Fallacy of denying the hypothesis

Question

I need help with this question Im not sure what a fallacy of denying the hypothesis is.

Use a truth table to show that $p\to q$ and $\neg p$, $\therefore \neg q$ is not a valid rule of inference. It is called the fallacy of denying the hypothesis.

Answer 1

See also Denying the antecedent.

It is a fallacy exactly because from the two premisse (or : assumptions, or hypothesis) :

$p \rightarrow q \ $ and $\ \lnot p$

it is not possible to validly conclude with : $\lnot q$.

The truth-functional properties of the conditional : $\rightarrow$ state that when $p$ is $FALSE$ and $q$ is $TRUE$, the conditional $p \rightarrow q$ is $TRUE$.

In this case, both the premises are $TRUE$ but the conclusion is $FALSE$; thus, the argument is not valid.

Answer 2

It is called a fallacy because it is not a valid argument. What is a valid argument, then?

A valid argument is an argument where the conclusion can never be false when the premises are true.

Let's now build the truth table of your argument: \begin{array}{cc|ccc}p&q&p\to q&\neg p&\neg q\\\hline T&T&T&F&F\\T&F&F&F&T\\F&T&T&T&F\\F&F&T&T&T\end{array}

As you can see, there are two assignments of truth values to the sentences letters that make the premises true (last two lines of the table), yet only one of them also makes the conclusion true. Therefore, there is a case where the premises are true and the conclusion is false which, by definition, means that the argument is invalid.

Take a simple example, where $p$ means "it has rained" and $q$ means "the ground is wet". Assuming the first premise is true, i.e. "if it has rained, then the ground is wet", knowing $\neg p$, i.e. knowing that "it hasn't rained", isn't enough to conclude about whether "the ground is wet or not" (the ground might be wet for some other reason). This is the core idea of the fallacy.