09 Slides--Proofs

09 Slides--Proofs - CS103 HO#9 Slides-Proofs April 7 2010...

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CS103 HO#9 Slides--Proofs April 7, 2010 1 An argument is valid if the conclusion must be true in any circumstances in which the premises are all true . In that case, we say that the conclusion is a logical consequence of the premises and that we have proved the conclusion. Arguments Arguments If you overslept, you'll be late. You aren’t late. Therefore: (a) You did oversleep. (b) You didn’t oversleep. (c) You're late. (d) None of these follows. If you overslept, you'll be late. You didn't oversleep. Therefore: (a) You're late. (b) You aren't late. (c) You did oversleep. (d) None of these follows. If you overslept, you'll be late. You aren’t late. Therefore: You didn’t oversleep. If you are in France, you're in Europe. You aren’t in Europe. Therefore: You aren’t in France. If p, then q. Not-q. Therefore: Not-p Valid Arguments Modus tollens Invalid Arguments If you overslept, you'll be late. You didn't oversleep. Therefore: You aren't late. If you are in France, you're in Europe. You aren’t in France. Therefore: You aren’t in Europe. If p, then q. Not-p. Therefore: Not-q. (a) If my glasses are on the kitchen table, then I saw them at breakfast. (b) I was reading the newspaper in the living room or I was reading the newspaper in the kitchen. (c) If I was reading the newspaper in the living room, then my glasses are on the coffee table. (d) I did not see my glasses at breakfast. (e) If I was reading my book in bed, then my glasses are on my bed table. (f) If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table. Intermediate conclusions: Conclusion: My glasses are on the coffee table. (a) If my glasses are on the kitchen table, then I saw them at breakfast. (b) I was reading the newspaper in the living room or I was reading the newspaper in the kitchen. (c) If I was reading the newspaper in the living room, then my glasses are on the coffee table. (d) I did not see my glasses at breakfast. (e) If I was reading my book in bed, then my glasses are on my bed table. (f) If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table. Intermediate conclusions: (g) The glasses are not on the kitchen table. [From (a), (d)] Conclusion: My glasses are on the coffee table.

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CS103 HO#9 Slides--Proofs April 7, 2010 2 (a) If my glasses are on the kitchen table, then I saw them at breakfast. (b) I was reading the newspaper in the living room or I was reading the newspaper in the kitchen. (c) If I was reading the newspaper in the living room, then my glasses are on the coffee table. (d) I did not see my glasses at breakfast. (e) If I was reading my book in bed, then my glasses are on my bed table. (f) If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table. Intermediate conclusions: (g) The glasses are not on the kitchen table. [From (a), (d)] (h) I did not read the newspaper in the kitchen. [From (f), (g)] (i) I read the newspaper in the living room. [From (b), (h)] Conclusion: My glasses are on the coffee table. [From (c), (i)] p 1 p 2 ...
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09 Slides--Proofs - CS103 HO#9 Slides-Proofs April 7 2010...

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