# Introduction to Logic

## 8.1 Truth Tables and Calculation of truth-values

So far we’ve see truth tables used to define the operators. They have other uses as well: they make it possible to classify and to compare statements to appreciate their logical properties, to test arguments for validity, and to define rules of deduction and replacement. A truth table exhibits all the truth-values that it is possible for a given statement or set of statements to have. But let’s back up just a bit.

In the previous section we introduced the truth-functional definitions of the operators. With that information (exhibited on truth-tables, which showed all the possible values “p” and “q” could have), we have enough information that we can “calculate” or figure out the truth-value of compound statements as long as we know the truth-values of the simple statements that make them up.

For instance, since we know that

“Bananas are fruit” is true

and “Apples are fruit” is true

and “Pears are fruit” is true,

we can figure out that this statement:

B ⊃ (A ∙ ~P)

is false.

How? List the truth values under the letters, and then combine the values according to the definitions of the five operators, starting at the smallest unit and working up to the largest.

B | ⊃ | (A | ∙ | ~P) |

T | T | FT |

This table shows us the values of these three statements. Each is true, so we have a “T” under each statement; and since the negation of “Pears are fruit” occurs (“Pears are not fruit”), we have an “F” under the tilde.

The simplest or smallest level at which any “calculation” can be done is that negation of a simple statement. The next level is in the conjoining of the negated statement with “Apples are fruit.” The claim that “Apples are fruit but Pears are not” is false, so an “F” goes under the dot.

B | ⊃ | (A | ∙ | ~P) |

T | T | F | FT |

That dot statement is the consequent of the conditional, and the antecedent of the conditional is true, so the conditional itself is false; an “F” goes under the horseshoe. I’ve colored it red to make it more noticeable.

B | ⊃ | (A | ∙ | ~P) |

T | F | T | F | FT |

Now, in this one-line format, the only way to get these symbols to line up straight is to present them in a table. But the table showing us that B ⊃ (A ∙ ~P) is false is **not** what we’ll call a “Truth Table.” A truth table shows **all** the possible truth values that the simple statements in a compound or set of compounds can have, and it shows us a result of those values; it is always at least two lines long. The example we are looking at here is simply calculating the value of a single compound statement, not exhibiting all the possibilities that the form of this statement allows for.

The tables we used to define the operators, repeated below, are truth tables. There are no combinations of truth values for these statements that have not been shown.

Negation

p | ~p |

T | F |

F | T |

Conjunction

p | ∙ | q |

T | T | T |

T | F | F |

F | F | T |

F | F | F |

Disjunction

p | v | q |

T | T | T |

T | T | F |

F | T | T |

F | F | F |

Material Conditional

p | ⊃ | q |

T | T | T |

T | F | F |

F | T | T |

F | T | F |

Biconditional

p | ≡ | q |

T | T | T |

T | F | F |

F | F | T |

F | T | F |

Truth tables provide the truth-functional definition of the five operators. With those definitions, we can calculate the truth value of compound statements once we know the truth values of the simple ones that make them up. Here are some examples, and some exercises you can practice at.

James Farmer taught *The History of the Civil Rights Movement* at Mary Washington.

Martin Luther King taught theology at Mary Washington.

Since the first of these is true and the second is false, the conjunction made up of them (F . K) is false.

If King taught theology or Farmer taught *Civil Rights* at Mary Washington, then a major figure of the Civil Rights Movement was a member of the UMW faculty.

This conditional has a disjunction for its antecedent:

(K v F) ⊃ M

K = King taught at UMW

F = Farmer taught at UMW

M = a major figure of the CRM was on the faculty.

(K v F) ⊃ M

(K | v | F) | ⊃ | M |

F | T |
T | T |
T |

I’ve marked the truth values: the first one we can enter is the bolded one, because K v F is the smallest unit; the second one is the slanted one, which combines the value from K v F with the value of M. By the way, don’t infer from this example that the first value you can calculate will always be the left-most one. The last value you can do is the one for what’s called the main operator: this statement is a conditional, its main operator is the horseshoe.

Here are some more you can practice with:

1. Obama and Clinton are Democrats if Christie is a Republican.

2. Either Clinton will run or Christie is a Democrat.

3. If Clinton runs then she is at least 35 years old.

4. Obama is commander in chief if and only if he is President.

5. Being born in America is a necessary condition for being president.

6. If being born here is a necessary condition for running, then Schwarzenegger cannot run.

7. Being sound is a sufficient condition for being valid and being valid is a necessary condition for being sound; in addition, being deductive is a necessary condition for being sound and for being valid.

8. If either Hume did not invent truth tables or Wittgenstein wrote the Tractatus, then Russell’s paradox was bad news to Frege; but Kant denied that “existence” was a predicate only if Aristotelian logic dominated for two thousand years.

9. Either Hume did not invent truth tables or else Wittgenstein wrote the Tractatus, and Russell’s paradox was bad news to Frege only if Kant denied that “existence” was a predicate, given that Aristotelian logic dominated for two thousand years.

10. If it is false both that Hume invented truth tables and that Kant denied “existence” was a predicate, then given that Aristotelian logic dominated for two thousand years, Wittgenstein’s writing the Tractatus implies that Russell’s paradox was bad news to Frege.

Answers:

1. Obama and Clinton are Democrats if Christie is a Republican. C > (O . C)

2. Either Clinton will run or Christie is a Democrat. C v D

3. If Clinton runs then she is at least 35 years old. C > T

4. Obama is commander in chief if and only if he is President. O ≡ C

5. Being born in America is a necessary condition for being president. P > A

6. If being born here is a necessary condition for running, then Schwarzenegger cannot run. (P > R) > ~S

7. Being sound is a sufficient condition for being valid and being valid is a necessary condition for being sound; in addition, being deductive is a necessary condition for being sound and for being valid. [(S > V) . (S > V)] . [(S v V) > D]

8. If either Hume did not invent truth tables or Wittgenstein wrote the Tractatus, then Russell’s paradox was bad news to Frege; but Kant denied that “existence” was a predicate only if Aristotelian logic dominated for two thousand years.

[(~H v W) > R] . (K > A)

H = f ; W = t; R = t; K= t; A= t

9. Either Hume did not invent truth tables or else Wittgenstein wrote the Tractatus, and Russell’s paradox was bad news to Frege only if Kant denied that “existence” was a predicate, given that Aristotelian logic dominated for two thousand years.

A > [(~H v W) . (R > K)]

10. If it is false both that Hume invented truth tables and that Kant denied “existence” was a predicate, then given that Aristotelian logic dominated for two thousand years, Wittgenstein’s writing the Tractatus implies that Russell’s paradox was bad news to Frege.

~(H . K) > [A > (W > R)]