinguish between sentences example the proposition that it is not snowing is the negation of. expressing propositions and other kinds of sentence logicians the proposition that it is snowing. sometimes call the former declarative sentences Always when I The letters P Q used here should be compared with the. speak of sentences it is declarative sentences I have in mind unless variables x y of algebra they may be considered as a kind of. there is some explicit denial Now if we select two English sentences variable and are frequently called by logicians propoJitionai variables. say it is raining and it is snowing then we may suitably place In introducing the minus sign I might say let x and y be any. if then and and either or to obtain two numbers then I shall write the result of subtracting y from x. the new English sentences if it is raining then it is snowing as x y In an analogous way I introduced above using. it is raining and it is snowing and either it is raining or it is propositional variables in place of numerical variables since in logic. snowing The two original sentences have merely been substituted we are concerned with propositions not numbers. for the two blanks in if then and and either Propositional variables will also help us to express the logical form. or Further if we select one English sentence say it is of complex propositions compare the use of schematic letters F. raining then we may suitably place not to obtain the new and G in 9 of Section I Consider for example the complex. English sentence it is not raining Thus grammatically speaking proposit ion. the effect of these words is to form new sentences out of one or. I If it is raining then it is not the case that if it is not. two given sentences Hence I call them sentence forming operators. snowing it is not raining, on sentences Other examples are although nevertheless. requiring two sentences to complete it because also Let us use P for the proposition that it is raining and Q for. requiring two and it is said that requiring only one the proposition thdt it is snowing Then with the aid of and. This book is written in English and so mentions English sentences we may write 1 symbolically as. and words but the above account could be applied by appropriate 2 P Q P. translation to all languages I know of There is nothing parochial. about logic despite this appearance to the contrary we introduce brackets here in an entirely obvious way 2 as well. In this section we are concerned with the rules for manipUlating as being a kind of shorthand for 1 with the advantages of brevity. if then and not and we begin by introducing special and c1arity once at least the feeling of strangeness associated with. logical symbols for these operators Suppose that P and Q are any novel symbolism has worn off succeeds in expressing the logical. The Proposi ional Calculus I Conditionals QJld Negarion. form of I We can see that 2 also gives the logical form of the proposition will appear the numbers of the original assumptions on. quite different proposition which the argument at that stage depends. 3 If there is a fire then it is not the case that if there is not. smoke there is not a fire Rule of Assumptions A, The first rule of derivatiou to be introduced is the rule ofassumptions.
here P is a stand in for the proposition that there is a fire and Q for which we call A This rule permits us 0 introduce at QJly stage of. the proposition that there is smoke an argument any proposition we choose as an assumption of the. argument We simply write the proposition down as a new line. When we argue we draw or deduce or derive a conclusion from. write A to the right of it and to the left of it we put its own. given premisses in logic we formulate rules called rules ofderivatioll. number to show that it depends on itself as an assumption Thus. whose object is so to control the activity of deduction as to ensure. we might begin an argument, that the conclusion reached is validly reached Another feature of. ordinary argumentation is that it proceeds in slages the conclusion I J P Q A. of one step is used as a premiss for a new step and so on until a. This means that our first step has been to assume the proposition. final conclusion is reached It will be helpful therefore if we. P Q by the rule of assumptions Or after nine lines of argument. distinguish at once between assumptions and premisses By an. we may proceed, assumption we shall understand a proposition which is in a given. stretch of argumentation the conclusion of no step of reasoning 10 10 Q A. but which is rather taken for granted at the outset of the total. This means that at the tenth line we assume the proposition Q. argument By a premiss we shall understand a proposition which. by the rule of assumptions, is used at a particular stage in the total argument to obtain a. It may seem dangerously liberal that the rule of assumptions. certain conclusion An assumption may be and characteristically. imposes no limits on the kind of assumptions we may make in. will be used as a premiss at a given stage in an argument in order. particular there is no question of ensuring that assumptions made. to obtain a certain conclusion This conclusion may itself then be. are true This is best understood by reminding ourselves that the. used as a premiss for a further step in the argument and so on. logician s concern is with the soundness of the argument rather than. Thus a premiss at a certain stage will be either an assumption of. the truth or fal sity of any assumptions made hence A allows us to. the argument as a whole or a conclusion of an earlier phase in the. make any assumptions we please the job of the logician is to make. argument At any given stage in the total argument we shall have. sure that any conclusion based on them is validly based lor to. a conclusion obtained ultimately from a certain assumption or. investigate their credentials, combination of assumptions and we shall say that this conclusion. rests on or depends on that assumption those assumptions. Roughly our procedure in setting out arguments will be as Modus ponendo pOf ens MPP. follows Each step will be marked by a new line and each line will The second rule of derivation concerns the operator We name it. be numbered consecutively On each line will appear either an modus ponendo ponens abbreviated to MPP which was the medieval. assumption of the argument as a whole or a conclusion drawn from term for a closely related principle of reasoning Given as premisses. propositions at earlier lines and based on these propositions as a conditional proposition and the antecedent of that conditional. premisses To the right of each proposition will be stated the rule of MPP permits us to draw the consequent of the conditional as a. derivation used to justify its appearance at that stage and where conclusion For example given P Q and p we can deduce Q. necessary the numbers of the premisses used To the left of each Or to take a more complicated example given Q P Q. The Proposilional Calculus 1 Conditionals and Negation. and Q we can deduce P Q Written more formally these conclusion R from 2 and 4 as premisses The numbers 2 and 4. two arguments become appear on the right accordingly In deciding what assumptions to. cite on the left we note that 4 rests on 1 and 3 whilst 2 rests. 1 1 I P Q A only on itself we pool these assumptions to obtain I 2 and 3. 2 2 P A Secondly I show that given P Q R P Q and P we. 1 2 3 Q 1 2MPP may validly conclude R,4 I P Q R A I t P Q p 11.
2 I Q P Q A,1 2 3 P Q 1 2MPP,1 3 4 Q R 1 3 MPP, On the first two lines of each of these arguments we make the 2 3 5 Q 2 3 MPP. required assumptions by the rule A numbering on the left accord. 1 2 3 6 R 4 5 MPP, ingly At line 3 we draw the appropriate conclusion by the rule. MPP the consequent of the conditional at line I given at line 2 At lines 4 and 5 the premisses used for the applications of. the antecedent of that conditional To the right at line 3 in both MPP are also assumptions so that the same pair of numbers. cases we note the rule used MPP together with the numbers of appears on the right and on the left But at line 6 the premisses. the premisses used in this application of the rule To the left at are the conditional 4 Q R and its antecedent 5 Q neither of. line 3 we mark the assumptions on which the conclusion rests which are assumptions of the argument as a whole in detennining. in this case again I and 2 which here are both premisses for the the numbers on the left therefore we pool the assumptions on. application of MPP and assump ions of the total argument which 4 and 5 rest t 3 and 2 3 respectively to obtain. Here are more complicated examples using on1y the two rules A I 2 and 3. and MPP I shall show first that given the assumptions P Q It should be obvious that MPP is a reliable principle of reasoning. Q R and P we may validly conclude R It can never lead us at least from true premisses to a false conclusion. YI f I CI Q l For it is a basic feature of our use of if then that if a. 3 1 I P Q A 1, conditional is true and if also its antecedent is true then its consequent. 2 2 Q R A must be true too and MPP precisely allows us to affirm as a con. 3 3 P A clusion the consequent of a conditional given as premisses the. 1 3 4 Q 1 3 MPP conditional itself and its antecedent. It will be a help to have an a bbreviation for the cumbersome. 1 2 3 5 R 2 4 MPP, expression given as assumptions we may validly conclude. The first three lines here merely make the necessary assumptions To this end I introduce the symbol. At line 4 we draw by MPP the conclusion Q given at line I the. conditional P Q and at line 3 its antecedent P Hence I and. 3 are mentioned to the right as premisses for the application of. called often but misleadingly in the literature of logic the assertion. the rule and to the left as the assumptions used at that stage At sign It may conveniently be read as therefore Before it we list. line 5 we use Q the conclusion at line 4 as a premiss for a new in any order our assumptions and after it we write the conclusion. application of MPP noting that Q is the antecedent of the con drawn Using this notation we may conveniently sum up the four. ditional Q R assumed at line 2 So we obtain the desired pieces of reasoning above from now on to be called proofs thus. The Propositional Calculw J Conditionals and Negation. 1 P Q Pf Q 6P Q R P M Q,1 Q P Q Q P Q 1 I P Q R A,3P Q Q R PfR 2 2 P A.
4 P Q R P Q PfR 3 3 R A, Results obtained in this form we shall call sequents Thus a sequent 1 2 4 Q R 1 2 MPP. is an argument frame containing a set of assumptions and a con 1 2 3 5 Q 3 4 MIT. clusion which is claimed to follow from them Effectively sequents. which we can prove embody valid patterns of argument in the sense For line 5 we notice that 3 R is the negation of the consequent. that if we take the P Q R in a proved sequent to be actual of the conditional 4 Q R so that by MIT we may conclude. propositions then reading I as therefore we obtain a valid the negation Q of the antecedent of 4 to the rigbt we cite 3. argument The propositions to the left of 1 become assumptions and 4 and to the left I and 2 the assumptions on which 4. of the argument and the proposition to the right becomes a con rests and 3 the assumption namely itself on which 3 rests. clusion validly drawn from those assumptions From this point of We may see the soundness of the rule MIT by ordinary examples. view in constructing proofs we are demonstrating the validity of The following are evidently sound arguments. patterns of argument which is one of the logician s chief concerns 4 If Napoleon was Chinese then he was Asiatic Napoleon. The sequent proved can be written down immediately from the was not Asiatic therefore be was not Chinese. last line of the p roof In place of the numbers on the left we write. 5 If Napoleon was French then he was European Napoleon. the propositions appearing on the corresponding lines then we. was not European therefore he was not French, place the assertion sign finally we add as conclusion the proposition. on the last line itself To see this the four sequents above should In both cases given a conditional and the negation of its consequent. be compared with the last lines of the corresponding proofs we deduce validly the negation of its antecedent in 4 the conclusion. is true and so are both premisses in 5 the conclusion is false but. Modlls tol cndo tollens MTT so is one premiss It should be clear that this pattern of reasoni ng. The third rule of derivation concerns both and Again we use will never lead from premisses which are alltrue to afalseconclusion. a medieval term for it modw tollendo tollens abbreviated to MIT. Given as premisses a conditional proposition and lhe negation of its. consequent MIT permits us to draw the negation of the antecedent Double neg tion ON. of the conditional as a conclusion The fourth rule of derivation purely concerns negation By the. Here are two simple examples of the use of MIT I set the double negation of a proposition P we understand the proposition. precedent of citing the sequent proved before the proof Po Intuitively to affirm that it is not the case that it is nOl the. case that it is raining is the same as to affirm that it is raining and. this holds for any piOposition whatsoever the double negation of a. 1 I P Q A proposition is identical with the proposition itself Hence from the. 2 2 Q A double negation of a proposition we can derive validly the propo. 1 2 3 P 1 2 MIT sition and vice versa This principle lies behind the rule of double. Thus we take a proof as a proof of a sequent but it is also natural to say In negation ON given as premiss the double negation of a propo. a different sense that in a proof a cone usion is proved from certain assumptioOJ sition ON permits us to draw the proposition itself as conclusion. This resultant ambiguity in the word prove is fairly harmless and given as premiss any proposition ON permits us to draw its. The Propositional Calculus I Conditionals and Negation. double negation as conclusion Uruike MPP and MIT DN Europeans how might we prove tbat if Napoleon was German. requires oruy one premiss for its application not two It s use is then he was European We naturally say suppose Napoleon was. exemplified in the following proofs German bere we take the antecedent of the conditional to be. proved as an extra assumption now all Germans are Europeans. 7P Q Q P the given assumption therefore Napoleon was European here we. 1 I P Q A derive the consequent as conclusion so if Napoleon was German. 2 2 Q A he was European here we treat the previous steps of the argument. as a proof of the desired conditional, The fifth rule of derivation the rule of conditional proof CP. 1 2 4 P 1 3 MIT imitates exactly this natural procedure and is our most general. device for obtaining conditional conclusions Its working is harder. Note especially that since the consequent of I P Q is Q. to grasp than that of the earlier rules but familiarity with it is. we need to obtain the negation of this i e Q before we can. indispensable I first state it then exemplify and discuss it. apply the rule MIT hence we require the step of ON from 2 to. Suppose some proposition call it B depends as one of its. 3 before the use of MIT at line 4, assumptions on a proposition call it A then CP permits us to. 8 P Q Q P derive the conclusion A B on the remaining assumptions if any. In other words at a certain stage in a proof we have derived the. conclusion B from assumption A and perhaps other assumptions. 2 2 Q A then CP enables us to take this as a proof of A B from the other. 1 2 3 P 1 2 MIT assumptions if any,1 2 4 P 3DN For example.
Note especially that from I and 2 by MIT we draw as conclusion 9 P QI Q P. the negation of the antecedent of 1 i e P hence we require I I P Q A. the step of ON from 3 to 4 in order to obtain the conclusion P. Note also that the conclusion of an application of ON rests on. exactly the same assumptions u its premiss 1 2 3 P 1 2 MIT. 4 Q P 2 3 CP, Conditional proof CP In attempting to derive the conditional Q P from P Q. The rules MPP and MIT enable us to use a conditiona l pumiss we first assume its antecedent Q at line 2 and derive its. together with either its antecedent or the negation of its consequent consequent P at line 3 CP at line 4 enables us to treat this. in order to obtain a certain conclusion either its consequent or as a proof of Q P from just assumption I On the right. the negation of its antecedent But how may we derive a conditional we give first the number ofthe assumed antecedent and second the. cone usion The most natural device is to take the antecedent of number of the concluded consequent On the left the assumption. the conditional we wish to prove as an extra assumption and aim 2 at line 3 disappears into the antecedent of the new conditional. to derive its consequent as a conclusion if we succeed we may take and we are left with I alone Always in an application of CP. this as a proof of the original conditional from the original the number of assumptions falls by one in this manner the one. assumptions if any For example given tbat all Germans are omitled being called the discharged assumption. The Predicate Calculus 1 Conditionals and Negation. 10 P Q R Q P R This proof uses all five rules of derivation introduced so far and. I I P Q R A, deserves study Aiming to prove a complex conditional we assume. its antecedent Q Pat line 2 and try to prove its consequent. 2 2 Q A P R Since this is conditional we assume its antecedent P at. 3 3 P A line 3 and after a series of steps using DN MIT and MPP we. 1 3 4 Q R 1 3 MPP derive its consequent R at line 7 Two steps of CP paralleling. the last two steps in the proof of 10 complete the job by discharging. 1 2 3 5 R 2 4 MPP,in turn the assumptions 3 and 2, 1 2 6 P R 3 5 CP Proofs 10 and 11 suggest a useful and important general method. 7 Q P R 2 6CP for discovering the proofs of sequents with complex conditionals as. conclusion After using the rule A for the assumptions given in the. A morc complicated example involving double use of CP in sequent we assume also the antecedent of the desired conditional. attempting to derive the conditional Q P R from conclusion and aim to prove its consequent if this is also a con. P Q R we first assume its antecedent Q at line 2 and ditional we assume its antecedent and aim to prove its consequent. aim to derive its consequent P R since this consequent is also we repeat this procedure until our target becomes to prove a non. conditional we assume its antecedent P at line 3 and aim to conditional conclusion If we can derive this from the assumptions. derive its consequent R This is achieved by two steps of MPP we now have the right number of CP steps applied in reverse order. lines 4 and 5 at line 6 we treat this by CP as a proof of will prove the original sequent. P R from assumptions 1 and 2 and we cite to the right line I end this section with a remark on two common fallacies so. 3 the assumption of the antecedent and line 5 the derivation common that they have received names In accordance with rule. of the consequent In turn we treat this at line 7 as a proof of MPP if a conditional is true and also its antccedent then we can. Q P R from assumption I alone and we cite to the right soundly derive its consequent If a conditional is true and also its. line 2 the assumption of its antecedent and line 6 the derivation consequent is it sOWld to derive its antecedent The following. of its consequent As before the assumptions on the left decrease example shows that it is not sound to do so it is true that if Napoleon. by one at each step ofep was German then he was European since all G ermans are Europeans. and it is true that Napoleon was European but it is fal se and so. II Q R Q P P R cannot soundly be deduced from these true premisses that Napoleon. was German To suppose that it is sound to derive the antecedent. I I Q R A of a conditional from tbe conditional and its consequent is to. 2 2 Q P A commit the fallac of a rming the conse uell Again in accordance. 3 3 P A with rule MIT if a conditional is true and also the negation of its. consequent tben we can soundly derive the negation of its antecedent. 3 4 P 3 DN, But it is not sound to derive the negationofa cond itional s consequem. 2 3 5 Q 2 4 MTI from the conditional itself and the negation of its antecedent and to. 2 3 6 Q 5DN suppose that it is sound is to commit the fallac of denying the. 1 2 3 7 R 1 6 MPP antecedent The same example may be used it is true that if. Napo eon was German then he was European and true also that. 1 2 8 P R 3 7 CP he was not German but it is not true that Napoleon was not. 9 Q P P R 2 8 CP European,Conjunction and Disjunction.
The Propositional Calculus J,a P Q PI Q,Put schematically the sequent. b P Q Q I P,1 P Q PI Qand,c P QI Q P,5 P Q Qf P, are sound patterns of reasoning as we have proved But the. 3 CONJUNCT I ON AND DISJUNCTION, 6P Q QfPand Of the four sentence forming operators on sentences mentioned in. 7P Q Pf Q the last section as being of importance to the logician only two have. so far been discussed if then and not In the present. are not sound patterns as we have shown by finding examples of section we introduce rules for arguments involving and. propositions P and Q such that the assumptions of 6 and 7 turn and either or. out true whilst their conclusions tum out false for it is a neces ijl l Let P and Q be any two propositions Then the proposition that. condition of a sound pattern of argument that it shall never lead us both P and Q is called the conjunction of P and Q and is written. from assumptions that are all true to a false conclusion 6 is in. fact the pattern of the fallacy of affirming the consequent and 7 P Q. that of the fallacy of denying the antecedent P and Q are called the conjunclS of the conjunction P Q Similarly. the proposition that either P or Q is called the disjunction of P and Q. and is written, Find proofs for the following sequents using the rules of derivation. introduced so far, a P P Q PI Q P and Q are called the disjuncts of the disjunction P v Q The.
b Q P R R Q I P symbol v is intended to remind classicists of the Latin vel as. opposed to aut for P v Q is understood not to exclude the. c P Q PI Q,possibility that both P and Q might be the case. d Q P Fe Q There are two rules or derivation concerning the rule of. e P Q QI P introduction and the rule of elimination and there are two rules. f P Q I Q P concerning v the rule o v in roduction and the rule olv elimination. g P QI Q P Introduction rules serve the purpose of enabling us to deri ve. h P Q I Q P conclusions containing or v whilst e limination rules serve the. purpose of enabling us to use premisses containing or v We. i P Q Q RI P R,discuss and exempl ify these rules in turn. j P Q R I P Q P R,J k P Q R S R HP Q S introduction 1. l P Q I Q R P R The rule of introduction 1 is exceptionally easy to master. m PI P Q Q Given any two propositions as premisses 1 permits us to derive. n PI Q R P R Q their conjunction as a conclusion The rule clearly corresponds to. a sound principle of reasoning for if A and B are the case separately. 2 Show that the following sequents are unsound patterns of argument it is obvious that A B must be the case The following proofs. by finding actual propositions for P and Q such that the assumption s e o emplify the use of 1. are true and the conclusion false, The Propositfonal Calculus 1 Conjunction and Disjunclioll. 12 P Q p Q 15P Q Q,I I P A I P Q A,2 2 Q A 2 Q I E.
1 2 3 P Q 1 2 1,16 P Q R P Q R, At line 3 by 1 we conclude the conjunction of the assumptions I I P Q R A. I and 2 To the right we cite 1 and 2 as the premisses for the 2 2 P Q A. application of 1 to tbe len we cite the pool of the assumptions on. which these premisses rest in this case themselves. 13 P Q R P Q R 1 2 5 Q R 1 3 MPP,I P Q R A 1 2 6 R 4 5 MPP. 2 2 P A I 7 P Q R 2 6 CP, We desire the conditional conclusion P Q R and so we. 2 3 4 P Q 2 3 1, assume its antecedent at line 2 and aim for R E is used at. 1 2 3 5 R 1 4MPP lines 3 nd 4 to obtain the conjuncts P and Q separately which. 1 2 6 Q R 3 5 CP e reqUJ re f r the MPP steps at lines 5 and 6 To the right. 7 P Q R 2 6CP In an appiJcatlOn of E we cite the conjunction employed as a. premiss and to the left tbe assumptions on which that conjunction. In attempting to prove the conditional P Q R we assume. first its antecedent P line 2 and second the antecedent of its The rules 1 and E are frequently used together in the same. proof For example, consequent Q line 3 A step of 1 at line 4 gives us the con.
junction of these assumptions enabling us to apply MPP at line. 5 to obtain R Two steps of CP complete the proof 17 P Q Q P. elimination E 2 P I E, The rule of elimination E is just as straightforward Given 3 Q I E. any conjunction as premiss E permits us to derive either conjunct 4 Q P 3 2 1. as a conclusion Again the rwe is evidently sound for if A B is. the case it is obvious that A separately and B separately must be. 18 Q R P Q P R,the case Here are examples,14 P Q P 2 2 P Q A. I P Q A 2 3 P 2 E,2 P I E 2 4 Q 2 E, The Propositiol al Calculus J Conjullclion alld Disjunction. 1 2 5 R 1 4 MPP assumptions apart from A itself on which C rests in its derivation. from A and any assumptions apart from B itself on which Crests. 1 2 6 P R 3 5 1, in its derivation from B Thus the typical situation for a step of vB. 7 P Q P R 2 6 CP is as follows we have a disjunction A v B as a premiss and wish to. derive a certain conclusion C we aim first to derive C from the. We desire the conditional conclusion P Q P R hence we first disjunct A and second to derive C from the second disjunct B. a ssume the antecedent P Q and aim for P R This aim is When these phases of the argument are completed we have the. translated into the aim for P and R separately from which P R situation described in a b and c above and can apply vB to. will follow by 1 P follows from P Q by E and so does Q obtain the conclusion C direct from A v B On the right we. which can be ust d in conjunction with line I to obtain R by MPP unfortunately need to cite five lines i the line where the disjunction. line 5 When 1 is used at line 6 the premisses are 3 and 5 A v B appears ii the line where A is assumed iii the line where. and these rest respectively on assumption 2 and assumptions I C is derived from A iv the line where B is assumed v the line. and 2 Hence the pool of Ihese I and 2 is cited to the lefl where C is derived from B And on the left the conclusion may. rest on rather a complex pool of assumptions derived from three. v introdrtctioll vI sources i any assumptions on which A v B rests ii any assump. The rule of v introduction wt name vI Given any proposition as tions on which C rests in its derivation from A though not A. premiss vI permits us to derive the disjunction of that proposition itself iii any assumptions on which C rests in its derivation from. and any proposition as a conclusion Thus from P as premiss we B though not B itself. may derive P v Q as a conclusion or Q v P as a conclusion and Though involved to state exactly the rule vE corresponds to an. here it makes no difference what proposition Q is Clearly the entirely natural principle of reasoning Suppose it is the case that. conclusion will in general be much weaker than the prcmiss in an either A or B i e that one of A or B is true and suppose that on the. application of vI It may that is be the case that either P or Q assumption A we can show C to be the case Le that if A holds C. even when it is no t the case that P None the less the rule is accept holds suppose also that on the assumption B we can still show that. able in the sense that when P is the case it must be also the case C holds i e that if B holds C also holds then C holds either way. that either P or Q For examplc it is the case that Charles I was For example you agree that either it is raining o r it is fine A v B. beheaded It follows that either he was beheaded or he was sent given that it is raining then it is not fit to go for a walk from A. to the electric chair even though of course he was not scnt to the we derive C given that it is fine then it must be very hot so that. electric chair A disjunction Pv Q is true if 01 least one of its again it is not fit to go for a walk from B we derive q Hence. disjuncts is true so that rule v cannot lead from a true premiss to either way it is not fit to go for a walk we conclude C. a fal se conclusion though it may lead to a dull one. 19PvQI QvP,v elimination vE I I Pv Q A, The rule of v elimination vE is rather more complex I first slate 2 2 P A.
it then explain and justify it and finally exemplify both it and vI. Let A B and C be a ny three propositions and suppose a that 2 3 QvP 2 vI. we are given that A v B b that from A as an assumption we can 4 4 Q A. derive C as conclusion c that from B as an assumption we can 4 5 Q v P 4 vI. derive C as conclusion then vE permits us to draw C as a conclusion. from any assumptions on which A v B rests together with any 6 Q v P 1 2 3 4 5 vE. The Proposilionai Calculus 1 Conjunction and Disjunction. On line I we assume Pv Q since this is a disjunction we aim to line 8 are those on which the disjunction P v Q rests itself 2. derive the conclusion Q v P from the first disjunct P assumed at together with any used to obtain P v R from 3 apart from 3. line 2 and aJso from the second disjunct Q assumed at line 4 itself none as line 4 reveals and any used to obtain P v R from. This is achieved on lines 3 and 5 by steps of vI which should be 5 apart from 5 itself namely 1 as line 7 reveals A step of. obvious At Hne 6 we conclude Q v P from assumption I CP completes the proof from 1 of the desired conditional. dircrctly since it follows from each disjunct separately On the right. we cite line I the disjunction line 2 assumption of first disjunct ZlP v QvR Qv PvR. line 3 derivation of conclusion from that disjunct line 4 1 I Pv QvR A. assumption of second disjunct and line 5 derivation of con. clusion from that disjunct To the left we cite any assumptions. on which the disjunction rests here 1 rests on itself which is 2 3 Pv R 2 vi. therefore cited together with any assumptions used to derive the 2 4 Qv PvR 3 vI. conclusion from the disjuncts apart from the disjuncts themselves. 5 5 Q v R A, inspection of the citations to the left of line 3 and 5 shows that. there are none such This proof should reveal the importance of 6 6 Q A. keeping accurate assumption records on the left of proofs lines 3 6 7 Qv Pv R 6 vI. and 5 here give indeed the right conclusion Q v P but not from. the right assumption which is 1 this is achieved on1y at line 6. which differs from lines 3 and 5 in the annotation on the left 8 9 PvR 8 vI. 8 10 Q v Pv R 9 vI,20 Q R PvQ PvR 5 II Qv Pv R 5 6 7 8 10 vB. 1 1 Q R A 12 Qv Pv R 1 2 4 5 11 vE, 3 J P A This proof deserves detailed study in the use both of vI and of vE. 4 Pv R Careful attention to bracketing is required The assumption is a. disjunction the second of whose disjuncts is a disjunction itself. 5 5 Q A The proof falls into two distinct parts lines 2 4 and Jines. 1 5 6 R 1 5 MPP 5Hll the first part establishes the desired conclusion from the. 1 5 7 PvR 6 vI first disjunct of the original disjunction line 4 and the second. 1 2 8 Pv R part establishes the same conclusion from the second disjunct. 2 3 4 5 7 vE, line 11 lhis should explain the final step of vE at line 12. 9 Pv Q Pv R 2 8 CP The second part lines 5HII which begins with a disjunctive. assumption also falls into two sub parts and involves a subsidiary. The desired conclusion here is conditional so we assume its step of vE at line 11 Lines 6 7 obtain the conciusion from the. antecedent P v Q line 2 and aim to derive P v R this assumption fir t disjunct Q of 5 and lines 8 IO obtain the conclusion. is a disjunction so we assume each disjunct in turn lines 3 and 5 from its second disjunct R Hence the tinaJ conclusion is obtained. and derive the conclusion P v R from each lines 4 and 7 Hence no less than five times in the proof from different assumptions each. the citation on the right at line 8 is 2 3 4 5 7 The assumptions at time. The Propositional Calculus I,Conjunclion and Disjunclion.
Reductio ad absurdum RAA 23P Pl P, The last rule to be introduced at this stage is in many ways the most I I P P A. powerful and the most useful it is easy to understand though a 2 2 P A. little difficult to state precisely We shall call it the rule of reductio. 1 2 3 P 1 2 MPP, ad absurdum RAA First we define a contradiction A conlradiction. is a conjunction the second conjunct of which is the negation of the 1 2 4 P P 2 3 1. first conjunct thusP P R R P Q P Q areall 5 P 2 4 RAA. contradictions Now suppose that from an assumption A together. perhaps with other assumptions we can derive a contradiction as a Again desiring P we assume P line 2 and obtain a contr adiction. conclusion then RAA permits us to derive A as a conclusion line 4 Given I therefore we conclude P by RAA The. from those other assumptions if any This rule rests on the sequent proved is striking and perhaps unexpected given that if. natural principle that if a contradiction can be deduced from a a proposition is the case then so is its negation we can conclude that. proposition A A cannot be true so that we are entitled to affirm its negation is true This is the first surprising result to be established. its negation A by our rules but there will be more. The rule RAA is particularly useful when we wish to derive. Here afe examples, negative conclusions It suggests that instead of attempting a. 22 P Q P QJ P direct proof we should assume the corresponding ajfirmalil e. proposition and aim to derive a contradiction thus indirectly. I 1 P Q A establishing the negative It can also be used however to establish. 2 2 P Q A affirmatives themselves via ON If we want to derive A we may. 3 3 P assume A and obtain a contradiction Hence by RAA we can. conclude A the negation of what we assumed and so by ON. 1 3 4 Q 1 3 MPP we obtain A It is a good general tip for proof discovery that when. 2 3 5 Q 2 3 MPP direct attempts fail often an RAA proof will succeed. So far ten rules of derivation have been introduced we shall need. 1 2 3 6 Q Q 4 5 1,no new ones until Chapter 3,1 2 7 P 3 6 RAA. Thi s is a typical example of the use of RAA Aiming at the con Find proofs for the following sequents. clusion P we assume line 3 P and hope to derive from it a. contradiction for if P leads to a contradiction we can conclude. b P Q R f Q P R, P by RAA We obtain the contradiction Q Q at line 6 and.
so conclude P at line 7 On the right we cite the assumption e P Q P R P Q R. which we are blaming for the contradiction the one whose negation d QfPvQ. we conclude in the RAA step here 3 and the contradiction e P Q PvQ. itself here 6 On the left as in a CP step the number of assump f P R Q R f P v Q R. tions naturally falls by one there being omitted the one which we g P Q R Sf P R Q S. blame for the cont radiction,ft S II P Q R st P v R Q v S.

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