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.