![]() We might want to combine this complex sentence with other sentences. One last thing needs to be observed, however. The most commonly used such symbol is “→”. It will be useful, however, to replace the English phrase “if…then…” by a single symbol in our language. Then, the whole expression could be represented by writing We could thus represent this sentence by lettingīe represented in our logical language by The sentence, “If Lincoln wins the election, then Lincoln will be President” contains two atomic sentences, “Lincoln wins the election” and “Lincoln will be President”. Thus, it would be useful if our logical language was able to express these kinds of sentences in a way that made these elements explicit. To make these relations explicit, we will have to understand what “if…then…” and “not” mean. ![]() And the second sentence above will, one supposes, have an interesting relationship to the sentence, “The Earth is the center of the universe”. For example, the first sentence tells us something about the relationship between the atomic sentences “Lincoln wins the election” and “Lincoln will be President”. We could treat these like atomic sentences, but then we would lose a great deal of important information. The Earth is not the center of the universe. If Lincoln wins the election, then Lincoln will be President. “If…then….” and “It is not the case that….” 2.1 The ConditionalĪs we noted in chapter 1, there are sentences of a natural language, like English, that are not atomic sentences. If Amy plays soccer then Bill plays hockey.2. Which of the following is a conditional statement? If you make a mistake, choose a different button. Feedback to your answer is provided in the RESULTS BOX. Select your answer by clicking on its button. The conditional is defined to be true unless a true hypothesis leads to a false conclusion.ĭirections: Read each question below. Summary: A conditional statement, symbolized by p q, is an if-then statement in which p is a hypothesis and q is a conclusion. Also, in item 6, the hypothesis is the negation of r. Note that in item 5, the conclusion is the negation of p. If Harrison Ford is not an American actor, then 7 2 is equal to 49. If Harrison Ford is an American actor, then 7 2 is not equal to 49. If a rectangle does not have 4 sides, then a square is not a quadrilateral. If 7 2 is equal to 49, then Harrison Ford is an American actor. If a rectangle does not have 4 sides, then Harrison Ford is an American actor. If 7 2 is equal to 49, then a rectangle does not have 4 sides. Write each conditional below as a sentence. Solution: Since hypothesis s is true and conclusion r is false, the conditional s r is false. Solution: Since hypothesis r is false and conclusion s is true, the conditional r s is true. In the following examples, we are given the truth values of the hypothesis and the conclusion and asked to determine the truth value of the conditional. Solution: The conditional x y represents, "If Gisele has a math assignment, then David owns a car. However, intuitively, we know that this is false because the sun and the number three have nothing to do with one another! Therefore, the logical conditional allows implications to be true even when the hypothesis and the conclusion have no logical connection. The implication of a b is that: since the sun is made of gas, this makes 3 a prime number. In logic, the conditional is defined to be true unless a true hypothesis leads to a false conclusion. Note that the logical meaning of this conditional statement is not the same as its intuitive meaning. In Example 2, "The sun is made of gas" is the hypothesis and "3 is a prime number" is the conclusion. Solution: The conditional a b represents "If the sun is made of gas, then 3 is a prime number." a Then construct a truth table for this conditional. Thus, the conditional p q represents the hypothetical proposition, "If I do my homework, then I get an allowance." However, as you can see from the truth table above, doing your homework does not guarantee that you will get an allowance! In other words, there is not always a cause-and-effect relationship between the hypothesis and conclusion of a conditional statement. Solution: In Example 1, the sentence, "I do my homework" is the hypothesis and the sentence, "I get my allowance" is the conclusion. Now that we have defined a conditional, we can apply it to Example 1. ![]() Note that a conditional is a compound statement. In the truth table above, p q is only false when the hypothesis (p) is true and the conclusion (q) is false otherwise it is true. The conditional is defined to be true unless a true hypothesis leads to a false conclusion. The logical connector in a conditional statement is denoted by the symbol. Symbolized by p q, it is an if-then statement in which p is a hypothesis and q is a conclusion. ![]()
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