CRITICAL THINKING PHIL 1010
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Date Created: 09/21/15
CHAPTER 4 SUPPLEMENTAL MATERIAL PROPOSITIONAL DEDUCTIVE ARGUMENTS ELEMENTARY ARGUMENT FORMS Now that we have discussed simple and compound statements we are prepared to examine a common type of argument propositional deductive arguments In Chapter Two we saw that all arguments are either deductive or inductive This supplement is concerned with how the good form test applies to one kind of deductive argument propositional deductive arguments Some of the most famous arguments ever made are propositional deductive arguments You may have heard of this one from the French philosopher Descartes l I think Therefore 2 I am Propositional deductive arguments are particularly crucial in some elds For example they made possible the development of the computer and they are frequently used in computer programming Propositional deductive arguments are deductive arguments that rely on the logical relationships between statements The term propositional deduction comes from the fact that statements are propositions BOX Connections In Chapter One we said that a statement is a proposition that is either true or false The best way to identify propositional arguments is to look for one of the eight argument forms that we will discuss later in this supplement This is not a perfect guide to identifying propositional arguments because there are an infinite number of propositional argument forms Fortunately the vast majority of them will never be made by any human being When it comes to the propositional arguments that you are likely to find in your courses the eight kinds of propositional arguments we will study here will cover the vast majority of cases BOX Technical Term Sentential Logic TruthFunctional Logic Sentential logic and truthfunctional logic are other names for propositional deduction We chose to use propositional logic instead of sentential logic because the former more emphatically expresses the distinction between sentences and statements that was introduced in Chapter One We chose to use propositional logic instead of truth functional logic because it is the name you are most likely to see in other courses Five of the eight propositional argument forms are valid They pass the good form test The other three propositional argument forms are invalid They fail the good form test When you compose your own arguments knowing which of these common forms are valid pass the good form test and which are invalid fail the good form test will help you think more clearly write more clearly and be a more convincing speaker The first three of the eight forms contain a premise that is a disjunction We will refer to these first three as disjunctive argument forms The next five contain at least one premise that is a conditional We will refer to these five as conditional argument forms BOX Technical Term Syllogism In addition to being the most common forms found in propositional deduction these eight argument forms share another feature They all have precisely two premises and a conclusion Arguments with this form with two premises and a conclusion are sometimes called syllogisms 1 Disjunctive Forms The first step in understanding the three disjunctive argument forms is to note the surprising fact that the English word or has two different meanings Or is ambiguous Let us return to two examples discussed above Recall that Bret said d I bet Jaime had either a bagel sandwich a hamburger or a chicken sandwich This is a disjunctive compound statement that contains three other statements d1 Jaime had a bagel sandwich d2 Jaime had a hamburger d3 Jaime had a chicken sandwich Unless there is some odd context it is reasonable to assume that Bret thinks that Jaime had only one sandwich for lunch She had either a bagel sandwich a hamburger or a chicken sandwich and she did not 1 have a bagel sandwich and a hamburger 2 have a bagel sandwich and a chicken sandwich 3 have a hamburger and a chicken sandwich 4 have a bagel sandwich a hamburger and a chicken sandwich In other words when Bret says d I bet Jaime had either a bagel sandwich a hamburger or a chicken sandwich he is using or to mean that Jaime had one and only one of the three sandwiches This use of or is called an exclusive or and it is the most common use of or Suppose that you order a corned beef sandwich in a restaurant and the waiter says e That comes with fries or a salad Which would you like The waiter is using an exclusive or His or means one or the other but not both The corned beef sandwich comes with fries or a salad but not both If you want both fries and a salad you will have to pay more Recall the case of the acceptable forms of identification that one can use to establish one s identity when one goes to vote in the state of Georgia In the state of Georgia there are seventeen different forms of acceptable identification Here again is the list from the web site of the Secretary of State of Georgia Voters are required to present identification at their polling place prior to casting their ballot Proper identification shall consist of any one of the following 1 A valid Georgia driver s license 2 A valid identification card issued by a branch department agency or entity of the State of Georgia any other state or the United States authorized by law to issue personal identification 3 A valid United States passport 4 A valid employee identification card containing a photograph of the elector and issued by any branch department agency or entity of the United States government this state or any county municipality board authority or other entity of this state 5 A valid employee identi cation card containing a photograph of the elector and issued by any employer of the elector in the ordinary course of such employer39s business 6 A valid student identi cation card containing a photograph of the elector from any public or private college university or postgraduate technical or professional school located within the State of Georgia 7 A valid Georgia license to carry a pistol or revolver 8 A valid pilot39s license issued by the Federal Aviation Administration or other authorized agency of the United States 9 A valid United States military identi cation card 10 A certi ed copy ofthe elector s birth certi cate 11 A valid social security card 12 Certi ed naturalization documentation 13 A certi ed copy of court records showing adoption name or sex change 14 A current utility bill or a legible copy thereof showing the name and address of the elector 15 A bank statement or a legible copy thereof showing the name and address of the elector 16 A government check or paycheck or a legible copy thereof showing the name and address of the elector or 17 A government document or a legible copy thereof showing the name and address of the elector1 Look at the end of line 16 You will see the word or This or in called an inclusive or because a voter in Georgia may present more than one of these seventeen items There is no rule that one must present one and only one of the seventeen items above Suppose that this is a list of possible items that are available at a pizza restaurant Anchovies Artichoke Hearts Bacon Banana Peppers Black Olives Broccoli BBQ Chicken Chicken Extra Cheese Feta Cheese Fresh Basil Green Olives Green Peppers Ground Beef Ham Italian Sausage Jalapenos Jerk Chicken Meatballs Minced Garlic Mushrooms Onions Pepperoncini Pepperoni Pesto Pineapple Portobello Roasted Red Peppers Spinach Sundried Tomatoes Tempeh Tofu Tomatoes When you visit you ask the waitress what you can have on your pizza and having memorized the list she rattles off the whole list f You can have anchovies or artichoke hearts or bacon or banana peppers or The or that the waitress is using an inclusive or You can have any combination of toppings that you want on your pizza assuming that you are willing to pay for them When or is used to mean one or the other but not both it is called an exclusive or When or is used to mean one or the other or bot it is called an inclusive or The or in the corned beef example is an exclusive or The or in the pizza example is an inclusive or When someone is speaking or writing the ambiguity of or is usually resolved by context It is not likely that you had any problems understanding that you only get fries exclusive or salad with your corned beef sandwich but that you can have anchovies inclusive or artichoke hearts inclusive or bacon inclusive or banana peppers inclusive or on your pizza The context of ordering in various different things in restaurants resolves the ambiguity of or However there are cases in which people misunderstand each other because one person is using an exclusive or and another person is thinking of an inclusive or BOX Hint for Future Courses Or in Other Languages Some languages do not have the ambiguity problems found in the English word or For example Latin has two words for or The Latin word vel means or in its inclusive sense and the Latin word out means or in its exclusive sense 1a Denying a Disjunct Now that we have covered this surprising feature of or in English we are in a position to look at the argument forms that reply use or the disjunctive argument forms The first disjunctive form we will look at is called denying a disjunct An argument that denies a disjunct has one of the following two forms al 1 or 82 b lSl or 2 2 Not 81 2 Not 82 Therefore Therefore 3 2 3 81 Remember that Sn is a variable that stands for statements 81 and 2 are statements Each of these two forms contains two premises and a conclusion The first premise of each is a disjunction The second premise asserts that one of the disjuncts is false It denies one of the disjuncts This is what gives this argument form its name The only difference between these two forms is which of the disjuncts is denied Arguments that deny a disjunct are valid They pass the good form test You can see that it is a valid form because you know that in order for a disjunction to be true one of its disjuncts must be true The first premise of an argument that denies a disjunct tells us that a disjunction is true So we know that at least one of the disjuncts is true The second premise of an argument that denies a disjunct tells us that one of the disjuncts is false This guarantees that the other disjunct will be true and this is precisely what is said in the conclusion of an argument with this form BOX Technical Terms Disjunctive Syllogism Alternative Syllogism Disjunctive syllogism and alternative syllogism are both names sometimes used for an argument that denies the disjunct Suppose that the catalog for your university states All students must take Math 1113 or higher Let also suppose that you have a friend Irene who is enrolled at your university In that case Irene must take Math 1113 or a math course with a number higher than 1113 Let us call a math course with a number higher than 1113 a higherlevel math course In that case the following statement is true h Irene must take Math 1113 or a higherlevel math course If we let S1 Irene must take Math 1113 S2 Irene must take a higherlevel math course then h is in the form of the first premise of an argument that denies a disjunct Now let us make a further supposition Suppose that we know that Irene will not take Math 1113 She is planning to graduate after next semester and Math 1113 is not offered next semester So we know Not S1 This is the second premise in form a above We can conclude that Irene will take a higher level math course We can conclude S2 We have made an argument that denies a disjunct Here is the standardization of our argument 1 Irene must take Math 1113 or a higherlevel math course 2 Irene cannot take Math 1113 Therefore 3 Irene must take a higher math course It has the form a noted above a 1 S1 or S2 2 Not S 1 Therefore 3 S2 This argument about Irene has the form of denying the disjunct Because we know that this is a good form we know that the argument about Irene passes the good form test Be careful not to forget about the true premises test Any argument that denies a disjunct is a valid argument But not all valid arguments are sound Not all valid arguments pass the true premises test If either premise of an argument that denies a disjunct is false the argument is not a good argument even though it is still a valid argument 1b Fallacy Affirming an Inclusive Disjunct Arguments that affirm an inclusive disjunct are invalid They fail the good form test Here are the two forms of an argument that affirms a disjunct al 1 or 82 b lSl or 2 2 1 2 82 Therefore Therefore 3 Not 82 3 Not 81 This form differs in two ways from denying a disjunct First the second premise affirms one of the disjuncts instead of denying it Second the conclusion denies one of the disjuncts instead of affirming it It is easy to see why the two forms are frequently confused Both forms make use of three similar looking statements In the invalid form the argument assumes that one and only one disjunct can be true The invalid form erroneously uses the exclusive or Suppose that your local pizza place is not good It only offers two toppings pepperoni and mushroom Consider the following argument 1 You can have pepperoni or mushrooms on your pizza 2 You have ordered pepperoni on your pizza Therefore 3 You cannot order mushrooms on your pizza This invalid argument is an instance of form a of affirming an inclusive disjunct Sl You have pepperoni on your pizza SZ You have mushrooms on your pizza 1 1 or 2 2 S 1 Therefore 3 Not 82 This argument says that if you order pepperoni on your pizza then this guarantees that you will not order mushrooms This is not a valid argument You could have a pizza with pepperoni and mushrooms 1c Affirming an Exclusive Disjunct In English exclusive disjunctions are much more common than inclusive disjunctions Therefore it is important to note that while affirming an inclusive disjunct results in an invalid argument form affirming an exclusive disjunct is a valid argument form An argument affirms an exclusive disjunct when it has one of the following forms a l 81 or 2 and not both b 1 1 or 2 and not both 2 1 2 82 Therefore Therefore 3 Not 82 3 Not 81 Notice that the first premise of these forms is the statement of an exclusive or An exclusive means one or the other and not both Let us suppose that in a particular state it happens that there are only two political parties the Democrats and the Republicans There are no other political parties in this state Then suppose that someone made the following argument 1 Either the Democrats or the Republicans won the election 2 The Democrats won the election Therefore 3 The Republicans did not win the election Elections are not like pizza Only one party can win So the or in the rst premise is an exclusive or This argument is an instance of version a of af rming an exclusive disjunct Sl The Democrats win the election SZ The Republicans win the election 1 1 or 2 and not both 2 S 1 Therefore 3 Not 82 This is a valid argument form You can now see why it is important to determine whether or not the or that an author is using is inclusive or exclusive If it is exclusive then af rming a disjunct is a valid argument form It is an instance of af rming an exclusive disjunct If an author s or is inclusive then af rming a disjunct is an invalid argument It is an instance of affirming an inclusive disjunct You need to read carefully to determine which or the author intends to use When you are evaluating arguments with disjunctive forms do not forget that the order that the premises are stated is irrelevant to the quality of an argument For this reason the disjunctive statements that we have as the rst premise in the forms above may not come rst in the argument as it is written in English Fallacy False Dichotomy Our focus in this supplement is on the good form test as it applies to propositional arguments But there is a fallacy involving the true premises test that appears suf ciently frequently in arguments with a disjunctive form that we will mention it here The Fallacy of False Dichotomy occurs when the rst premise of an argument with disjunctive form is false because there are other alternatives besides the two presented in this premise Consider the following argument 1 Dwight is either a biology major or a nance major 2 Dwight is not a biology major Therefore 3 Dwight is a nance major This argument is a perfectly valid instance of an argument that denies a disjunct It passes the good form test But it fails the true premises test The rst premise is false In most colleges and universities there are many different disciplines in which one could major It is extremely unlikely that you could nd a school that had only two majors Even if by chance one did nd a particular school say a technical college where there were only two majors it is extremely unlikely that the two would be as different as biology and nance So the disjunctive premise is mentioning only two alternatives when in fact there are almost certainly many more Therefore the fact that the student in question is not a biology major is not suf cient for us to conclude that he is a nance major The argument is valid It is valid because if the two premises were both true then the conclusion would also be true However it is unlikely that the disjunctive premise is true Arguments with a disjunctive form sometimes commit the Fallacy of False Dichotomy because people sometimes see things in black and white They often overlook alternatives As you are evaluating disjunctive arguments to see whether they are valid make sure that the disjunctive premise is not a False Dichotomy The most interesting subjects for argument are probably too complex to be easily represented by only two alternatives 2 Conditional Forms Now we are ready to consider some deductive arguments with a form that relies on a conditional All these arguments have at least one premise that is a conditional statement We will consider four conditional forms that contain one conditional premise In two of these four the second premise either affirms or denies the antecedent In the other two the second premise either affirms or denies the consequent There is one conditional form that has two premises and a conclusion all of which are conditionals At this point you may want to go back and review Section I of Chapter 4 to make sure that you remember the difference between the antecedent and the consequent of a conditional If you do not have that rmly in mind the following discussion will be difficult to follow BOX Technical Term Hypothetical Syllogisms The five argument forms that we call conditional arguments forms are also called hypothetical syllogisms 2a Affirming the Antecedent An argument with this form has one premise that is a conditional statement and a second premise that affirms the antecedent of that conditional On the basis of that affirmation a conclusion is drawn that affirms the consequent Here is the form 1 If S1 then 82 2 S 1 Therefore 3 2 This conditional argument form is valid Let us look again at our example about snow 1 If it is snowing then the temperature is below 32 degrees 2 It is snowing Therefore 3 The temperature is below 32 degrees This argument is an instance of affirming the antecedent Any argument with this form is valid Any argument with this form passes the good form test If the two premises are true then the conclusion must be true Affirming the antecedent is a commonly used form in many disciplines Rodrigo Enrique ElizondoOmana and three fellow researchers recently conducted an experiment to determine whether the pace of a course had any effect on student learning in anatomy courses 2 Anatomy is the study of the structure and organs of living things The experiment was conducted on two groups of anatomy students who were given an anatomy test before and after they took the course One group took the course in onehour class periods over twenty weeks The other group took the same course in twohour class periods over nine weeks The same material was covered and the same tests were administered The researchers reasoned that if a difference emerged in the groups s scores there would be reason to deduce that pace was a factor in learning It turned out that there was a signi cant difference in scores The group of students in the twentyweek course scored significantly higher than the group who took the course in nine weeks ElizondoOmana concluded that the pace of a course is an in uential factor in how well students learn anatomy in anatomy courses The argument may be paraphrased as follows 1 If there is a significant difference in test scores of groups of students who took the course for different lengths of time then the pace at which the students took the course must have been a factor in their learning 2 There was a significant difference in test scores of groups of students who took the course for different lengths of time Therefore 3 The pace at which the students took the course must have been a factor in their learning Let Sl stand for the antecedent of the conditional l and let 82 stand for the consequent of this conditional ElizondoOmana s argument has this form 1 If S1 then 82 2 S 1 Therefore 3 2 This argument form is one of most simple and most basic argument forms It is perhaps the most common of all argument forms Arguments with this form pass the good form test BOX Technical Term M odus Ponens The Latin name for an argument that affirms its antecedent is modus ponens It is frequently abbreViated MP 2b Fallacy Denying the Antecedent Suppose that the results of ElizondoOmana s study had been different Suppose that there had been no significant difference among the students group scores The researchers might then have made the following argument 1 If there is a significant difference in test scores of groups of students who took the course for different lengths of time then the pace at which the students took the course must have been a factor in their learning 2 There was not a significant difference in test scores of groups of students who took the course for different lengths of time Therefore 3 The pace at which the students took the course must not have been a factor in their learning Using 81 and 2 as we did just above this argument has the following form 1 If S1 then 82 2 Not S 1 Therefore 3 Not S Arguments that have this form are called arguments that deny the antecedent Arguments that deny the antecedent are similar to arguments that af rm the antecedent They both have the same conditional premise However the second argument has a premise that denies the antecedent instead of af rming it However this second argument form fails the good form test It is not valid In order to understand why this argument form is not a valid argument recall our example about snow Suppose that someone were to deny the antecedent like this 1 If it is snowing then the temperature is below 32 degrees 2 It is not snowing Therefore 3 The temperature is not below 32 degrees This example shows that arguments that deny the antecedent are invalid Comparing the snow argument and our modi ed version of ElizondoOmana s argument illustrates the value of standardization Until you standardize the modi ed version of Elizondo Omana s argument it is hard to see that fails the good form test But once it is standardized it is easier to see that it is an instance of denying the antecedent The snow argument just above allows us to see that arguments that deny the antecedent have bad form Important ElizondoOmana did not make this fallacious argument We modi ed his argument to illustrate an argument with this bad argument form Let us look back at our modi ed version of ElizondoOmana s argument There are many ways to measure student achievement The study we have been talking about tested students at the beginning and at the end of the course Such tests are called pretests and posttests They can be valuable evidence about student learning but this method of measuring student achievement is fallible It does not provide certain knowledge For example it is possible that the two classes had different instructors and that the instructor of the longer course was not as good as the instructor of the shorter course It is possible that the worse instructor canceled out the positive effects of the longer course format This shows that it is possible for the conclusion of this argument 1 If there is a signi cant difference in test scores of groups of students who took the course for different lengths of time then the pace at which the students took the course must have been a factor in their learning 2 There was not a signi cant difference in test scores of groups of students who took the course for different lengths of time Therefore 3 The pace at which the students took the course must not have been a factor in their learning to be false even though both premises are true If there is the slightest possibility that the conclusion of a deductive argument can be false when its premises are true then that argument has bad form 2c Denying the Consequent Arguments that deny the consequent have this form 1 If S1 then 82 2 Not 82 Therefore
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