Applying vacuous truths to real-world implications

Question

Here is an implication that confuses me when I think about it:

$\qquad$ I am holding a pen $\implies$ It is raining outside.

This implication seems to say that it will rain outside whenever I hold a pen.

If I am not holding a pen, the implication is true. But how can this be so if I can just hold a pen and see that it does not rain? My guess is that implications can be true sometimes and false sometimes, so me holding a pen and seeing it does not rain does not prove that the implication is always false. But if this is the case, what does it even mean for the implication to be true, for the times when I do not hold a pen?

$$$$ I can see that the implication

$\qquad$ I am holding a pen $\implies$ False,

would be true when I do not hold a pen, even though its consequence is never true. And anytime I hold a pen, the consequence in the implication will be false.

Answer 1

I'm sure other people have said this somewhere on math stackexchange, but reading the proposed duplicate answers, I see plenty of room for confusion, so I think it's probably easiest to just write a clarification here.

The logical definition of implication doesn't really line up with the colloquial definition of implication unless you include a universal quantifier.

In your example, we wouldn't colloquially say that the statement "If you are holding a pen, then it is raining outside" is true, even though it is sometimes true logically, because what matters is whether or not it's always true.

The right way to translate the statement into a logical statement is to say "At all times, if you are holding a pen, then it is raining outside." The "at all times" portion of the sentence (which is a universal quantifier) ensures that, in order for the sentence to be true, we are not only interested in now in particular, but rather all possible moments. So to say the statement is false, we only need to find one counterexample - one particular time when you were holding a pen and it was a clear day outside. Which, as you said, can easily be accomplished by simply picking up your pen on a clear day.

Answer 2

As not so much a proof as an informal justification, let us fill in the truth table for $A\Rightarrow B$. We start with:

$$\begin{array}{c|c|c|c} \space&A&B&A\Rightarrow B\\\hline 1&T&T&\\ 2&T&F&\\ 3& F&T&\\ 4&F&F& \end{array}$$

Consider two cases.

Case 1

Suppose $A \Rightarrow B$ is true.

Consider two sub-cases.

Sub-case 1

Suppose $A$ is true. Since $A \Rightarrow B$ is true, then $B$ must also be true. We can now fill in line 1:

$$\begin{array}{c|c|c|c} \space&A&B&A\Rightarrow B\\\hline 1&T&T&T\\ 2&T&F&\\ 3& F&T&\\ 4&F&F& \end{array}$$

Sub-case 2

Suppose $A$ is false (your vacuous case). The we cannot conclude anything about $B$ from $A \Rightarrow B$. $B$ could be true, or it could be false. We can now fill in lines 3 and 4:

$$\begin{array}{c|c|c|c} \space&A&B&A\Rightarrow B\\\hline 1&T&T&T\\ 2&T&F&\\ 3& F&T&T\\ 4&F&F&T \end{array}$$

Case 2

Suppose $A\Rightarrow B$ is false. Then $A$ would have to be true and $B$ would have to be false. Now we can fill in the remaining line 2:

$$\begin{array}{c|c|c|c} \space&A&B&A\Rightarrow B\\\hline 1&T&T&T\\ 2&T&F&F\\ 3& F&T&T\\ 4&F&F&T \end{array}$$

Answer 3

what does it even mean for the implication to be true ?

Here are the four types of ‘imply’:

1. $P$ implies $R$

   • Some complex number is real implies that every positive number is real.

2. for each $x, Px$ implies $Rx\qquad\leftarrow$universal implication

   • Every multiple of $6$ is even.

3. $P$ logically implies $R$

   • $\big(A\to\exists y\,By\big)\,$ logically implies $\,\exists y\big(A\to By\big).$

4. $Px$ logically universally implies $Rx$

   • $x\not=x\:$ logically universally implies $\,Rx.$

(Roughly speaking, a logical truth is a sentence that is true regardless of how its symbols are interpreted. For examples 1 & 2, the context is mathematical analysis.)

$\qquad$ I am holding a pen $\implies$ It is raining outside.$\qquad(1)$

This implication seems to say that it will rain outside whenever I hold a pen.

No it does not: this Type 1 example is a synthetic implication (its specific context might be right now, in Ximending, Taipei), not an analytic implication or general truth that suggests a prediction or "will / will not".

me holding a pen and seeing it does not rain does not prove that the implication is always false.

It does: your description means precisely that implication $(1)$ is false.

Which is not to say that it is logically false: clearly, it is true in some other scenario/context.

In contrast, the implication

$\qquad$ "I am holding a pen and every empty-handed person is holding no object implies that I am not empty-handed"

is logically true (Type 3 above). When you're writing the following and wanting to analyse implications across varying contexts, you are probably thinking of Types 2-4 above, whereas vacuous truth in the context of Type 1 is really just a matter of definition:

the implication is always false

the implication to be true, for the times when I do not hold a pen

the implication would be true when I do not hold a pen

anytime I hold a pen, the consequence in the implication will be false.