Therefore, B.

These two rules suffice to prove the conclusion about starting the engine:

1. If the brakes are on and the switches are on then the engine is ready to start.

2. The brakes are on.

3. The switches are on.

4. Therefore, the brakes are on and the switches are on [The rule for introducing "and" applied to sentences 2 and 3]

5. Therefore, the engine is ready to start. [Modus ponens applied to sentences 1 and 4].

Table I presents a set of formal rules of inference for the sentential calculus. With such rules, you can construct a formal proof, as in the preceding example, with each step in the proof warranted by one of the rules of inference.

Your knowledge of the meaning of the connectives helps you to understand the validity of the rules in Table I. However, the rules do not rely on these meanings. They work in a formal way, allowing you to write patterns of symbols given other patterns of symbols. A proof in a formal calculus is accordingly like a computer program. A computer predicts the weather, for example, but it has no idea of what rain or sunshine is or of what it is doing. It slavishly shifts "bits," which are symbols made up from patterns of electricity, from one memory store to another, and

Table I

Formal Rules of Inference for the Sentential Calculus"

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