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Chapter 3: Problem 24

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) He is intelligent or an overachiever. He is not intelligent. \(\therefore\) He is an overachiever.

Short Answer

Expert verified
The argument that 'If he is not intelligent, then he must be an overachiever' is a valid argument based on the rule of disjunctive syllogism in propositional logic.

Step by step solution

01

Translation to Symbolic Form

Translate each statement into propositional logic. Let 'P' denote 'he is intelligent' and 'Q' denote 'he is an overachiever'. The given statements can be translated as follows: \n 'He is intelligent or an overachiever' translates to 'P \(\vee\) Q'.\n 'He is not intelligent' translates to '\(\neg\)P'. \n 'Therefore, he is an overachiever' translates to 'Q'.
02

Formulate the Argument Symbolically

The argument in symbolic form, using our defined propositions, becomes 'If P \(\vee\) Q and \(\neg\)P, then Q'. This argument follows the rule of disjunctive syllogism.
03

Validate the Argument

By the rule of disjunctive syllogism in logic, the argument is valid. The rule states that 'If P \(\vee\) Q is true and P is false, then Q must be true'. Hence based on this rule, if 'he is intelligent or he is an overachiever' is true and 'he is not intelligent' is true, then 'he is an overachiever' must also be true.

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Most popular questions from this chapter

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. If I am a full-time student, I cannot work. If I cannot work, I cannot afford a rental apartment costing more than \(\$ 500\) per month. Therefore, ...

In this section, we used a variety of examples, including arguments from the Menendez trial, the inevitability of Nixon's impeachment, Spock's (fallacious) logic on Star Trek, and even two cartoons, to illustrate symbolic arguments. a. From any source that is of particular interest to you (these can be the words of someone you truly admire or a person who really gets under your skin), select a paragraph or two in which the writer argues a particular point. (An intriguing source is What Is Your Dangerous Idea?, edited by John Brockman, published by Harper Perennial, 2007.) Rewrite the reasoning in the form of an argument using words. Then translate the argument into symbolic form and use a truth table to determine if it is valid or invalid. b. Each group member should share the selected passage with other people in the group. Explain how it was expressed in argument form. Then tell why the argument is valid or invalid.

Write an original argument in words for the contrapositive reasoning form.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If The Graduate and Midnight Cowboy are shown, then the performance is sold out. Midnight Cowboy was shown and the performance was not sold out. \(\therefore\) The Graduate was not shown.

Use Euler diagrams to determine whether each argument is valid or invalid. All dogs have fleas. Some dogs have rabies. Therefore, all dogs with rabies have fleas.
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