Complex Propositions
We've noted that there are many ways simple sentences can be modified or connected to form complex propositions. When an operator modifies one sentence, we call it a unary operator. When an operator connects two sentences, we call it a binary operator. Here are some examples of unary and binary operators
| sentence | operator | unary or binary? |
| Ignat hoped that Abner would win the race. | Ignat hoped that | unary |
| Martha is a skier but her husband is not. | but | binary |
| Fred will go to the opera if and only if John agrees. | if and only if | binary |
| I won't go on that rollercoaster. | not | unary |
| Arthur will see to it that everyone has a good time. | Arthur will see to it that | unary |
How many different operators are there? If "Ignat hoped that" is a different operator than "Martha hoped that," then there are many more operators than there are names!
In PL, however, we'll be interested in just a few operators. In
fact, PL deals with only the following five operators:
| not |
| and |
| or |
| if __ then ___ |
| if and only if |
Why five, and why just these five? Propositional logic deals only with a class of propositions which are truth-functional. A truth-functional proposition is a proposition where the truth value of the whole proposition is completely determined by the truth value of its simple propositional components. This definition will make more sense when you've seen how we can compute truth values of complex propositions when we know the truth value of its parts. The five connectives listed above can be used to create truth-functional propositions, and it turns out that any complex truth-functional proposition can constructed from these five operators.
In the next section you'll be introduced to the symbols we'll use as
abbreviations for these five operators.
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