Chapter 3: Truth Tables

3.1: Introduction

With the definitions given we can work out the truth-value for all possible combinations of the truth-values of the letters in the formula.

We use truth-tables for this.

3.2: Matrix Rows and Columns

Here's how to build a truth-table for ~ ~ p.

We need two rows because there are two possible assignments of truth-value to p.

1.  The matrix has one column

     p        ~ ~ p

     1

     0

2.  Mark the main operator of the formula.

     p        ~ ~ p

     1

     0

              Ý

The main column of values will be above this arrow.

Begin to fill in the table.

3. Put values under the letters.

     p        ~ ~ p

     1            1

     0            0

              Ý

4. Fill in the next column beside the ~.

     p        ~ ~ p

     1          0 1

     0          1 0

              Ý

5. Fill in the main column.

     p        ~ ~ p

     1        1 0 1

     0        0 1 0

              Ý

Here's how to build a truth-table for (p É ~ ~ p).

We still need two rows because there are two possible assignments of truth-value to p.

Note that the calculation s in these truth tables follow the steps of assembling the formula.

Here is the assembly sequence for (p É ~ ~ p).

1.   p

2.   ~ p

3.   ~ ~ p

4.  (p É ~ ~ p)

We'll put the numbers under the operators to show the order in which the table is to be filled.

     p        ( p É ~ ~ p )

     1       

     0       

                  Ý

                1 4 3 2 1

Values above 2 are calculated from values above 1 according to the definition for ~ because assembly stage 2 is the negation of the asembly stage 1.

Values above 3 are calculated from values above 2 similarly.

Values above 4 are calculated from values above 3 and 1 according to the definition for ~ because assembly stage 4 is the conditional with 1 and 3 as parts.

     p        ( p É ~ ~ p )

     1          1 1 1 0 1

     0          0 1 0 1 0

                  Ý

                1 4 3 2 1

As the number of letters in formulas increases so does the number of rows in the truth table because the number of posible combinations of truth-values for the letters increases.

Here's how to build a truth-table for (p É ~ ~ q).

We need four rows because there are four possible assignments of truth-value to p and q.

Here is the assembly sequence for (p É ~ ~ q).

1.   p

2.   q

3.   ~ q

4.   ~ ~ q

5.  (p É ~ ~ q)

We'll put the numbers under the operators to show the order in which the table is to be filled, and fill the table using the same procedure as before.

     p q       ( p É ~ ~ q )

     1 1         1 1 1 0 1   

     1 0         1 0 0 1 0   

     0 1         0 1 1 0 1   

     0 0         0 1 0 1 0   

                   Ý

                 1 5 4 3 2

3.3: Classifying Formulas

Any formula which has a truth-table with a main column all '1's is a tautology. It is always true.

Tautologies are said to be analytically true, or necessarily true.

Any formula which has a truth-table with a main column all '0's is a contradiction. It is always false.

Contradictions are said to be self-contradictory, analytically false, or necessarily false.

Any formula which has a truth-table with a main column with at least one '1' and at least one '0'  is a contingency.

Contingencies are said to be true or false contingently, or synthetic.

3.4: Truth-Tabular Relations between Formulas

Pairs of Formulas

It is possible to draw up a truth-table for more than one formula

Here is a truth table for the pair of formulas ( p É  q ) and ( p & ~  q )

     p q       ( p É q )       ( p & ~ q )

     1 1           1               0

     1 0           0               1

     0 1           1               0

     0 0           1               0

                   Ý               Ý

A pair of propositions are truth-table contradictories if and only if, on every row of their truth-table they have opposite truth values.

A pair of propositions are truth-table contraries if and only if, on every row of their truth-table they are not both true, and on at least one row they are both false.

A pair of propositions are truth-table equivalences if and only if, on every row of their truth-table they both have the same truth-value.

A pair of propositions are truth-table sub-contraries if and only if, on every row of their truth-table they are not both false, and on at least one row they are both true.

Given a pair of formulas, the first of them tautologically implies the second, if and only if, on any row of their truth-table on which the first is true, so is the second.

(Note that if X and Y are formulas, if X and Y are equivalent, then X tautologically implies Y and Y tautologically implies X.)

X tautologically implies Y is sometimes abbreviated as X Þ Y.

X is equialent to Y is sometimes abbreviated as X Û Y.

Sets of Formulas

More than two formulas can be evaluated on a single truth table, in the obvious way.

A set of formulas is consistent if and only if all the formulas in the set are true on at least one row of their common truth-table.

A set of formulas is inconsistent if and only if there is no row of their common truth-table on which all the formulas in the set are true.