Section 3.5 Symbolic Arguments

What You Will Learn Symbolic arguments Standard forms of arguments

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.

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.

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

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 3.

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.

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.

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.

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.

Example 2: Determining the Validity of an Argument with a Truth Table Solution Construct a truth table. p q [(p → q) ⋀ ~ q] → ~p T F T F T F F T F T T F T 1 3 2 5 4 Since the answer, column 5, has all T’s, the argument is valid.

Standard Forms of Valid Arguments Law of Detachment Law of Contraposition Law of Syllogism Disjunctive Syllogism

Standard Forms of Invalid Arguments Fallacy of the Converse Fallacy of the Inverse

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.

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.

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.

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.

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.

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.

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.

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

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.

Example 6: An Argument with Three Premises Solution

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.