LOGIC GUIDE 2 To better understand VALIDITY, we now turn to the topic of logical form. LOGICAL FORM The logical form of a statement or argument is the skeleton, or structure. If you retain only the words that name logical relationships or quantity, and replace all the other words with placeholders, the logical form is revealed. Suppose we take the argument ~ All people are mortal. Socrates is a person. Therefore, Socrates is mortal. Which words would we replace with placeholders? All the words that do not name relationships or quantities. Those words are indicated in red below. All people are mortal. Socrates is a person. Therefore, Socrates is mortal.
You can use any placeholders you please, since they have no literal meaning they are just holding a place! Two rules to follow: 1. Always use the same placeholder to stand for the same word. 2. Avoid getting fancy with your choice of placeholders. Reduce the risk of confusing yourself. Suppose we use the following placeholders for our example argument. Let X = people. Let Y = mortal. Let Z = Socrates. This argument All people are mortal. Socrates is a person. Therefore, Socrates is mortal. has this logical form All X are Y Z is an X Therefore, Z is a Y LOGICAL FORM AND VALIDITY Logical form is the only thing that makes an argument valid or invalid! Consider validity is a relationship of certainty between the premises and conclusion. Since the logical form of an argument is the set of relationships that exist within the argument, it is the logical form that makes an argument valid or invalid. Many different arguments may have the same logical form. For example Suppose we use the following placeholders for our example argument. Let X = alligators. Let Y = purpality (the state of being purple.) Let Z = Shakira. This argument All alligators are purple. Shakira is an alligator. Therefore, Shakira is purple. has this logical form All X are Y Z is an X Therefore, Z is a Y As we saw in Logic Guide 1, this is a valid argument.
All alligators are purple. Shakira is an alligator. Therefore, Shakira is purple. Everything in the set alligators has the property of being purple, and Shakira is in that set, so she must have that property. All people are mortal. Socrates is a person. Therefore, Socrates is mortal. Everything in the set people has the property of being mortal, and Socrates is in that set, so he must have that property. The Socrates argument and the Shakira argument are identical in logical form The Socrates argument is valid and sound. The Shakira argument is valid, but not sound. CHECK YOUR SELF ~ 1. If an argument is valid, any other argument with the same logical form will be valid. 2. A sound argument must be valid AND have all true premises. So if an argument is sound, any other argument with the same logical form will be valid, but may not or may not be sound. Remember, formal errors and factual errors are independent of one another. GOODBYE, CATEGORICAL LOGIC The examples we have been using (the Socrates and Shakira arguments) are both examples of categorical logic. What I am calling the Socrates argument was first used by Aristotle in the 4 th century B.C. It is still widely used to teach beginning logic, just because it is so intuitive and simple.
Aristotle was the first person in the Western world to systematically set down the rules for this style of logic, so it is sometimes called Aristotelian logic. Aristotle was not Greek, but studied in Greece at lato s Academy for over 20 years. He had come to Greece because he was fascinated with what lato had written about lato s own mentor, Socrates. We will hear more about Aristotle later in the course, when we are studying metaphysics (ideas about the nature of reality) and epistemology (ideas about the nature of knowledge.) Categorical logic is very powerful, but only for a very limited range of problems. In modern times, many people, such as Bertrand Russell, did pioneering work in developing new, more powerful systems of logic. In fact, in the 20 th century, more was written about logic than in all previous centuries combined. We will now turn to truth-functional logic, a style of logic more common to modern thought, although a group of philosophers known as the Stoics had already begun exploring truthfunctional arguments in the 4th century B.C.! HELLO, TRUTH-FUNCTIONAL ARGUMENTS Consider the following statement. If I am human, then I am mortal. Notice that it asserts an if-then relation. Such statements are called conditional statements. When they are used as premises in an argument, they are called conditional premises. The logical form of the statement would be, If then. Notice that we keep the terms if and then, since together they name a logical relationship, but replace all other terms with placeholders. In this conditional statement, we are asserting that whenever is true, is also true. We are not saying the causes. We are not saying is never true unless is true. Consider ~ If I study, I will get a good grade on the test. In this example I am not saying I will not get a good grade unless I study. I may get lucky, I may be really smart, the test may be easy, etc.
I am not saying that studying will necessarily be the cause of the good grade, since there may be other reasons why I get a good grade. I am only saying that studying is sufficient for me to get a good grade. Notice also that this sentence is not, by itself, an argument. I have shown no reason why studying entails (logically) getting a good grade. I have only asserted that if the first condition (studying) is true, the second condition (good grade) will also be true. ANTECEDENT AND CONSEUENT In any conditional () premise, the first condition is called the antecedent. The second condition is called the consequent. For example ~ If I am human, then I am mortal. The antecedent is I am human, the consequent is I am mortal. Notice that we DO NOT include the if as part of the antecedent, or the then as part of the consequent. The if and then indicate the relationship between the antecedent and the consequent. CONDITIONAL ARGUMENTS Here is a typical argument using a conditional first premise. If I am a fish, then I can swim. I am a fish. Therefore, I can swim. The logical form would be, Therefore, Notice that in the second premise, we asserted the antecedent, in order to end up with the consequent as the conclusion. We can easily form four, superficially similar arguments by asserting or denying either the antecedent or the consequent in the second premise. Therefore, Therefore, Not Therefore, Not Not Therefore, Not
Whatever happens in the second premise, that is the name of the argument. There is no reason why it had to be that way, but that is the commonly-used convention. So, the names of the four arguments we have formed are as follows. 1. 2. 3. 4. Therefore, Therefore, Not Therefore, Not Not Therefore, Not Affirming the antecedent Affirming the consequent Denying the antecedent Denying the consequent We may sometimes be misled by invalid arguments superficially resembling valid arguments. In fact, two of these arguments are valid, and two are invalid! The table below shows which arguments are valid, and which are invalid, and explains why. 1. 2. 3. 4. Therefore, Therefore, Not Therefore, Not Not Therefore, Not Affirming the antecedent This form is valid. Affirming the consequent This form is invalid. Denying the antecedent This form is invalid. Denying the consequent This form is valid. The first premise states that whenever is true, is also true. The second premise states that is presently true. If both those premises were true, there is no way the conclusion could be false. Since you have, you have. The first premise states that whenever is true, is also true. The second premise states that is presently true. But no premise states that is never true unless is true. Therefore, even if both premises are true, the conclusion could be false. For all you know, based on the premises, you can have without. The first premise states that whenever is true, is also true. The second premise states that is presently false. But no premise states that is never true unless is true. Therefore, even if both premises are true, the conclusion could be false. For all you know, based on the premises, you can have without. The first premise states that whenever is true, is also true. The second premise states that is presently false. If both those premises were true, there is no way the conclusion could be false. If you have, you d have. If then means that whenever you have, you will also have. It does not mean that you cannot have without. Both of the invalid forms illustrated above make the same mistake they confuse sufficient with necessary. What is the difference between sufficient and necessary?
means that is sufficient to guarantee. It does not mean that is necessary for. For example, if Godzilla stomps me, that is sufficient to kill me. Does that mean that if Godzilla never stomps me, I will live forever? Of course not! It is sufficient, but it is not necessary. If we had said Only if, then, we would be saying is necessary for, but not that is sufficient for. For example, fuel is necessary for fire, but it is not sufficient (you also need heat and oxygen.) If we had said If and Only, we would be saying is both necessary and sufficient for. For example, the motion of atoms or molecules is necessary for heat, and sufficient (because that s what heat is!) Godzilla is also not necessary for heat, but may be sufficient. CHECK YOURSELF! Referring to the previous table, do the following four things for the four argument forms given. For each of the four arguments (A, B, C, D) below, do the following four things (1, 2, 3, 4.). 1. Show the logical form. (Be sure to always include the first premise, since the first premise defines which place holder is the antecedent, and which placeholder is the consequent!) 2. Give the name of the argument (affirming or denying the antecedent or consequent). 3. Tell whether it is a valid or invalid form. 4. Explain WHY it is valid or invalid. When you finish, check your own answers against the answers provided below. A. If I am a fish, then I will swim underwater. I am not a fish. Therefore, I will not swim underwater. B. If I am a fish, then I will swim underwater. I am a fish. Therefore, I will swim underwater. C. If I am a fish, then I will swim underwater. I will swim underwater. Therefore, I am a fish.
. D. If I am a fish, then I will swim underwater. I will not swim underwater. Therefore, I am not a fish. ANSWERS A. If I am a fish, then I will swim underwater. I am not a fish. Therefore, I will not swim underwater. I. Not Therefore Not II. Denying the antecedent. III. IV. Invalid The first premise says that is sufficient for, not that it is necessary. Something other than a fish may also swim underwater. B. If I am a fish, then I will swim underwater. I am a fish. Therefore, I will swim underwater. I. Therefore II. Affirming the antecedent. III. IV. Valid The first premise says that is sufficient for, the second says we have. In that case, we would have to have. C. If I am a fish, then I will swim underwater. I will swim underwater. Therefore, I am a fish. I. Therefore II. Affirming the consequent III. IV. Invalid The first premise says that is sufficient for, not that it is necessary. Something other than a fish may also swim underwater. D. If I am a fish, then I will swim underwater. I will not swim underwater. Therefore, I am not a fish. I. Not Therefore Not II. Denying the consequent III. IV. Valid The first premise says that is sufficient for. Since we don t have (2 nd premise), we don t have ANYTHING that would be sufficient for.