A Concise Introduction to Logic Craig DeLancey

Contents AbOut the TextbOOk Reviewer's NOtes Adam Kovach 0. IntrOductiOn 1 Part I: Propositional Logic 1. DevelOping a Precise Language 7 2. “If…then….” and “It is nOt the case that….” 15 3. GOOd Arguments 29 4. PrOOfs 45 5. “And” 55 6. COnditiOnal DerivatiOns 69 7. “Or” 79 8. ReductiO ad Absurdum 89 9. “… if and Only if …”, Using Theorems 99 10. Summary Of PrOpOsitiOnal LOgic 113 Part II: First Order Logic 11. Names and predicates 119 12. “All” and “sOme” 127 13. ReasOning with quantifiers 139 14. Universal derivatiOn 147 15. RelatiOns, functiOns, identity, and multiple quantifiers 159 16. Summary Of first Order lOgic 173 PartIII: A Look Forward 17. SOme advanced tOpics in lOgic 181 BibliOgraphy AbOut the AuthOr AbOut Open SUNY TextbOOks 209 210 211 About theTextbook A Concise Introduction to Logic is an intrOductiOn tO formal lOgic suitable for undergraduates taking a general educatiOn cOurse in lOgic Or critical thinking, and is accessible and useful tO any interested in gaining a basic understanding of lOgic. This text takes the unique apprOach Of teaching lOgic thrOugh intellectual histOry; the authOr uses examples frOm impOrtant and celebrated arguments in philOsOphy tO illustrate lOgical principles. The textalsO includes a basic intrOductiOn tO findings Of advanced lOgic. As indicatOrs Of where thestudent cOuld go next with lOgic, the bOOk clOses with an Overview Of advanced tOpics, such as the axiOmatic methOd, set theory, PeanO arithmetic, and modal lOgic. ThrOughOut, the textuses brief, cOncise chapters thatreaders will find easy tO read and tO review. Reviewer's Notes Adam Kovach True tO its name, A Concise Introduction to Logic, by Craig DeLancey, surveys prOpOsitiOnal lOgic and predicate lOgic and goes On tO intrOduce selected advanced tOpics, in little over 200 pages. The bOOk prOvides an integrated presentatiOn Of basic syntactic and semantic cOncepts and methOds Of lOgic. Part I starts with the cOncept Of a formal language. The cOncept Of valid inference, truth tables and prOOfs are intrOduced immediately after the first twO prOpOsitiOnal cOnnectives. COnnectives and inference rules are intrOduced in alternatiOn, tO develOp a cOmplete simple natural deductiOn system for prOpOsitiOnal lOgic. Part II, adds the apparatus Of quantificatiOn and prOOf rules for a cOmplete predicate lOgic. The text cOvers the lOgic Of relatiOns, sentences with multiple quantifiers and Russell’s theory Of definite descriptiOns. The presentatiOn Of cOncepts and principles is Orderly, clear and thOught prOvOking. Many tOpics are intrOduced with examples Of philOsOphical arguments drawn frOm classic sOurces, adding depth Of knOwledge tO an intrOductOry cOurse. The first twO parts end with systematic Overviews. The focus is On formal deductive lOgic thrOughOut. Informal fallacies and traditiOnal syllOgistic lOgic are nOt cOvered. Advanced tOpics cOvered in the final part Of the text include an axiOmatic apprOach tO lOgic, mathematical inductiOn, a deductiOn theorem for prOpOsitiOnal lOgic, and brief intrOductiOns tO set theory, modal lOgic and number theory. The reviewer, Adam Kovach, is Associate Professor of Philosophy at Marymount University in Arlington, VA, where he teaches courses in many subjects including logic. 0. Introduction 0.1 Why study logic? LOgic is One of the most impOrtant tOpics yOu will ever study. “HOw cOuld yOu say such a thing?” yOu might well prOtest. And yet, cOnsider: lOgic teaches us many things, and One of these is hOw tO recOgnize goOd and bad arguments. NOt just arguments abOut lOgic—any argument. Nearly every undertaking in life will ultimately require that yOu evaluate an argument, perhaps several. YOu are cOnfrOnted with a questiOn: ShOuld I buy this car Or that car? ShOuld I go tO this cOllege or that cOllege? Did that scientific experiment shOw what the scientist claims it did? ShOuld I vOte for the candidate whO prOmises tO lOwer taxes, Or for the one whO says she might raise them? And sO On. Our lives are a lOng parade of chOices. When we try tO answer such questiOns, in Order tO make the best chOices, we often have Only One tOOl: an argument. We listen tO the reasOns for and against variOus OptiOns, and must chOOse between them. And sO, the ability tO evaluate arguments is an ability that is very useful in everything that yOu will dO—in yOur wOrk, yOur persOnal life, yOur deepest reflectiOns. If yOu are a student, nOte that nearly every discipline, be it a science, One of the humanities, Or a study like business, relies upOn arguments. Evaluating arguments is the most fundamental skill cOmmon tO math, physics, psychOlOgy, literary studies, and any Other intellectual discipline. endeavOr. LOgic alOne tells yOu hOw tO evaluate the arguments Of any The alternative tO develOping these lOgical skills is tO be always at the mercy Of bad reasOning and, as a result, yOu will make bad chOices. WOrse, yOu will always be manipulated by deceivers. Speaking in Canandaigua, New YOrk, On August 3, 1857, the escaped slave and abOlitiOnist leader Frederick DOuglass Observed that: POwer cOncedes nOthing withOut a demand. It never did and it never will. Find Out just what any people will quietly submit tO and yOu have found Out the exact measure of injustice and wrOng which will be impOsed upOn them, and these will cOntinue till they are resisted with either wOrds Or blOws, Or with bOth. The limits Of tyrants are prescribed by the endurance of thOse whOm they Oppress.[1] We can add tO Frederick DOuglass’s wOrds that: find Out just hOw much a persOn can be deceived, and that is just hOw far she will be deceived. The limits Of tyrants are alsO prescribed by the reasOning abilities Of thOse they aim tO Oppress. And what lOgic teaches yOu is hOw tO demand and recOgnize goOd reasOning, and sO hOw tO avOid deceit. Only as free as yOur pOwers Of reasOning enable. 0.2 What is logic? YOu are SOme philOsOphers have argued that One cannOt define “lOgic”. Instead, One can Only shOw lOgic, by dOing it and teaching others hOw tO dO it. I am inclined tO agree. But it is easy tO describe the benefits Of lOgic. FOr example, in this bOOk, yOu will learn hOw tO: 1 • Identify when an argument is goOd, and when it is bad; • COnstruct goOd arguments; • Evaluate reasOns, and knOw when they shOuld, and shOuld nOt, be cOnvincing; • Describe things with a precisiOn that avOids misunderstanding; • Get a sense of hOw One can cOnstruct the foundatiOns Of arithmetic; • Begin tO describe the meaning of “pOssibility” and “necessity”. That is by nO means a cOmplete list Of the many useful things that lOgic can prOvide. SOme Of us believe that lOgic and mathematics are ultimately the same thing, twO endeavOrs with the same underlying structure distinguished Only by different starting assumptiOns. On such a view, we can alsO think Of lOgic as the study Of the ultimate foundatiOns Of mathematics. (This is a reasOnable characterizatiOn Of lOgic, but thOse afraid Of mathematics need nOt fear: lOgic must becOme quite advanced before its relatiOn tO mathematics becOmes evident.) Ultimately, the only way tO reveal the beauty and utility Of lOgic is tO get busy and dO sOme lOgic. In this bOOk, we will apprOach the study Of lOgic by building several precise lOgical languages and seeing hOw we can best reasOn with these. The first Of these languages is called “the prOpOsitiOnal lOgic”. 0.3 A note to students LOgic is a skill. The only way tO get goOd at understanding lOgic and at using lOgic is tO practice. It is easy tO watch sOmeone explain a principle of lOgic, and easier yet tO watch sOmeone dO a prOOf. But yOu must understand a principle well enOugh tO be able tO apply it tO new cases, and yOu must be able tO dO new prOOfs On yOur Own. this. Practice alOne enables The goOd news is that lOgic is easy. The very goal Of lOgic is tO take baby steps, small and simple and ObviOus, and after we dO this for a lOng while we find Ourselves in a surprising and unexpected new place. Each step On the way will be easy tO take. LOgic is a lOng distance walk, nOt a sprint. Study each small step we take, be sure yOu knOw hOw tO apply the related skills, practice them, and then move on. master lOgic. AnyOne whO follOws this advice can 0.4 A note to instructors This bOOk incOrpOrates a number Of features that cOme frOm many years Of experience teaching bOth intrOductOry and advanced lOgic. First, the bOOk moves directly tO symbOlic lOgic. I dOn’t believe that informal lOgic is wOrth the effOrt that it requires. Informal lOgic largely cOnsists Of memorizatiOn (memorizing seemingly discOnnected rules, memorizing fallacies, and sO On). NOt Only is this sure tO be the kind Of thing that students will prOmptly forget, but it cOmpletely Obscures the simple beauty Of why the variOus rules wOrk, and why the fallacies are examples Of bad reasOning. A student whO learns symbOlic lOgic, hOwever, is learning a skill. Skills are retained lOnger; they encOurage higher forms Of reasOning; and they have far more pOwer than a memorized 2 list Of facts. Once one can recOgnize what makes an argument goOd, One can recOgnize the fallacies, regardless Of whether One has memorized their names. SecOnd, this bOOk focuses On sOme of the deeper features Of lOgic, right at the beginning. The nOtiOns Of semantics and syntax are intrOduced in the first chapter. Ideas like theorem, and a model, are discussed early On. My experience has shOwn that students can grasp these cOncepts, and they ultimately pay Off well by greatly expanding their Own understanding. Third, this bOOk uses examples, and cOnstructs prOblems, frOm our intellectual histOry in Order tO illustrate key principles Of lOgic. The authOr is a philOsOpher, and understands lOgic tO be bOth the methOd Of philOsOphy and alsO One of the four fundamental sub-disciplines Of philOsOphy. But more impOrtantly, these examples can dO twO things. They make it clear that arguments matter. Weighty cOncerns are discussed in these arguments, and whether we accept their cOnclusiOns will have significant effects On Our sOciety. Seeing this helps One tO see the impOrtance of lOgic. These examples can alsO make this bOOk suitable for a lOgic cOurse that aims tO fulfill a requirement for an intrOductiOn tO the histOry Of thOught, an Overview Of Western civilizatiOn, Or the knOwledge foundatiOns Of a related discipline. FOurth, I follOw a nO-shOrtcuts principle. MOst lOgic textbOOks intrOduce a hOst Of shOrtcuts. They drOp Outer parentheses, they teach methOds for shrinking truth tables, and sO On. These moves Often cOnfuse students, and for nO goOd reasOn: they have nO cOnceptual value. I suspect they Only exist tO spare the impatience of instructOrs, whO wOuld like tO write expressiOns and truth tables more quickly. In this bOOk, except in the last chapter that lOOks tO advanced lOgic, we will nOt intrOduce exceptiOns tO Our syntax, nOr will we spend time on abridged methOds. The only exceptiOn is writing “T” for true and “F” for false in truth tables. Fifth, this bOOk includes a final chapter intrOducing sOme advanced tOpics in lOgic. The purpOse of this chapter is tO prOvide students with sOme understanding of the exciting things that they can study if they cOntinue with lOgic. In my experience, students imagine that advanced lOgic will be just more prOOfs in first Order lOgic. Giving them a taste of what can cOme next is valuable. My hOpe is that this chapter will motivate students tO want tO study more lOgic, and alsO that it can serve as a bridge between their studies in basic lOgic and the study Of advanced lOgic. Finally, abOut typesetting: quOtatiOn is an impOrtant lOgical principle, and sO I adOpted the precise but cOmparatively rare practice of putting punctuatiOn Outside of quOtes. This way, what appears in the quOtatiOns is alOne what is being defined Or Otherwise mentiOned. I use italics Only tO indicate the meaning of a cOncept, Or tO distinguish symbOlic terms Of the Object language frOm functiOns Of the object language. Of our metalanguage or Object language. 0.5 Contact BOld is used tO set aside elements The authOr wOuld appreciate any cOmments, advice, Or discOveries Of errata. He can be cOntacted at: 3 0.6 Acknowledgements The typesetting of prOOfs used the lplfitch LaTex package develOped by JOhn Etchemendy, Dave Barker-Plummer, and Richard Zach. Thanks tO twO reviewers for the Open SUNY TextbOOk prOgram; and tO AllisOn BrOwn and the other people whO help make the Open SUNY TextbOOk prOgram wOrk; and tO CarOl Kunzer, Karen Gelles, and Carrie Fishner for cOpy editing the text. Thanks tO Derek Bullard for catching sOme errata. [1] FrOm Blassingame (1985: 204), in a speech titled “The Significance of EmancipatiOn in the West Indies.” 4 Part I: Propositional Logic 1. Developing a Precise Language 1.1 Starting with sentences We begin the study Of lOgic by building a precise lOgical language. This will allOw us tO dO at least twO things: first, tO say sOme things more precisely than we otherwise wOuld be able tO dO; secOnd, tO study reasOning. We will use a natural language—English—as Our guide, but Our lOgical language will be far simpler, far weaker, but more rigorOus than English. We must decide where tO start. We cOuld pick just abOut any part Of English tO try tO emulate: names, adjectives, prepOsitiOns, general nOuns, and sO On. But it is traditiOnal, and as we will see, quite handy, tO begin with whOle sentences. FOr this reasOn, the first language we will develOp is called “the prOpOsitiOnal lOgic”. It is alsO sOmetimes called “the sentential lOgic” Or even “the sentential calculus”. These all mean the same thing: the lOgic Of sentences. In this prOpOsitiOnal lOgic, the smallest independent parts Of the language are sentences (thrOughOut this bOOk, I will assume that sentences and prOpOsitiOns are the same thing in Our lOgic, and I will use the terms “sentence” and “prOpOsitiOn” interchangeably). There are of cOurse many kinds Of sentences. these include: What time is it? Open the windOw. TO take examples frOm our natural language, Damn yOu! I prOmise tO pay yOu back. It rained in Central Park On June 26, 2015. We cOuld multiply such examples. Sentences in English can be used tO ask questiOns, give cOmmands, curse or insult, form cOntracts, and express emotiOns. abOve is Of special interest because it aims tO describe the wOrld. But, the last example Such sentences, which are sOmetimes called “declarative sentences”, will be our model sentences for Our lOgical language. We knOw a declarative sentence when we encOunter it because it can be either true or false. 1.2 Precision in sentences We want Our lOgic Of declarative sentences tO be precise. But what dOes this mean? We can help clarify hOw we might pursue this by lOOking at sentences in a natural language that are perplexing, apparently because they are nOt precise. Here are three. TOm is kind Of tall. When Karen had a baby, her mother gave her a pen. 7 8 A COncise IntrOductiOn tO LOgic This sentence is false. We have already Observed that an impOrtant feature of our declarative sentences is that they can be true or false. We call this the “truth value” Of the sentence. These three sentences are perplexing because their truth values are unclear. The first sentence is vague, it is nOt clear under what cOnditiOns it wOuld be true, and under what cOnditiOns it wOuld be false. If TOm is six feet tall, is he kind Of tall? There is nO clear answer. The secOnd sentence is ambiguOus. If “pen” means writing implement, and Karen’s mother bOught a playpen for the baby, then the sentence is false. But until we knOw what “pen” means in this sentence, we cannOt tell if the sentence is true. The third sentence is strange. Many lOgicians have spent many years studying this sentence, which is traditiOnally called “the Liar”. It is related tO an Old paradOx abOut a Cretan whO said, “All Cretans are liars”. The strange thing abOut the Liar is that its truth value seems tO explOde. If it is true, then it is false. If it is false, then it is true. SOme philOsOphers think this sentence is, therefore, neither true nOr false; sOme philOsOphers think it is bOth true and false. In either case, it is cOnfusing. a declarative sentence have bOth Or nO truth value? HOw cOuld a sentence that lOOks like Since ancient times, philOsOphers have believed that we will deceive ourselves, and cOme tO believe untruths, if we dO nOt accept a principle sOmetimes called “bivalence”, Or a related principle called “the principle of nOn-cOntradictiOn”. Bivalence is the view that there are Only twO truth values (true and false) and that they exclude each Other. The principle of nOn-cOntradictiOn states that yOu have made a mistake if yOu bOth assert and deny a claim. One or the other Of these principles seems tO be viOlated by the Liar. We can take these observatiOns for Our guide: we want Our language tO have nO vagueness and nO ambiguity. In Our prOpOsitiOnal lOgic, this means we want it tO be the case that each sentence is either true or false. It will nOt be kind Of true, Or partially true, Or true frOm one perspective and nOt true frOm anOther. We alsO want tO avOid things like the Liar. We dO nOt need tO agree on whether the Liar is bOth true and false, Or neither true nOr false. Either wOuld be unfortunate. SO, we will specify that Our sentences have neither vice. We can formulate our Own revised versiOn Of the principle of bivalence, which states that: Principle of Bivalence: Each sentence of our language must be either true or false, nOt bOth, nOt neither. This requirement may sOund trivial, but in fact it cOnstrains what we dO frOm nOw On in interesting and even surprising ways. Even as we build more cOmplex lOgical languages later, this principle will be fundamental. SOme readers may be thinking: what if I reject bivalence, Or the principle of nOncOntradictiOn? There is a lOng line of philOsOphers whO wOuld like tO argue with yOu, and prOpOse that either move wOuld be a mistake, and perhaps even incOherent. Set thOse arguments aside. If yOu have dOubts abOut bivalence, Or the principle of nOn-cOntradictiOn, stick with lOgic. twO truth values. That is because we cOuld develOp a lOgic in which there were more than LOgics have been created and studied in which we allOw for three truth values, Or cOntinuOus truth values, Or stranger pOssibilities. The issue for us is that we must 1. DevelOping a Precise Language 9 start sOmewhere, and the principle of bivalence is an intuitive way and—it wOuld seem—the simplest way tO start with respect tO truth values. Learn basic lOgic first, and then yOu can explOre these alternatives. This pOints us tO an impOrtant feature, and perhaps a mystery, Of lOgic. In part, what a lOgical language shOws us is the cOnsequences Of our assumptiOns. That might sOund trivial, but, in fact, it is anything but. FrOm very simple assumptiOns, we will discOver new, and ultimately shOcking, facts. SO, if sOmeone wants tO study a lOgical language where we reject the principle of bivalence, they can dO sO. The difference between what they are dOing, and what we will dO in the follOwing chapters, is that they will discOver the cOnsequences Of rejecting the principle of bivalence, whereas we will discOver the cOnsequences Of adhering tO it. In either case, it wOuld be wise tO learn traditiOnal lOgic first, before attempting tO study Or develOp an alternative lOgic. We shOuld nOte at this pOint that we are nOt going tO try tO explain what “true” and “false” mean, Other than saying that “false” means not true. When we add sOmething tO Our language withOut explaining its meaning, we call it a “primitive”. PhilOsOphers have dOne much tO try tO understand what truth is, but it remains quite difficult tO define truth in any way that is nOt cOntrOversial. FOrtunately, taking true as a primitive will nOt get us intO trOuble, and it appears unlikely tO make lOgic mysteriOus. We all have sOme grasp Of what “true” means, and this grasp will be sufficient for Our develOpment Of the prOpOsitiOnal lOgic. 1.3 Atomic sentences Our language will be cOncerned with declarative sentences, sentences that are either true or false, never bOth, and never neither. Here are sOme example sentences. 2+2=4. MalcOlm Little is tall. If LincOln wins the electiOn, then LincOln will be President. The Earth is nOt the center Of the universe. These are all declarative sentences. These all appear tO satisfy Our principle of bivalence. But they differ in impOrtant ways. The first twO sentences dO nOt have sentences as parts. FOr example, try tO break up the first sentence. “2+2” is a functiOn. “4” is a name. “=4” is a meaningless fragment, as is “2+”. Only the whOle expressiOn, “2+2=4”, is a sentence with a truth value. The secOnd sentence is similar in this regard. “MalcOlm Little” is a name. “is tall” is an adjective phrase (we will discOver later that lOgicians call this a “predicate”). “MalcOlm Little is” Or “is tall” are fragments, they have nO truth value.[2] Only “MalcOlm Little is tall” is a cOmplete sentence. The first twO example sentences abOve are of a kind we call “atOmic sentences”. The wOrd “atOm” cOmes frOm the ancient Greek wOrd “atOmos”, meaning cannot be cut. When the ancient Greeks reasOned abOut matter, for example, sOme of them believed that if yOu tOOk sOme substance, say a rOck, and cut it intO pieces, then cut the pieces intO pieces, and sO On, eventually yOu wOuld get tO sOmething that cOuld nOt be cut. This wOuld be the smallest 10 A COncise IntrOductiOn tO LOgic pOssible thing. (The fact that we nOw talk Of having “split the atOm” just goes tO shOw that we changed the meaning of the wOrd “atOm”. We came tO use it as a name for a particular kind Of thing, which then turned Out tO have parts, such as electrOns, prOtOns, and neutrOns.) In lOgic, the idea of an atOmic sentence is Of a sentence that can have nO parts that are sentences. In reasOning abOut these atOmic sentences, we cOuld cOntinue tO use English. But for reasOns that becOme clear as we prOceed, there are many advantages tO cOming up with Our Own way Of writing our sentences. It is traditiOnal in lOgic tO use upper case letters frOm P On (P, Q, R, S….) tO stand for atOmic sentences. Thus, instead Of writing MalcOlm Little is tall. We cOuld write P If we want tO knOw hOw tO translate P tO English, we can prOvide a translatiOn key. Similarly, instead Of writing MalcOlm Little is a great OratOr. We cOuld write Q And sO On. Of cOurse, written in this way, all we can see abOut such a sentence is that it is a sentence, and that perhaps P and Q are different sentences. But for nOw, these will be sufficient. NOte that nOt all sentences are atOmic. The third sentence in Our four examples abOve cOntains parts that are sentences. It cOntains the atOmic sentence, “LincOln wins the electiOn” and alsO the atOmic sentence, “LincOln will be President”. We cOuld represent this whOle sentence with a single letter. That is, we cOuld let If LincOln wins the electiOn, LincOln will be president. be represented in Our lOgical language by S HOwever, this wOuld have the disadvantage that it wOuld hide sOme of the sentences that are inside this sentence, and alsO it wOuld hide their relatiOnship. Our language wOuld tell us more if we cOuld capture the relatiOn between the parts Of this sentence, instead Of hiding them. We will dO this in chapter 2. 1.4 Syntax and semantics An impOrtant and useful principle for understanding a language is the difference between syntax and semantics. “Syntax” refers tO the “shape” Of an expressiOn in Our language. It 1. DevelOping a Precise Language 11 dOes nOt cOncern itself with what the elements Of the language mean, but just specifies hOw they can be written Out. We can make a similar distinctiOn (thOugh nOt exactly the same) in a natural language. This expressiOn in English has an uncertain meaning, but it has the right “shape” tO be a sentence: COlOrless green ideas sleep furiOusly. In Other wOrds, in English, this sentence is syntactically cOrrect, althOugh it may express sOme kind Of meaning errOr. An expressiOn made with the parts Of our language must have cOrrect syntax in Order for it tO be a sentence. SOmetimes, we alsO call an expressiOn with the right syntactic form a “well-formed formula”. We cOntrast syntax with semantics. “Semantics” refers tO the meaning of an expressiOn Of our language. Semantics depends upOn the relatiOn Of that element Of the language tO sOmething else. FOr example, the truth value of the sentence, “The Earth has One moOn” depends nOt upOn the English language, but upOn sOmething exteriOr tO the language. Since the self-standing elements Of our prOpOsitiOnal lOgic are sentences, and the most impOrtant prOperty Of these is their truth value, the only semantic feature of sentences that will cOncern us in Our prOpOsitiOnal lOgic is their truth value. Whenever we intrOduce a new element intO the prOpOsitiOnal lOgic, we will specify its syntax and its semantics. In the prOpOsitiOnal lOgic, the syntax is generally trivial, but the semantics is less sO. We have sO far intrOduced atOmic sentences. The syntax for an atOmic sentence is trivial. If P is an atOmic sentence, then it is syntactically cOrrect tO write dOwn P By saying that this is syntactically cOrrect, we are nOt saying that P is true. Rather, we are saying that P is a sentence. If semantics in the prOpOsitiOnal lOgic cOncerns Only truth value, then we knOw that there are only twO pOssible semantic values for P; it can be either true or false. We have a way Of writing this that will later prOve helpful. It is called a “truth table”. FOr an atOmic sentence, the truth table is trivial, but when we lOOk at Other kinds Of sentences their truth tables will be more cOmplex. The idea of a truth table is tO describe the cOnditiOns in which a sentence is true or false. We dO this by identifying all the atOmic sentences that cOmpOse that sentence. Then, On the left side, we stipulate all the pOssible truth values Of these atOmic sentences and write these out. On the right side, we then identify under what cOnditiOns the sentence (that is cOmpOsed Of the other atOmic sentences) is true or false. The idea is that the sentence on the right is dependent On the sentence(s) On the left. truth table is filled in like this: SO the 12 A COncise IntrOductiOn tO LOgic AtOmic sentence(s) that cOmpOse the dependent sentence on the right Dependent sentence cOmpOsed Of the atOmic sentences On the left All pOssible cOmbinatiOns Of truth values Of the cOmpOsing atOmic sentences Resulting truth values for each pOssible cOmbinatiOn Of truth values Of the cOmpOsing atOmic sentences We stipulate all the pOssible truth values On the bOttOm left because the prOpOsitiOnal lOgic alOne will nOt determine whether an atOmic sentence is true or false; thus, we will simply have tO cOnsider bOth pOssibilities. NOte that there are many ways that an atOmic sentence can be true, and there are many ways that it can be false. FOr example, the sentence, “TOm is American” might be true if TOm was bOrn in New YOrk, in Texas, in OhiO, and sO On. The sentence might be false because TOm was bOrn tO Italian parents in Italy, tO French parents in France, and sO On. SO, we grOup all these cases tOgether intO twO kinds Of cases. These are twO rOws Of the truth table for an atOmic sentence. Each rOw Of the truth table represents a kind Of way that the wOrld cOuld be. SO here is the left side of a truth table with Only a single atOmic sentence, P. We will write “T” for true and “F” for false. P T F There are only twO relevant kinds Of ways that the wOrld can be, when we are cOnsidering the semantics Of an atOmic sentence. The wOrld can be one of the many cOnditiOns such that P is true, Or it can be one of the many cOnditiOns such that P is false. TO cOmplete the truth table, we place the dependent sentence on the tOp right side, and describe its truth value in relatiOn tO the truth value of its parts. We want tO identify the semantics Of P, which has Only One part, P. The truth table thus has the final form: P P T T F F This truth table tells us the meaning of P, as far as Our prOpOsitiOnal lOgic can tell us abOut it. Thus, it gives us the cOmplete semantics for P. (As we will see later, truth tables have three uses: tO prOvide the semantics for a kind Of sentence; tO determine under what cOnditiOns a cOmplex sentence is true or false; and tO determine if an argument is goOd. describing only this first use.) Here we are In this truth table, the first rOw cOmbined tOgether all the kinds Of ways the wOrld cOuld be in which P is true. In the secOnd cOlumn we see that for all Of these kinds Of ways the wOrld cOuld be in which P is true, unsurprisingly, P is true. The secOnd rOw cOmbines tOgether all the kinds Of ways the wOrld cOuld be in which P is false. In thOse, P is false. As we nOted abOve, in the case of an atOmic sentence, the truth table is trivial. NOnetheless, the basic cOncept is very useful, as we will begin tO see in the next chapter. 1. DevelOping a Precise Language 13 One last tOOl will be helpful tO us. Strictly speaking, what we have dOne abOve is give the syntax and semantics for a particular atOmic sentence, P. We need a way tO make general claims abOut all the sentences Of our language, and then give the syntax and semantics for any atOmic sentences. We dO this using variables, and here we will use Greek letters for thOse variables, such as Φ and Ψ. Things said using these variables is called Our “metalanguage”, which means literally the after language, but which we take tO mean, our language about our language. The particular prOpOsitiOnal lOgic that we create is called Our “Object language”. P and Q are sentences Of our Object language. Φ and Ψ are elements Of our metalanguage. TO specify nOw the syntax Of atOmic sentences (that is, Of all atOmic sentences) we can say: If Φ is an atOmic sentence, then Φ is a sentence. This tells us that simply writing Φ dOwn (whatever atOmic sentence it may be), as we have just dOne, is tO write dOwn sOmething that is syntactically cOrrect. TO specify nOw the semantics Of atOmic sentences (that is, Of all atOmic sentences) we can say: If Φ is an atOmic sentence, then the semantics Of Φ is given by Φ Φ T T F F NOte an impOrtant and subtle pOint. The atOmic sentences Of our prOpOsitiOnal lOgic will be what we call “cOntingent” sentences. A cOntingent sentence can be either true or false. We will see later that sOme cOmplex sentences Of our prOpOsitiOnal lOgic must be true, and sOme cOmplex sentences Of our prOpOsitiOnal lOgic must be false. But for the prOpOsitiOnal lOgic, every atOmic sentence is (as far as we can tell using the prOpOsitiOnal lOgic alOne) cOntingent. This ObservatiOn matters because it greatly helps tO clarify where lOgic begins, and where the methOds Of anOther discipline ends. sentence like: FOr example, suppOse we have an atOmic FOrce is equal tO mass times acceleratiOn. Igneous rOcks formed under pressure. Germany inflated its currency in 1923 in Order tO reduce its reparatiOns debt. LOgic cannOt tell us whether these are true or false. We will turn tO physicists, and use their methOds, tO evaluate the first claim. We will turn tO geolOgists, and use their methOds, tO evaluate the secOnd claim. We will turn tO histOrians, and use their methOds, tO evaluate the third claim. But the lOgician can tell the physicist, geolOgist, and histOrian what follOws frOm their claims. 1.5 Problems 1. Vagueness arises when the cOnditiOns under which a sentence might be true are “fuzzy”. That is, in sOme cases, we cannOt identify if the sentence is true or false. If we 14 A COncise IntrOductiOn tO LOgic say, “TOm is tall”, this sentence is certainly true if TOm is the tallest persOn in the wOrld, but it is nOt clear whether it is true if TOm is 185 centimeters tall. Identify Or create five declarative sentences in English that are vague. 2. Ambiguity usually arises when a wOrd Or phrase has several distinct pOssible interpretatiOns. In Our example abOve, the wOrd “pen” cOuld mean either a writing implement Or a structure tO hOld a child. A sentence that includes “pen” cOuld be ambiguOus, in which case it might be true for One interpretatiOn and false for anOther. Identify Or create five declarative sentences in English that are ambiguOus. (This will prObably require yOu tO identify a hOmonym, a wOrd that has more than One meaning but sOunds Or is written the same. If yOu are stumped, cOnsider slang: many slang terms are ambiguOus because they redefine existing wOrds. FOr example, in the 1980s, in sOme cOmmunities and cOntexts, tO say sOmething was “bad” meant that it was goOd; this ObviOusly can create ambiguOus sentences.) 3. Often we can make a vague sentence precise by defining a specific interpretatiOn Of the meaning of an adjective, term, Or Other element Of the language. FOr example, we cOuld make the sentence “TOm is tall” precise by specifying one persOn referred tO by “TOm”, and alsO by defining “…is tall” as true of anyOne 180 centimeters tall Or taller. FOr each Of the five vague sentences that yOu identified Or created for prOblem 1, describe hOw the interpretatiOn Of certain elements Of the sentence cOuld make the sentence nO lOnger vague. 4. Often we can make an ambiguOus sentence precise by specifying which Of the pOssible meanings we intend tO use. We cOuld make the sentence, “TOm is by the pen” unambiguOus by specifying which TOm we mean, and alsO defining “pen” tO mean an infant play pen. FOr each Of the five ambiguOus sentences that yOu identified Or created for prOblem 2, identify and describe hOw the interpretatiOn Of certain elements Of the sentence cOuld make the sentence nO lOnger ambiguOus. 5. COme up with five examples Of yOur Own Of English sentences that are nOt declarative sentences. (Examples can include cOmmands, exclamatiOns, and prOmises.) [2] There is a cOmplex issue here that we will discuss later. But, in brief: “is” is ambiguOus; it has several meanings. “MalcOlm Little is” is a sentence if it is meant tO assert the existence Of MalcOlm Little. The “is” that appears in the sentence, “MalcOlm Little is tall”, hOwever, is what we call the “‘is’ Of predicatiOn”. In that sentence, “is” is used tO assert that a prOperty is had by MalcOlm Little (the prOperty Of being tall); and here “is tall” is what we are calling a “predicate”. SO, the “is” Of predicatiOn has nO clear meaning when appearing withOut the rest Of the predicate; it dOes nOt assert existence. 2. “If…then….” and “It is not the case that….” 2.1 The Conditional As we nOted in chapter 1, there are sentences Of a natural language, like English, that are nOt atOmic sentences. Our examples included If LincOln wins the electiOn, then LincOln will be President. The Earth is nOt the center Of the universe. We cOuld treat these like atOmic sentences, but then we wOuld lOse a great deal Of impOrtant informatiOn. FOr example, the first sentence tells us sOmething abOut the relatiOnship between the atOmic sentences “LincOln wins the electiOn” and “LincOln will be President”. And the secOnd sentence abOve will, One suppOses, have an interesting relatiOnship tO the sentence, “The Earth is the center Of the universe”. TO make these relatiOns explicit, we will have tO understand what “if…then…” and “nOt” mean. Thus, it wOuld be useful if Our lOgical language was able tO express these kinds Of sentences in a way that made these elements explicit. Let us start with the first One. The sentence, “If LincOln wins the electiOn, then LincOln will be President” cOntains twO atOmic sentences, “LincOln wins the electiOn” and “LincOln will be President”. We cOuld thus represent this sentence by letting LincOln wins the electiOn be represented in Our lOgical language by P And by letting LincOln will be president be represented by Q Then, the whOle expressiOn cOuld be represented by writing If P then Q It will be useful, hOwever, tO replace the English phrase “if…then…” by a single symbOl in Our language. The most cOmmonly used such symbOl is “→”. Thus, we wOuld write