3.5 Symbolic Arguments 145 These statements in the following form constitute what we will call a symbolic argument. Premise 1: If we go to the Bahamas, then we will go snorkeling. Premise 2: We go to the Bahamas. Conclusion: We will go snorkeling. A symbolic argument consists of a set of premises and a conclusion. The argument is called a symbolic argument because we generally write the argument in symbolic form to determine its validity. Learning Catalytics Keyword: Angel-SOM-3.5 (See Preface for additional details.) Definition: Valid and Invalid 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. An argument that is not valid is invalid. The argument just presented is an example of a valid argument, as the conclusion necessarily follows from the premises. Now we will discuss a procedure to determine whether an argument is valid or invalid. We begin by writing the argument in symbolic form. To write the argument in symbolic form, we let p and q be p q : We go to the Bahamas. : We will go snorkeling. Symbolically, the argument is written p q p q Premise 1: Premise 2: Conclusion: (The three-dot-triangle is read “therefore.”) → ∴ Write the argument in the following form. premise 1 premise 2 conclusion p q p q and then If [ ] [( ) ] → ∧ → Then construct a truth table for the conditional statement p q p q [( ) ] → ∧ → (Table 3.33). If the truth table answer column is true in every case, then the statement is a tautology, and the argument is valid. If the conditional statement is not a tautology, then the argument is invalid. Recall conditional statements that are tautologies are called implications. Since the conditional statement p q p q [( ) ] → ∧ → is a tautology (see column 5), the conditional statement is also an implication. Thus, the conclusion necessarily follows from the premises and the argument is valid. Table 3.33 p q p q [( ) → ∧ p] → q T T T T T T T T F F F T T F F T T F F T T F F T F F T F 1 3 2 5 4
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