Step 1. Figure out how many individual claim variables are in the argument or compound claim you will analyze–i.e. how many individual letters representing claims.
Example 1. Here is an argument: A -> (B v C)
~B
~C / ~A
The argument contains three individual claim variables: A, B, C.
Step 2. Every claim variable has two truth values: True or False.
Here is a useful formula to tell know how many rows you will need for the truth table:
Take the total number of independent claim variables you need to plot and make that an exponent of 2. The value for that exponent is the number of rows needed.
One claim variable: P = two rows, 21 .
Two variables: P, Q = four rows, 22.
Three variables: P, Q, R = eight rows, 23.
Four variables: P, Q, R, S = sixteen rows, 24 and so forth.
Example 2. The argument in Example 1 has three individual claim variables.
Hence its truth table will have eight rows.
Step 3. Make a column for every separate claim and compound claim in the statement or argument you analyze.
Make separate columns for: i) individual claims represented by the letters of the claim variables, ii) parenthetical claims (claims within a claim), iii) all premises and iv) the conclusion of the argument.
Example 3. The argument in example one has eight separate claims: A, B, C, (B v C), A -> (B v C), ~B, ~C and ~A.
Step 4. Your truth table must lay out the complete truth possibilities for all the claims. To do this you need to list the truth possibilities for every claim variable.
Take the first claim variable, assign it a value of T for the top half of the rows and F for the bottom half.
Then take the second claim variable and assign it a value of T for the first and third quarters of the rows.
For the third claim variable, assign T and F values for alternating eighths of the rows, and alternating sixteenths for the fourth variable, alternating thirty-seconds and sixty-fourths for the fifth and sixth individual claim variables.
Keep assigning truth values alternating by the appropriate exponents of two for each additional individual claim variable.
Example 4. Truth value assignments for individual claim variables for the argument in example 1.
|
A |
B |
C |
B v C |
A -> (B v C) |
~B |
~C |
~A |
|
T |
T |
T |
|
|
|
|
|
|
T |
T |
F |
|
|
|
|
|
|
T |
F |
T |
|
|
|
|
|
|
T |
F |
F |
|
|
|
|
|
|
F |
T |
T |
|
|
|
|
|
|
F |
T |
F |
|
|
|
|
|
|
F |
F |
T |
|
|
|
|
|
|
F |
F |
F |
|
|
|
|
|
Step 5. Assign appropriate truth values to the compound claims based on: a) the truth values of the individual claim variables and b) the logical relationship between the simpler claims in a component claim. Work from the most simple compound claims to the most complex.
Example 5. Truth value assignments for all claims variables for the argument in example 1.
|
A |
B |
C |
B v C |
A -> (B v C) |
~B |
~C |
~A |
|
T |
T |
T |
T |
T |
F |
F |
F |
|
T |
T |
F |
T |
T |
F |
T |
F |
|
T |
F |
T |
T |
T |
T |
F |
F |
|
T |
F |
F |
F |
F |
T |
T |
F |
|
F |
T |
T |
T |
T |
F |
F |
T |
|
F |
T |
F |
T |
T |
F |
T |
T |
|
F |
F |
T |
T |
T |
T |
F |
T |
|
F |
F |
F |
F |
T |
T |
T |
T |
Step 6. If you are plotting a single compound claim on a truth table, compare it with other claims you have plotted. If the claim's truth values are equivalent for the same truth assignments of the individual claim variables then the two compound claims are equivalent.
Step 7. To determine the validity of an argument, establish the possible truth values for all of the arguments individual claim variables, compound claims, premises and conclusion. Cross out all rows in which the conclusion is true. Then cross out all rows in which any or all of the premises are false. If there are no rows remaining, then the argument is valid. If there are one or more rows that contain all premises assigned a value of T while the conclusion is false, the argument is invalid.
Example 7a. Eliminate rows in which the conclusion is true.
|
A |
B |
C |
B v C |
P1: A -> (B v C) |
P2: ~B |
P3: ~C |
Conc.: ~A |
|
T |
T |
T |
T |
T |
F |
F |
F |
|
T |
T |
F |
T |
T |
F |
T |
F |
|
T |
F |
T |
T |
T |
T |
F |
F |
|
T |
F |
F |
F |
F |
T |
T |
F |
Example 7b. Eliminate rows in which one or all of the premises
are false.
|
|
P1: A -> (B v C) |
P2: ~B |
P3: ~C |
Conc.: ~A |
|
X |
T |
F |
F |
F |
|
X |
T |
F |
T |
F |
|
X |
T |
T |
F |
F |
|
X |
F |
T |
T |
F |
All the remaining lines contain at least one false premise, therefore the argument is valid.
Step 8. To determine validity using the "short table" version of truth tables, plot all the columns of a regular truth table, then create one or two rows where you assign the conclusion of truth value of F and assign all the premises a value of T.
Example 8. Two rows with a false conclusion.
|
A |
B |
C |
B v C |
P1: A ->(B v C) |
P2: ~B |
P3: ~C |
Conc.: ~A |
|
|
|
|
|
T |
T |
T |
F |
Step 9. Work backwards from the conclusion and the premises.
Can you assign truth values to the compound claims and individual claim variables that are consistent with the truth values assigned to the premises and the conclusion?
If you can, you demonstrate that the argument is invalid. If you cannot, that indicates that argument is valid.
Some trial and error work may be needed here.
Example 9. If you know that ~A is false, what else can you derive? Assign a T truth value to the premises. Will this work?
|
A |
B |
C |
B v C |
P1: A -> (B v C) |
P2: ~B |
P3: ~C |
Conc.: ~A |
|
T |
F |
F |
xxxx |
T |
T |
T |
F |
(B v C) needs to be true for P:1 to be true. (B v C) cannot be true if B and C are both false.
