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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.
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:
- have a bagel sandwich and a hamburger.
- have a bagel sandwich and a chicken sandwich.
- have a hamburger and a chicken sandwich.
- 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:
- A valid Georgia driver's license;
- 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;
- A valid United States passport;
- 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;
Onions Pepperoncini Pepperoni Pesto Pineapple Portobello Roasted Red Peppers Spinach Sun-dried 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 both ” 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 “ aut ” 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: (a) (1) S1 or S2. (b) (1) S1 or S2. (2) Not S1. (2) Not S2. Therefore, Therefore, (3) S2. (3) S1. Remember that “Sn” is a variable that stands for statements. S1 and S2 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 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 S This is the second premise in form (a) above. We can conclude that Irene will take a higher level math course. We can conclude: S 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.
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
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 firmly 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 S2. (2) S1. Therefore, (3) S2. 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 Elizondo-Omana 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 one-hour class periods over twenty weeks. The other group took
(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,
(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.
you will see that this argument is an instance of the tri-conditional form. This argument form is one you will encounter frequently, both in ordinary conversations as well as in college courses. It is not usually confused with the other conditional argument forms because, unlike the other four, it has three conditional statements (not one). BOX, Technical Terms : Hypothetical Syllogism, Pure HS, Pure Conditional Other names for the tri-conditional argument form include “the hypothetical syllogism,” “the pure hypothetical syllogism,” and the “pure conditional argument.” It is frequently abbreviated HS. BOX, Guide for Finding, Standardizing, and Evaluating Propositional Deductive Arguments This Guide is an amplification of the “Guide for Finding, Standardizing, and Evaluating Arguments” that is in Chapter Two. The numbered sentences are merely copies from the Guide in Chapter Two. The paragraphs with “Pr” (for “Propositional”) in front of them are additional materials that apply only to propositional moral arguments. Finding Arguments
- Look for an attempt to convince. Pr. Look for an attempt to convince that relies on the logical relationships between statements.
- Find the conclusion.
- Find the premises.
- Review the following to make sure that you have correctly identified the conclusion and the premises: imperfect indicator words, sentence order, premises and/or conclusion not in declarative form, unstated premises and/or conclusion, text with multiple arguments.
- Review the following to make sure that you have not incorrectly identified something as a premise or a conclusion when in fact it is not part of an argument: assertions, questions, commands, descriptions, and explanations. Standardizing Arguments
- Rewrite the premises and the conclusion as declarative sentences. Make sure that each premise and the conclusion is a grammatically correct declarative sentence. Rewrite the premises and conclusion as necessary to make them clearer but do not change the meaning of the passage. Remove pronouns from the sentences and replace them with the nouns or noun phrases to which they refer.
- Review any phrases you have omitted to be sure that they are not premises or a conclusion.
- Number the premises and the conclusion. Place the premises before the conclusion and insert “Therefore,” between the premises and the conclusion. Pr. Assign statement letters and formalize the argument.
- Compare your standardization to the original passage to make sure that you have not omitted any arguments found in the passage and to be sure that you have correctly identified the premises and the conclusion. Pr. Check to be sure that your formalization correctly captures the meaning of the argument as expressed in English. Evaluating Arguments: The True Premises Test
- Check to see if the premises are accurate descriptions of the world.
- Consider whether or not the premises are appropriate for the argument’s audience.
- Review the assumed premises to be sure that the assumptions are reasonable. Evaluating Arguments: The Good Form Test
- Determine whether the argument is a deductive argument or inductive argument. A deductive argument claims that the truth of the premises shows that conclusion must be true. An inductive argument is an argument which claims that the truth of the premises shows that the conclusion is likely to be true.
- Determine whether the premises are positively relevant to the conclusion. Look at each premise individually to see if the truth of the premise provides some evidence for the truth of the conclusion. Look at the premises as a group to see if the truth of all of them provides some evidence for the truth of the conclusion. Pr. Determine the form of the propositional argument. Compare it to the eight propositional argument forms discussed in this chapter. Here they are: Disjunctive Forms 1a. Denying a Disjunct (a) (1) S1 or S2. (b) (1) S1 or S2. (2) Not S1. (2) Not S2. Therefore, Therefore, (3) S2. (3) S1. 1b. Fallacy: Affirming an Inclusive Disjunct (a) (1) S1 or S2. (b) (1) S1 or S2. (2) S1. (2) S2. Therefore, Therefore, (3) Not S2. (3) Not S1. 1c. Affirming an Exclusive Disjunct (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. 18 Conditional Forms 2a. Affirming the Antecedent (1) If S1, then S2. (2) S1. Therefore, (3) S2. 2b. Fallacy: Denying the Antecedent (1) If S1, then S2. (2) Not S1. Therefore, (3) Not S. 2c. Denying the Consequent (1) If S1, then S2. (2) Not S2. Therefore,
