Question

Some of the following arguments are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers.

1. c) If a product of two numbers is 0, then at least one of the numbers is 0. I pick a number x such that neither (2x + 1) nor (x − 7) equals 0. Therefore, the product (2x + 1)(x − 7) is not 0.

2. d) All cheaters sit in the back row. Monty sits in the back row. So, Monty is a cheater.

Answer 1

Question C) Assume that (2x + 1) * (x - 7) = 0. As (x + 7) is not zero, we can divide both sides of equation by (x + 7). (2x + 1) = 0 / (x - 7) => 2x + 1 = 0(1). We knew that 2x + 1 was not equal to 0, but we got (1) equation, which can't be true, so (2x + 1) * (x - 7) can't be 0 in case we pick x so that (2x + 1) and (x - 7) are not 0.

Question D) All the cheaters sitting in the back row does not mean, that Monty is a cheater, because in the back row are sitting all the cheaters and maybe other people, so we are not sure that Monty is a cheater. If they said "all the people sitting in the back row are cheaters", than Monty would be a cheater but this sentence is converse of the given sentence, so this means that it cant' be true. In conclusion, it's not obvious that Monty is a cheater, so we can't tell for sure.

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