Truth tables, also known as logic tables are an important part of symbolic logic, also known as propositional logic or sentential logic. Propositional logic is mainly concerned with finding the "truth value" of statements, in order to assess the vailidity of arguments.
In propositional logic there are a few operational symbols:
~ NOT (negation)
& AND (conjunction)
v OR (disjunction)
V exclusive OR
→ IF....THEN (implication)
↔ IF AND ONLY IF (biconditional)
that are used to show the relationships of sentences. Variables (A,B,C,P,Q,R...) are used to represent the sentence.
Example:
A - Max is a dog.
B - Max is hungry.
so the sentence Max is a dog and hungry. would be represented as A & B.
On to truth tables. On the left side of the truth table is all the possible true-false combinations for the variables we are using. For one variable there are two possible truth combinations : T (true) or F (false). For two variables there are 4 combinations, for 3 variables there are 8 combinations and so on. On the right side of the table there is the truth value for the statement.
~ - Negation
A | ~A
--------
T | F
F | T
The negation serves simply to reverse the truth values. True becomes false, false becomes true.
& - Conjuction:
A B | A&B
------------
T T | T
T F | F
F T | F
F F | F
This truth table tells us that the conjunction of A and B is only true when both A and B are true.
v - Disjunction
A B | A v B
------------
T T | T
T F | T
F T | T
F F | F
This truth table says that as long as one of the variables is true, the disjunction is true (either A or B is true). It was argued by some logiticians that the
conversational implication of saying either-or is that we do not know which one is true, just that one of them is. If we knew A and B were both true, we would say A&B not A v B. Because of this, the special exclusive disjunction (represented by the symbol V (capital v) should be used. The truth table for V is below:
V - Exclusive Disjunction
A B | A V B
------------
T T | F
T F | T
F T | T
F F | F
This truth table says that the statement is true only when A or B have different truth values. When A and B are both true or both false, the sentence is false.
→ - Implication, If-Then
A B| A→B
------------
T T | T
T F | F
F T | T
F F | T
This is often the hardest truth table to grasp. For most of the other operational symbols, their truth tables very closely resemble or match what their
engligh translations mean. The If-Then table is not so clear. It says that if A is true and A→B is true, then B is true. Or, if we look at just the first two lines, a true statement cannot imply a false statment. If it does, the whole thing is false. The last two lines on the table say that if we start with a false statment or premise, it doesn't matter what the second part is, the whole thing is true (ie:
If I gave birth to puppies than you would be their father. is true, because I am never going to give birth to puppies. The statement
I gave birth to puppies. is always going to be false.)
↔ - Biconditional
A B | A↔B
------------
T T | T
T F | F
F T | F
F F | T
The biconditional states that when both sentences have the same truth value, the statment is true. So, when both A and B are true, or when they are both false, the biconditional true. (this truth table is the
same as for ~(A V B), the negation of the exclusive or).
An example of how to work a more complex truth table:
Either Sam and Mark go running or Pat goes running.
S - Sam goes running
M - Mark goes running
P - Pat goes running
This gives us the formula (S&M) v P. Within this, there is a sub-formula S&M. The operational symbol & applies only to S and M, not to the whole formula.
S M P | (S&M) v P
----------------------
T T T |
T T F |
T F T |
T F F |
F T T |
F T F |
F F T |
F F F |
Here is our truth table for this sentence. To start, we will have to assess the truth value of the sub-formula S&M. To do this, we just look at the truth values for S and M, we are ignoring P for now. We will record the truth value for each line under the & symbol. By placing it there, we know that that column in our table applies to S&M. Here is the table with that column completed:
S M P | (S&M) v P
----------------------
T T T | T
T T F | T
T F T | F
T F F | F
F T T | F
F T F | F
F F T | F
F F F | F
Now that we have the truth value of our sub-formula we can find the truth value for the whole sentence. If there were more sub-formulas you would continue finding thier truth values and in essence work "outward" until you had the truth value for the whole sentence. For the next step we are going to treat S&M as a single variable. We are going to nickname
S&M - Q for simplicity. Now we can asess the truth value of Q v P. The table for Q v P would look like:
Q P | Q v P
------------------
T T | T
T F | T
F T | T
F F | F
F T | T
F F | F
F T | T
F F | F
The column for Q (on the left) is the same as the column for S&M on the above truth table. The column for P in this table is also the same as the column for P above.
It is not nessisary to make an entirely new truth table for this step. I only did so for clarity. It is preferable to write this final column on the original table, placing it under the disjunction (v) operator like so:
S M P | (S&M) v P
----------------------
T T T | T T
T T F | T T
T F T | F T
T F F | F F
F T T | F T
F T F | F F
F F T | F T
F F F | F F
The far right column under v is the truth value for the entire statement given the contions of truth or falsehood given on the left side of the table. We can see that this table is accurate just by thinking about what it represents. For example, on line 4, Sam runs, but Mark doesn't and neither does Pat. So the statement
Either Sam and Mark go running or Pat goes running. is false in this case, and that is what the truth table tells us. We can think about all 8 lines this way, keeping in mind that we are not using the exclusive or (V) - even if everyone goes running the statment is still true.
It should be noted that there are many other variations on how to write out a truth/logic table. I prefer this method because involves the least amount of writing, and I think it is fairly straight forward. The left-right division of variables and formulas is not nessisary, but it makes it easier for other people who are looking at your work to read.