From the founder of logic, Aristotle, onwards logicians have analyzed formal patterns of valid inference. A deduction is valid if its conclusion must be true given that its premises are true. The original aim of logic, as Leibniz remarked, was to replace rhetoric with calculation. Modern formal logic began during the last quarter of the 19th century, but nowadays logicians draw a sharp distinction between formal systems of logic, which they refer to as proof theory, and semantic systems of logic, which they refer to as model theory. The distinction is clearest in the case of the sentential calculus. This calculus concerns implications that depend on sentential negation, as expressed by "not," and various sentential connectives, such as "if," "and," and "or," which are treated in an idealized way. The following inference is an example of a valid deduction that can be proved in the sentential calculus:

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

The brakes are on.

The switches are on.

Therefore, the engine is ready to start.

The inference is based on three atomic sentences (i.e., sentences that contain neither negation nor any connectives): the brakes are on, the switches are on, and the engine is ready to start. The inference is valid and has the form

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