# Section 3.5. Symbolic Arguments. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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1 Section 3.5 Symbolic Arguments

2 What You Will Learn Symbolic arguments Standard forms of arguments 3.5-2

3 Symbolic Arguments A symbolic argument consists of a set of premises and a conclusion. It is called a symbolic argument because we generally write it in symbolic form to determine its validity

4 Symbolic Arguments An argument is valid when its conclusion necessarily follows from a given set of premises. An argument is invalid or a fallacy when the conclusion does not necessarily follow from the given set of premises

5 Law of Detachment Also called modus ponens. Symbolically, the argument is written: Premise 1: Premise 2: Conclusion: p q p q If [premise 1 and premise 2] then conclusion [(p q) p ] q 3.5-5

6 To Determine Whether an Argument is Valid 1. Write the argument in symbolic form. 2. Compare the form of the argument with forms that are known to be either valid or invalid. If there are no known forms to compare it with, or you do not remember the forms, go to step

7 To Determine Whether an Argument is Valid 3. If the argument contains two premises, write a conditional statement of the form [(premise 1) (premise 2)] conclusion 4. Construct a truth table for the statement above

8 To Determine Whether an Argument is Valid 5. If the answer column of the truth table has all trues, the statement is a tautology, and the argument is valid. If the answer column of the table does not have all trues, the argument is invalid

9 Example 2: Determining the Validity of an Argument with a Truth Table Determine whether the following argument is valid or invalid. If you watch Good Morning America, then you see Robin Roberts. You did not see Robin Roberts. You did not watch Good Morning America

10 Example 2: Determining the Validity of an Argument with a Truth Table Solution Let p: You watch Good Morning America. q: You see Robin Roberts. In symbolic form, the argument is p q ~p ~p The argument is [(p q) ~q] ~p

11 Example 2: Determining the Validity of an Argument with a Truth Table Solution Construct a truth table. p q [(p q) ~ q] ~p T T F F T F T F T F T T F F F T F T F T T T T T F F T T Since the answer, column 5, has all T s, the argument is valid

12 Standard Forms of Valid Arguments Law of Detachment p q p q Law of Syllogism p q q r p r Law of Contraposition p q ~q ~ p Disjunctive Syllogism p q ~ p q

13 Standard Forms of Invalid Arguments Fallacy of the Converse p q q p Fallacy of the Inverse p q ~ p ~q

14 Example 4: Identifying a Standard Argument Determine whether the following argument is valid or invalid. If you are on Facebook, then you see my pictures. If you see my pictures, then you know I have a dog. If you are on Facebook, then you know I have a dog

15 Example 4: Identifying a Standard Argument Solution Let p: You are on Facebook. q: You see my pictures. r: You know I have a dog. In symbolic form, the argument is p q q r p r It is the law of syllogism and is valid

16 Example 5: Identifying Common Fallacies in Arguments Determine whether the following argument is valid or invalid. If it is snowing, then we put salt on the driveway. We put salt on the driveway. It is snowing

17 Example 5: Identifying Common Fallacies in Arguments Solution Let p: It is snowing. q: We put salt on the driveway. In symbolic form, the argument is p q q p It is in the form of the fallacy of the converse and it is a fallacy, or invalid

18 Example 5: Identifying Common Fallacies in Arguments Determine whether the following argument is valid or invalid. If it is snowing, then we put salt on the driveway. It is not snowing. We do not put salt on the driveway

19 Example 5: Identifying Common Fallacies in Arguments Solution Let p: It is snowing. q: We put salt on the driveway. In symbolic form, the argument is p q ~p ~q It is in the form of the fallacy of the inverse and it is a fallacy, or invalid

20 Example 6: An Argument with Three Premises Use a truth table to determine whether the following argument is valid or invalid. If my cell phone company is Verizon, then I can call you free of charge. I can call you free of charge or I can send you a text message. I can send you a text message or my cell phone company is Verizon. My cell phone company is Verizon

21 Example 6: An Argument with Three Premises Solution Let p: My cell phone company is Verizon. q: I can call you free of charge. r: I can send you a text message. In symbolic form, the argument is p q q r r p p

22 Example 6: An Argument with Three Premises Solution Write the argument in the form (p q) (q r) (r p)] p. Construct a truth table

23 Example 6: An Argument with Three Premises Solution

24 Example 6: An Argument with Three Premises Solution The answer, column 7, is not true in every case. Thus, the argument is a fallacy, or invalid

25 Section 3.6 Euler Diagrams and Syllogistic Arguments

26 What You Will Learn Euler diagrams Syllogistic arguments

27 Syllogistic Arguments Another form of argument is called a syllogistic argument, better known as syllogism. The validity of a syllogistic argument is determined by using Euler (pronounced oiler ) diagrams

28 Euler Diagrams One method used to determine whether an argument is valid or is a fallacy. Uses circles to represent sets in syllogistic arguments

29 Symbolic Arguments Versus Syllogistic Arguments Symbolic argument Syllogistic argument Words or phrases used and, or, not, if-then, if and only if all are, some are, none are, some are not Methods of determining validity Truth tables or by comparison with standard forms of arguments Euler diagrams

30 Example 3: Ballerinas and Athletes Determine whether the following syllogism is valid or invalid. All ballerinas are athletic. Keyshawn is athletic. Keyshawn is a ballerina

31 Example 3: Ballerinas and Athletes Solution All ballerinas, B, are athletic, A. Keyshawn is athletic, so must be placed in the set of athletic people, which is A. We have a choice, as shown above. The conclusion does not necessarily follow from the set of premises. The argument is invalid

32 Example 4: Parrots and Chickens Determine whether the following syllogism is valid or invalid. No parrots eat chicken. Fletch does not eat chicken. Fletch is a parrot

33 Example 4: Parrots and Chickens Solution The first premise tells us that parrots and things that eat chicken are disjoint sets that is, sets that do not intersect. Fletch is not a parrot, the argument is invalid, or is a fallacy

34 Example 5: A Syllogism Involving the Word Some Determine whether the following syllogism is valid or invalid. All As are Bs. Some Bs are Cs. Some As are Cs

35 Example 5: A Syllogism Involving the Word Some Solution The premise All As are Bs is illustrated

36 Example 5: A Syllogism Involving the Word Some Solution The premise Some Bs are Cs means that there is at least one B that is a C

37 Example 5: A Syllogism Involving the Word Some Solution The first illustrations shows that the conclusion Some As are Cs, does not follow, the argument is invalid

38 Example 6: Fish and Cows Determine whether the following syllogism is valid or invalid. No fish are mammals. All cows are mammals. No fish are cows

39 Example 6: Fish and Cows Solution The first premise tells us that fish and mammals are disjoint sets. The second tells us that the set of cows is a subset of the set of mammals. The conclusion necessarily follows from the premises and the argument is valid

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