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.
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).
style="voice: laurence-fishburne"..
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Commented
Nov 27, 2016 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.
"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.
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.
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.
S, "green" as G and "grass" as GG changes anything at all.
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Commented
Nov 27, 2016 at 11:20
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.
