Truth Tables and Validity

With a truth table, we can determine whether or not an argument is validAn argument is valid if, but only if, whenever the premises of the argument are true, the conclusion is also true.  Whenever this is the case, the conclusion of the argument follows logically from, is a logical consequence of,  the premises.   Validity can be established with a truth table in the following manner:  construct a column for each premise and a column for the conclusion.  Examine each row of the truth table looking for an invalidating row, that is, a row in which each of the premises is true, and the conclusion is false.  If such a row exists, the argument is not valid.  If no such row exists, then the argument is valid.

Suppose we wish to determine whether the argument  '(P Q) and P, therefore  Q' is valid.  The first thing to do is to construct a truth table that will allow us to test the argument for validity.  In the examples on this page. I am collapsing several steps, so if you are a bit puzzled, review the basic pages on truth tables and equivalence tests.

(STEP 1) Construct the basic truth table for the argument, including the necessary guide columns.  Since the argument contains only 2 statement letters, P and Q, we need a 4 row truth table.

 1 2 3 4 5 6 7 P Q (P → Q) P ∴Q T T T ⊥ ⊥ T ⊥ ⊥

Ultimately we will be examining each row to see whether there is any row in which the value for  Columns 4 and 6 is True while the value for Column 7 is false.  Notice that I have used a thicker line between columns 5 and 6 and between columns 7 and 8  to indicate the break between the between the premises and the conclusion.  The turnstile () before the Q indicate that Q is the conclusion of the argument.

(STEP ) We now want to construct a truth table for our particular sentence. The first step is to assign truth values to each sentence letter in our premises--values that come directly from the guide columns.  We do the same for the conclusion.

 1 2 3 4 5 6 7 P Q (P → Q) P ∴Q T T T T T T T ⊥ T ⊥ T ⊥ ⊥ T ⊥ T ⊥ T ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

(STEP 4) All that is left to do now is to fill in the remaining values for the compound statements and then make a determination as to the validity of the argument.

 1 2 3 4 5 6 7 P Q (P → Q) P ∴Q T T T T T T T T ⊥ T ⊥ ⊥ T ⊥ ⊥ T ⊥ T T ⊥ T ⊥ ⊥ ⊥ T ⊥ ⊥ ⊥

Since there is no row win which all of the premises are true and the conclusion false, this argument is valid.

Compare the previous argument to a similar looking argument, and one that many people who have not had the benefit of a class in formal logic mistakenly believe to be valid:

(P Q) and not P, therefore  not Q

(STEP 1) Construct the basic truth table for the argument, including the necessary guide columns.  Since the argument contains only 2 statement letters, P and Q, we need a 4 row truth table.

 1 2 3 4 5 6 7 8 9 P Q (P → Q) ~ P ∴~ Q T T T ⊥ ⊥ T ⊥ ⊥

Ultimately we will be examining each row to see whether there is any row in which the value for  Columns 4 and 6 is True while the value for Column 8 is false.  Notice that I have used a thicker line between columns 5 and 6 and between columns 7 and 8  to indicate the break between the between the premises and the conclusion.  Since the conclusion of the argument is the negation of a statement, Column 8 is the column for the dominant operator in the conclusion.

(Step 2)  Fill in the values for the simple statements in the premises and the conclusion:

 1 2 3 4 5 6 7 8 9 P Q (P → Q) ~ P ∴~ Q T T T T T T T ⊥ T ⊥ T ⊥ ⊥ T ⊥ T ⊥ T ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

(Step 3)  Fill in the values for the compound statements and determine whether the argument is valid.

 1 2 3 4 5 6 7 8 9 P Q (P → Q) ~ P ∴~ Q T T T T T ⊥ T ⊥ T T ⊥ T ⊥ ⊥ ⊥ T T ⊥ ⊥ T ⊥ T T T ⊥ ⊥ T ⊥ ⊥ ⊥ T ⊥ T ⊥ T ⊥

In Row 3 of this truth table, all of the premises are true and the conclusion is false, so the argument is NOT VALID.