Given two statements, $P$ and $Q$, and the logical connective, $\implies$, the truth table for $P \implies Q$ is:
$$\begin{array}{ c | c || c | }
P & Q & P\Rightarrow Q \\ \hline
\text T & \text T & \text T \\
\text T & \text F & \text F \\
\text F & \text T & \text T \\
\text F & \text F & \text T
\end{array}$$
Lines one and two are quite clear. What is ambiguous is lines three and four.
One explanation as to why $P \implies Q$ is true when $P$ is false, provided by Velleman goes:
Let $P(x)$ be the statement $x>2$ and $Q(x)$, $x^2 > 4$. When $x=3$, $P$ is true, and $Q(x) = 9$ thus $Q$ is true. If $P(x) = 1$ then $Q(x) = 1$ and they are both false. If $P(x) = -3$ then $Q(x) = 9$ and so the statement is true.
This explanation was quite unsatisfactory to me. Looking at Enderton, we have:
For example, we might translate the English sentence, ”If you're telling the truth then I'm a monkey's uncle,” by the formula ($V \implies M$). We assign this formula the value $T$ whenever you are fibbing. In assigning the value $T$, we are certainly not assigning any causal connection between your veracity and any simian features of my nephews or nieces. The sentence in question is a conditional statement. It makes an assertion about my relatives provided a certain condition — that you are telling the truth — is met. Whenever that condition fails, the statement is vacuously true.
Very roughly, we can think of a conditional formula ($p\implies q$) as expressing a promise that if a certain condition is met (viz., that $p$ is true), then q is true. If the condition $p$ turns out not to be met, then the promise stands unbroken, regardless of $q$.
Though a significant improvement over the Velleman explanation, I still feel uncomfortable with it.
Really, it seems we can conjure up as many silly counter-examples as we like, such as:
If pigs can fly, then I can walk on water.
Though following the above truth table, the implication is that my ability to walk on water is true.
After considering it, it seems to me that $\implies$ is only meaningful when $P$ is true, then we can look at it in relation to $Q$. However, if $P$ is false, then we actually know nothing about the relationship between $P$ and $Q$. This would give a truth table of:
$$\begin{array}{ c | c || c | }
P & Q & P\Rightarrow Q \\ \hline
\text T & \text T & \text T \\
\text T & \text F & \text F \\
\text F & \text T & \text ? \\
\text F & \text F & \text ?
\end{array}$$
where the $?$ denotes that given $P$ we actually don't know anything about $ \implies $.
Thus. one way to clear this up would be to assume that $? = T$. Thus the vacuous truth is a "definition of convenience" in a sense.
The above is my take on the matter.
Could someone provide some clarification on the conditional logical connective?
Best Answer
One can, correctly, assign the truth-value of true to the statement $P\implies Q$ whenever $P$ is false, or whenever $Q$ is true. $P\implies Q\,$ is false if and only if both $P$ is true and $Q$ is false. That covers all the cases. So we can say that $P\implies Q$ is true, unless "proven false", by which I mean to say:
What we can also say is that in classical logic and in math, it is a mistake to attribute any sort of causal relationship between $P$ and $Q$ when writing or reading an implication $P\implies Q$. Put differently, $P\implies Q$, by itself, does not imply any causal relationship between $P$ and $Q$: It is defined to convey nothing more, and nothing less, than is conveyed by the statement: $\;\lnot P \lor Q$, or if you prefer, it tells us nothing more (and nothing less) than what is conveyed by the statement: $\;\lnot(P \land \lnot Q)$.
Your concern is not trivial, nor are you alone in being "bothered" by that lack of some stronger relationship between $P$ and $Q$. There are logics, such as relevance logic which aim to capture aspects of implication that are ignored by the "material implication" operator in classical truth-functional logic, requiring some sort of relevance between antecedent and conditional of a true implication. See also the Wikipedia entry entitled: Paradoxes of material implication for more on "alternate" non-classical logics.