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CHAPTER SUMMARY

Connectives and Truth Values

  • In propositional logic we use symbols to stand for the relationships between statements—that is, to indicate the form of an argument. These relationships are made possible by logical connectives such as conjunction (and), disjunction (or), negation (not), and conditional (If . . . then . . .). Connectives are used in compound statements, each of which is composed of at least two simple statements. A statement is a sentence that can be either true or false.
  • To indicate the possible truth values of statements and arguments, we can construct truth tables, a graphic way of displaying all the truth value possibilities.
  • A conjunction is false if at least one of its statement components (conjuncts) is false. A disjunction is still true even if one of its component statements (disjuncts) is false. A negation is the denial of a statement. The negation of any statement changes the statement’s truth value to its contradictory (false to true and true to false). A conditional statement is false in only one situation—when the antecedent is true and the consequent is false.

Checking for Validity

  • The use of truth tables to determine the validity of an argument is based on the fact that it’s impossible for a valid argument to have true premises and a false conclusion. A basic truth table consists of two or more guide columns listing all the truth value possibilities, followed by a column for each premise and the conclusion. We can add other columns to help us determine the truth values of components of the argument.
  • You can check the validity of arguments not only with truth tables but also with the short method. In this procedure we try to discover if there is a way to make the conclusion false and the premises true by assigning various truth values to the argument’s components.

Proof of Validity

·        The method of proof is a way to confirm the validity of an argument by deducing its conclusion from its premises using simple, valid argument forms. Most valid complex arguments consist of several of these valid sub-arguments (most of which you may already know). Determining the validity of the larger argument then is a matter of moving step by step from premises to conclusion, identifying the valid, component arguments along the way.

·        The method of proof uses nine rules of inference and nine rules of replacement. By properly applying them, you can confirm an argument’s validity.