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Chapter 21: Truth Tables

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Chapter 21:Truth Tables

- 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.

- 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.

- 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 |

- 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.

- 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).

_p q p ~qqp

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.

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).

p q r p (qvr) ~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

- 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.

- 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.

p q r p qq ~rp & 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

- 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.

- 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.

- 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.

- 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

- p q is a contingent statement
_p q p q

T T T

T F F

F T T

F F T

- 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.

- The statements “p q” and “~p v q” are logically equivalent.
_p q (p q) (~pvq)

T T T T F T

T F F T F F

F T T T T T

F F T T T T