Valid and Invalid Arguments, Modus Ponens, Modus Tollens

  1. Valid Arguments.

    1. Definition: An argument is a sequence of statements in which all of the statements except the last one are premises (aka assumptions, aka hypotheses), and the last one is the conclusion.

    2. Definition: An argument form is a sequence of statement forms in which all of the statement forms except the last one are premises (aka assumptions, aka hypotheses), and the last one is the conclusion.

    3. Examples:

      • p implies q
        p
        therefore q

      • If Sally is a freshman, then Sally has not declared a major.
        Sally is a freshman.
        therefore Sally has not declared a major.

    4. Definition: An argument form is valid if no matter what particular statements are substituted for the statement variables in its premises, the conclusion is true whenever all of the premises are true.

    5. Definition: An argument is valid if its form is valid.

  2. Testing an Argument Form for Validity.

    1. To test an argument form for validity:

      1. Identify the premises and conclusion of the argument form.
      2. Construct a truth table showing the truth values of all premises and the conclusion.
      3. If the truth table contains any rows in which all of the premises are true and the conclusion is false, then the argument form is invalid. Otherwise, the argument form is valid.
        (So, the crucial rows to check the conclusion are the ones in which all of the premises are true.)

    2. Examples:

      • p or (q or r)
        not r
        therefore p or q

        Show that the above argument form is valid.

      • p implies (q or not r)
        q implies (p and r)
        therefore p implies r

        Show that the above argument form is invalid.

  3. Modus Ponens (method of affirming) and Modus Tollens (method of denying).

    1. Argument forms consisting of two premises and a conclusion are called syllogisms.
      The first and second premises are called the major premise and the minor premise.
      A very important and commonly used form of syllogism is called modus ponens, which is Latin for "method of affirming." It has this classic form:
      p implies q
      p
      Therefore, q
      You may construct a truth table to prove the validity of this argument form.

    2. Another important and commonly used form of syllogism is called modus tollens, which is Latin for "method of denying." It has this classic form:
      p implies q
      not q
      Therefore, not p.
      You may construct a truth table to prove the validity of this argument form.

      This is sometimes referred to as proof by contradiction.

    3. Examples:

      • If today is Monday, then I will go to MA 125.
        Today is Monday.
        Therefore, I will go to MA 125.

        Is the conclusion always true whenever both premises are true?
        Is this a valid argument?
        What happens on Wednesday? I go to MA 125 (but it is not Monday).
        The argument is valid every day of the week, but it is only a sound argument on Monday. Every other day of the week it is still a valid argument, but it is an unsound argument since one or more premises is false.

      • If my car is out of gas, then it will not run.
        My car runs [is running].
        Therefore, my car is not out of gas.