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Lesson_Seven_and_Test

# Lesson_Seven_and_Test - LESSON 7 Truth-functional Logic...

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Section 13: Simple Truth Tables, Barker, pp. 77-83. ***Read the section, then the following additional comments. In this section we explore what happens to our basic relationships when the simple sentences fall into various combinations of truth or falsehood. We will express these possibilities in “truth tables” which will capture all possible combinations. To keep things compact, we will express truth with a capital “T,” and falsehood with a capital “F.” 1. Negation What happens when you negate a false sentence? It becomes true. What if you negate a true sentence? It becomes false. Thus here is a very simple truth table. P -P T F F T The first column exhausts the possibilities for truth and falsehood of P, the second column indicates what happens when they are negated. As we can see, their truth value is reversed. 2. Conjunction Under what conditions can a conjunction be true? Remember, a conjunction tells us that both statements are true. Thus, a conjunction as a whole can only be true if, in fact, both statements are true. So we get this truth table: P Q P & Q T T T F T F T F F F F F There is only one line on this truth table where the conjunction is true, namely where both propositions are true. In all other combinations, the conjunction is false. 3. Disjunction When is a disjunction true? Whenever at least one of the propositions is true. This fact can be summarized with the following truth table: 1 LESSON 7 Truth-functional Logic, Part 2

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P Q P v Q T T T F T T T F T F F F Thus we see that a disjunction is true unless it turns out that both propositions are false. The last line may cause you to be puzzled a bit. If a statement says “or,” doesn’t that mean that one must be true and the other one false? Sometimes, but not always. If I say, The Braves or the Reds will win tonight. B v R We need to understand from the context how we are to interpret this statement. If the Braves play against the Reds, then one will be true, the other false. But if they are playing against other teams and not each other both statements may in fact be true. We call this distinction the difference between an exclusive disjunction where both cannot be true and an inclusive disjunction where both may be true. Logic is cautious by nature—it never wants to rule out something unless there is good reason to. That’s why the normal assumption for a disjunction is inclusive. A disjunction is false only if both propositions are false. 4. Conditional In what combination of truth and falsehood is a conditional as a whole true? The results turn out to be a little surprising, but make sense when you think about the meaning of a conditional. This time, let us follow the lines of the basic truth table with an example. If it’s a rose then it has thorns.
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Lesson_Seven_and_Test - LESSON 7 Truth-functional Logic...

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