An Introduction To Formal Logic

1. WHAT IS LOGIC?2In this case the premises at least involve the same concepts as the conclusion. But thisargument still isn’t as good as (1) from earlier: unlike the earlier example, this argumentisn’t valid — its conclusion doesn’t follow from its premises. But what exactly does validity,or “following from,” consist in? What’s wrong with argument (2) as compared to (1)?One thing that’s worse about the second argument is that its conclusion is false: Poohisn’t a rabbit, he’s a bear! But that isn’t what distinguishes valid from invalid arguments ingeneral, because there are valid arguments with false conclusions, and invalid argumentswith true conclusions. For example:(3) All rabbits are birds.Winnie the Pooh is a rabbit. Winnie the Pooh is a bird.(4) All rabbits are mammalsBugs Bunny is a mammal. Bugs Bunny is a rabbit.Argument (3) is valid, but has a false conclusion. And argument (4) has a true conclusion,as well as true premises, but it still isn’t valid, because its conclusion doesn’t follow fromits premises: that all rabbits are mammals and that Bugs Bunny is a mammal doesn’t yetguarantee that Bugs Bunny is a rabbit.Validity isn’t determined by whether the premises or the conclusion are as a matter offact true. It rather has to do with the relationship between the premises and the conclusion.When we ask about validity we want to know whether, if all the premises were true, theconclusion would also have to be true. Put another way:An argument is VALID if and only if it is impossible for all of its premisesto be true but its conclusion false.Let’s unpack some of the concepts involved in the two definitions we’ve encountered a littlebit more. Exercises 1.1As you’ve seen, we always put the conclusion at the end of an argument and indicate itusing the three “therefore dots” Informally presented arguments don’t always have theconclusion at the end, however — it can appear at the beginning, or even in the middle. Ineach of the following arguments, highlight the phrase which expresses the conclusion:1. It is sunny. So I should take my sunglasses.2. It must have been sunny. I did wear my sunglasses, after all.3. No one but you has had their hands in the cookie-jar. And the scene of the crime islittered with cookie-crumbs. You’re the culprit!4. Miss Scarlett and Professor Plum were in the study at the time of the murder. AndReverend Green had the candlestick in the ballroom, and we know that there is noblood on his hands. Hence Colonel Mustard did it in the kitchen with the lead-piping.Recall, after all, that the gun had not been fired.

1. WHAT IS LOGIC?1.23Background ConceptsFirst, we said that an argument is a collection of STATEMENTS. Statements are sentencesthat are either true or false. Truth and falsity are called TRUTH - VALUES. The truth-valueof a statement is determined by what the world is like. A statement like ‘Syracuse is inNew York State’ describes the world as being a certain way. This statement happens to betrue because the world in fact is as the statement describes it. ‘Syracuse is in Alaska’, bycontrast, describes the world incorrectly, and is therefore false. As the ancient philosopher(and logician!) Aristotle put it in his book Metaphysics:“To say of what is that it is not, or of what is not that it is, is false, while to sayof what is that it is, and of what is not that it is not, is true.” (1011b25)It’s important to notice that not all English sentences count as statements in this sense.For example, none of the following sentences can be assessed as true or false: Welcome to the Syracuse Airport! Please have your ID ready. Are there any liquids in your bag?A sentence like ‘please have your ID ready’ isn’t meant to describe the world, but to askyou to do something. Similarly, ‘Welcome to Syracuse Airport’ is just a greeting, and isn’tmeant to offer an accurate or inaccurate description of the world. And although the answerto the last question on this list has a truth-value, the question itself doesn’t. In general, thingslike greetings, requests, orders, and questions don’t have truth-values, therefore don’t countas statements, and for that reason can’t be premises or conclusions of arguments. Thoughit’s important to keep this point in mind, moving forward we’ll generally use the words“statement” and “sentence” interchangeably.Next, notice that because a statement’s truth value depends on what the world is like,its truth-value could have been different if the world had been different. For example, thesentence ‘Rieppel is a philosopher’ is in fact true, but if I had taken up a different career itwould have been false. Conversely, ‘Rieppel is a professional juggler’ is false, but if I hadgone to juggling school instead of continuing with philosophy, it would have been true.Philosophers often invoke the notion of a POSSIBLE WORLD in this connection. The ideais that besides the actual world, there are various other possible worlds, other ways thingscould have been — alternative histories, or alternative universes, if you like. ‘Rieppel is aprofessional juggler’ is false in the actual world, but it is true in other possible worlds, oneswhere I went to juggling school or joined the circus. Similarly, ‘Rieppel is a philosopher’is as a matter of fact true, but it is false in other possible worlds where I didn’t pursuephilosophy. Sentences like this, which are true in some possible worlds and false in others,are said to be CONTINGENT.Other sentences are not contingent. For example, ‘Syracuse either is or is not in NewYork State’ isn’t just true in the actual world, it’s true in every possible world, that is, it’s aNECESSARY TRUTH . Mathematical truths are another example: ‘2 2 4’ is again true inevery possible world, and therefore a necessary truth. At the other extreme, sentences like‘Syracuse both is and is not in New York State’ and ‘2 2 5’ are false in every possibleworld, or NECESSARILY FALSE.

1. WHAT IS LOGIC?4Returning to arguments, you can think of the notion of validity in terms of possible worldstoo. We said that an argument is valid just in case it’s impossible for all of its premises to betrue but its conclusion false. Phrased in terms of possible worlds, this becomes:An argument is VALID if and only if there is no possible world where allof its premises are true but its conclusion is false.Equivalently put: an argument is valid if its conclusion is true in every possible world inwhich all of its premises are true.This gives us an informal way to test whether an argument is valid: we imagine a worldwhere all the premises are true, and then ask ourselves whether the conclusion would haveto be true as well at that world. If so, the argument is valid. On the other hand, if you canimagine a world where all the premises are true but the conclusion is still false, the argumentisn’t valid. So again, whether an argument is valid or not isn’t determined by whether itspremises and conclusion are actually true or false. It’s about the connection between them— whether there’s any way for the premises to be true but the conclusion false.There are other logical concepts that we’ll encounter in this class that involve the notionsof necessity and possibility, besides validity. Some we’ve already mentioned:. A sentence is CONTINGENT if and only if it is possible for it to betrue, and also possible for it to be false. A sentence is a NECESSARY TRUTH if and only if it is not possiblefor it to be false. A sentence is a NECESSARY FALSEHOOD if and only if it is not possible for it to be true. Two sentences are CONTRADICTORY if and only if they necessarilyhave opposite truth values. Two sentence are EQUIVALENT if and only if they necessarily havethe same truth value. A collection of sentences is JOINTLY CONSISTENT if and only if itis possible for all of them to be true together, and JOINTLY INCON SISTENT otherwise.Notice that these concepts apply to different things. Whereas the first three concern properties of single sentences, the next two concern properties of pairs of sentences, and the lastones concern properties of whole collections of sentences. Validity is again slightly different, because it is a property had (or lacked) by only those collections of sentences that alsohave a designated conclusion, i.e. by those collections that are arguments. Exercises 1.2A. For each of the following: is it necessarily true, necessarily false, or contingent?1. Caesar crossed the Rubicon.2. Someone once crossed the Rubicon.3. No one has ever crossed the Rubicon.

1. WHAT IS LOGIC?54. If Caesar crossed the Rubicon, then someone has.5. Even though Caesar crossed the Rubicon, no one has ever crossed the Rubicon.6. If anyone has ever crossed the Rubicon, it was Caesar.B. Consider the following sentences:G1.G2.G3.G4.There are at least four giraffes at the zoo.There are exactly seven gorillas at the zoo.There are not more than two martians at the zoo.Every giraffe at the zoo is a martian.Now, for each of the following, determine if the sentences in question are jointly consistent or jointly inconsistent:1.2.3.4.G2, G3, and G4G1, G3, and G4G1, G2, and G4G1, G2, and G3C. Could there be:1. Jointly consistent sentences, one of which is necessarily false?2. Jointly consistent sentences, one of which is a necessary truth?3. Jointly inconsistent sentences, one of which is a necessary truth?In each case: if so, give an example; if not, explain why not.1.3Good and Bad ArgumentsBeing valid is certainly one thing that makes for a good argument, intuitively speaking. Butthere’s more to being a good argument than that.First off, if an argument has an obviously false premise, then even if it is valid, it remainsof limited interest because it doesn’t establish its conclusion. By contrast, if an argument isvalid and all of its premises are true, then we know that its conclusion has to be true too.Arguments like this are said to be sound:An argument is SOUND if and only it (i) is valid, and (ii) has premises thatare in fact true.Arguments are generally intended to be not just valid, but sound. So if you’re faced withan argument, in a philosophy class or elsewhere, whose conclusion you want to resist, youhave two options: you can either try to show that the argument is not valid, or you can tryto show that one of its premises is false (and the argument therefore isn’t sound). Whatyou can’t do is accept it as valid, and concede that its premises are true, but still reject theconclusion as false: if it’s valid, and has true premises, its conclusion has to be true too.Although it’s important in practice to determine whether or not the premises of an argument are in fact true, it is (for the most part) not the job of logic to do this. The job of logic

1. WHAT IS LOGIC?6is just to determine whether or not an argument is valid. The task of determining whetherthe argument’s premises are in fact true (and the argument sound) is usually best left toexperts in the relevant field: biologists, historians, philosophers, physicists, economists, orwhomever.A second way in which validity is not all there is to good argumentation comes out if youconsider the following:In January 2016, it snowed in Syracuse.In January 2017, it snowed in Syracuse.In January 2018, it snowed in Syracuse.In January 2019, it snowed in Syracuse.In January 2020, it snowed in Syracuse.So: It snows every January in Syracuse.This argument generalizes from observations about several past cases to a conclusionabout all cases. The argument isn’t valid in our sense, because even if it snowed in Januaryin many recent years, that doesn’t mean it’s impossible for it not to snow in some futureyear. The argument could be made stronger by adding additional premises, about othersnowy Syracuse Januaries in the past. But however many premises of this sort we add, theargument will remain invalid.That doesn’t mean that it’s a bad argument. Arguments like this one are called INDUC TIVE arguments, and they are often used legitimately and with great success in science andeveryday life. In this book, we will set aside the difficult question of what makes for a goodinductive argument. What logic studies is the different notion of DEDUCTIVE validity —where the truth of the premises has to guarantee the truth of the conclusion — and this willbe the focus of our concern. Exercises 1.3Here are some exercises to test your understanding of deductive validity and related concepts we’ve discussed. For these questions, you don’t need to worry about the distinctionbetween validity and “validity in virtue of logical form” to be discussed in §1.4 below. Youshould just use the definition of validity we gave in §1.1 and §1.2 above.A. Which of the following arguments are valid? Which are invalid?1.Socrates is a man.All men are carrots. Socrates is a carrot.2.Abe Lincoln was either born in Illinois or he was once president.Abe Lincoln was never president. Abe Lincoln was born in Illinois.3.If I pull the trigger, Abe Lincoln will die.I do not pull the trigger. Abe Lincoln will not die.

1. WHAT IS LOGIC?74.Abe Lincoln was either from France or from Luxemborg.Abe Lincoln was not from Luxemborg. Abe Lincoln was from France.5.If the world were to end today, then I would not need to get up tomorrow morning.I will need to get up tomorrow morning. The world will not end today.6.Joe is now 19 years old.Joe is now 87 years old. Bob is now 20 years old.B. Could there be:1.2.3.4.5.6.7.8.9.10.11.12.13.14.A valid argument that has one false premise and one true premise?A valid argument that has only false premises but a true conclusion?A valid argument with only false premises and a false conclusion?A valid argument with only true premise but a false conclusion?An invalid argument with only true premises and a true conclusion?An invalid argument with only false premises but a true conclusion?A sound argument with a false conclusion?A sound argument with at least one false premise?An invalid argument that can be made valid by the addition of a new premise?A valid argument that can be made invalid by the addition of a new premise?A valid argument, the conclusion of which is necessarily false?An invalid argument, the conclusion of which is necessarily true?A valid argument whose premises are jointly inconsistent?A valid argument with only one premise?In each case: if so, give an example; if not, explain why not.1.4Formal ValidityThere’s one last complication we have to address before setting out on our investigation oflogic. Consider the following arguments:(5) This beach ball is green all over. This beach ball is not red all over.(6) Reihan is a bachelor. Reihan is not married.In both cases it is impossible for the premise to be true and the conclusion false: if something’s green all over it can’t be any other color, and being unmarried is part of what it is tobe a bachelor. Both arguments are therefore valid.But there’s an important difference between valid arguments like these and one like thefollowing:

1. WHAT IS LOGIC?8(7) Jenny is either happy or sad.Jenny is not happy. Jenny is sad.This argument is also valid, but there’s more. It has a special structure, or logical form, thatwe might represent as follows:A or Bnot-A BThis is an excellent structure for an argument to have, because any argument of this formwill be valid, no matter what sentences we put in place of A and B! Or consider our BugsBunny argument, which has the structure represented to the right:(1) All rabbits are mammals.Bugs Bunny is a rabbit. Bugs Bunny is a mammal.All F are Ga is F a is GAgain, this is a great structure, because any argument of this form will be valid, no matterwhat predicates we put in for F and G or what name we put in for a.The general point is that arguments like (7) and (1) are valid simply in virtue of theirlogical form. They each exhibit a logical structure which renders any argument with thatstructure valid. By contrast, arguments (5) and (6), though valid, are not valid in virtue oftheir logical form. For example, the form of (6) could be represented as follows:(6) Reihan is a bachelor Reihan is not married.a is F a is not-G.Here the premise ascribes a certain property (being a bachelor) to an individual, and theconclusion then denies another property (being married) of that individual. However, thereare other arguments that share this same structure but aren’t valid:Reihan is a runner. Reihan is not married.This isn’t valid because it’s trivial to imagine a world where Reihan is a runner but alsomarried. What made argument (6) valid wasn’t its logical form, but the specific meaningsof the words ‘bachelor’ and ‘married’ that occur in its premise and its conclusion. Otherarguments that have the same form but involve words with different meanings (e.g. ‘runner’)may no longer be valid. Logic is all about identifying patterns that make arguments valid.So it only cares about FORMALLY VALID arguments like (1) and (7), not arguments like (5)and (6) that are valid for reasons other than their logical form.Due to logic’s concern with form, we will approach the task of distinguishing valid frominvalid arguments in an indirect way. We will first introduce a formal language in whichwe can symbolize English arguments. Doing this lets us represent the logical forms of thosearguments. We will then give a precise definition of validity for arguments cast in this formalnotation. And this will in turn give us our indirect means of distinguishing valid from invalid

1. WHAT IS LOGIC?9arguments in English: if an English argument can be symbolized as a valid argument in ourformal notation, then that English argument is formally valid.In fact, we will study two systems of logic, involving two different formal languages.These systems will differ in what words they treat as logical constants, that is, which wordsthey treat as indicative of logically significant structure, or form. The first system we willstudy is Truth-Functional Logic (or TFL). It will let us represent the structure of argumentslike (7) via the symbolization to the right:(7) Jenny is either happy or sad.Jenny is not happy. Jenny is sad.(A B) A BThis language treats words like ‘either . . . or’ and ‘not’ as logical constants (represented by‘ ’ and ‘ ’ respectively), and will use upper-case letters to represent complete statements(like ‘A’ for ‘Jenny is happy’, and ‘B’ for ‘Jenny is sad’). TFL is the topic of Part 1 of thisbook.In Part 2 of the book, we will turn to First-Order Logic (or FOL). It will let us representthe structure of things like the Bugs Bunny argument via the symbolization to the right:(1) All rabbits are mammals.Bugs Bunny is a rabbit. Bugs Bunny is a mammal. x(Fx Gx)Fa GaThis system extends Truth-Functional Logic by treating words like ‘all’ as logical constants(represented by ‘ ’).1 It also lets us represent some of the internal structure of a simplestatement like ‘Bugs Bunny is a rabbit’, showing that it is formed by combining the name‘Bugs Bunny’ (represented as ‘a’) with the predicate ‘is a rabbit’ (represented by ‘F’). Withthis very short preview out of the way, let’s get started with logic!1 Atthis point you might be wondering how logicians decide which English words to treat as logical constants, and represent by special logical symbols that indicate “logical form.” This is a diffi

I first learned formal logic from Michael Byrd at UW-Madison, using Lemmon’s Beginning Logic, and first taught logic in 2008 as a teaching assistant for Branden Fitelson at UC Berkeley, using Forbes’ Modern Logic. When I s