Why, in Logic, Does “False” Imply Anything?

In a class on symbolic logic, students are taught the truth tables that define the “logical connectives” ∧ (and), ∨ (or), ¬ (not), and → (if  … then). Everything makes sense until they are told that if p is false, then \(p\rightarrow q\) is true whether or not q is true. How can we say that “If pigs fly, then 2 is even” is a true statement? Or, for that matter, “If pigs fly, then there’s a kangaroo in my pocket”? This is especially troublesome when students, naturally wanting brevity, read \(p\rightarrow q\) as “p implies q “. How can nonsense imply anything?

A quick survey

Before I get into full answers, let’s look at some examples of the question. First, in 1994, in the infancy of Ask Dr. Math, we got this question:

8th Grade Logic

Why in a conditional statement if the "p" in the hypothesis is false, then the entire statement is true?  Why isn't it undecided?

In 1997, in answer to a more general question about logic, Doctor Mike included an answer to the question, knowing it is common:

A False Statement Implies Any Statement

... In case you still are wondering about why a False implies anything, try this explanation on for size.

Like Doctor Ken in the 1994 answer, Doctor Mike here focused on an example, pointing out that if the condition of such a statement is false, then whatever happens, you couldn’t be convicted of lying, because you made no promises about what would happen in any situation that actually happens. In 2004, Jay asked a follow-up question that was added to the same page:

Why are the logical statements "false implies true" and "false implies false" always considered "true"?

I've read the previous note, but could you please give a more "formal" explanation?

Doctor Schwa replied with an answer relating the idea to set theory:

More formally, I'd say "implies" means the same as "subset" in set theory.  That is, when you say, "if it rains, then the ground gets wet," you mean, "the set of times when it rains is a subset of the set of times when the ground gets wet."

So, since the empty set is a subset of any set, a false statement implies any statement.

Formally, this is a good way to think of it; but it may not satisfy everyone – particularly since it is not obvious why the empty set can be a subset at all. (That’s another question we get from time to time.)

In 2005, Doctor Achilles answered this one:

Logic and Conditional Sentences

I have a question about conditional statements.  I am having a hard time understanding why two false statements in a conditional makes it true.

I tried to use different statements to create a truth table but I get stuck on the same concept.  I tried a sentence like "If a polygon is a square, then the sides are equal."  If I assume a rectangle, then it seems to me that the statement is undefined.  Being true or false does not even apply.

This is much like the 1994 question, and is entirely reasonable. Be sure to read his answer; for the sake of space I will be focusing on two more recent questions whose answers include much of what others have said, while going beyond.

A deeper look

In 2007, Keven asked:

The Logic behind Conditional Statements

I've read questions of the same title "Why is "false implies true/false" always "true"?" and I can understand the reason in the "subset way".

Could you explain the reason just in the logic way?  Because when we say "work the same as subsets", we must prove it does work the same...

When I trust the set theory, I must trust logic first.  So I want to understand the reason just on the way logic goes.

"False implies true/false" is true...why?

So Keven doesn’t like Doctor Schwa’s approach, because it doesn’t directly relate to logic. Having recently taught the subject and thought about this issue, I had several things to say.

1. “If … then” doesn’t mean “implies”

First, don't use the word "implies" to talk about a conditional statement; A->B should be read merely as "if A then B".  "Implies" suggests a cause-and-effect relationship, or at least a logical connection of some sort.  But the conditional statement is not meant to suggest that (even though many examples given in texts look that way).  The statement "if A then B" says nothing more than "if A is true, then B is true"--not "if A is true, then it CAUSES B to be true".  It means that whenever we find that A is true, then we can know that B is true (or else A->B would have been false).

The problem is that we naturally tend to see a conditional statement as something more, because everyday usage leaks over into our logic, and the word “implies” reinforces that tendency. Think of logical statements as merely observations about what things happen to be together, not about causality, or underlying reasons, or even necessary connections (“these things always go together”). Although mathematicians can use the word “implication” for this, it is misleading if you don’t pay close attention to the definition.

In fact, a logical statement is not even an assertion that we make for some reason, so that we have some stake in its truth. It is just a statement that may be true or false in any given situation. If A is true but B is not, then the statement “if A, then B” is false, because if it were true, then B would have to be true.

But what if we don’t have enough evidence to judge whether the statement is true?

2. It is a mathematical definition, not everyday reasoning

Second, this conditional statement is in a sense just something mathematicians define for their own purposes, not something that necessarily agrees with the natural-language use of the phrase.  And what we need in logic (at least in traditional two-valued logic, as opposed to a logic that might include an "undetermined" value) is for every statement to be either true or false.  In this context, we have to choose what A->B will mean in every case--in order to define a complete truth table.

Some of the earlier questioners felt that “If FALSE, then …” should just be “undetermined” or “I don’t know”; and they are correct, in real life. But symbolic logic requires a logical truth value of T or F for every statement, so we don’t have that option. We need some choice, which could be purely arbitrary (like rounding up on 5), or might have a specific reason based on how we plan to use it.

Here I gave a familiar answer:

Traditionally, mathematicians subscribe to a sort of "innocent until proven guilty" rule: we can't say something is false just because there is no evidence; instead, when there is no evidence of truth or falsity, we say it is true.  This is what lies behind the related facts about sets: we say that the null set is a subset of any set because there is no evidence that it is not--there is no element in the null set that is NOT an element of the other set!

But still, why not say “guilty until proven innocent”?

3. This is what works in describing logical arguments

Ultimately, I realized, the reason for the choice comes from our application, not from the real world. One application of symbolic logic is to validate arguments, and here, if the truth value of a conditional statement were not defined as it is, then a valid argument would fail the test:

Let's consider an argument like this:

  I have a cold.
  If I have a cold, then my nose is running.
  ------------------------------------------
  Therefore, my nose is running.

This has the form

  A
  A->B
  ----
  B

To show that this is a valid argument, we write it as a single statement:

  ((A) ^ (A->B)) -> B

That is, the whole argument is a big conditional.  If its truth table is ALWAYS true, no matter what the truth values of A and B are, then we consider the argument valid.

That is, an argument is considered valid if the equivalent statement is a tautology.

I made truth tables for this (valid) argument, using every possible definition of the conditional, and showed that the only definition for which it becomes a tautology when it should is the accepted one.

Now the truth table accurately reflects the validity of the argument. And that, I think, is why we make this definition: it works.

Why does it work?  Because we want to say an argument is valid when the conclusion follows from the premise: if A is really true, then B had better be true.  We DON'T CARE what happens if the premise is false; the argument is still valid because it doesn't tell you what happens then.  There's the meaning behind that "innocent until proven guilty" idea.

An example

In 2008, I answered another question, which gave me a chance to fill out a couple areas:

Logic Statement False Implies True

I am well familiar with the linguistic arguments which clarifies this somehow confusing concept.  Is there a deeper philosophical argument that touches on the underlying logic of this concept {logic axiom}?

The most disturbing thing about this logic axiom is that it eliminates {defeats} the logical contingency of the conclusion on the premise, which somehow goes against the very essence of logic! By layman definition, logic is something that allows "naturally" the consequence to flow from a premise.  When the same conclusion happens no matter what the premise is, the connectedness of the logic in between the conclusion and its premise loses significance, just like the definition of a function is defeated when one certain value from the domain point {maps} into two or more different values from the range. 

If the moon is made of cheese, then I will go to the movie next week can rather best describe a sarcastic {insane} mode of thinking than a flow of "natural" logic especially when it can be said equally that if the moon is NOT made of cheese, I still go to the movie!  The dilemma of this concept, though I use it myself to prove some propositions like the empty set is a subset of every set, is that it kills the "natural" connectedness inherent in logic.
 
Unfortunately, the very definition of logic itself is so intuitive and vague in the same way the set or sanity is defined, otherwise undefined!  Even though I trained myself to live with this concept and I use it in my formal proofs, I try to avoid using it as much as possible.  It is like employing proof by contradiction.  I would rather prove directly.

Abe was thinking of conditional statements in cause-and-effect terms, as identifying “natural consequences”. He thought of logic as philosophy, and his vague conception led to confusion. (Logic is considered a part of philosophy not because it is deep in itself, but because we need to think clearly when we get to the deep stuff.) I repeated many of the ideas in the previous answer (which had not yet been archived so I couldn’t refer him to it): The conditional statement is not about cause-and-effect; the decision about truth value in the disputed cases depends on context; and one important context is judging the validity of an argument, which works if we take the “innocent until proven guilty” approach.

He responded with a comment that overstated the conclusion:

I found it rather interesting that there is indeed some philosophical underlying mode of thinking built into the definition of "P implies Q," that is, if there is no evidence that something is false, then we must assume that it is true.  This we may cast as the "default rule of the truth."
...
Now comes the interesting case which P is false and Q is true, yet we must assume that the implication is True ONLY for lack of better knowledge or evidence pointing to the other direction.  This is to me a philosophy or a mode of thinking which I accept as a sound one though it has some serious implications beyond math.

Since, as I have said, math has its own truth which need not agree with the real world, it doesn’t tell us anything about deeper philosophical ideas: The definition of the conditional statement is not about truth in general, but just about what we want a conditional statement to mean. He also tried to restate my reasoning, which led me to clarify it with a fully stated example:

Let's take an example.  I have a piece of paper here that is coated with a chemical that changes color.  I claim that if the paper is wet, it is red.  That is,

  WET -> RED

(Note that I didn't say "wet implies red"; the word "implies", as I said before, is not really appropriate for this connective, as you'll see in a moment.)

Now let's consider what you might see when I show you the paper, taking the four cases in your order.

1. It's wet, and it's red.  That agrees with my statement, so you say my statement is true.  (You do not have enough evidence to conclude that my statement is ALWAYS true; you've just seen one case.  Maybe tomorrow it will be cooler, and you'll find that the paper is only red if it's wet AND warm.  That's why you can't say it's true that wet implies red, only that it is true in this instance that "if it's wet, then it's red."  Do you see the difference?)

2. It's dry, and it's blue.  You don't know that it would be red if it were wet; there's no evidence one way or the other.  So, simply by convention, you say that my statement is true, meaning that the evidence is consistent with that conclusion.  But you can't say that you've proved that wetness IMPLIES redness; all you can say is that it might.

3. It's wet, and it's blue.  That disproves my statement; we have a case where it is wet but NOT red.  My statement is definitely false. (This case IS enough to disprove the stronger claim that wet implies red; you have a counterexample.)

4. It's dry, and it's red.  Hmmm ... maybe it's ALWAYS red, and my statement was technically true but misleading; or maybe it's red for some other reason than wetness.  Or maybe it actually turns blue when it gets wet, and I just lied.  Again, you really don't know!  The evidence at hand deals only with the case where it's dry, and my statement is about what would be true if it were wet.  So you have to say that it's true, because you haven't disproved it, just like in case 2.

So your cases 2 and 4 are both "true" for the same reason, not for different reasons.  The evidence in both cases is consistent with my statement, so we call it true.

He concluded with a concise statement:

In sum, P implies Q is nothing more than a claim or a proposition.  We may uphold the rest of the logic table for P implies Q since the logic equivalence (truth value) for the remaining three cases does NOT contradict our claim about P implies Q, although not useful statements in some cases.  Thanks again for the great example.

One might say that “truth”, in the sense we need here, just means “non-contradiction”.