Knj

Rule for conditional proof A (a supposition)

Rule for modus ponens If A then B A

aThe rules in the left-hand column introduce negation and the sentential connectives into inferences; those in the right-hand column eliminate them from inferences.

displays symbols that meteorologists can interpret as maps of weather. Indeed, proofs and computer programs are intimately related, and certain programs can prove inferences in logical calculi. Likewise, certain programming languages, such as PROLOG, are akin to a logical calculus.

Formal proofs establish that inferences are valid, but validity is not a concept that is defined within proof theory. Its definition hinges on truth, which underlies the semantics of the calculus (i.e., its model theory). In the model theory of the sentential calculus, the truth or falsity of compound sentences depends only on the truth or falsity of their constituent sentences. Thus, an assertion of the form, A or B or both, is true if A is true, B is true, or both of them are true. Otherwise it is false. Logicians lay out these definitions in truth tables, as shown in Table II. Each row in a truth table is a "model" of a possibility and presents the truth value of the compound sentence—in this case, A or B or both—in that possibility. The first row in the table, for instance, presents the case in which A is true and B is true, and so the disjunction is true in this possibility.

One problematic connective is "if." Its everyday usage sometimes departs from its idealized logical meaning in the sentential calculus. An assertion such as

If that patient has malaria then she has a fever is, in fact, compatible with three possibilities: The patient has malaria and a fever, she has no malaria and fever, and she has no malaria and no fever. It is false in only one case: She has malaria and no fever. The assertion is therefore equivalent to

If that patient has malaria then she has a fever, and if she does not have malaria then she either has or does not have a fever.

Logical license exists just as much as poetic license: Logicians make simplifying assumptions about the meanings of logical terms.

Table II

A Truth Table for the Disjunction A or B or Both, Which Shows Its Truth Value for the Four Possibilities Depending on the Truth or Falsity of A and of B

Table II

A Truth Table for the Disjunction A or B or Both, Which Shows Its Truth Value for the Four Possibilities Depending on the Truth or Falsity of A and of B

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