# Vacuous truths

I dont understand the concept (or need for) of vacuous truths/implications. Why is it that if say a statement a is false then we can conclude that any implication a => b is 'true'?

Ive been reading online on this but everything has been vague so far, the most sensible explanation Ive seen so far goes like this: All implications in logic are either true or false (and not both) so an implication of the form a => b (where a is false) is either true or false but the only case where such an implication is false is when a is true and b is false which is not the case therefore we must conclude that it is true.

Now this almost elucidates the issue for me but then it evokes another issue for me. If we can argue in the manner that is done above then we can essentially argue that implications between completely unrelated things are true. Consider for example the statement "If the derivative of sinx is cosx then quadratic equations have a general solution/formula". Now clearly this is either true or it is false, it clearly is not false as both (the derivative of sinx is cosx) and (quadratic equations have a general solution/formula) are true so therefore this implication is true. In general we can use such an argument to conclude that any implication a => b (where it is not true that a is true and b is false) is true. But this seems completely counterintuitive to me because there appears to be no casual link between the a and b in my concrete example and this idea suggests that implications can be true for statements that are completely unrelated and have no casual relationship between them which appears to contradict what I thought was the definition of an implication (namely that a entails b, or that a CAUSES b).

In fact what do we mean when we say then that an implication statement a => b is "true", do we mean that the truth of a actually necessarily causes the truth of b or that is possible for a to cause b even if we dont know there is a casual relationship?

By the way I havent had any exposure to undergraduate Maths at all yet so please no heavy use of set theory or formal logic etc as Im just a beginner

It's interesting that I made it this far in math (final year PhD student) without thinking about this. Of course, I'm not a logician.

You can have silly implications like that where there's no causal link. The reason why you never see them is precisely because they are silly. They aren't really worth mentioning. If you have already proved that both a and b are true, the fact that a implies b is not really worthy of note. So, the causality comes into play because without that, no one would be interested.

If you think about it, if you are trying to prove a implies b, you are allowed to assume anything that you have already proven, as well as the fact that a is true. There's no rule that says you must necessarily USE a in the proof. But if you don't use a, it would be silly to make the statement.

Implication means IF a is true, then b is true. The key word is if. So, you don't really care what happens if a is false. You could declare the implication to be false if you wanted, if a is false. It's just a matter of convention. Someone has to make up definitions (and then the mathematical community agrees on them). They aren't written in the sky. The accepted convention is to define it to be true. In our everyday language, it doesn't really even come up because we normally only deal with implications where a is true.

It's interesting that I made it this far in math (final year PhD student) without thinking about this. Of course, I'm not a logician.

You can have silly implications like that where there's no causal link. The reason why you never see them is precisely because they are silly. They aren't really worth mentioning. If you have already proved that both a and b are true, the fact that a implies b is not really worthy of note. So, the causality comes into play because without that, no one would be interested.

If you think about it, if you are trying to prove a implies b, you are allowed to assume anything that you have already proven, as well as the fact that a is true. There's no rule that says you must necessarily USE a in the proof. But if you don't use a, it would be silly to make the statement.

Implication means IF a is true, then b is true. The key word is if. So, you don't really care what happens if a is false. You could declare the implication to be false if you wanted, if a is false. It's just a matter of convention. Someone has to make up definitions (and then the mathematical community agrees on them). They aren't written in the sky. The accepted convention is to define it to be true. In our everyday language, it doesn't really even come up because we normally only deal with implications where a is true.

Ok that makes sense but then I dont really see why we call it implication at all since its not really implication in the normal sense (where a is a sufficient condition for b), but just a seemingly arbitrary convention...

Ok that makes sense but then I dont really see why we call it implication at all since its not really implication in the normal sense (where a is a sufficient condition for b), but just a seemingly arbitrary convention...

No, it is implication in the normal sense. It's just that the normal sense is sort of incomplete. It doesn't cover all possible truth values for a and b.

I'm not a logician either but I have studied a fair amount of it and I believe we have these things called disjunctive normal form and conjunctive normal form, and if I am recalling this correctly this has something to do with the fact that all statements in classical logic can be reduced to or's and and's, and negation of course. The conditional connective adds no extra meaning beyond its representation in terms of or's and and's, the fact that it is the formalization of our experience of causation makes it a useful, but ultimately superfluous, psychological crutch, as classical logic admits no temporal component.

Yes, implication can be phrased as (a AND b) OR NOT a. So, yes, it's a psychological crutch. Not strictly needed.

So, actually, not only is it implication in the normal sense, but the whole point is that it is implication in the normal sense, but since it's a technical version of it, we have to decide on what it means precisely, and that leads to weird things that we wouldn't normally say outside of a math/logic context.

I'm not a logician either but I have studied a fair amount of it and I believe we have these things called disjunctive normal form and conjunctive normal form, and if I am recalling this correctly this has something to do with the fact that all statements in classical logic can be reduced to or's and and's, and negation of course. The conditional connective adds no extra meaning beyond its representation in terms of or's and and's, the fact that it is the formalization of our experience of causation makes it a useful, but ultimately superfluous, psychological crutch, as classical logic admits no temporal component.

No, it is implication in the normal sense. It's just that the normal sense is sort of incomplete. It doesn't cover all possible truth values for a and b.

Why cant we just not assign truth values for the implication for the truth values of a and b that arent the actual case (since they are pretty much meaningless as the state of affairs is not the actual case), i.e., leave it incomplete?

I suppose that if we actually want to assign truth values to the implication with all possible truth values for a and b and that we assume that the only case where a => b can be false is if a is true and b false, then it obviously follows that, with the exception of a being true and b false, all truth values for the implication are "vacuously" true since they cant be false.

I think I now understand what homeomorphic means by when he says its just silly because it seems to be pointless anyways.

If pigs can fly then I'm the King of Siam.

That's a true statement. When you understand that you will be enlightened.

Hint: Tell me how that statement could possibly be false.

If pigs can fly then I'm the King of Siam.

That's a true statement. When you understand that you will be enlightened.

Hint: Tell me how that statement could possibly be false.

Well before I may have objected that how does pigs flying cause the fact that you are the King of Siam. Well instead I say your statement is either true or false just for the sake of assigning it a truth value then it cant be false because it is false that pigs are flying and you are not the King of Siam so its true. Is this correct?

Stephen Tashi
I think the simplest way to understand vacuuous implications is to consider what constitutes a valid counterexample. If someone claims "A implies B" is true then what must we do to have a counterexample? We must show a situation where A is true and B is false.

So the negation of "A implies B" must be "A and (not B))". The negation of that negation must be equivalent to the original statement "A implies B". By some uncontroversial logical manipulations we can show "not ( A and (not B))" equivalent to "(not A) or B".

What is sufficient to make "(not A) or B" true? One thing that works is to have A be false. This is why vacuuous implication works. If you don't like vacuuous implication, you could try to define a new form of implication, but what are you going to do for its truth table? You have to say whether "A implies B" is true or false in the cases when A is false.

Suppose you say "A implies B" is false when A is false. Then when you are being considered for the Fields Medal because of your masterful proof of the theorem that "X implies Y", someone will point out that your theorem is false. His countexample will be to show a case where X is false.

D H
Staff Emeritus
Implication means IF a is true, then b is true. The key word is if. So, you don't really care what happens if a is false. You could declare the implication to be false if you wanted, if a is false. It's just a matter of convention.
It's a lot more than a matter of convention. It is a very necessary concept. Suppose we know that (a) P implies Q and (b) P is true. We therefore also know Q is true; it cannot be false. This is the rule of modus ponens:

If P then Q
P
Therefore Q.

This gives us half of the truth table for implication:

Code:
    |  Q
P→Q| 0  1
---|-----
0| ?  ?
P  |
1| 0  1

Knowing that P→Q tells us nothing about Q in the case that P is false. Logically it has to be this way; just because P causes Q doesn't mean other things can't cause Q as well, even if P is false. Thus those two (so far) unknown values in the truth table must be the same value; otherwise P→Q would say something in the case that P is false. In other words, we have

Code:
    |  Q
P→Q| 0  1
---|-----
0| X  X
P  |
1| 0  1

So how to fill in that upper half? The answer lies in looking at what P→Q tells us about P when we know Q. Knowing that Q is false is the key. If P being true always causes Q to be true and we know that Q is not true then we can logically conclude that P is false. This is the rule of modus tollens:

If P then Q
Not Q
Therefore not P.

Thus the P=0, Q=0 element of the P→Q truth table is 1. Since the P=0, Q=1 element of the table must have the same value as the P=0, Q=0 element, the truth table becomes

Code:
    |  Q
P→Q| 0  1
---|-----
0| 1  1
P  |
1| 0  1

It's a lot more than a matter of convention.

I just said that because, in principle, you are free to assign any truth values you want to it, since it is a definition. It just doesn't make much sense, unless you use the usual definition.

Deveno
i like to view P→Q as: Q contains P.

so what if you're not in P? (P is false). that doesn't change the relationship of P to Q. the only way we can falsify P→Q is to be inside of P, but outside of Q. when we're outside of P, we might be outside of Q, we might not be, it's irrelevant to whether or not Q contains P.

for example: if you are the President of the United States, you are an American citizen.

the set of Presidents is a subset of American citizens. considering people who aren't President, doesn't change the truth of that relationship.

Well before I may have objected that how does pigs flying cause the fact that you are the King of Siam. Well instead I say your statement is either true or false just for the sake of assigning it a truth value then it cant be false because it is false that pigs are flying and you are not the King of Siam so its true. Is this correct?

Are you familiar with the truth table for implication? That's a good place to start.

Another key element here is that you used the word "cause." AHA, this is the source of your psychological block. Logical implication is not causality. Nothing to do with causality. Put causality far, far out of your mind; and study the truth table for implication.

What would be evidence that "If pigs can fly then I am the King of Siam?" is false? What would have to happen in order for that statement to be true? You almost sort of have it in your paragraph above.

http://en.wikipedia.org/wiki/Vacuous_truth

Honestly, just read this thoroughly, it's actually a really good page on it (which is strange for wiki).

I think the most satisfying reason for this convention (and it is a convention, unlike what some people are suggesting) is the following (taken from the above page):

Making vacuous implications "true" makes many mathematical propositions that people tend to think are true come out as true. For example, most people would say that the statement

For all integers x, if x is even, then x + 2 is even.

is true. Now suppose that we decide to say that all vacuously true statements are false. In that case, the vacuously true statement

If 3 is even, then 3 + 2 is even

is false. But in this case, there is an integer value for x (namely, x = 3), for which it does not hold that

if x is even, then x + 2 is even

Therefore our first statement isn’t true, as we said before, but false. This does not seem to be how people intuitively use language, however.

and also this:

First, calling vacuously true sentences false may extend the term "lying" to too many different situations. Note that lying could be defined as knowingly making a false statement. Now suppose two male friends, Peter and Ned, read this very article on some June 4, and both (perhaps unwisely) concluded that "vacuously true" sentences, despite their name, are actually false. Suppose the same day, Peter tells Ned the following statement S:

If I am female today, i.e., June 4, then I will buy you a new house tomorrow, i.e., June 5.

Suppose June 5 goes by without Ned getting his new house. Now according to Peter and Ned’s common understanding that vacuous sentences are false, S is a false statement. Moreover, since Peter knew that he was not female when he uttered S, we can assume he knew, at that time, that S was vacuous, and hence false. Since Peter has spoken a falsehood, then Ned has every right to accuse Peter of having lied to him. On the face of it, this line of reasoning appears to be suspect.

What would be evidence that "If pigs can fly then I am the King of Siam?" is false? What would have to happen in order for that statement to be true? You almost sort of have it in your paragraph above.

Steve, just replace "true" with "false" in your sentence. Vacuous truth is a convention- it isn't clear that we must assign such things truth values as you seem to be suggesting. It just turns out that it makes more sense to use that convention rather than a different one (for many reasons, some listed above) but you could still have a perfectly legitimate logical system if you considered such statements as false.

This makes it very hard to introduce people to it when they first see it- they ask "why?" and the answer is "it's convenient".

i like to view P→Q as: Q contains P.

so what if you're not in P? (P is false). that doesn't change the relationship of P to Q. the only way we can falsify P→Q is to be inside of P, but outside of Q. when we're outside of P, we might be outside of Q, we might not be, it's irrelevant to whether or not Q contains P.

for example: if you are the President of the United States, you are an American citizen.

the set of Presidents is a subset of American citizens. considering people who aren't President, doesn't change the truth of that relationship.

Yes, this is all in some way an equivalent question to: "is the empty set a subset of all other sets?". Asking that is a vacuously true sort of statement: the empty set is contained in any other set if for all elements x in the empty set, they are also in the other set. There are no such elements, so this is vacuously true. Conversely, if you consider your logic as a sort of set theory, we will consider vacuously true statements as true precisely when we consider the empty set as being a subset of all others (note, if you disagree with vacuous truth, then you can't consider the empty set as being a subset of all others, which I imagine would be rather inconvenient).

HallsofIvy
Homework Helper
Suppose your teacher tells you, at the beginning of the course, "If you get an "A" on every test, you will get an "A" for the course".

You do NOT get an "A" on every test but do pretty well. Now consider these outcomes:
1) You receive an "A" for the course. Was the teacher lying at the beginning of the course?

2) You receive a "B" for the course. Was the teacher lying at the beginning of the course?

Note that the teacher's statement, at the beginning of the course says nothing at all about what would happen if you did not get an "A" on every test.

disregardthat
It's important to note though in Hall's example that "If you get an "A" on every test, you will get an "A" for the course" as a logical proposition is not a statement about the teachers intentions. In that case he could be lying no matter of your performance if he had no intentions of giving you an A for the course even though you get A on every test.

Thanks for the replies guys, I do see that it makes sense now and follows from the basic axioms of logic. I guess the problem was just that I was trying to reconcile it with my intuitive/concrete understanding of implication (an understanding based on casuality hence problematic).

Thanks for the replies guys, I do see that it makes sense now and follows from the basic axioms of logic. I guess the problem was just that I was trying to reconcile it with my intuitive/concrete understanding of implication (an understanding based on casuality hence problematic).

No! It doesn't follow from the axioms, that's the point. If you read the wiki page I posted, there is actually a case for not considering vacuous truths as always being true.

No! It doesn't follow from the axioms, that's the point. If you read the wiki page I posted, there is actually a case for not considering vacuous truths as always being true.

It does (see Arguments for taking all vacuously true statements to be true on wiki) if we want to assign truth values to an implication (a => b) for all possible truth values of a and b. The axiom that it follows from is that an implication is false if and only if a is true but b is false and since we actually want to assign truth values to the implication, it follows from that that it must be true although it tells us nothing (hence vacuous). What is wrong with this analysis?

Edit: Oh I see you could argue that this argument begs the question but I see no reason to be that pedantic

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Saying something follows from the axioms is rather strong, but if you see on that page, you certainly can carry on without assuming that vacuous statements are true. In fact, you can assume they are all false fine- it shows that in that case, the implies sign is equivalent to the AND operator.

There is plenty of reason to object to this, but you can't say that vacuous truth follows from the axioms (unless it is one of them).

If pigs can fly then I'm the King of Siam.
This is similar to a Tosh.0 joke where he joked about wanting to "fornicate" with a "mythical" baby (the hypothetical offspring of Brad Pitt and David Beckham). He argued that the idea of such an act is not offensive, because it depends on the false premise that two men can produce a child.

I saw 'the Enigmatic Giant', an anime episode, on tv the other day that describes the mechanics of an interresting dilemma. The giant in the title guards a bridge and only allows people to cross if they answer his question correctly.

His question is "If you lie, I will run you through with my sword, but if you tell the truth, I will strangle you with my bare hands, what do you say?"

The correct response is "You will run me through with your sword".

If the giant runs the answerer through with his sword then, by his own stated rules, he implies that the answerer lied but if that was the case then the answerer was telling the truth in the original answer and then should be strangled by the giants bare hands.

Ive been reading online on this but everything has been vague so far, the most sensible explanation Ive seen so far goes like this: All implications in logic are either true or false (and not both) so an implication of the form a => b (where a is false) is either true or false but the only case where such an implication is false is when a is true and b is false which is not the case therefore we must conclude that it is true.

...Now this almost elucidates the issue for me but then it evokes another issue for me. If we can argue in the manner that is done above then we can essentially argue that implications between completely unrelated things are true....

I neither know what you have been reading, nor have I received any schooling in formal logic. However, I think common sense tells you there is something screwy with the question itself.

It would seem to me that 'A implies B' will always be based on a set of axioms. That is to say, we look at A and then we look at B and cannot determine whether there is a commonality or causality implied unless we do so within a set of axioms. The 'act of logic' is always embedded in the frame of our stated set of axioms, and whether one thing implies another (and whether they are true) is wholly dependent on those.

Therefore, I think it would be in error to declare something is 'universally true' independent of axioms, because you're indirectly asking for a definition of 'ultimate truth' and I suspect we'd get into a big sticky mess looking at that.

[There is also a question around causality or co-incidence. Not so relevant in maths but in science we may see "A=>B", but actually what we might not be realising is "A iif C" and "B iif C" and we are just seeing incidents of C. But I am unclear if this might be another subject.]

Deveno
Yes, this is all in some way an equivalent question to: "is the empty set a subset of all other sets?". Asking that is a vacuously true sort of statement: the empty set is contained in any other set if for all elements x in the empty set, they are also in the other set. There are no such elements, so this is vacuously true. Conversely, if you consider your logic as a sort of set theory, we will consider vacuously true statements as true precisely when we consider the empty set as being a subset of all others (note, if you disagree with vacuous truth, then you can't consider the empty set as being a subset of all others, which I imagine would be rather inconvenient).

i like you.

indeed, vacuous truth is intimately linked with the empty set, which has a long list of descriptive properties, but unfortunately precious few elements to carry them. one of my favorites is: the empty set consists of those elements whose existence is impossible.

disregardthat
One source of confusion of the logical statement "P --> Q" can occur when it is interpreted causally. That a proposition logically implies something, does not mean that the truth of a proposition would causally imply the truth of another. Logic does not deal in causation, only logical causation.

A lot of "If A, then B" is falsely recognized as a logical implication when A and B are falsely recognized as logical propositions. E.g. "If you let go of this rock, then it falls down to the ground" can quite easily be thought of as logical implication of two propositions. But is "To let go of this rock" a proposition? No, it's an action.

Rather, "You have let go of this rock" is a proposition. Or "You will let go of this rock" is a proposition. Also, "This rock will fall to the ground" is a proposition. So the statement "If you have let go of this rock, then this rock will fall to the ground" is the (in my opinion) closest to an interpretation of the original statement as a logical proposition. This isn't to be interpreted as a causal relation between the two situations of letting go of a rock, and the rock falling to the ground.

There is a pure logical relation between the two propositions which can be true or false depending on the actual truth value of the propositions. Note that the temporal aspect of the propositions matter quite heavily to their interpretations.

Can I recommend the following?:
http://www.dpmms.cam.ac.uk/~wtg10/implication.html
(and anything else on Gower's page, who is an excellent writer).

He addresses many of the issues we've raised, as well as this causality issue (for example, on the grounds that the Riemann Hypothesis is true, he points out the issue with saying that the "Riemann hypothesis implies Fermat's Last theorem". Given both being true, and given what we've said, we'd probably have to concede this as being a true statement, although pretty much no mathematician would try to argue that FLT is a consequence of the RH).

Deveno
One source of confusion of the logical statement "P --> Q" can occur when it is interpreted causally. That a proposition logically implies something, does not mean that the truth of a proposition would causally imply the truth of another. Logic does not deal in causation, only logical causation.

A lot of "If A, then B" is falsely recognized as a logical implication when A and B are falsely recognized as logical propositions. E.g. "If you let go of this rock, then it falls down to the ground" can quite easily be thought of as logical implication of two propositions. But is "To let go of this rock" a proposition? No, it's an action.

Rather, "You have let go of this rock" is a proposition. Or "You will let go of this rock" is a proposition. Also, "This rock will fall to the ground" is a proposition. So the statement "If you have let go of this rock, then this rock will fall to the ground" is the (in my opinion) closest to an interpretation of the original statement as a logical proposition. This isn't to be interpreted as a causal relation between the two situations of letting go of a rock, and the rock falling to the ground.

There is a pure logical relation between the two propositions which can be true or false depending on the actual truth value of the propositions. Note that the temporal aspect of the propositions matter quite heavily to their interpretations.

but for some propositions, there's a high degree of correlation between logical implications and cause-and-effect. for example, when you are counting, you might enumerate by amount, or by time (the two distinct but related meanings of "next"). the english word "then" reflects this ambiguity, (like "next" does), we see P→Q as meaning "first P, then Q" and for cause-and-effect relationships, that's what happens. that is, if there IS a cause-and-effect relationship between P and Q ("P causes Q") then if P occurs, we will expect Q at some point. even in proofs, one often sees P→Q phrased as "Q is a consequence of P" note the time-implication of the word "consequence" (Q follows P, it's after it in time).

so there's a difference in the pure logic, and the way we talk about it. but the pure logic is abstracted from the way we talk and think. i mean, it's very easy to confuse one's abstraction of something, with the actual thing itself, the way someone might confuse my name with ME (just what is the difference between a "variable constant" and a "constant variable", and why on earth would we ever devise such a beastly thing?).

moreover, there's often a logical modelling of cause-and-effect systems, whereby a physical situation is put into logical language, the rules of logic applied, and then the resulting deduction used to reason about the physical system, especially in systems where time is a variable. so although "you let go of this rock" may not be a logical proposition, if one casts it in terms where "if you let go of this rock, then it will fall to the ground" becomes a valid logical implication, the rules of logic itself (is it modus ponens? i forget) tells us, that if that (properly formulated) logical implication is true, in the real world, said rock WILL fall to the ground, if it is "let go of" (P and P→Q, therefore: Q). which is why, among other things, buildings have roofs on them (ok, maybe rain doesn't quite qualify as "rocks", but whatever).