It is helpful at this point to change our example. So, if we are going to respect the principle of bivalence, then we have to put either T or F in for each of the last two rows. But our principle of bivalence requires that-in all kinds of situations-every sentence is either true or false, never both, never neither. If we did, we would be saying that sometimes a conditional can have no truth value that is, we would be saying that sometimes, some sentences have no truth value. Note now that our principle of bivalence requires us to fill in these rows. Some students, however, find it hard to determine what truth values should go in the next two rows. So the first rows of our truth table are uncontroversial. Would the sentence “If Lincoln wins the election, then Lincoln will be President” still be true? Most agree that it would be false now. Similarly, suppose that Lincoln wins the election, but Lincoln will not be President. Then, would I have spoken truly if I said, “If Lincoln wins the election, then Lincoln will be President”? Most people agree that I would have. Suppose the world is such that Lincoln wins the election, and also Lincoln will be President. For the first row of the truth table, we have that P is true and Q is true. Let us consider each kind of way the world could be. But we want to capture as much of the meaning of the English “if…then…” as we can, while remaining absolutely precise in our language. What matters is that once we define the semantics of the conditional, we stick to our definition. Now, we must decide upon what the conditional means. In the case of a conditional formed out of two atomic sentences, like our example of (P→Q), our truth table will have 2 2 rows, which is 4 rows. Thus, for n atomic sentences, our truth table must have 2 n rows. Note that, since there are two possible truth values (true and false), whenever we consider another atomic sentence, there are twice as many ways the world could be that we should consider. There are four kinds of ways the world could be that we must consider. Thus, the left hand side of our truth table will look like this: P We must consider when P is true and when it is false, but then we need to consider those two kinds of situations twice: once for when Q is true and once for when Q is false. That means, we have to consider four possible kinds of situations. Note that either atomic sentence could be true or false. But now, our sentence has two parts that are atomic sentences, P and Q. We know how to write the conditional, but what does it mean? As before, we will take the meaning to be given by the truth conditions-that is, a description of when the sentence is either true or false. The second sentence (the one after the arrow, which in this example is “ Q”) is called the “consequent”. The first constituent sentence (the one before the arrow, which in this example is “ P”) is called the “antecedent”. It is also sometimes called a “material conditional”. This kind of sentence is called a “conditional”. There are several ways to do this, but the most familiar (although not the most elegant) is to use parentheses. In that case, we need a way to identify that this is a single sentence when it is combined with other sentences. 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.
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