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1. If you take Art 151 in the fall, you will be eligible to take Art 151 in the spring. You were not eligible to take Art 152 in the spring. Therefore, you did not take Art 151 in the fall. 2. He will attend Stanford or Yale. Hi did not attend Yale. Therefore, he attended Stanford. 3. If I had a nickel for every logic problem I have solved, then I would be rich. I have not received a nickel for every logic problem I have solved. Therefore, I am not rich.

1. If you take Art 151 in the fall, you will be eligible to take Art 151 in the spring. You were not eligible to take Art 152 in the spring. Therefore, you did not take Art 151 in the fall. 2. He will attend Stanford or Yale. Hi did not attend Yale. Therefore, he attended Stanford. 3. If I had a nickel for every logic problem I have solved, then I would be rich. I have not received a nickel for every logic problem I have solved. Therefore, I am not rich.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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LEARNING ACTIVITY 4.3 In exercises 1-6, determine whether the argument is valid or invalid by comparing its symbolic form into standard forms. For each valid argument, state the name of its standard form. 1. If you take Art 151 in the fall, you will be eligible to take Art 151 in the spring. You were not eligible to take Art 152 in the spring. Therefore, you did not take Art 151 in the fall. 2. He will attend Stanford or Yale. Hi did not attend Yale. Therefore, he attended Stanford. 3. If I had a nickel for every logic problem I have solved, then I would be rich. I have not received a nickel for every logic problem I have solved. Therefore, I am not rich. 4. If it is a dog, then it has fleas. It has fleas. Therefore, it is a dog. 5. If we serve salmon, then Vicky will join us for lunch. If Vicky will join us for lunch, then Marilyn will not join us for lunch. Therefore, if we serve salmon, Marilyn will not join us for lunch. 6. If I go to college, then I will not be able to work for my Dad. I did not go to college. Therefore, I went to work for my Dad.
Transcribed Image Text:LEARNING ACTIVITY 4.3 In exercises 1-6, determine whether the argument is valid or invalid by comparing its symbolic form into standard forms. For each valid argument, state the name of its standard form. 1. If you take Art 151 in the fall, you will be eligible to take Art 151 in the spring. You were not eligible to take Art 152 in the spring. Therefore, you did not take Art 151 in the fall. 2. He will attend Stanford or Yale. Hi did not attend Yale. Therefore, he attended Stanford. 3. If I had a nickel for every logic problem I have solved, then I would be rich. I have not received a nickel for every logic problem I have solved. Therefore, I am not rich. 4. If it is a dog, then it has fleas. It has fleas. Therefore, it is a dog. 5. If we serve salmon, then Vicky will join us for lunch. If Vicky will join us for lunch, then Marilyn will not join us for lunch. Therefore, if we serve salmon, Marilyn will not join us for lunch. 6. If I go to college, then I will not be able to work for my Dad. I did not go to college. Therefore, I went to work for my Dad.
STANDARD FORMS Arguments can be shown to be valid if they have the same symbolic form as an argument that is known to be valid. For instance, we have shown that the argument h-m h am is valid. This symbolic form is known as direct reasoning. All arguments that have this symbolic form are valid. Table 4.1 Standard Form of Four (4) Valid Arguments Direct Reasoning p-q P Aq Contrapositive Reasoning p-q -9 p-q -9 -p Transitive Reasoning Disjunctive Reasoning g-r r-m Note that transitive reasoning can be extended more than two (2) conditional premises. Say, if we have p→q, q→r, and r→s then a valid conclusion for the arguments is p→s. Example 4.3 Use standard form to determine a valid conclusion for the following arguments. 1. If Kim is a lawyer (p), then she will be able to help us (q). Kim is not able to help us (-p). Therefore: Solution: The symbolic form of the premises is If they had a good time (g), they will return (r). If they return (r), we will make more money (m). Therefore: p-q Solution: The symbolic form for the premises is q-r -p-r This matches the standard form known as contrapositive reasoning. Thus the valid conclusion is -p (Kim is not a lawyer.) p-q q AP p pvq -P Aq Table 4.2 Standard Form of Two (2) Invalid Arguments On the other hand, this matches the standard form called transitive reasoning thus, the valid conclusion is gm (If they had a good time, they will make more money). Fallacy of the Converse Fallacy of the Inverse pvq -q -p p-q P Any argument that matches either of the two forms as illustrated is invalid.
Transcribed Image Text:STANDARD FORMS Arguments can be shown to be valid if they have the same symbolic form as an argument that is known to be valid. For instance, we have shown that the argument h-m h am is valid. This symbolic form is known as direct reasoning. All arguments that have this symbolic form are valid. Table 4.1 Standard Form of Four (4) Valid Arguments Direct Reasoning p-q P Aq Contrapositive Reasoning p-q -9 p-q -9 -p Transitive Reasoning Disjunctive Reasoning g-r r-m Note that transitive reasoning can be extended more than two (2) conditional premises. Say, if we have p→q, q→r, and r→s then a valid conclusion for the arguments is p→s. Example 4.3 Use standard form to determine a valid conclusion for the following arguments. 1. If Kim is a lawyer (p), then she will be able to help us (q). Kim is not able to help us (-p). Therefore: Solution: The symbolic form of the premises is If they had a good time (g), they will return (r). If they return (r), we will make more money (m). Therefore: p-q Solution: The symbolic form for the premises is q-r -p-r This matches the standard form known as contrapositive reasoning. Thus the valid conclusion is -p (Kim is not a lawyer.) p-q q AP p pvq -P Aq Table 4.2 Standard Form of Two (2) Invalid Arguments On the other hand, this matches the standard form called transitive reasoning thus, the valid conclusion is gm (If they had a good time, they will make more money). Fallacy of the Converse Fallacy of the Inverse pvq -q -p p-q P Any argument that matches either of the two forms as illustrated is invalid.
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