Propositional Deductive Arguments: Understanding Elementary Argument Forms, Study notes of Philosophy

Download Propositional Deductive Arguments: Understanding Elementary Argument Forms and more Study notes Philosophy in PDF only on Docsity! 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. (1) I think. Therefore, (2) I am. Propositional deductive arguments are particularly crucial in some fields. 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, Truth-Functional Logic Sentential logic and truth-functional 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. 1 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; 2 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 higher-level math course. In that case, the following statement is true: (h) Irene must take Math 1113 or a higher-level math course. If we let S1 = Irene must take Math 1113. S2 = Irene must take a higher-level 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 higher-level 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 S1. 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. 5 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: (a) (1) S1 or S2. (b) (1) S1 or S2. (2) S1. (2) S2. Therefore, Therefore, (3) Not S2. (3) Not S1. 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. S1 = You have pepperoni on your pizza. S2 = You have mushrooms on your pizza. (1) S1 or S2. (2) S1. Therefore, (3) Not S2. 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) (1) S1 or S2 and not both. (b) (1) S1 or S2 and not both. (2) S1. (2) S2. Therefore, Therefore, (3) Not S2. (3) Not S1. 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. 6 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 first premise is an exclusive or. This argument is an instance of version (a) of affirming an exclusive disjunct. S1 = The Democrats win the election. S2 = The Republicans win the election. (1) S1 or S2 and not both. (2) S1. Therefore, (3) Not S2. 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 affirming a disjunct is a valid argument form. It is an instance of affirming an exclusive disjunct. If an author’s or is inclusive, then affirming 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 first premise in the forms above may not come first 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 sufficiently frequently in arguments with a disjunctive form that we will mention it here. The Fallacy of False Dichotomy occurs when the first 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 finance major. (2) Dwight is not a biology major. Therefore, (3) Dwight is a finance 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 first 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 find a school that had only two majors. Even if by chance one did find 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 finance. 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 sufficient for us to conclude that he is a finance major. The 7 (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 affirm the antecedent. They both have the same conditional premise. However, the second argument has a premise that denies the antecedent instead of affirming 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 modified version of Elizondo-Omana’s argument illustrates the value of standardization. Until you standardize the modified 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: Elizondo-Omana did not make this fallacious argument. We modified his argument to illustrate an argument with this bad argument form.) Let us look back at our modified version of Elizondo-Omana’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 pre-tests and post-tests. 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 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. 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 S2. (2) Not S2. Therefore, 10 (3) Not S1. The first premise is a conditional statement. The second premise denies the consequent of that conditional statement. In other words, the second premise claims that the consequent of the first premise is false. The conclusion drawn is that the antecedent of the first premise is false. Let us use our snow example again. (1) If it is snowing, then the temperature is below 32 degrees. (2) The temperature is not below 32 degrees. Therefore, (3) It is not snowing. This is valid. If the true premises are true, then the conclusion must be true. Knight Steel and T. Franklin Williams wrote an editorial in the Journal of the American Geriatrics Society. 3 They are concerned “that so little has been accomplished in the field of geriatric medicine.” They are also concerned about the fact that few doctors are being trained as geriatric specialists. They believe that the lack of specialists in geriatrics is the reason that there has been little progress in the field. Paraphrasing part of their reasoning yields the following. (1) In order for us to make good progress in the field of geriatric medicine, there needs to be a sufficient number of geriatric specialists trained to conduct research. (2) There is not a sufficient number of new geriatric specialists being trained to conduct research. Therefore, (3) We are not making good progress in the field of geriatric medicine. This is a valid argument. It is an instance of denying the consequent and so this argument passes the good form test. BOX, Technical Term: Modus Tollens The Latin name for an argument that denies the consequent is modus tollens. It is frequently abbreviated MT. 2d. Fallacy: Affirming the Consequent Arguments that affirm the consequent have the following form: (1) If S1, then S2. (2) S2. Therefore, (3) S3. Imagine that the call for more geriatric specialists was answered. Say that in the next five years, a significant number of new doctors trained in that specialty. Would that necessarily solve the problem that occupied Steel and Williams? Consider the following modified version of Steel and Williams’s argument. (1) In order for us to make good progress in the field of geriatric medicine, there needs to be a sufficient number of geriatric specialists trained to conduct research. (2) There is a sufficient number of new geriatric specialists being trained to conduct research. 11 Therefore, (3) We are making good progress in the field of geriatric medicine. Does this argument have a valid form? Let us look again at our snow example. (1) If it is snowing, then the temperature is below 32 degrees. (2) The temperature is below 32 degrees. Therefore, (3) It is snowing. This argument form is invalid. You have all experienced days when the temperature was below 32 degrees but it did not snow. So the modified version of Steel and Williams’s argument is also invalid. Both of the previous two arguments are instances of affirming the consequent. Returning our modified version of Steel and Williams’s argument, ask yourself whether there is any possibility that there could be a large number of specialists trained in a field without there being significant accomplishment in that field. Keep in mind that it is the possibility you are considering. For instance, it is possible that newly trained specialists cannot complete research for many years. Research takes time, experience, and financial investment. If a thousand new geriatric specialists trained and were then sent to work as on-call physicians throughout the country, would it be likely that such employment would enhance the probability of research in the field? It is not likely. Even when a sufficient number of trained specialists are practicing in a field, that fact does not guarantee accomplishment of research goals. The forms 2a, 2b, 2c, and 2d are easily confused. The patterns of affirming and denying antecedents and consequents can start to spin in one’s head. If you think that an argument falls into one of these argument forms, you have to carefully examine the argument to be sure that you have correctly identified the right conditional argument form. 2e. Tri-Conditional The tri-conditional argument form contains three conditional statements. It looks like this: (1) If S1, then S2. (2) If S2, then S3. Therefore, (3) If S1, then S3. This is a valid form of argument. If the premises are true, then the conclusion must be true. A recent article in Prevention magazine explains why it is a good thing for older people to go back to school. Experts cited there made the following argument: 4 (1) If people who go back to school, then they will get more exercise and stimulation. (2) If they get more exercise and stimulation, then they experience improvements in overall health. Therefore, (3) Therefore, if people go back to school, they will experience improvements in overall health. If you make S1 = People go back to school. S2 = People get more exercise and stimulation. S3 = People experience improvements in overall health. 12