Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Analyzing ArgumentsSection 1.5Valid arguments•An argument consists of two parts: the hypotheses (premises) and the conclusion.•An argument is valid if the conclusion of the arguments is guaranteed under the given set of hypotheses.Conditional Representation of an Argument•An argument having n hypotheses, h1, h2, …, hn and conclusion c can be represented by the conditional [h1 ^ h2 ^ … ^ hn] c.•If the above conditional is always true, (regardless of the truthfulness of the individual statements) the argument is valid.Tautologies•A tautology is a statement that is always true.•What this means is that if every entry for a particular column in a truth table has a value of true, then that statement is a tautology.•An argument having n hypotheses h1, h2, …, hn and conclusion c is valid if and only if the conditional [h1 ^ h2 ^ … ^ hn] c is a tautology.Example•Is this argument valid?If you listen to rock and roll, you do not go to heaven. If you are a moral person, you go to heaven. Therefore, you are not a moral person if you listen to rock and roll.•Step 1: Identify the hypotheses and the conclusion.•Step 2: Identify the simple statements in the hypotheses and conclusion.•Step 3: Write the hypotheses and conclusion in symbolic form.•Step 4: Construct a truth table.•Step 5: Verify if the conditional [h1 ^ h2 ^ … ^ hn] c is a tautology.Example – Step 1•Step 1:Identify the hypotheses and conclusion. h1: If you listen to rock and roll, you do not go to heaven.h2: If you are a moral person, you go to heaven.c: Therefore, you are not a moral person if you listen to rock and roll.Example – Step 2•Step 2: Identify the simple statements.p: You listen to rock and roll. q: You go to heaven. r: You are a moral person.Note: 3 simple statements implies 8 rows in the truth table.Example – Step 3•Step 3: Write the hypotheses and conclusion in symbolic form.•h1 : p ~q•h2 : r q• c : p ~r (remember the conclusion is using the if connective ~r if p.)Example – Step 4•Step 4: Construct the truth table.TTTTTTTFFFTTFFTFTTFFTTTTTTFFTFTTTTTFFTTFTTTTTTTFFTTFFFTFTTFTTTFTFTFFTTTFFTFFFTTT(h1 ^ h2) cp ~rh1 ^ h2rqp ~q~r~qrqpch2h1Example – Step 5•Step 5: Is h1 ^ h2 c a tautology.•Yes. Looking at the last column of the truth table, we see that all the values are TRUE.•So the argument is
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