Validity — Predicate Logic (some invalid forms), Study notes of Logic

Download Validity — Predicate Logic (some invalid forms) and more Study notes Logic in PDF only on Docsity! Philosophy 101 (2/22/11) • HW #2 to be returned today (end of class) • I will be grading on a “curve” after all. [more soon] • Solutions to HW #2 posted (later today) • HW #3 assigned last week (due next Thursday) • Quiz #2 on this Thursday (on rational belief) • Today: Chapter 3, Continued • Two subtle aspects of formal validity • Cogency (of invalid arguments) • Next: Chapter 4 — Strong Arguments Chapter 3: Well-Formed Arguments 32 • Validity — Predicate Logic (some invalid forms) 74 Chapter 3 Well-Formed Arguments Table 3.4 Some Patterns of invalid Arguments in Predicate logic Pattern I Example L All As are Bs. 1. All men are mortaL 1. All As are Bs. 1. All men are mortal. 2. Fido is mortal. 3. Fido is a man. As are Bs is called a quantifier. Many different quantifiers are used in generalizations, including "lots of," "nearly all," "hardly any," "few," and countless others. Generalizations figure prominently in valid arguments from predicate logic. Table 3.3 displays some of the more common patterns. The common patterns in Table 3.4 are invalid. EXERCISES AND STUDY QUESTIONS Each of the following arguments follows one of the patterns identified in Tables 3.3 and 3.4. For each argument, use circles and boxes to identifY its key parts. Then state the pattern for each argument and state whether or not it is valid. *1. 1. All logicians are dull. 2. Irving is a logician. 3. Irving is dull. 2. 1. All logicians are dull. 2. Irving is not a logician. 3. Irving is dulL *3. 1. All logicians are dull. 2. Irving is dull. 3. Irving is a logician. 4. 1. All logicians are dulL 2. All (who are) dull are party animals. 3. All logicians are party animals. *5. 1. No logicians are dull. 2. Irving is a logician. 3. Irving is not dull. 6. 1. All logicians are dull. 2. Irving is not dull. 3. Irving is not a logician. 7. 1. All bearded logicians wear glasses. 2. Irving is a bearded logician. 3. Irving wears glasses. II. Well-Formed Arguments 75 AS. A Modification of the Dtifinition tif Validity Arguments such as the following one raise a question about our definition Argument 3.12 1. Jones is a mother. 2. jones is female. Is Argument 3.12 valid? You might think it is. Since there is no way the premise could be true and the conclusion false, it seems that the truth of the premise does guarantee the truth of the conclusion. On the other hand, you might think that Argument 3.12 is not valid. There is no recognizable valid pattern of argu-ment here, and we've said that validity has to do with the pattern or form of argument. The correct answer to this question is somewhat complicated, for there are two very different ways in which the premises of an argument can be said to guarantee the truth of the argument's conclusion. One way depends only on the form or pat-tern of the argument. Argument 3.4, for example, is valid no matter who the term "Boris" refers to and no matter what is meant by "student" or "State U." In contrast, the premise of Argument 3.12 appears to guarantee the truth of its conclusion, but this depends in part on the fact that "mother" means "female parent," so anything that is a mother is also female. Consequently, if (1) is true, then (2) will be true as well. There are, then, two ways to think about validity: one concerns the form of arguments alone and a second takes the meanings of the key terms of the argument into account. For our purposes, we will interpret validity in the first way, that is, the validity of an argument will not depend on extra assumptions about the meanings of terms. Valid arguments are ones whose pattern or structure all by itself assures that the premises are properly related to the conclusion. To avoid confusion, we can refine our earlier definition of validity as follows: 03.1 b: An argument is valid if and only the argument follows a pattern such that it is impossible for any argument following that pattern to have true premises and a false conclusion. According to this definition, if an argument is valid, then it follows a pattern such that all arguments following that same pattern are also valid. On this new understanding of validity, Argument 3.12 is not valid. However, Argument 3.12 is very closely connected to another argument that is valid. The premise of Argument 3.12 could be replaced by la. Jones is a female and Jones is a parent. Notice that and are equivalent. When is put into the argument, we get Another important Example: Most As are Bs. x is an A. -------------------- x is a B. Chapter 3: Well-Formed Arguments 33 • Validity — A Clarification of the Definition • Our initial definition of validity was a bit unclear. You can see this unclarity in certain more subtle examples, such as: 74 Chapter 3 Well-Formed Arguments Table 3.4 Some Patterns of invalid Arguments in Predicate logic Pattern I Example L All As are Bs. 1. All men are mortaL 1. All As are Bs. 1. All men are mortal. 2. Fido is mortal. 3. Fido is a man. As are Bs is called a quantifier. Many different quantifiers are used in generalizations, including "lots of," "nearly all," "hardly any," "few," and countless others. Generalizations figure prominently in valid arguments from predicate logic. Table 3.3 displays some of the more common patterns. The common patterns in Table 3.4 are invalid. EXERCISES AND STUDY QUESTIONS Each of the following arguments follows one of the patterns identified in Tables 3.3 and 3.4. For each argument, use circles and boxes to identifY its key parts. Then state the pattern for each argument and state whether or not it is valid. *1. 1. All logicians are dull. 2. Irving is a logician. 3. Irving is dull. 2. 1. All logicians are dull. 2. Irving is not a logician. 3. Irving is dulL *3. 1. All logicians are dull. 2. Irving is dull. 3. Irving is a logician. 4. 1. All logicians are dulL 2. All (who are) dull are party animals. 3. All logicians are party animals. *5. 1. No logicians are dull. 2. Irving is a logician. 3. Irving is not dull. 6. 1. All logicians are dull. 2. Irving is not dull. 3. Irving is not a logician. 7. 1. All bearded logicians wear glasses. 2. Irving is a bearded logician. 3. Irving wears glasses. II. Well-Formed Arguments 75 AS. A Modification of the Dtifinition tif Validity Arguments such as the following one raise a question about our definition Argument 3.12 1. Jones is a mother. 2. jones is female. Is Argument 3.12 valid? You might think it is. Since there is no way the premise could be true and the conclusion false, it seems that the truth of the premise does guarantee the truth of the conclusion. On the other hand, you might think that Argument 3.12 is not valid. There is no recognizable valid pattern of argu- ment here, and we've said that validity has to do with the pattern or form of argument. The correct answer to this questio is somewhat complicated, for there are two very different ways in which the premises of an argument can be said to guarantee the truth of the argument's conclusion. One way depends only on the form or pat- tern of the argument. Argument 3.4, for example, is valid no matter who the term "Boris" refers to and no matter what is meant by "student" or "State U." In contrast, the premise of Argument 3.12 appears to guarantee the truth of its conclusion, but this depends in part on the fact that "mother" means "female parent," so anything that is a mother is also female. Consequently, if (1) is true, then (2) will be true as well. There are, then, two ways to think about validity: one concerns the form of arguments alone and a second takes the meanings of the key terms of the argument into account. For our purposes, we will interpret val d ty in the first way, that is, the validity of an argument will not depend on extra assumptions about the meanings of terms. Valid arguments are ones whose pattern or structure all by itself assures that the premises are properly related to the conclusion. To avoid confusion, we can refine our earlier definition of validity as follows: 03.1 b: An argument is valid if and only the argument follows a pattern such that it is impossible for any argument following that pattern to have true premis s and a f lse conclusion. According to this definition, if an argument is valid, then it follows a pattern such that all arguments following that same pattern are also valid. On this new understanding of validity, Argument 3.12 is not valid. However, Argument 3.12 is very closely connected to another argument that is valid. The premise of Argument 3.12 could be replaced by la. Jones is a female and Jones is a parent. Notice that and are equivalent. When is put into the argument, we get • Is this argument valid? One might think it is, because it might seem that it would be a logical contradiction for the premise of this argument to be true while its conclusion is false. • But, strictly speaking, we will not classify this argument as valid. ➡This is because we have no logical theory (sentential or predicate) according to which this argument has a valid form. • This leads to an important clarification of our definition. Chapter 3: Well-Formed Arguments 34 • Validity — A Clarification of the Definition • Here is a clarified defintion of validity: • D3.1b: An argument is valid iff the argument has some logical form such that it is impossible for any argument with that form to have true premises and a false conclusion. • That is, all valid arguments must have valid logical forms. • The following two (“equivalent”) arguments are valid: 1. Jones is a female and Jones is a parent. --------------------------------------------------- 2. Jones is a parent. 1. Jones is a mother. 2. All mothers are females. ---------------------------------- 3. Jones is a female. Chapter 3: Well-Formed Arguments 35 • Validity — A Further Clarification • In our revised definition of validity, we only require that the argument instantiate some valid logical form. • Because we have two different notions/theories of logical form (sentential-logical form and predicate-logical form), we must be careful about certain cases where they come apart. • Consider the following argument: 1. All men are mortal. 2. Socrates is a man. 3. Therefore, Socrates is mortal. • As we have seen, this argument has a valid predicate-logical form, and so it is (according to our refined definition) valid. Chapter 3: Well-Formed Arguments 36 • Validity — A Further Clarification 1. All Xs are Ys. [X = men, Y = mortals] 2. s is an X. [s = Socrates] 3. Therefore, s is a Y. • But, ask yourself: what is its sentential-logical form? • Remember, from the point of view of sentence logic, this argument contains three distinct “atomic” sentences! • This is because none of the sentences in the argument contains any of the 5 sentential connectives (and, or, ~, if, iff). So, we have: 1. P. 2. Q. 3. Therefore, R. Chapter 3: Well-Formed Arguments 37 • Validity — A Further Clarification • The moral of this story is that some arguments do not have a valid sentential form, but they do have a valid predicate form. • Such arguments are (still) valid, but in order to see that they are valid, one needs to look at predicate-logical form. • Sentence-logical form is “coarse grained” or “zoomed out”. It is not capable of “seeing” subject-predicate structure. • Predicate-logical form is “finer-grained” or “zoomed in”. It is capable of “seeing” subject-predicate structure. • On the other hand, if an argument does have a valid sentence- logical form, it must have a valid predicate-logical form as well. Chapter 3: Well-Formed Arguments 38 • Cogency • Some invalid arguments are better than others (from a logical point of view). Some are cogent, while others are not. • In fact, cogency comes in degrees. Here is an example: 1. Boris is a student at State U. 2. Almost all students at State U. voted. ------------------------------------------------- 3. Boris voted. • This argument is invalid, because it is possible for its conclusion to be false, even given the truth of its premises. • But, this argument is cogent, since its conclusion is probable, given the truth of all of its premises. That leads to our definition.