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Math in Society (Lippman)
17: Logic
17.10: Evaluating Deductive Arguments with Truth Tables

17.10: Evaluating Deductive Arguments with Truth Tables

Last updated

Jul 18, 2022
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- 17.6: Truth Tables: Conditional, Biconditional
- 17.11: Forms of Valid Arguments

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Arguments can also be analyzed using truth tables, although this can be a lot of work.

Analyzing arguments using truth tables

To analyze an argument with a truth table:

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

Example 34

Consider the argument

$\begin{array} {ll} \text{Premise:} & \text{If you bought bread, then you went to the store.} \\ \text{Premise:} & \text{You bought bread.} \\ \text{Conclusion:} & \text{You went to the store.} \end{array}$

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:

$\begin{array} {ll} \text{Premise:} & b \rightarrow s \\ \text{Premise:} & b \\ \text{Conclusion:} & s \end{array}$

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

$\begin{array}{|c|c|c|} \hline b & s & b \rightarrow s \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|} \hline b & s & b \rightarrow s & (b \rightarrow s) \wedge b \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|c|} \hline b & s & b \rightarrow s & (b \rightarrow s) \wedge b & {[(b \rightarrow s) \wedge b] \rightarrow s} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}$

Since the truth table for $[(b \rightarrow s) \wedge b] \rightarrow s$ is always true, this is a valid argument.

Try it Now 13

Determine whether the argument is valid:

$\begin{array} {ll} \text{Premise:} & \text{If I have a shovel, I can dig a hole.} \\ \text{Premise:} & \text{I dug a hole.} \\ \text{Conclusion:} & \text{Therefore, I had a shovel.} \end{array}$

Answer

Let $S=$ have a shovel, $D=\operatorname{dig}$ a hole. The first premise is equivalent to $S \rightarrow D$ . The second premise is $D$ . The conclusion is $S$ . We are testing $[(S \rightarrow D) \wedge D] \rightarrow S$

$\begin{array}{|c|c|c|c|c|} \hline S & D & S \rightarrow D & (S \rightarrow D) \wedge D & {[(S \rightarrow D) \wedge D] \rightarrow S} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}$

This is not a tautology, so this is an invalid argument.

Example 35

$\begin{array} {ll} \text{Premise:} & \text{If I go to the mall, then I’ll buy new jeans.} \\ \text{Premise:} & \text{If I buy new jeans, I’ll buy a shirt to go with it.} \\ \text{Conclusion:} & \text{If I go to the mall, I’ll buy a shirt.} \end{array}$

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:

$\begin{array} {ll} \text{Premise:} & m \rightarrow j \\ \text{Premise:} & j \rightarrow s \\ \text{Conclusion:} & m \rightarrow s \end{array}$

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

$\begin{array}{|c|c|c|c|c|c|c|c|} \hline m & j & s & m \rightarrow j & j \rightarrow s & (m \rightarrow j) \wedge(j \rightarrow s) & m \rightarrow s & {[(m \rightarrow j) \wedge(j \rightarrow s)] \rightarrow(m \rightarrow s)} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \end{array}$

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

Analyzing arguments using truth tables

Example 34

Solution

Try it Now 13

Example 35

Solution

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