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Unformatted text preview: Introduction to Logic Lecture 8 Brian Weatherson, Department of Philosophy September 28th, 2009 Logic 201 (Section 5) Lecture 8 1 / 41 Representing Truth Conditions The truth conditions of a complex depend on the parts. Truth tables are used to display the dependencies Logic 201 (Section 5) Lecture 8 2 / 41 Our aim is to, for each row, work up from the atomic sentences to the full sentence, saying what the truth value of that sentence is in each row. The columns on the left tell us the truth value of the atomic sentences. On the right, we write the truth value of each table, in each of the possible conditions, under its main connective. Logic 201 (Section 5) Lecture 8 3 / 41 Our technical term for a sentence with T s in every row is a Tautology . This is the window that we get when we click on the “Assessment” button in Boole. Logic 201 (Section 5) Lecture 8 4 / 41 Every tautology is a logical truth. But some sentences that have to be true are not tautologies. Logic 201 (Section 5) Lecture 8 5 / 41 Some sentences are tautologically equivalent . This just means that they have the same truth table. That is, they have T s and F s in the same rows as each other. Logic 201 (Section 5) Lecture 8 6 / 41 Validity Logic 201 (Section 5) Lecture 8 7 / 41 Validity We can use truthtables, and hence Boole, to work out whether some arguments are valid. Remember that a valid argument is one where it is impossible for the premises to be true and the conclusion false. So if there is no row where the premises are true and the conclusion false, then the argument is valid. It is harder to use truth tables to see whether an argument is invalid. Any invalid argument will have a row where the premises are true and the conclusion false. But to see whether the argument is really invalid, we have to check whether that row represents a real possibility. We’ll illustrate each of the points on this slide with several examples. Logic 201 (Section 5) Lecture 8 8 / 41 ( A ∧ B ) ∨ ( A ∧ C ) , therefore ( A ∨ B ) ∧ ( A ∨ C ) . We’re first going to test whether this argument is valid. ( A ∧ B ) ∨ ( A ∧ C ) ( A ∨ B ) ∧ ( A ∨ C ) We do this by building a truth table for the premise and the conclusion. If there is no row where the premise is true and the conclusion false, then the argument is valid. Logic 201 (Section 5) Lecture 8 9 / 41 ( A ∧ B ) ∨ ( A ∧ C ) , therefore ( A ∨ B ) ∧ ( A ∨ C ) . On every line where the premise is T , the conclusion is also T . So the argument is valid. Logic 201 (Section 5) Lecture 8 10 / 41 ( A ∨ B ) ∧ ( A ∨ C ) , therefore ( A ∧ B ) ∨ ( A ∧ C ) ....
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 Fall '11
 JonWinterbottom
 Logic, Conclusion, consequence, AnaCon, FOCon

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