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Truth tables.
Truth Tables are used to define and illustrate all the possible outcomes of syllogisms. Two statements serve as our given premises, the third is the conclusion. These three may ONLY be true or false. So, we may begin right away with constructing all the possibilities. We will use the
symbols S1 and S2 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, S1 and S2 for short.
We now add a third column for the conclusion. This column represents
"statement 1and statement 2.". Unless BOTH are true, C (our conclusion) will be false. Read the table row by row. if (S1 is false) and (S2 is false) then (S1 and S2) will be false.
| S1 |
S2 |
S1 and S2 |
| 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 (S1) be: "Joe went shopping", our second statement (S2) be: "Mary went shopping", and our conclusion (C) be: The garage is empty. Symbolic logic simply uses symbols for statements and
operators (conjunctions and negation) to simplify the writing. Instead of writing: "If Joe went shopping and Mary went shopping then,
the garage is empty," we use symbols and write: "S1 and S2 → C". The → symbol means implies and takes
care of the "if-then" portion of our compound statement.
That represented the truth table for the
conjunction and. We continue with the others...
(ALL the possibilities!) We begin with modifying the one we just did. To say not C instead of C. Think of it like this, if we had written our conclusion statement as: "the garage contains a car", then
we could write "NOT C" to mean "the garage does NOT contain a car".
| S1 |
S2 |
Not (S1 and S2) |
| F |
F |
T |
| F |
T |
T |
| T |
F |
T |
| T |
T |
F |
Next we make a table for or. If either premise statement
is true, then the conclusion is true. This is called the inclusive or as it includes the case where both statements are true.
| S1 |
S2 |
S1 or S2 |
| 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.)
| S1 |
S2 |
S1 Xor S2 |
| F |
F |
F |
| F |
T |
T |
| T |
F |
T |
| T |
T |
F |
This now covers all the possible conjunctions, and the remaining
tables need less explanation. The three possible conjunctions are and or and, Xor. It is also possible
to use the word not to change statements from true (T) to false (F). Logic is simple this way. It is up to the logician
to construct more complicated tables involving just these simple concepts. One should understand them well enough to 'construct' them on their own.
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.
| S1 |
S2 |
NOT (S1 or S2) |
| F |
F |
T |
| F |
T |
F |
| T |
F |
F |
| T |
T |
F |
This is NOT exclusive or.
| S1 |
S2 |
NOT (S1 Xor S2) |
| 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 and it with statement 2. almost the same as in our first table. This is like changing from "Ed went away from there" to "Ed stayed there".
| S1 |
S2 |
(NOT S1) and S2 |
| F |
F |
F |
| F |
T |
T |
| T |
F |
F |
| T |
T |
F |
We now, instead, use the inverse of the second premise (S2), in constructing a table
and-ing together our premises.
| S1 |
S2 |
S1 and (NOT S2) |
| F |
F |
F |
| F |
T |
F |
| T |
F |
T |
| T |
T |
F |
And yet another table which negates both premises.
| S1 |
S2 |
(NOT S1) and (NOT S2) |
| 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 negating (placing 'NOT' before) them.
| S1 |
S2 |
NOT [(NOT S1) and S2] |
| F |
F |
F |
| F |
T |
T |
| T |
F |
F |
| T |
T |
F |
We are in constructing in building this table
Nand-ing together our premises, one of which is itself also negated.
| S1 |
S2 |
NOT [S1 and (NOT S2)] |
| F |
F |
F |
| F |
T |
F |
| T |
F |
T |
| T |
T |
F |
And yet another table which negates both premises and the conclusion.
| S1 |
S2 |
NOT [(NOT S1) and (NOT S2)] |
| F |
F |
F |
| F |
T |
T |
| T |
F |
T |
| T |
T |
T |
The last six tables used and as the conjunction for our two premises.
The next twelve tables repeat this pattern using or for the first six then Xor for the next six.
| S1 |
S2 |
(NOT S1) or S2 |
| F |
F |
T |
| F |
T |
T |
| T |
F |
F |
| T |
T |
T |
We now, instead, use the inverse of the second premise (S2), in constructing a table
or-ing together our premises.
| S1 |
S2 |
S1 or (NOT S2) |
| F |
F |
T |
| F |
T |
F |
| T |
F |
T |
| T |
T |
T |
And yet another table which negates both premises.
| S1 |
S2 |
(NOT S1) or (NOT S2) |
| 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 negating (placing 'NOT' before) them.
| S1 |
S2 |
NOT [(NOT S1) or S2] |
| F |
F |
F |
| F |
T |
F |
| T |
F |
T |
| T |
T |
F |
We are in constructing in building this table
Nor-ing together our premises, one of which is itself also negated.
| S1 |
S2 |
NOT [S1 or (NOT S2)] |
| F |
F |
F |
| F |
T |
T |
| T |
F |
F |
| T |
T |
F |
And yet another table which negates both premises and the conclusion.
| S1 |
S2 |
NOT [(NOT S1) or (NOT S2)] |
| F |
F |
F |
| F |
T |
F |
| T |
F |
F |
| T |
T |
T |
Now come six using Xor as the conjunction (or operator.)
| S1 |
S2 |
(NOT S1) Xor S2 |
| F |
F |
T |
| F |
T |
F |
| T |
F |
F |
| T |
T |
T |
We now, instead, use the inverse of the second premise (S2), in constructing a table
Xor-ing together our premises.
| S1 |
S2 |
S1 Xor (NOT S2) |
| 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.
| S1 |
S2 |
(NOT S1) Xor (NOT S2) |
| 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 negating (placing 'NOT' before) them.
| S1 |
S2 |
NOT [(NOT S1) Xor S2] |
| F |
F |
F |
| F |
T |
T |
| T |
F |
T |
| T |
T |
F |
We are in constructing in building this table
X-Nor-ing together our premises, one of which is itself also negated.
| S1 |
S2 |
NOT [S1 Xor (NOT S2)] |
| F |
F |
F |
| F |
T |
T |
| T |
F |
T |
| T |
T |
F |
And yet another table which negates both premises and the conclusion.
| S1 |
S2 |
NOT [(NOT S1) Xor (NOT S2)] |
| 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.
| S1 |
S2 |
C = S1 |
| 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.
| S1 |
S2 |
C = S2 |
| 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.
| S1 |
S2 |
C = NOT ( S1 ) |
| F |
F |
T |
| F |
T |
T |
| T |
F |
F |
| T |
T |
F |
Here the conclusion is the negation of Statement 2.
| S1 |
S2 |
C = NOT ( S2 ) |
| 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.
| S1 |
S2 |
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.
| S1 |
S2 |
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:
(S1 and S2) and NOT (S1 or S2) = C (Our conclusion.)
This table will have six columns, one for each of the following terms:
S1
S2
S1 and S2
S1 or S2
NOT (S1 or S2)
(S1 and S2) and NOT (S1 or S2) = C
| S1 |
S2 |
S1and S2 |
S1or S2 |
NOT (S1or S2) |
(S1 and S2) and NOT (S1 or S2) = 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.
And, or, Xor, These three are the conjunctions which are used to make more complex or compound statements. Together
with the ability to change true to false and vice versa of the negation (not), they are comprise all of the operators.
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!
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