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# Superman - Superman does not exist A valid argument from...

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Superman does not exist: A valid argument from Kalish/Montague Tim Davis, COT 3100, Spring 2012 October 28, 2011 Abstract An argument is valid if its conclusion must follow from its premises. However, even a valid argument may have a false premise, in which case we are not compelled to accept its conclusion. In contrast, an argument containing a logical fallacy is invalid , and its conclusion does not follow from its premises, even if all its premises are accepted as true. Two related arguments are considered: an invalid one from the COT 3100 textbook ( Discrete Mathematics and Its Applications , Rosen, 7th Edition), and the valid argument on which it is based (from Logic: Techniques of Formal Reasoning , by Kalish and Montague). 1 Overview Section 2 provides the definition of a valid argument . An argument from Rosen is shown to contain a logical fallacy (Section 3), although a “proof” is given if we ignore this fallacy. Rosen’s argument is a simplification of a more interesting valid argument by Kalish and Montague (Section 4). Section 5 provides a rigorous proof of the Kalish/Montague argu- ment, and shows that their argument is valid. However, Section 6 presents the first of two radically different worldviews that would discard one or more of its premises as false. The second worldview is presented in Section 7 which discusses the reasons why I believe the Kalish/Montague argument contains a false premise. 2 What is a valid argument? An argument is a sequence of statements that starts with a set of premises, or assumptions, and ends with a conclusion. If the argument is valid , then the conclusion must follow from the preceding statements. That is, an argument is valid if and only if it is impossible for the premises to be true and the conclusion to be false. In other words, when faced with a valid argument, IF you believe the premises to be true, THEN you must accept the conclusion. Otherwise, the rules of logic themselves must be discarded. With a valid argument, however, if you do not accept the truth of one or more premises, then the conclusion may or may not be true. You are not compelled to believe the conclusion is true or false (assuming your objections to the premise(s) are reasonable). The valid argument does not apply, and the conclusion has not been shown to be either true or false. 1

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3 A supposedly valid argument from the textbook, but with a logical fallacy Problem 35 in Section 1.6 of the COT 3100 textbook (Rosen, 7th Edition), is given below. It appears as problem 35 in Section 1.5 in the 6th edition as well. I have enumerated the statements in the argument to refer to them later. *35. Determine if whether this argument, taken from Kalish and Montague (1964) is valid: 1. If Superman were able and willing to prevent evil, he would do so.
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