Truth Tables are used to define and illustrate all the possible outcomes of

These three may ONLY be *true* or *false*. So, we may begin right away with constructing all the possibilities. We will use the
symbols S_{1} and S_{2} for the two premise statements, and C for the conclusion.

First, let us simply draw the table we need, and fill
it in above with headings for the columns as we go. We will make
this first heading "statement 1", and the next heading "statement 2", or, S_{1} and S_{2} for short.

S_{1} |
S_{2} |

F | F |

F | T |

T | F |

T | T |

We now add a third column for the conclusion. This column represents "statement 1

S_{1} |
S_{2} |
S_{1} and S_{2} |

F | F | F |

F | T | F |

T | F | F |

T | T | T |

It may help a bit to put this into words. We can let our first statement (S

That represented the truth table for the conjunction

S_{1} |
S_{2} |
Not (S_{1} and S_{2}) |

F | F | T |

F | T | T |

T | F | T |

T | T | F |

Next we make a table for

S_{1} |
S_{2} |
S_{1} or S_{2} |

F | F | F |

F | T | T |

T | F | T |

T | T | T |

In our normal speech we use the same word ("or") for two completely different meanings. We might say "Bob went to get some gas or a meal" and would still say that was true if; Bob both got some gas and had a meal. On the other hand we could say "Turn left or right" and expect this will be understood as meaning one way or the other but not both.

We use just plain old "*or*" to represent the inclusive case, (and "*Xor*" to express the exclusive case.)

We follow this with the other *or*, the *exclusive or* which we write as *Xor* to distinguish it from the other or, the inclusive one. This *Xor* means Statement 1 or Statement 2 but NOT both. (Mary drove north, or Mary drove south, but not both directions at once.)

S_{1} |
S_{2} |
S_{1} Xor S_{2} |

F | F | F |

F | T | T |

T | F | T |

T | T | F |

These next two are found by inverting the output of the last two tables (like was done to the first *and* table to create a *NOT and* or *Nand* table.) This one being a *Nor* or *NOT or* table.

S_{1} |
S_{2} |
NOT (S_{1} or S_{2}) |

F | F | T |

F | T | F |

T | F | F |

T | T | F |

This is

S_{1} |
S_{2} |
NOT (S_{1} Xor S_{2}) |

F | F | T |

F | T | F |

T | F | F |

T | T | T |

It is possible to express a change in one of the premises from true to false or vice versa also. Here we invert Statement 1 and

S_{1} |
S_{2} |
(NOT S_{1}) and S_{2} |

F | F | F |

F | T | T |

T | F | F |

T | T | F |

We now, instead, use the inverse of the second premise (S

S_{1} |
S_{2} |
S_{1} and (NOT S_{2}) |

F | F | F |

F | T | F |

T | F | T |

T | T | F |

And yet another table which negates both premises.

S_{1} |
S_{2} |
(NOT S_{1}) and (NOT S_{2}) |

F | F | T |

F | T | F |

T | F | F |

T | T | F |

The next three tables simply take the last three tables and negate their conclusions by

S_{1} |
S_{2} |
NOT [(NOT S_{1}) and S_{2}] |

F | F | F |

F | T | T |

T | F | F |

T | T | F |

We are in constructing in building this table

S_{1} |
S_{2} |
NOT [S_{1} and (NOT S_{2})] |

F | F | F |

F | T | F |

T | F | T |

T | T | F |

And yet another table which negates both premises and the conclusion.

S_{1} |
S_{2} |
NOT [(NOT S_{1}) and (NOT S_{2})] |

F | F | F |

F | T | T |

T | F | T |

T | T | T |

The last six tables used

S_{1} |
S_{2} |
(NOT S_{1}) or S_{2} |

F | F | T |

F | T | T |

T | F | F |

T | T | T |

We now, instead, use the inverse of the second premise (S

S_{1} |
S_{2} |
S_{1} or (NOT S_{2}) |

F | F | T |

F | T | F |

T | F | T |

T | T | T |

And yet another table which negates both premises.

S_{1} |
S_{2} |
(NOT S_{1}) or (NOT S_{2}) |

F | F | T |

F | T | T |

T | F | T |

T | T | F |

The next three tables simply take the last three tables and negate their conclusions by

S_{1} |
S_{2} |
NOT [(NOT S_{1}) or S_{2}] |

F | F | F |

F | T | F |

T | F | T |

T | T | F |

We are in constructing in building this table

S_{1} |
S_{2} |
NOT [S_{1} or (NOT S_{2})] |

F | F | F |

F | T | T |

T | F | F |

T | T | F |

And yet another table which negates both premises and the conclusion.

S_{1} |
S_{2} |
NOT [(NOT S_{1}) or (NOT S_{2})] |

F | F | F |

F | T | F |

T | F | F |

T | T | T |

Now come six using

S_{1} |
S_{2} |
(NOT S_{1}) Xor S_{2} |

F | F | T |

F | T | F |

T | F | F |

T | T | T |

We now, instead, use the inverse of the second premise (S

S_{1} |
S_{2} |
S_{1} Xor (NOT S_{2}) |

F | F | T |

F | T | F |

T | F | F |

T | T | T |

Did you notice that the last two tables had exactly the same result? When this happens, it allows us to conclude that they are equivalent. That is, they have the same meaning, merely expressed differently. If one is true, the other will be true also and vice versa, and if one is false the other MUST also be false.

The next table negates both premises then *Xor*-ed together.

S_{1} |
S_{2} |
(NOT S_{1}) Xor (NOT S_{2}) |

F | F | F |

F | T | T |

T | F | T |

T | T | F |

The next three tables simply take the last three tables and negate their conclusions by

S_{1} |
S_{2} |
NOT [(NOT S_{1}) Xor S_{2}] |

F | F | F |

F | T | T |

T | F | T |

T | T | F |

We are in constructing in building this table

S_{1} |
S_{2} |
NOT [S_{1} Xor (NOT S_{2})] |

F | F | F |

F | T | T |

T | F | T |

T | T | F |

And yet another table which negates both premises and the conclusion.

S_{1} |
S_{2} |
NOT [(NOT S_{1}) Xor (NOT S_{2})] |

F | F | T |

F | T | F |

T | F | F |

T | T | T |

There are some simple tables which we should also consider. Sometimes the conclusion (C) has little to do with both premises, instead being related to only one of them as in the following case where it is the same as the first statement.

S_{1} |
S_{2} |
C = S_{1} |

F | F | F |

F | T | F |

T | F | T |

T | T | T |

Or if it simply is the same as Statement 2 then this would be its table.

S_{1} |
S_{2} |
C = S_{2} |

F | F | F |

F | T | T |

T | F | F |

T | T | T |

We may also make the conclusion to be the negation of Statement 1.

S_{1} |
S_{2} |
C = NOT ( S_{1} ) |

F | F | T |

F | T | T |

T | F | F |

T | T | F |

Here the conclusion is the negation of Statement 2.

S_{1} |
S_{2} |
C = NOT ( S_{2} ) |

F | F | T |

F | T | F |

T | F | T |

T | T | F |

Even more trivial is where the conclusion is shown to have nothing at all to do with the premises. Here the conclusion is always true.

S_{1} |
S_{2} |
C = True |

F | F | T |

F | T | T |

T | F | T |

T | T | T |

Here we illustrate a table where the conclusion is always false.

S_{1} |
S_{2} |
C = False |

F | F | F |

F | T | F |

T | F | F |

T | T | F |

Truth tables are used usually to illustrate more complex statements.

We can have any number of columns and rows with as many variables or premises as we desire and any number of conclusive statements.

This next table will be used to illustrate how we can work out a problem. We will find the conclusion (C) to the following complex statement:

(S_{1} and S_{2}) and NOT (S_{1} or S_{2}) = C (Our conclusion.)

This table will have six columns, one for each of the following terms:

S_{1}

S_{2}

S_{1} and S_{2}

S_{1} or S_{2}

NOT (S_{1} or S_{2})

(S_{1} and S_{2}) and NOT (S_{1} or S_{2}) = C

S_{1} |
S_{2} |
S_{1}and S_{2} |
S_{1}or S_{2} |
NOT (S_{1}or S_{2}) |
(S_{1} and S_{2}) and NOT (S_{1} or S_{2}) = C |

F | F | F | F | T | F |

F | T | F | T | F | F |

T | F | F | T | F | F |

T | T | T | T | F | F |

The result is interesting here in that it has been seen before. Note that it is always false. We have just seen that this means that the result has little meaning and thus our equation can not be used as means of proving anything. You can use any statements you like and the result will always be false. In the end we can say that the statement "It is true that Joe went fishing and Ed went shopping, and not true that Joe went fishing or Ed went shopping." is always false.

The important tables to remember or understand are the simple ones.

In the end analysis this is just another way of writing a syllogism. One method of making use of the syllogisms of Aristotle to solve complex logical equations.

So perhaps you noted that with just two premises, and one conclusion,
there will be only sixteen unique tables we can construct? That is, there are only sixteen conclusions. You may have also noted that this system has only two *states*. Any statement may only be true or false, assuming it is in fact a *statement*. This *logic*,
works well for any system having only two states. As we move on,
we will consider this from a mathematical point of view, using a
one for true and a zero for false.

Boolean algebra next!