Chapter 21: Truth Tables

1
Chapter 21Truth Tables
2
What Truth Tables Do (p. 209)

• Truth tables provide a systematic way to examine
  all possible combinations of truth values for a
  statement or for the statements in an argument.
• Truth tables allow you to
• determine whether an argument form is valid.
• determine whether a statement is a tautology, a
  contradiction, or a contingent statement.
• determine whether two statements are logically
  equivalent.

3
Truth Tables for Arguments Setting Up the
Tables (pp. 210-211)

• An argument form is valid if it is impossible for
  all its premises to be true and the conclusion
  false.
• To need to construct guide columns that present
  all the possible combinations of truth values of
  the simple statements in the argument.
• If there is one simple statement, p, there are
  two rows p is true in one and false in the
  other.
• If there are two simple statements, p and q,
  there are four rows.
• If there are three simple statements, there are
  eight rows.
• In general, there are 2n rows, where n is the
  number of simple statements in the argument.

4
Truth Tables for Arguments Setting Up the
Tables (pp. 210-211)

• The truth table guide columns for two, three, and
  four simple statements look like this
• p q p q r p q r s
• T T T T T T T T T
• T F T T F T T T F
• F T T F T T T F T
• F F T F F T T F F
• F T T T F T T
• F T F T F T F
• F F T T F F T
• F F F T F F F
• F T T T
• F T T F
• F T F T
• F T F F
• F F T T
• F F T F
• F F F T
• F F F F

5
Truth Tables for Arguments Setting Up the
Tables (pp. 210-211)

• Notice that in the left most guide column the
  truth value changes from true to false after half
  the rows have been marked true.
• As you move from left to right, the variation in
  truth values is twice as frequent as in the
  previous column
• If there are eight rows (three statements), for
  example, the left column will be four Ts followed
  by four Fs, the next column varies the truth
  value every two rows, and the right-most guide
  column varies the truth value every-other row.

6
Evaluating an Argument (pp. 211-215)

• Once you have the guide columns set up, you use
  them to determine the truth values of the
  statement in the argument on the basis of the
  definitions of the symbols.
• Assume you are given the following argument form
• p ? q
• q /? p
• You construct the truth table as follows,
  including a column for every compound statement
  in the argument (truth values for compound
  statements are placed beneath the connective in
  the statement).

7
Evaluating an Argument (pp. 211-215)

• _p q ? p ? q q p
• T T ? F F T T
• T F ? T T F T
• F T ? T F T FJ
• F F ? T T F FJ

• Notice that in rows three and four the conclusion
  is false It is only those rows that could show
  that the argument is invalid, if it is invalid.
• A checkmark has been placed by those rows to
  remind us that it is those rows we need to check.
• The column for q was constructed only so that we
  could construct the column for the premise.
• The row showing invalidity has been circled.

8
Evaluating an Argument (pp. 211-215)
Assume you are given the argument form p ? (q
v r) q /? p ? r You construct the truth
table as follows, including a column for every
compound statement in the argument (truth values
for compound statements are placed beneath the
connective in the statement).
9
Evaluating an Argument (pp. 211-215)

• p q r ? p ? (q v r) q p ? r
• T T T ? T T F T
• T T F ? T T F FJ
• T F T ? T T T T
• T F F ? F F T FJ
• F T T ? T T F T
• F T F ? T T F T
• F F T ? T T T T
• F F F ? T F T T

10
Evaluating an Argument (pp. 211-215)

• Notice that in rows two and four the conclusion
  is false It is only those rows that could show
  that the argument is invalid, if it is invalid.
• A checkmark has been placed by those rows to
  remind us that it is those rows we need to check.
• The column for q v r was constructed only so
  that we could construct the column for the
  premise.
• A line has been drawn through it to remind us
  that it does not play a role in evaluating the
  argument.
• In both rows in which the conclusion is false, at
  least one premise is also false. So, the
  argument form is valid.

11
Evaluating an Argument (pp. 211-215)

• Now consider the following argument form
• p ? q
• q ? r /? p r
• You proceed as you did before, setting up the
  guide columns and using the definitions of the
  symbols to determine the truth values of the
  component statements.

12
Evaluating an Argument (pp. 211-215)

• p q r ? p ? q q ? r p r
• T T T ? T F F T
• T T F ? T T T FJ
• T F T ? F T F T
• T F F ? F T T FJ
• F T T ? F F F FJ
• F T F ? F T T FJ
• F F T ? T T F FJ
• F F F ? T T T FJ

13
Evaluating an Argument (pp. 211-215)

• A line was drawn through the column for r, since
  that column was completed only so we could
  determine the truth values for the second
  premise.
• Notice that in rows two, seven, and eight, all
  the premises are true and the conclusion is
  false.
• Whenever there is a row in which all the
  premises are true and the conclusion is false,
  that shows that the argument form is invalid.
• Rows showing that the argument form is invalid
  are circled.

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Evaluating an Argument Summary (pp. 211-215)

• Truth tables can be constructed for either
  arguments, in which statements are represented by
  upper-case letters, or for argument forms, which
  is stated in terms of the statement variables p,
  q, r, s, t,
• Construct the guide columns for the truth table.
• Construct a column of the truth table for every
  compound statement.
• Attending only to the columns for the premises
  and conclusion, determine whether there is a row
  in which all the premises are true and the
  conclusion is false.
• If there is at least one row in which all the
  premises are true and the conclusion is false,
  the argument form is invalid.
• Circle all rows in which the premises are true
  and the conclusion is false.
• If there is no row in which all the premises are
  true and the conclusion is false, the argument
  form is valid.

15
Tautologies, Contradictions, and Contingent
Statements (pp. 217-218)

• A tautology is a statement that is true in virtue
  of its form.
• The column of a truth table for a tautology has a
  T in every row.
• A contradiction is a statement that is false in
  virtue of its form.
• The column of a truth table for a contradiction
  has an F in every row.
• A contingent statement is a statement that is
  sometimes true and sometimes false.
• The column of a truth table for a contingent
  statement contains at least one T and at least
  one F.

16
Tautologies, Contradictions, and Contingent
Statements (pp. 217-218)

• p ? p is a tautology
• p ? p ? p
• T ? T
• F ? T
• p ? p is a contradiction
• p ? p ? p
• T ? F F
• F ? F T

17
Tautologies, Contradictions, and Contingent
Statements (pp. 217-218)

• p ? q is a contingent statement
• _p q ? p ? q
• T T ? T
• T F ? F
• F T ? T
• F F ? T

18
Logical Equivalence (pp. 218-219)

• Two statements are logically equivalent if they
  are true under the same circumstances.
• If two statements are logically equivalent, then
  a biconditional in which those statements flank
  the double-arrow will be a tautology.

19
Logical Equivalence (pp. 218-219)

• The statements p ? q and p v q are logically
  equivalent.
• _p q ? (p ? q) ? (p v q)
• T T ? T T F T
• T F ? F T F F
• F T ? T T T T
• F F ? T T T T