How would one, as intuitively as possible, explain why the following implication is false
I understand that for either the statement or must be true. Suppose the statement is true; the falsity of the implication is quite apparent then. If the statement is true then the veracity is apparent. Thus the implication is only true for one statement of the two within .
However, the following implication is surprisingly true:
How come? Suppose on the right-hand side of the implication is true; that implication is false! The implication is only true if on the right-hand side. Once again, the implication is only true for one statement of the two within . In spite of this, we still claim that the implication as a whole is true whereas the first one is not.
I know it can be explained if we consider either side as equations with different number of roots, but I'd rather want to understand this as intuitively as possible.
It is true because either one of the statements is true or both. Since the statement satisfies this condition, it's true. Same thing goes for the two following statements.
Regarding the last statement: it is false because neither of the statements are true and therefore the statement as a whole is false.
the statement x ≥ 4 => x > 4 is false for the following reason:
it may be the case that x = 4, in which case x ≥ 4 is true, and x > 4 is false. in other words, we cannot be sure that x > 4, when x ≥ 4.
you cannot, in general, derive a stronger statement from a weaker one.
the statement x > 4 => x ≥ 4 is true, you can usually weaken a statement without fear of consequence.
consider the following english statements:
"i am older than you, so six months from now i will still be at least your age"
"i am at least as old as you, so six months from now i will be older than you"
the first one is true. if in fact, i am not as old as you, i am "spouting nonsense" which is still valid reasoning, but irrelevant. this is the "x = 4" case in:
x > 4 => x ≥ 4. we don't have a good term for this kind of statement in english: logically true, but invalid (because it doesn't apply). sometimes it is paraphrased as: "you can derive anything from a false premise". in general, in mathematics, we are only concerned with CONDITIONAL truth: what is true given some assumptions. it used to be hoped that we could derive all of mathematics from "self-evident statements" such as: "a line is the shortest distance between two points". unfortunately, this doesn't seem to be the case, and there are two problems:
1) finding suitable "self-evident statements" isn't as easy as in sounds. for example, on a sphere, the shortest distance between two points is NOT a straight line, but an arc of a great circle.
2) if we restrict our statements severely enough, we can sure that our theory is sound...but if we want to capture even something so simple as arithmetic (and we surely do), there are limits to what we can prove (we might suspect something is always true, but be unable to prove it, at least in our theory).
so when you have a statement like:
A implies B
the tacit interpretation is: if A is true, then we can derive the truth of B. very few go about trying to prove such a statement by proving A is false (although in some strange cases, this is done).
i like to think of it like so: "A => B" means: IF A, THEN B. if not A, who knows, who cares?
By definition of implication. Would you say that the statement "if n is divisible by 4, then n is divisible by 2" is true for all integer n? What about n = 3? For another example, if a person is the president of the United States, he/she is at least 35 years old. There are no exceptions. What about John, who is 50 years old and is not a president? Does his existence invalidate the law?
people struggle with the F => T is true part of logical implication. there is a popular song that sort of pokes fun of this that begins:
"When it's night-time in Italy, then it's Wednesday over here".
you can have a lot of fun with this:
"I am the king of Siam, if and only if the sky is green" is a true statement (unless, somehow, i happen to actually become the king of Siam, OR the sky turns suddenly green, without these two events being inextricably linked. let's not go there). even better, one can write such drivel as:
"2 is an odd number if and only if 9 is prime", even though 2 being odd and 9 being prime have nothing to do with each other. it appears that i am saying they DO have something to do with each other, because that is the way we try to make sense of things. our brains seem to automatically reject "false thoughts" once we know they are untrue. we seem to draw a distinction in our thought between a hypothetical (possibly true) statement, and a statement which we KNOW is either true or false.
2 = 3 implies 4 = 6 is logically valid:
if 2 = 3, then
2*2 = 2*3, so 4 = 6.
but 2 = 3 isn't TRUE (at least with our ordinary meaning of these as natural numbers). fortunately, this means i don't have to worry about having "proved" 4 = 6, but the question remains:
why would i even go there? ("even", get it?). formal logic does not map well to "everyday life". linguistic truth, in a natural language (such as english) is not the same concept as "logical truth". the use of "true" for both concepts, while at times illuminating, can cause confusion as well. informal (english) proofs can sometimes exacerbate this problem, by assuming the reader knows "the rules of the game". formalizing this eliminates the vagueness of what we mean by "true", but it also makes it darn hard to explain what we mean!