# About

Truth tables are useful for checking up on the validity of arguments formalized in propositional logic and for finding necessary true propositions and necessary false propositions. They cannot be used to discover contingent propositions.^{1}

Truth tables are constructed by typing out all the possible (and sometimes impossible too) truth values that propositions might have, T and F. This is most easily done in tables, hence the name *truth tables*. The first thing to do when constructing a such table is to calculate how many rows is needed. The number of needed rows is 2^{n} at most where n is the number of simple sentence variables to be evaluated. Here’s a table that reveals how many rows are needed for first few number of sentence variables

Number of sentence variables |
Number of rows needed at most |

1 | 2 |

2 | 4 |

3 | 8 |

4 | 16 |

5 | 32 |

10 | 1024 |

In the truth table âTâ means true and âFâ means false. In the table each possible combination of T’s and F’s is there.

# Constructing truth tables

Here is an easy way to construct truth tables. First we wright each of the sentence variables in a column on it’s own. Here’s an example with 3 sentence variables:

P |
Q |
R |

Then we calculate the total number of rows needed at most. In our example it is 2^{3}=8=m. (Refer to the total number of rows needed as m.) So we need 8 rows at most to reveal all the possibilities. The rows are filled out like this:

- Under the first sentence variable fill in m*1/2 rows with T and then the same number of F rows.

In our example the number is 4

P |
Q |
R |

T | ||

T | ||

T | ||

T | ||

F | ||

F | ||

F | ||

F |

- Under the second sentence variable fill in m*1/4 T’s and then the same number of F’s and keep on alternating between T and F with this number until all the rows are filled.In our example the number is 2

P |
Q |
R |

T | T | |

T | T | |

T | F | |

T | F | |

F | T | |

F | T | |

F | F | |

F | F |

- Under the third sentence variable fill in m*1/8 T’s and then the same number of F’s and keep on alternating with these number until the rows are filled.In our example the number is 1

P |
Q |
R |

T | T | T |

T | T | F |

T | F | T |

T | F | F |

F | T | T |

F | T | F |

F | F | T |

F | F | F |

- And so on.

One can generalize this method to this

- Under the b
^{th}sentence variable write first m*b*1/2 T’s and then the same number of F’s. Continue with alternating between T’s and F’s until all the rows are filled.

# The basic parts of the first five truth tables

## 1 sentence variable

P |

T |

F |

## 2 sentence variables

P |
Q |

T | T |

T | F |

F | T |

F | F |

## 3 sentence variables

P |
Q |
R |

T | T | T |

T | T | F |

T | F | T |

T | F | F |

F | T | T |

F | T | F |

F | F | T |

F | F | F |

## 4 sentence variables

P |
Q |
R |
S |

T | T | T | T |

T | T | T | F |

T | T | F | T |

T | T | F | F |

T | F | T | T |

T | F | T | F |

T | F | F | T |

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 sentence variables

P |
Q |
R |
S |
T |

T | T | T | T | T |

T | T | T | T | F |

T | T | T | F | T |

T | T | T | F | F |

T | T | F | T | T |

T | T | F | T | F |

T | T | F | F | T |

T | T | F | F | F |

T | F | T | T | T |

T | F | T | T | F |

T | F | T | F | T |

T | F | T | F | F |

T | F | F | T | T |

T | F | F | T | F |

T | F | F | F | T |

T | F | F | F | F |

F | T | T | T | T |

F | T | T | T | F |

F | T | T | F | T |

F | T | T | F | F |

F | T | F | T | T |

F | T | F | T | F |

F | T | F | F | T |

F | T | F | F | F |

F | F | T | T | T |

F | F | T | T | F |

F | F | T | F | T |

F | F | T | F | F |

F | F | F | T | T |

F | F | F | T | F |

F | F | F | F | T |

F | F | F | F | F |

### References

N. Swarts and R. Bradley, 1979, p. 282.

### Notes

1This is because the inner structure of the proposition may not be captured by the chosen formalization or by propositional logic in general. Predicate logic will reveal more necessary true propositions and necessary false propositions, but it too has limits. See N. Swartz and R. Bradley, 1979, chapter 5, section 4.