# If false imples true then you can't make statements about real life? Confused [closed]

I might have rushed into this and have insufficient knowledge about logic, however i have yet to see anything online go against this. Though i am not foolish enough to believe I have 'broken' logic, I am merely curious as to where I am being confused.

I realised i havent really posted a question so i'll have to do that:
I don't understand how false implies true, I'll try to explain below

People seem to be okay with making normal sentences out of A -> B.

If i have a cold, then my nose is runny.
I have a cold.
My nose is runny.

A = i have a cold
B = Nose is runny
If A then B

So far so good, this seems perfectly reasonable.

Also, false implies true exists. "If the sun is green then the grass is yellow". This statement is true. This is(almost) fine, since i can never prove nor disprove it since the sun will never be green. The only contention i have is that logic assumes that false implies true. False would(in my and some other peoples heads) imply:

"We dont know anything if false, the IF statement says something about it if it is true, not false". If the sun is green THEN the grass is yellow, we dont know anything about it if the sun is not green, the statement does not say anything about that.

It is an axiom of convenience that (False implies true) the grass is yellow is true. Logic needed that to function as far as i understood it, or it was much more convenient this way, and this leads me to my confusion, because now logic cannot be applied to statements about real life anymore.

What i mean is:

A -> B
If a is false, then B is true, which also means:

A = I have a cold
B = I have a runny nose

So if A is false then B is true. This means i have a runny nose... which is not necessarily true

The thing i could be missing would be that i can actually disprove this case of If A then B if we assume that A is cold and B is runny nose. I could disprove it just by saying I don't have a cold and my nose is not runny.

Is the precondition then that B will always be true only if A can never be true?

Which would mean that "if the sun is green then the grass is yellow" is true but "if i have a cold then my nose is runny" is not necessarily true, in which case i am satisfied.

Also i am just reading wiki articles and other questions about logic but it seems like i am making slow progress in understanding anything. If anyone has suggestions as to where a good place/course to further my knowledge would be, that would also be highly appriciated.

Edit: Edited the question so that it actually points to what i am wondering about, not just 'read my text please'.

Edit 2: My confusion was this: If the sun is green then the cows are chickens. This statement is true in logic. My contention was that if we have a false statement first(The sun is green) then the second part would be true(Cows are chickens). I Thought i could use this format to make a false statement first(i do not have a cold) and then make another statement(my nose is runny) and i thought this would be true. However as far as i understand it, i can only make such statements if the first premise(i do not have a cold) is ALWAYS false and "i do not have a cold" is not always false, however, "the sun is green" is always false. So because "the sun is green" is always false, then the following statement: "Cows are chickens" is true, because we cannot possibly disprove it(because the sun is green can never happen, we may never know if cows are chickens if the sun indeed was green and therefore cannot disprove it), and in logic, if we cannot disprove something we assume it to be true. Anyways this was a silly question and i figured it out earlier when talking to someone, but either way thanks to you all for your answers!

What i mean is:

A -> B
If a is false, then B is true, which also means:

A = I have a cold
B = I have a runny nose

So if A is false then B is true. This means i have a runny nose... which is not necessarily true

What you are doing in this particular passage makes little sense.

First of all: A -> B means "If A is true, then B is true"

It does not mean that "If A is false, then B is true"

And it does certainly not mean that A is false, and B is true.

Your confusion is this. Yes, the statement A -> B is considered true (in logic), if it is the case that A is False and B is true. However, that does not mean that if A -> B is true, then A is false and B is true. That is, you cannot go the other way around.

More formally:

From ~A and B you can infer A->B

But from A->B you cannot infer ~A and B

So, you can not infer that you have a runny nose (or, for that matter, that you do not have a cold) just because it is true that if you have a cold, you have a runny nose. And this, as you rightly point out, is exactly as it should be ... so there is no problem here.

• Concise and to the point, I am not sure why OP hasn't accepted your answer. Perhaps because "a little knowledge is a dangerous (In OP's case stupid) thing." Nov 9, 2019 at 9:19
• @BertrandWittgenstein'sGhost To be frank, I was wondering the same thing :P Nov 9, 2019 at 12:16
• I wouldn't worry too much about it, you got my +1. Plus, considering OPs logic skills, I would have been worried if they had liked your answer. Hehe. Regards Nov 9, 2019 at 12:24

"If P, then Q" is defined as being false only if P is true and Q is false. This means that the truth value of the proposition as a whole is decided by the truth value of P and Q.

It must be stressed that logic is not concerned with "metaphysical possibility" in some weak sense. For example "the sun is green" takes, under one truth value assignment, the value true, and it takes the value false under another truth value assignment. For every argument with n propositional variables, you will have 2^n possible ways of assigning truth values to the propositional variables (the ones that have no logical connectives) and these will define the value of the complex propositions built up from them. I would recommend studying truth tables online or from a logic textbook to get a handle on this.

However there is no way of assigning truth values to P in (P and not-P) where the proposition as a whole will be true . Let ⊥ stand for any proposition that is of this sort. Then if we have the proposition "if ⊥, then Q", this proposition is always true, because its antecedent is never true, thus not allowing the possibility of the antecedent being true and the consequent false. . This does not, however, mean that Q is necessarily true. In fact, "if ⊥, then Q" is always true, regardless of whether Q is true OR false.

There is another idea that you are hinting at called the principle of explosion in classical logic. The idea is that from a set of inconsistent premises, one can deduce any proposition Q. Now while this is a feature of classical logic, it's not exactly useful to start with a set of inconsistent premises if you want to use the premises to support the conclusion. Because one is justified in accepting the conclusion of a valid argument only if one has reason to believe that all of the premises are true, which will never happen if your premises are inconsistent.

For Those Who Get It

/A->B/=T iff {/A/=T & /B/=T} OR (a big OR here) /A/=F.

That means, Given {/A/=F and A->B}, we can not say {/B/=T}.

{/A/=F}->{/A->B/=T}.

In Layman's Terminology

Simply put your understanding is seriously skewed. Saying "if A is false then B is true" in itself is a statement you need to proof. It's not an axiom.

If A is false then B is true is this: {/A/=F}->{/B/=T} and this is, most definitely, not that same as A->B. Or in other words, given A->B, and /A/=F it is IMPOSSIBLE to deduce /B/=F.

From the fact that it doesn't rain and the sun is shining, I can definitely not infer that the fact that it rains implies that the sun is shining.

So, no, from ¬A and B, we cannot infer that A → B.

I figured it out and here is what i figured out, also it was an obscure question, i don't think i managed to convey exactly where i was getting lost properly. I don't know if that is reason enough to delete it or if it will ever be helpful to anyone, so if someone suggests deleteion i am happy to do so. If anyone cares here is what i was confused about(articulated to the best of my ability):

My confusion was this: If the sun is green then the cows are chickens. This statement is true in logic. My contention was that if we have a false statement first(The sun is green) then the second part would be true(Cows are chickens). I Thought i could use this format to make a false statement first(i do not have a cold) and then make another statement(my nose is runny) and i thought this would be true. However as far as i understand it, i can only make such statements if the first premise(i do not have a cold) is ALWAYS false and "i do not have a cold" is not always false. However, "the sun is green" is always false. So because "the sun is green" is always false, then the following statement: "Cows are chickens" is true, because we cannot possibly disprove it(because the sun is green can never happen, we may never know if cows are chickens in the universe where the sun indeed was green and therefore cannot disprove it), and in logic, if we cannot disprove something we assume it to be true.