2.1 The Conditional

As we noted in chapter 1, there are sentences of a natural language, like English, that are not atomic sentences. Our examples included

If Lincoln wins the election, then Lincoln will be President.

The Earth is not the center of the universe.We could treat these like atomic sentences, but then we would lose a great deal of important information. For example, the first sentence tells us something about the relationship between the atomic sentences “Lincoln wins the election” and “Lincoln will be President”. And the second sentence above will, one supposes, have an interesting relationship to the sentence, “The Earth is the center of the universe”. To make these relations explicit, we will have to understand what “if…then…” and “not” mean. 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. Let us start with the first one.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”. We could thus represent this sentence by letting

Lincoln wins the election

be represented in our logical language by

P

And by letting

Lincoln will be president

be represented byQ

Then, the whole expression could be represented by writing

If P then Q

It will be useful, however, to replace the English phrase “if…then…” by a single symbol in our language. The most commonly used such symbol is “→”. Thus, we would writeP→Q

One last thing needs to be observed, however. We might want to combine this complex sentence with other sentences. In that case, we need a way to identify that this is a single sentence when it is combined with other sentences. There are several ways to do this, but the most familiar (although not the most elegant) is to use parentheses. Thus, we will write our expression