# Why is A implies B true if A is false and B is false?

It seems to me that the 'implies' in English language does not mean the same thing as the logical operator 'implies', in a similar way how 'OR' word in most cases means 'Exclusive OR' in our everyday language use.

Let's take two examples:

If today is Monday then tomorrow is Tuesday.

This is true.

But if we say:

If the sun is green then the grass is green.

This is also considered true. Why? What is the 'logic' in natural English behind this? It blows my mind.

• Because implication is about truth-preservation. If $A$ is false, there is no truth to be preserved. – Rodrigo de Azevedo Nov 25 '16 at 20:50
• Boolean logic has nothing to do with the English language. – Yuval Filmus Nov 25 '16 at 21:13
• Already covered on Math Stack Exchange in this thread and other related ones: math.stackexchange.com/questions/48161/… – Nayuki Nov 26 '16 at 4:12
• This Philosophy Stack Exchange take on the question is also relevant: Why are conditionals with false antecedents considered true? – duplode Nov 26 '16 at 7:14
• @MHH ah, right. "If x > 5 then x> 3" is non vacuous true, "if 2> 5 then 2>3" is a true implication (false premise) but non vacuous because there is no empty set involved. – eques Nov 27 '16 at 18:39

Humans are bad at logic until they have to employ it to figure out human affairs. Think of "if $A$ then $B$" as a kind of promise: "I promise to you that if you do $A$ then I will do $B$". Such a promise says nothing about what I might do if you fail to do $A$. In fact, I might do $B$ anyhow, and that would not make me a liar.

For instance, suppose your mother tells you:

If you clean up your room I will make pancakes.

And let us say that you did not clean up your room, but when you walked into the kitchen your mom was making pancakes. Ask yourself, whether this makes your mom a liar. It does not! She would be a liar only if you cleaned the room but she refused to make pancakes. There might be other reasons that she decided to make pancakes (perhaps your sister cleaned up her room). Your mom did not tell you "If you do not clean up the room I will not make pancakes," did she?

So, if I say

"If the sun is green then the grass is green."

that does not make me a liar. The sun is not green (you did not clean up the room), but the grass turned out to be green anyhow (but your mom made pancakes anyhow).

• It wouldn't make you a liar, but it wouldn't make you a truthteller either. Why don't you just say the honest truth, which is that it's purely a convention? Everybody on the planet seems to be afraid to say it (except for the user who posted the other answer on this page)... – user541686 Nov 26 '16 at 11:16
• What are you referring to when you sais "it is purely a convention"? The meaning of implication? Sure, but you are wrong when you say it is purely a convention, as if the meaning of implication were some sort of an arbitrary garbage that a bureaucrat came up with. Conventions (if you want to call them that) in mathematics are there for a good reason. They are useful, and they help explain things. They are far from arbitrary, which is why it is intelectually dishonest to take the position that "everything is just purely a convention". It makes you a troll. – Andrej Bauer Nov 26 '16 at 13:00
• Breathing is merely a convention. ;-) – jpaugh Nov 26 '16 at 20:07
• <span style="voice: samuel-jackson">You think that's air you're breathing?</span> – Andrej Bauer Nov 26 '16 at 20:18
• @AndrejBauer - ...uh, I think you meanstyle="voice: laurence-fishburne".. – Mark Rogers Nov 27 '16 at 20:11

It's a convention -- we could use a different one, but this one is convenient. Here's what Terence Tao says:

This is discussed in Appendix A.2 of my book [Analysis 1]. The notion of implication used in mathematics is that of material implication, which in particular assigns a true value to any vacuous implication. One could of course use a different convention for the notion of implication, however material implication is very useful for the purpose of proving mathematical theorems, as it allows one to use implications such as “if A, then B” without having to first check whether A is true or not. Material implication also obeys a number of useful properties, such as specialisation: if for instance one knows for every x that P(x) implies Q(x), then one can specialise this to a specific value of $x$, say 3, and conclude that P(3) implies Q(3). Note though that by doing so a non-vacuous implication can become a vacuous implication. For instance, we know that $x \geq 5$ implies $x^2 \geq 25$ for any real number $x$; specialising this to the real number 3, we obtain the vacuous implication that $3 \geq 5$ implies $3^2 \geq 25$.

The way I like to think of material implication is as follows: the assertion that A implies B is just saying that “B is at least as true as A”. In particular, if A is true, then B has to be true also; but if A is false, then the material implication allows B to be either true or false, and so the implication is true no matter what the truth value of B is.

• That statement sounds nice until you realize the intuition it's invoking is actually not true. Think about something like "If aliens roam Earth then I am an alien"... I'd be much more inclined to believe that aliens roam Earth than that I myself am an alien... – user541686 Nov 26 '16 at 11:21
• "If aliens roam Earth then I am an alien" is not a true implication; that is, q does not follow from p ordinarily. That is distinct from if p is false the implication is true – eques Nov 26 '16 at 17:05
• @Mehrdad shouldn't that be "If I am an alien, then aliens roam Earth"? – Paŭlo Ebermann Nov 27 '16 at 23:44
• @eques: "If the sun rises tomorrow then I will get up in the morning"... I'd bet if the sun didn't rise tomorrow I'd still get up in the morning (barring other effects of the sun disappearing). But people say stuff like that anyway. – user541686 Nov 28 '16 at 0:00
• @Mehrdad people say things that aren't logically rigorous all the time; that doesn't mean the rules of logic aren't good. And if someone still gets up in the morning even though the sun did not rise, they didn't counter their implication. The implication is still true – eques Nov 28 '16 at 1:48

"A implies B" means (short) "if A is true then B is true".

It means (a bit longer) "if A is true then I claim that B is true; if A is false then I don't make any claim about B whatsoever".

Now take "If the sun is green then the grass is green".

In the long form it is translated to "If the sun is green then I claim that the grass is green; if the sun is not green then I make no claim about the color of grass whatsoever". The sun is not green, so I make no claim about the color of grass whatsoever.

• So if you do not make any claim regarding the grass that means that everything is true for the grass... but how is this equivalent to " I do not make any claim towards the grass" ? – yoyo_fun Nov 25 '16 at 20:44
• Can the 'imply' logic operator be modeled using sets like the rest of the operators ? – yoyo_fun Nov 25 '16 at 20:47
• @yoyo_fun $A \rightarrow B$ is equivalent to $\neg A \vee B$ and you can model it the same. – hobbs Nov 26 '16 at 6:59
• @yoyo_fun Making no claim about the grass does not mean that everything wrt. grass is true! (The grass is alive; the grass is dead cannot both be true.) In context, what it means is, "If the sun is not green, then the original statement gives us no information about the grass whatsoever." – jpaugh Nov 26 '16 at 20:11

Let's take an example. Suppose that we want to express that $a$ is the only element of the set $S$ that satisfies property $P$. Then we can write $$\forall x \in S \;\; P(x) \Rightarrow x = a$$ This states that any element of $x$ that satisfies $P$ must be equal to $a$. It doesn't claim anything about elements not satisfying $P$. If $b$ doesn't satisfy $P$ and is different from $a$ then $P(b)$ is false and $b = a$ is false, and so $P(b) \Rightarrow b = a$ is true, just as in your example.

• This I think is the best answer. As an example: the claim "if an animal is a cat, then it is a mammal" is true even though there are animals that are mammals but not cats, and animals that are neither cats nor mammals. – jadhachem Nov 27 '16 at 1:27

It's important to note that many forms of logic have no concept of chronology or causality. If something is true, then it will--within its context--have been and continue to be true forever. Saying that X implies Y does not mean in any sense that X will in any way cause Y to be true. It merely means that X cannot be true without Y also being true, and Y cannot be false without X also being false.

To usefully describe causal relationships in the real world requires something beyond the constructs used in "timeless" logic. A concept like "For any action Y such that X would cause Y to be reasonable, Y shall be deemed reasonable" can be useful in a causal universe even if X might be false, but the implication operator completely blows up in such cases. If one were to say "X implies that Y shall be deemed reasonable" and it turned out that X was never true, that would imply that all actions shall be deemed reasonable.

I'm not sure what forms of logic include the constructs necessary to allow statements involving one-way causality, but recognizing that the logical definition of "implies" does not recognize the concepts of time and causality should make it easier to understand why they behave in counter-intuitive fashion.

While using Implication In English it not about the things or objects we consider.

Like in your given example which is blowing you mind is that If the $sun$ is $green$ and then $grass$ is $green$.

Sun is just is an object here, don't make any emotional attachments to it, that a sun can't be green.

You can just replace sun with a book or a letter $S$, green with $G$ and grass with $GG$. Now see the sentence If the S is G then GG is G.

{{S->G} $->$ {GG->G}}

This seems less confusing then while writing in English.

• What does "emotional attachment" have to do with anything? And how has spelling the objects any differently answered the question? – Lightness Races in Orbit Nov 26 '16 at 21:49
• @LightnessRacesinOrbit It's just for some students they see things emotionally rather than being logic oriented. And I'm sorry which spelling is mistaken ?? – iambruv Nov 27 '16 at 1:35
• I didn't say your spelling was mistaken. I'm asking why respelling "sun" as S, "green" as G and "grass" as GG changes anything at all. – Lightness Races in Orbit Nov 27 '16 at 11:20
• @LightnessRacesinOrbit Oh, it is just for convince, nothing more.sometime we get confused when sentence are given like some pens are pencil, all pencils are parrots, no parrot is a bird. So I prefer using these kind of symbol to make my mind to stop visualising how all pencils are related to being bird, because they are just object having no significance with either pencil or bird. – iambruv Nov 27 '16 at 12:21
• Yeah I still don't see how that answers the question but okay – Lightness Races in Orbit Nov 27 '16 at 12:38

To put your head in the right place for my answer, I want to mention what I like to call the Flying Monkeys Theorem, or what Wikipedia likes to call the Principle of Explosion, which states:

$$(p\wedge\neg p) \rightarrow q$$

Or, in English, this says "given a contradiction, monkeys might fly out of my butt (NSFW audio)", or alternately "from falsehood, anything follows". One way to think about this is that if $2+2=4$ and $2+2=5$ then $4=5$, which means that $0=1$, or it could mean that $16=25$, etc., and you can basically generate any equality you want. This is why there are so many tricks that result in $1=0$ or $1=-1$ by abusing a hidden division by zero, because you are not allowed to divide by zero so you can make anything you want true.

Once we're in this realm where we know $p$ is false, we're no longer in reality. We're in some alternate dimension where the Babel Fish is real, black is white and watch out for that Zebra crossing. So given that we're no longer in reality, of course the statement could be true. Specifically, I can use my false thing that I'm assuming to prove anything I want. So of course $F \rightarrow T$ and $F \rightarrow F$ are both true statements.

• I don't buy this argument. You're saying that, by writing $P\to Q$ where we know $P$ to be false, we're talking about some alternate reality where $Q$ could be true. If that's the case, why do you then go on to assume that $Q$ is true in that alternative reality? That seems philosophically unsatisfactory. Also, the whole "alternative reality" setup completely contradicts the formal semantics of logics: the truth or falsity of a formula in a particular model is determined with respect to that model, not with respect to some other model that the reader dreams up. – David Richerby Nov 27 '16 at 13:28
• @DavidRicherby let $r = \neg q$. Clearly $(p \wedge \neg p) \rightarrow q$ is just as valid as $(p \wedge \neg p) \rightarrow r$. From falsehood, anything follows, including another contradiction. – durron597 Nov 27 '16 at 23:48