Chapter 6.5 Notes PDF

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Chapter 6.5 Notes Fall 2018...

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6.5 Indirect Truth Tables ●

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If the conclusion of an argument is a tautology ○ Argument is automatically valid (tautologies can be proven from any set of premises) If the premises of an argument are inconsistent ○ then the argument is automatically valid (anything will logically follow from inconsistent beliefs)

Methods for Indirect Truth Table Method ● As the number of the simple propositions used in the argument increaes, the truth table requires more lines ● 4 lines for the argument with 2 simple propositions, 8 lines for 3, 16 for 4 ○ With indirect truth tables we can avoid this ● Determining validity ultimately boils down to this question: is there a possible counter example (a case where the premises are true but the conclusion is false?) ● So, to determine validity, we need not really think about a case where the conclusion is true or the premises false ● Basic idea ○ Is there a way to assign T to the premises and F to the conclusion consistently? ○ If yes, then it means there’s a counter example (aka invalid) ○ If there is no way to make this happen, then the argument is valid ● Determining the answer to this question requires the skill to “work backward”; i.e. based on the truth value assigned to the whole statement, compute the possible truth values that its components can have ○ Based on information that F is assigned to conditional, need to know how to compute truth values to components ○ May be several ways to compute from the truth value to the whole ■ Example on slide is good

1. Assign T to the premises, and F to the conclusion 2. Then, working backward, assign truth values to the components 3. If truth values can be assigned to the components without any contradiction, then the argument is invalid. If truth values cannot be assigned to the components without any contradiction, then the argument is valid 4. When a contradiction is detected, mark it with a circle and highlight it...