Arguments in Context Unit IV An Introduction to Deductive Arguments Chapter 11 The Language of Deduction

CHAPTER 11 The Language of Deduction SECTION INTRODUCTION In the last chapter , we identified some distinctive features arguments , and saw that the unique logical strength of deductive arguments is a result of their form . We then looked at a number of common deductive forms , and closed the chapter by identifying two argument forms that are often mistakenly taken to be deductive . In this chapter , we will focus on the language of deduction , specifically on the language of . As we saw in the last chapter , strict play an important role in a number of mon deductive argument forms . However , we can express conditional relationships in a variety of ways , and this means that we can express common deductive arguments in a variety of ways . As a result , we will be in a better position to spot deductive argument patterns when we run across them . Moreover , some of the lessons we will take from our considerations here will be relevant when we turn to a consideration of the language of inductive arguments . SECTION WHAT STRICT CONDITIONAL STATEMENTS SAY ( AND WHAT THEY DO ) As we have seen , are common in everyday reasoning . We can use in all kinds of ways , to make promises , threats , or offers , to talk about causal relationships , or to talk about the way things might have been ( if the Broncos had won Super Bowl Given this , let us between which are used to describe existing them conditional that are used for other purposes ( conditional promises , threats , conditional offers , For the sake of thinking about deductive arguments , we will set aside these other kinds and focus solely on conditional statements . Thus , we will focus on like the following If sugar is put in the water , then it will dissolve . Ifyou are driving without a license , then you are breaking the law . Ifthe bird is a mallard , then it is a member ofthe genus Arias . And not on like Ifyou clean the garage , then I lend you the car . Ifyou do hand over your wallet , then I take it from you . 123

124 THADDEUS To this point we have discussed conditional statements as if they were always expressed in the following way if then ' We have called this the Standard Form for , and have identified the if part as the antecedent , and the then part as the consequent . However , everyday language is a rich vehicle for expression , and gives us many ways to convey or articulate ideas . Conditional claims are no exception . Because oftheir importance to deductive arguments , we will identify and discuss some of the most common alternative ways of expressing . Towards this end , let us begin by thinking about the relationship ( that a strict conditional statement expresses . Let start with an example we seen before If a shape is a square , then it is a rectangle . This strict conditional is claiming that anytime the antecedent is true , and there is a shape that is a square , the consequent is also true the shape is a rectangle as well . This to all strict they claim that the truth of the antecedent is always accompanied by the truth of the . Let us say that a strict conditional claims that the antecedent is sufficient for the consequent because the conditional is saying that knowing the antecedent is true is sufficient for knowing that the consequent is true too , since the one is always accompanied by the other . If the relationship between the antecedent and the consequent in a strict conditional is one of sufficiency , what is the relationship between the consequent and the antecedent ?

It can be tempting to think that the relationship from the consequent to the antecedent is the same as the antecedent to the consequent , but the example shows us that this is not the case . If a shape is a square , then it is a rectangle , butjust because it is a rectangle does mean it is a square . After all , a square is only one kind of rectangle . squares and rectangles by So what is the relationship that the consequent has to the antecedent ?

Go back to the conditional a shape is a square , then it is a This tells us that squares are rectangles , but it also tells us that shapes that are not rectangles also are not squares . After all , since a square is a kind of rectangle , if you knew that shape was not a rectangle you would automatically know that it could be a square . In other words , it tells us that being a rectangle is a crucial to being a square . We can say , then , that in this case , the truth of the consequent is required , or is necessary for , the truth of the antecedent . This conclusion to all strict . In light ofthis , we will say that in a strict conditional the consequent is necessary for the antecedent .

THE LANGUAGE OF DEDUCTION 125 The fact that the relationship the antecedent has to the consequent is not the same as the relationship the consequent has to the antecedent it is worth emphasizing . According to psychologists and linguists who study reasoning , we have a tendency to misread conditional claims . The problem is that in certain stances we read as if the consequent were sufficient for the antecedent . We tend to read If A , then as saying also that If , then A . Of course nobody reads the conditional above about squares and rectangles in this way , because we know that would be false . It is in cases where we are so sure about the antecedent and the consequent that we become prone to making this That said , sufficiency and necessity are , in a sense , two sides of the same coin . From the perspective of the antecedent the relation is one of sufficiency for the consequent , while from the perspective of the the relation is one of necessity to the antecedent . More specifically A strict conditional of the form ifA , then simultaneously expresses two relations . It says that A is sufficient for AND that is necessary for A . The idea that a conditional expresses two relationships strikes some people as at first . However , we deal with this situation all the time in everyday life . Consider a different relation to the left of . When I say that I am standing to the left of you , I am simultaneously saying that you are standing to my right . So too , to say that A is the grandparent of is at the same time to say that is the grandchild of A . The same goes for say that A is sufficient for is to say simultaneously that is necessary for Antecedent Consequent If Then is necessary for Two Sides of the Same Coin Necessary and Conditions SECTION NECESSARY AND SUFFICIENT CONDITIONS We began Section by noting that there are many ways to express other than the standard ( if then ) form . Our discussion of the meaning of allows us to easily see two alternatives . To say that one thing is sufficient for another , or that one is necessary for the other , is to express a conditional relationship . Consider Ex . A person being legally eligible to buy alcohol in this state is sufficient for that person being at least 21 years of age .

126 THADDEUS Because this sentence asserts a sufficiency relation we can express it with a conditional . We can put it into standard form as follows If a person is legally eligible to buy alcohol in this state , then they are at least 21 years of age . There is nothing unique to this example . Using A ' and as variables , we can generalize on this and say that any sentence with the form A is a sufficient condition for can be expressed using the conditional , then or A is sufficient for IfA , then The same goes for necessary conditions . Consider the following example Ex . A person being at least 21 years of age is a necessary condition for a person being legally eligible to buy hol in this state . Because this sentence asserts a necessity relation we can express it with a conditional . Since a being at least 21 years of age is the necessary condition , then it will serve as the consequent in the form conditional , as follows If a person is legally eligible to buy alcohol in this state , then they are at least 21 years of age . Again , there is nothing unique to this example . We can generalize on this and say that for a person to say that A is a necessary condition for isjust to express the conditional If , then A or A is a necessary condition for If , then A The conditional above tells us that everybody who is eligible to legally buy alcohol in this state is at least exceptions . Given this , we can formulate an equivalent variant of this conditional as follows . sider somebody who is not at least 21 years of age , say they are 19 . What do we know about them given the truth ofthe conditional above ?

We know that they are not eligible to legally buy alcohol , and we can express this with the following conditional If a person is not at least 21 years of age , then they are not legally eligible to buy alcohol in this state , Being at least 21 years of age is a necessary condition for being legally eligible , so ifthis necessary condition is not met , it will also be true that a person will not be legally eligible . Again , there is nothing distinctive about this conditional expressed above , and we can say in general that IfA , then , then notA There is a special name for this equivalent variant of A , then it is called the contrapositive . SECTION ALTERNATIVE WAYS OF EXPRESSING There are ways of expressing beyond the language of necessary and sufficient conditions . haps the most obvious alternative to the standard form occurs when we use the term if out of order . For example , a person might say The bird is a member ofthe genus , if it is a mallard . The presence ofthe if here is a big clue that we got a conditional , and we can reformulate it into standard form as follows If the bird is a mallard , then it is a member of the genus . We can generalize on this

THE LANGUAGE OF DEDUCTION 127 , ifA ifA , then The term only if is another way that we sometimes express . The use of only if to express is especially common in legal language . The word only can be tricky , however , and it is important to see that the addition of only to an if fundamentally changes the meaning of the sentence . Consider the following example ( i ) A person is breaking the law ifthey are speeding . ii ) A person is breaking the law only are speeding . These two sentences say very different things . The first sentence says that ifa person is speeding , then they are breaking the law . That true . The second sentence is telling us that the only way a person might break the law is by speeding . But this is certainly false , since we can break the law by littering , blowing through a stop sign , etc . Adding an only to the if ' significantly changes the meaning of the sentence , and so we not transform only if sentences into in the same way we would an if sentence . So how do we do it ?

Let look at a new example Ex . A person is considered an active member of the club only if they have paid their yearly dues . This sentence tells us that paying yearly dues is a crucial part of being qualified as an active member . Put otherwise if a person has paid their yearly dues , they are not considered an active member . This gives us a conditional , and we can generalize on this result . A only , then notA Now that we also are aware of the contrapositive , we can see that A only if is equivalent to ifA , then as follows A only , then notA ifA , then Perhaps surprisingly all and every claims like all dogs are mammals or every student at this school has been issued an ID card express conditional claims as well . In order to see this , think about what they say . To say that every student has been issued an ID card is to say that every single student has an ID card , no exceptions , and so being a student is sufficient for having an ID card . Putting this into the standard form for requires modifying the sentence a bit ( since in standard form the antecedent and consequent are independent clauses ) and this can be a little awkward . We can transform the sentence every student at this school has been issued and ID card as follows If somebody is a person at this school , then they are a person who has an ID card . This pattern and we can say that All As are If something is an A , then something is a Claims like all As are tell us about membership , namely that things that are As are also members of . Similarly , claims about exclusion that use no or none can also express . For example , to say that no reptiles are is to say that all reptiles are not exceptions . Put , being a reptile is sufficient for not being . Like all claims , translating a no claim into

128 THADDEUS ROBINSON a conditional requires modifying the sentence a bit . The in this case is If something is a reptile , then it is not . The general pattern here is No As are If something is an A , then it is not a There are other ways of expressing , but this list captures the most common ways . In the next section , we will see how these alternatives figure into the deductive argument forms we learned . SECTION RECOGNIZING DEDUCTIVE FORMS Being able to recognize will allow us to recognize instances ofthe argument forms discussed in Chapter 10 . Consider the following argument Ex . Do worry , you will graduate . I know that the registrar cleared you , and this is sufficient for graduating . The claim that being cleared by the registrar is sufficient for graduating expresses a conditional . ing this into a conditional we can see that this argument is an instance of the form Modus . The registrar cleared you . Ifthe registrar clears you , then you will graduate . So , you will graduate . Here is another example Ex . Look , I told battery does have a charge , and the car will start only if the battery has a charge . So , I am sorry to say , the car wo start . We know that the claim car will start only if the battery has a charge expresses the if the battery does not have a charge , then the car wo start . Given this , we can see that this argument is actually an instance of Modus , and is consequently a deductive argument . If the battery does not have a charge , then the car wo start . The battery does not have a charge . So , the car wo start . Wait . We have seen that a strict conditional is equivalent to its contrapositive . Thus , in this case although we transformed the claim the car will start only if the battery has a charge into the the battery does not have a charge , then the car wo start we could have transformed it into a different ( but equivalent ) conditional . We might have transformed it to say the car starts , then the battery has a Had we transformed it this way , we would have standardized the argument as follows

THE LANGUAGE OF DEDUCTION 129 . Ifthe car starts , then the battery has a charge . The battery does not have a charge . So , the car wo start . But this is a case of Modus . So which is it ?

Is this argument an instance of Modus or Modus ?

The answer , in short , is that it depends on how you transform the conditional claim into standard form . This might seem problematic , but there is no need to worry . How we transform the conditional into standard form will dictate what form the argument takes , but we will never be able to accurately represent a tive argument as if we transform our conditional claims correctly . EXERCISES Exercise Set Identify at least one necessary condition for being a citizen of the United States . You might need to do a little bit of research . Identify a necessary condition for getting an A in this class . The terms consequent and conclusion sound very similar . How are they different ?

Exercise Set Directions Transform each of the following conditional claims into the correct standard form conditional . The chemical will dissolve , if the theory is correct . Earning 93 ofall available points in this course is sufficient for earning an A ' for the course . You may enter only if you are exercising your first amendment rights . All submitted photos are the property of .

130 THADDEUS No current Naval Commanders are convicted felons . Having a social security number is a necessary condition for obtaining a drivers license . Exercise Set Directions For each of the following arguments standardize it using standard form and if the ment is one of the kinds we have studied , it as such . Modus , the Consequent , etc . Understanding is impossible if words refer only to private sensations in the minds of speakers . We clearly do understand each other . Therefore , words do not refer merely to private sensations . Getting a on your AP exam is sufficient for getting college credit . Since I got a on the exam , I will get college credit . Future presidents will be allowed to serve a third term only if the Amendment is repealed . The Amendment will not be repealed . Therefore , future presidents will not be allowed to serve a third term . Every time Louis is tired , he edgy . He edgy today , so he must be tired today . You can do well in math classes only if you keep up with the assignments . You keep up with the , so you do well in math classes . Most senior citizens go to bed before 11 so my probably goes to bed before 11 The alternator is not working properly if the ammeter shows a negative reading . The current reading of the ammeter is negative . So , the alternator is not working properly . Manuel will play only ifthe situation is hopeless . But the situation is hopeless . So Manuel will play .

THE LANGUAGE OF DEDUCTION 131 Candidate Flores will lose the election if he does win and County , but he win either ofthese counties , so Flores will lose the election . 10 I passed Go and passing Go is a sufficient condition for collecting 200 , so hand over 200 ! 11 Elephants have been known to bury their dead . Elephants would bury their dead only ifthey have a concept . So , elephants have a concept . 12 Having a good technical education is a necessary condition for being a good engineer . Given the quality of Charity technical education , she will surely make a good engineer . 13 We can not worship a god or gods ifwe do have a capacity to form a concept ofthe divine . And we can not have a capacity to form a concept of the divine if we do have a capacity to form concepts transcending sense perception . So we can not worship god or gods if we do we have a capacity to form concepts sense perception . 14 We know that Philip is a nonresident . Since no citizens are , it follows that Philip is not a citizen . Notes . 1972 ) Psychology of Reasoning Structure and content . Harvard . Press , 61 .

Unit Summary In this unit we focused on deductive arguments . As we have seen , deductive arguments are indefeasible and have maximal logical strength due to their form . Although there are many argumentative forms , we specifically identified four especially common ones . We also looked at two forms people commonly fuse for deductive forms . Because strict are such an important part of many deductive , we talked through what a strict conditional is really saying , and identified a variety of different ways we express . This led , finally , to a consideration of the many ways that deductive arguments , themselves , can be expressed . EY Deductive Arguments Disjunctive ( Defeasible Arguments Denying the Antecedent ( DA ) Indefeasible Arguments Affirming the Consequent ( AC ) Argument Form Conditional Statement Strict Sufficiency Relation Modus ( Necessity Relation Modus ( Contrapositive Hypothetical ( FURTHER READING People have been studying deductive arguments for over 2000 years going back at least to the work of in the century . The study of deductive arguments is known as deductive or formal logic and there are many solid introductory books on the topic . Two in particular that stand out are A Concise duction to Logic by Patrick Hurley and The Power by Frances and Daniel . 133