When we discussed conditions earlier, we discussed the type where we take an action based on the value of the condition. We are now going to talk about a more general version of a conditional, sometimes called an implication.
implications
Implications are logical conditional sentences stating that a statement [latex]p[/latex], called the antecedent, implies a statement [latex]q[/latex], called the consequence.
Implications are commonly written as [latex]p\rightarrow{q}[/latex] and is translated as “if [latex]p[/latex], then [latex]q[/latex] .”
Implications are similar to the conditional statements we looked at earlier; [latex]p\rightarrow{q}[/latex] is typically written as “if [latex]p[/latex] then [latex]q[/latex],” or “[latex]p[/latex] therefore [latex]q[/latex].” The difference between implications and conditionals is that conditionals we discussed earlier suggest an action—if the condition is true, then we take some action as a result. Implications are a logical statement that suggest that the consequence must logically follow if the antecedent is true.
The English statement “If it is raining, then there are clouds is the sky” is a logical implication. Is this a valid argument, why or why not?
It is a valid argument because if the antecedent “it is raining” is true, then the consequence “there are clouds in the sky” must also be true.
Notice that the statement tells us nothing of what to expect if it is not raining. If the antecedent is false, then the implication becomes irrelevant.
A friend tells you that “If you upload that picture to Facebook, you’ll lose your job.” Describe the possible outcomes related to this statement, and determine whether your friend’s statement is invalid.
There are four possible outcomes:
You upload the picture and keep your job.
You upload the picture and lose your job.
You don’t upload the picture and keep your job.
You don’t upload the picture and lose your job.
There is only one possible case where your friend was lying—the first option where you upload the picture and keep your job. In the last two cases, your friend didn’t say anything about what would happen if you didn’t upload the picture, so you can’t conclude their statement is invalid, even if you didn’t upload the picture and still lost your job.
In traditional logic, an implication is considered valid (true) as long as there are no cases in which the antecedent is true and the consequence is false. It is important to keep in mind that symbolic logic cannot capture all the intricacies of the English language.
truth values for implications
[latex]p[/latex]
[latex]q[/latex]
[latex]p → q[/latex]
T
T
T
T
F
F
F
T
T
F
F
T
Again, if the antecedent [latex]p[/latex] is false, we cannot prove that the statement is a lie, so the result of the third and fourth rows is true.
Construct a truth table for the statement [latex]\left(m\wedge\sim{p}\right)\rightarrow{r}[/latex]
We start by constructing a truth table for the antecedent.
[latex]m[/latex]
[latex]p[/latex]
[latex]\sim{p}[/latex]
[latex]m\wedge\sim{p}[/latex]
T
T
F
F
T
F
T
T
F
T
F
F
F
F
T
F
Now we can build the truth table for the implication.
In this case, when [latex]m[/latex] is true, [latex]p[/latex] is false, and [latex]r[/latex] is false, then the antecedent [latex]m\wedge\sim{p}[/latex] will be true but the consequence false, resulting in an invalid implication; every other case gives a valid implication.
For any implication, there are three related statements, the converse, the inverse, and the contrapositive.
related statements
The original implication is “if [latex]p[/latex] then [latex]q[/latex]”: [latex]p\rightarrow{q}[/latex]
The converse is “if [latex]q[/latex] then [latex]p[/latex]”: [latex]q\rightarrow{p}[/latex]
The inverse is “if not [latex]p[/latex] then not [latex]q[/latex]”: [latex]\sim{p}\rightarrow\sim{q}[/latex]
The contrapositive is “if not [latex]q[/latex] then not [latex]p[/latex]”: [latex]\sim{q}\rightarrow\sim{p}[/latex]
Consider again the valid implication “If it is raining, then there are clouds in the sky.” Write the related converse, inverse, and contrapositive statements.
The converse would be “If there are clouds in the sky, it is raining.” This is certainly not always true. The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise, this is not always true. The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This statement is valid, and is equivalent to the original implication.