3.8: Analyzing Arguments with Truth Tables

Learning Objectives

Students will be able to:

• Determine the validity of an argument using truth tables

The validity of arguments also be determined using truth tables.

Analyzing arguments using truth tables

To analyze an argument with a truth table:

1. Represent each of the premises symbolically.
2. Create a conditional statement, joining all the premises to form the antecedent and using the conclusion as the consequent.
3. Create a truth table for the statement. If it is always true, then the argument is valid.

Example \(\PageIndex{6}\)

Consider the argument

Premise: If you bought bread, then you went to the store.

Premise: You bought bread.

Conclusion: You went to the store.

Solution

While this example is fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then form a conditional statement showing that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid.

We’ll let b represent “you bought bread” and s represent “you went to the store”. Then the argument becomes:

Premise: b → s

Premise: b

Conclusion: s

To test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [(b → s) ⋀ b] → s ?

Table \(\PageIndex{1}\)

b	s	b → s
T	T	T
T	F	F
F	T	T
F	F	T

Table \(\PageIndex{2}\)

b	s	b → s	(b → s) ⋀ b
T	T	T	T
T	F	F	F
F	T	T	F
F	F	T	F

Table \(\PageIndex{3}\)

b	s	b → s	(b → s) ⋀ b	[(b → s) ⋀ b] → s
T	T	T	T	T
T	F	F	F	T
F	T	T	F	T
F	F	T	F	T

Since the truth table for [(b → s) ⋀ b] → s is always true, this is a valid argument.

Example \(\PageIndex{7}\)

Premise: If I go to the mall, then I’ll buy new jeans.

Premise: If I buy new jeans, I’ll buy a shirt to go with it.

Conclusion: If I go to the mall, I’ll buy a shirt.

Solution

Let m = I go to the mall, j = I buy jeans, and s = I buy a shirt.

The premises and conclusion can be stated as:

Premise: m → j

Premise: j → s

Conclusion: m → s

We can construct a truth table for [(m → j) ⋀ (j → s)] → (m → s). Try to recreate each step and see how the truth table was constructed.

Table \(\PageIndex{4}\)

m	j	s	m→j	j→s	(m→j) ⋀ (j→s)	m→s	[(m→j) ⋀ (j→s)] → (m→s)
T	T	T	T	T	T	T	T
T	T	F	T	F	F	F	T
T	F	T	F	T	F	T	T
T	F	F	F	T	F	F	T
F	T	T	T	T	T	T	T
F	T	F	T	F	F	T	T
F	F	T	T	T	T	T	T
F	F	F	T	T	T	T	T

From the final column of the truth table, we can see this is a valid argument.

Contributors and Attributions

• Saburo Matsumoto
  CC-BY-4.0