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Analyzing Arguments

Section 1.5. Analyzing Arguments. Valid 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.

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Analyzing Arguments

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  1. Section 1.5 Analyzing Arguments

  2. Valid 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.)

  9. Example – 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 valid.

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