Assumed True until proven False. The Curious Case of the Vacuous Truth

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?

Solution 1:

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:

$P \implies Q$ is true if and only if it is not the case that both $P$ is true and $Q$ is false.

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.

Solution 2:

"... the vacuous truth is a "definition of convenience" in a sense."? No, not mere convenience. There are strong pressures that push towards making this choice of truth-values in lines 3 and 4 of the truth-table for the conditional. Here's one that others haven't mentioned.

One thing mathematicians need to be very clear about is the use of statements of generality and especially statements of multiple generality – you know the kind of thing, e.g. the definition of continuity that starts for any $\epsilon$ ... there is a $\delta$ ... And the quantifier-variable notation serves mathematicians brilliantly to regiment statements of multiple generality and make them utterly unambiguous and transparent.

Quantifiers matter to mathematicians, then: that's uncontentious. OK, so now think about restricted quantifiers that talk about only some of a domain (e.g. talk not about all numbers but just about all the even ones). How might we render Goldbach's Conjecture, say? As a first step, we might write

$(\forall n \in \mathbb{N})$(if $n$ is even and greater than 2, then $n$ is the sum of two primes)

We restrict the quantifier here by using a conditional. So now think about the embedded conditional here. What if $n$ is odd, so the antecedent of the conditional is false??? If we say this instance of the conditional lacks a truth-value, or may be false, then the quantification would have non-true instances and so would not be true! But of course we can't refute Goldbach's Conjecture by looking at odd numbers!! So, if the quantified conditional is indeed to come out true when Goldbach is right, then we'll have to say that the irrelevant instances of the conditional with a false antecedent come out true by default. In other words, the embedded conditional will have to be treated as a material conditional.

So: to put it a bit tendentiously and over-briefly, if mathematicians are to deal nicely with expressions of generality using the quantifier-variable notation they have come to know and love, they will have to get used to using material conditionals too.

Solution 3:

You can interpret $P \Rightarrow Q$ as meaning "whenever I've written $P$ down in a proof, I can subsequently write $Q$ down in a proof." (In other words, we're defining implication as "the thing that satisfies modus ponens.") Certainly you can always write down a tautology in a proof, so $Q$ can always be $T$. Similarly, if you write down a false statement in a proof then you can prove anything, so $P$ can always be $F$. The only thing you can't do is start with true statements and end up with false statements, so $P$ can't be $T$ if $Q$ is $F$.

Solution 4:

There are many good answers here, but I think it's worthwhile to address your claim directly. The fact that the implication $P\implies Q$ is true does not mean that the consequent $Q$ is true.

You state:

Really, it seems we can conjure up as many silly counter-examples as we like, such as:

If pigs can fly, than I can walk on water.

Though following the above truth table, the implication is that my ability to walk on water is true.

However, the two relevant lines in the truth table (where $P$ is false) are:

$$\begin{array}{ c | c || c | } P & Q & P\Rightarrow Q \\ \hline \text F & \text T & \text T \\ \text F & \text F & \text T \end{array}$$

A clear interpretation of this would be "if the implication is true and the antecedent is flase, then the consequent may be true or may be flase, i.e. we don't know."

Solution 5:

Your quote from Velleman, as it stands, is irremediably confused and nonsensical; so you are absolutely right in finding it "quite unsatisfactory".

In mathematics, it is possible to derive constructively any statement, true or false, from a false statement. For example, from "$2+2=5$", by subtracting $4$ from each side, we get "$0=1$", from which it is fairly straightforward to derive pretty well any statement you like. Outside mathematics, a statement such as "If pigs can fly, then Elvis Presley is president of the United States" can never be shown to be false, because to do so would require finding a flying pig, which will never happen. We can put such statements into the category "unfalsifiable" in distinction from "true", but generally nothing bad will happen if we lump these two categories together. The extent to which one is prepared to identify the unfalsifiable with the true is largely a matter of philosophical taste.