🚧 Setup for 3.4
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@ -2613,3 +2613,396 @@ $X = g$
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c. $?\text{isabove}(g, X)$
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$X = b_1, X = w_1$
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---
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**Exercise Set 3.4**
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Page 179
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1. Let the following law of algebra be the first statement of an argument: For
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all real numbers $a$ and $b$,
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$$ (a + b)^2 = a^2 + 2ab + b^2 $$
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Suppose each of the following statements is, in turn, the second statement of
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the argument. Use universal instantiation or universal modus ponens to write the
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conclusion that follows in each case.
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a. $a = x$ and $b = y$ are particular real numbers.
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b. $a = f_i$ and $b = f_i$ are particular real numbers.
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c. $a = 3u$ and $b = 5v$ are particular real numbers.
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d. $a = g(r)$ and $b = g(s)$ are particular real numbers.
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e. $a = \log(t_1)$ and $b = \log(t_2)$ are particular real numbers.
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Use universal instantiation or universal modus ponens to fill in valid
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conclusions for the arguments in 2-4.
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2.
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$$
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\text{If an integer} n \text{ equals } 2 \cdot k \text{ and } k \text{ is an integer, then } n \text{ is even.} \\
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0 \text{ equals } 2 \cdot 0 \text{ and } 0 \text{ is an integer.} \\
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\therefore \text{\_\_\_\_\_\_}
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$$
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3.
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$$
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\text{For all real numbers } a, b, c, \text{ and } d, \text{ if } b \neq 0 \text{ and } d \neq 0 \text{ then } \frac{a}{b} + \frac{c}{d} = \frac{(ad + bc)}{bd} \\
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a = 2, b = 3, c = 4, \text{ and } d = 5 \text{ are particular real numbers such that } b \neq 0 \text{ and } d \neq 0 \\
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\therefore \text{\_\_\_\_\_\_}
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$$
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4.
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$$
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\forall \text{ real numbers } r, a, \text{ and } b, \text{ if } r \text{ is positive, then } (r^q)^b = r^{ab} \\
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r = 3, a = \frac{1}{2}, \text{ and } b = 6 \text{ are particular real numbers such that } r \text{ is positive.} \\
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\therefore \text{\_\_\_\_\_\_}
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$$
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Use universal modus tollens to fill in valid conclusions for the arguments in 5
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and 6.
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5.
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$$
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\text{All irrational numbers are real numbers.} \\
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\frac{1}{0} \text{ is not a real number.} \\
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\therefore \text{\_\_\_\_\_\_}
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$$
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6.
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$$
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\text{If a computer program is correct, then compilation of the program does not produce error messages.} \\
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\text{Compilation of this program produces error messages.} \\
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\therefore \text{\_\_\_\_\_\_}
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$$
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Some of the arguments in 7-18 are valid by universal modus ponens or universal
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modus tollens; others are invalid and exhibit the converse or the inverse error.
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State which are valid and which are invalid. Justify your answers.
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7.
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$$
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\text{All healthy people eat an apple a day.} \\
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\text{Keisha eats an apple a day.} \\
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\therefore \text{Keisha is a healthy person.}
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$$
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8.
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$$
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\text{All freshmen must take a writing course.} \\
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\text{Caroline is a freshman.} \\
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\therefore \text{Caroline must take a writing course.}
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$$
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9.
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$$
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\text{If a graph has no edges, then it has a vertex of degree zero.} \\
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\text{This graph has at least one edge.} \\
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\therefore \text{This graph does not have a vertex of degree zero.}
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$$
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10.
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$$
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\text{If a product of two numbers is } 0 \text{, then at least one of the numbers is } 0 \text{.} \\
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\text{For a particular number } x \text{, neither } (2x + 1) \text{ nor } (x - 7) \text{ equals } 0 \text{.} \\
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\therefore \text{The product } (2x + 1)(x - 7) \text{ is not } 0 \text{.}
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$$
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11.
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$$
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\text{All cheaters sit in the back row.} \\
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\text{Monty sits in the back row.} \\
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\therefore \text{Monty is a cheater.}
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$$
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12.
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$$
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\text{If an 8-bit two's complement represents a positive integer, then the 8-bit two's complement starts with a 0.} \\
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\text{The 8-bit two's complement for this integer does not start with 0.} \\
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\therefore \text{This integer is not positive.}
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$$
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13.
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$$
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\text{For every student } x \text{, if } x \text{ studies discrete mathematics, then } x \text{ is good at logic.} \\
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\text{Tarik studies discrete mathematics.} \\
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\therefore \text{Tarik is good at logic.}
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$$
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14.
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$$
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\text{If compilation of a computer program produces error messages, then the program is not correct.} \\
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\text{Compilation of this program does not produce error messages.} \\
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\therefore \text{This program is correct.}
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$$
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15.
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$$
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\text{Any sum of two rational numbers is irrational.} \\
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\text{The sum } r + s \text{ is rational.} \\
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\therefore \text{The numbers } r \text{ and } s \text{ are both rational.}
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$$
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16.
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$$
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\text{If a number is even, then twice that number is even.} \\
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\text{The number } 2n \text{ is even, for a particular number } n \text{.} \\
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\therefore \text{The particular number } n \text{ is even.}
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$$
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17.
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$$
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\text{If an infinite series converges, then the terms go to 0.} \\
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\text{The terms of the infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ go to 0.} \\
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\therefore \text{The infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ converges.}
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$$
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18.
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$$
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\text{If an infinite series converges, then the terms go to 0.} \\
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\text{The terms of the infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ do not go to 0.} \\
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\therefore \text{The infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ does not converge.}
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$$
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19. Rewrite the statement "No good cars are cheap" in the form "$\forall x$ if
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$P(x)$ then $\neg Q(x)$." Indicate whether each of the following arguments
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is valid or invalid, and justify your answers.
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a.
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$$
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\text{No good car is cheap.} \\
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\text{A Rimbaud is a good car.} \\
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\therefore \text{A Rimbaud is not cheap.}
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$$
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b.
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$$
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\text{No good car is cheap.} \\
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\text{A Simbaru is not cheap.} \\
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\therefore \text{A Simbaru is a good car.}
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$$
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c.
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$$
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\text{No good car is cheap.} \\
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\text{A VX Roadster is cheap.} \\
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\therefore \text{A VX Roadster is not good.}
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$$
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d.
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$$
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\text{No good car is cheap.} \\
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\text{An Omnex is not a good car.} \\
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\therefore \text{An Omnex is cheap.}
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$$
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20.
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a. Use a diagram to show that the following argument can have true premises and
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a false conclusion.
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$$
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\text{All dogs are carnivorous.} \\
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\text{Aaron is not a dog.} \\
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\therefore \text{Aaron is not carnivorous.}
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$$
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b. What can you conclude about the validity or invalidity of the following
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argument form? Explain how the result from part (a) leads to this conclusion.
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$$
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\forall x, \text{ if } P(x) \text{ then } Q(x) \\
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\neg P(a) \text{ for a particular } a \\
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\therefore \neg Q(a)
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$$
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Indicate whether the arguments in 21-27 are valid or invalid. Support your
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answers by drawing diagrams.
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21.
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$$
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\text{All people are mice.} \\
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\text{All mice are mortal.} \\
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\therefore \text{All people are mortal.}
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$$
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22.
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$$
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\text{All discrete mathematics students can tell a valid argument from an invalid one.} \\
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\text{All thoughtful people can tell a valid argument from an invalid one.} \\
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\therefore \text{All discrete mathematics students are thoughtful.}
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$$
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23.
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$$
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\text{All teachers occasionally make mistakes} \\
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\text{No gods ever make mistakes.} \\
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\therefore \text{No teachers are gods.}
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$$
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24.
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$$
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\text{No vegetarians eat meat.} \\
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\text{All vegans are vegetarians.} \\
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\therefore \text{No vegans eat meat.}
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$$
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25.
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$$
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\text{No college cafeteria food is good.} \\
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\text{No good food is wasted.} \\
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\therefore \text{No college cafeteria food is wasted.}
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$$
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26.
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$$
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\text{All polynomial functions are differentiable.} \\
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\text{All differentiable functions are continuous.} \\
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\therefore \text{All polynomial functions are continuous.}
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$$
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27.
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[Adapted from Lewis Carrol.]
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$$
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\text{Nothing intelligible ever puzzles me.} \\
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\text{Logic puzzles me.} \\
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\therefore \text{Logic is unintelligible.}
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$$
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In exercises 28-32, reorder the premises in each of the arguments to show that
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the conclusion follows as valid consequence from the premises. It may be helpful
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to rewrite the statements in if-then form and replace some of them by their
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contrapositives. Exercises 28-30 refer to the kinds of Tarski words discussed in
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Examples 3.1.13 and 3.3.1. Exercises 31 and 32 are adapted from _Symbolic Logic_
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by Lewis Carroll.
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28.
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1. Every object that is to the right of all the blue objects is above all the triangles.
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2. If an object is a circle, then it is to the right of all the blue objects.
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3. If an object is not a circle, then it is not gray.
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$\therefore$ All the gray objects are above all the triangles.
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29.
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1. All the objects that are to the right of all the triangles are above all the circles.
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2. If an object is not above all the black objects, then it is not a square.
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3. All the objects that are above all the black objects are to the right of all the triangles.
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$\therefore$ All the squares are above all the circles.
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30.
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1. If an object is above all the triangles, then it is above all the blue objects.
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2. If an object is not above all the gray objects, then it is not a square.
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3. Every black object is a square.
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4. Every object that is above all the gray objects is above all the triangles.
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$\therefore$ If an object is black, then it is above all the blue objects.
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31.
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1. I trust every animal that belongs to me.
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2. Dogs gnaw bones.
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3. I admit no animals into my study unless they will beg when told to do so.
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4. All the animals in the yard are mine.
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5. I admit every animal that I trust into my study.
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6. The only animals that are really willing to beg when told to do so are dogs.
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$\therefore$ All the animals in the yard gnaw bones.
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32.
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1. When I work a logic example without grumbling, you may be sure it is one I understand.
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2. The arguments in these examples are not arranged in regular order like the ones I am used to.
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3. No easy examples make my head ache.
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4. I can't understand examples if the arguments are not arranged in regular order like the ones I am used to.
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5. I never grumble at an example unless it gives me a headache.
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$\therefore$ These examples are not easy.
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In 33 and 34 a single conclusion follows when all the given premises are taken
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into consideration, but it is difficult to see because the premises are jumbled
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up. Reorder the premises to make it clear that a conclusion follows logically,
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and state the valid conclusion that can be drawn. (It may be helpful to rewrite
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some of the statements in if-then form and to replace some statements by their
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contrapositives.)
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33.
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1. No birds except ostriches are at least 9 feet tall.
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2. There are no birds in this aviary that belong to anyone but me.
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3. No ostrich lives on mince pies.
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4. I have no birds less than 9 feet high.
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34.
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1. All writers who understand human nature are clever.
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2. No one is a true poet unless he can stir the human heart.
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3. Shakespeare wrote Hamlet.
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4. No writer who does not understand human nature can stir the human heart.
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5. None but a true poet could have written Hamlet.
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35. Derive the validity of universal modus tollens from the validity of
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universal instantiation and modus tollens.
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36. Derive the validity of universal form of part (a) of the elimination rule
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from the validity of universal instantiation and the valid argument called
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elimination in Section 2.3.
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@ -163,8 +163,141 @@ $y$ in $E$ anyone might choose to challenge you with.
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---
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Page 160
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**Negations of Statements with Two Different Quantifiers**
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$\neg(\forall x \text{ in } d, \exists y \text{ in } E \text{ such that } P(x, y)) \equiv \exists x \text{ in } D \text{ such that } \forall y \text{ in } E, \neg P(x, y)$
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$\neg(\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)) \equiv \forall x \text{ in } D, \exists y \text{ in } E \text{ such that } \neg P(x, y)$
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---
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Page 169
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**Universal Instantiation**
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If a property is true of _everything_ in a set, then it is true of _any
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particular_ thing in the set.
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---
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Page 170
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**Universal Modus Ponens**
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_Formal Version_
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$$
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\forall x, \text{ if } P(x) \text{ then } Q(x) \\
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P(a) \text{ for a particular } a \\
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\therefore Q(a)
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$$
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_Informal Version_
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$$
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\text{If } x \text{ makes } P(x) \text{true, then } x \text{ makes } Q(x) \text{ true.} \\
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a \text{ makes } P(x) \text{ true.} \\
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\therefore a \text{ makes } Q(x) \text{ true.}
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$$
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---
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Page 172
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**Universal Modus Tollens**
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_Formal Version_
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$$
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\forall x, \text{ if } P(x) \text{ then } Q(x) \\
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\neg Q(a) \text{ for a particular } a \\
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\therefore \neg P(a)
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$$
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_Informal Version_
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$$
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\text{If } x \text{ makes } P(x) \text{true, then } x \text{ makes } Q(x) \text{ true.} \\
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a \text{ does not make } Q(x) \text{ true.} \\
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\therefore a \text{ does not make } P(x) \text{ true.}
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$$
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---
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Page 173
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**Definition**
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To say that an _argument form_ is **valid** means the following: No matter what
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particular predicates are substituted for the predicate symbols in its premises,
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if the resulting premise statements are all true, then the conclusion is also
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true. An _argument_ is called **valid** if, and only if, its form is valid. It
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is called _sound_ if, and only if, its form is valid and its premises are true.
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---
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Page 176
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**Converse Error (Quantified Form)**
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_Formal Version_
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$$
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\forall x, \text{ if } P(x) \text{ then } Q(x) \\
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Q(a) \text{ for a particular } a \\
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\therefore \neg P(a) \text{ is an invalid conclusion}
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$$
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_Informal Version_
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$$
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\text{If } x \text{ makes } P(x) \text{true, then } x \text{ makes } Q(x) \text{ true.} \\
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a \text{ makes } Q(x) \text{ true.} \\
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\therefore a \text{ makes } P(x) \text{ true. } \text{ is an invalid conclusion}
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$$
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---
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Page 176
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**Inverse Error (Quantified Form)**
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_Formal Version_
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$$
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\forall x, \text{ if } P(x) \text{ then } Q(x) \\
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\neg P(a) \text{ for a particular } a \\
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\therefore \neg \neg Q(a) \text{ is an invalid conclusion}
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$$
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_Informal Version_
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$$
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\text{If } x \text{ makes } P(x) \text{true, then } x \text{ makes } Q(x) \text{ true.} \\
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a \text{ does not make } P(x) \text{ true.} \\
|
||||
\therefore a \text{ does not make } Q(x) \text{ true. } \text{ is an invalid conclusion}
|
||||
$$
|
||||
|
||||
---
|
||||
|
||||
Page 177
|
||||
|
||||
**Universal Transitivity**
|
||||
|
||||
_Formal Version_
|
||||
|
||||
$$
|
||||
\forall x P(x) \to Q(x) \\
|
||||
\forall x Q(x) \to R(x) \\
|
||||
\therefore \forall x P(x) \to R(x)
|
||||
$$
|
||||
|
||||
_Informal Version_
|
||||
|
||||
$$
|
||||
\text{Any } x \text{ that makes } P(x) \text{ true makes } Q(x) \text{ true.} \\
|
||||
\text{Any } x \text{ that makes } Q(x) \text{ true makes } R(x) \text{ true.} \\
|
||||
\therefore \text{Any } x \text{ that makes } P(x) \text{ true makes } R(x) \text{ true.} \\
|
||||
$$
|
||||
|
|
|
|||
|
|
@ -112,3 +112,28 @@ c. may be true or may be false.
|
|||
|
||||
c is the answer, it may be true or false depending on the nature of the property
|
||||
involving $x$, and $y$. In other words, it relies on what $P(x, y)$ states.
|
||||
|
||||
---
|
||||
|
||||
**Test Yourself**
|
||||
|
||||
Page 179
|
||||
|
||||
1. The rule of universal instantiation says that if some property is true for
|
||||
_______ in a domain, then it is true for _______.
|
||||
|
||||
2. If the first two premises of universal modus ponens are written as "If $x$
|
||||
makes $P(x)$ true, then $x$ makes $Q(x)$ true" and "For a particular value of
|
||||
$a$ _______ , " then the conclusion can be written as "______. "
|
||||
|
||||
3. If the first two premises of universal modus tollens are written as "If $x$
|
||||
makes $P(x)$ true, then $x$ makes $Q(x)$ true" and "For a particular value of
|
||||
$a$ _______ ," then the conclusion can be written as " _______. "
|
||||
|
||||
4. If the first two premises of universal transitivity are written as "Any $x$
|
||||
that makes $P(x)$ true makes $Q(x)$ true" and "Any $x$ that makes $Q(x)$ true
|
||||
makes $R(x)$ true," then the conclusion can be written as "_______."
|
||||
|
||||
5. Diagrams can be helpful in testing an argument for validity. However, if some
|
||||
possible configurations of the premises are not drawn, a person could
|
||||
conclude that an argument was _______ when it was actually _______.
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue