We developed propositional logic to model basic proof and truth. We did so by using propositional forms to represent sentences that were either true or false. We saw that propositional logic is consistent, sound, and complete. However, the sentences of mathematics involve ideas that cannot be fully represented in propositional logic. These sentences are able to characterize objects, such as numbers or geometric figures, by describing properties of the objects, such as being even or being a rectangle, and the relationships between them, such as equality or congruence. Since propositional logic is not well suited to handle these ideas, we extend propositional logic to a richer system.
Consider the sentence it is a real number. This sentence has no truth value because the meaning of the word it is undetermined. As noted on page 3, the word it is like a gap in the sentence. It is as if the sentence was written as
However, that gap can be filled. Let us make some substitutions for it:
5 is a real number. π/7 is a real number. Fido is a real number. My niece's toy is a real number.
With each replacement, the word that is undetermined is given meaning, and then the sentence has a truth value. In the examples, the first two propositions are true, and the last two are false.
Notice that changes, whereas is a real number remains fixed. ...
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