Why/How does the definition of implication in mathematics work?

[jaja1990]

I understand that we just have to fill the last two raws in the truth table with any value, and that we randomly chose True, and that the value True makes matters easier sometimes (I don't know an example of that, but I read that somewhere).


But the question is, since mathematics is tied to the real world, how does the random truth value not cause any problems?

I read the following, in a reply in some thread on the forums:-
"Note that we're essentially selecting axioms here; we're choosing the rules that govern the logical deductions we'll be making in the future (i.e. we're setting up a logical system, not making inferences based on that system)."

It sounds to be the type of answer I'm looking for, but for the lack of my knowledge, I still can't understand/imagine the full picture. Moreover, the explanation still doesn't explain how the theorems we make still apply in the real world. So, can you help me with this please? Particularly, can you explain how choosing the rules of inference randomly still let's theorems correspond to the real world?

 

[HallsofIvy]

I don't know where you got the idea that we "randomly" choose anything. I think of it rather as "innocent until proven guilty".

I imagine that you are taking the "hypothesis", u, say, to be T, F, T, F, in that order, and the conclusion, v, say, to be T, T, F, F, so that we get all four possible combinations, TT, TF, FT, and FF. In the first row, both u and v are true so the statement "If u is true then v is true" is certainly true. In the second case, u is true and v s false so the statement is certainly false. The last two that you are asking about are the last two statements, where the hypothesis, u, is false. Suppose your teacher, at the beginning of the course, tells you "if you get an A on every test, you will receive an A for the course". You don't get an A on every test! Because that statement only says what happens when you get an A on every test, it tells you nothing about what grade you will get. It is perfecftly reasonable that if, for example, you got an A on every one except one and a B on that test, that your teacher would still give you an A for the course. But if you got, say, C, on every test, you certainly should not get an A for the course. In either case, you certainly cannot say that what your teacher told you at the beginning of the course was not true.

 

[jaja1990]

"In either case, you certainly cannot say that what your teacher told you at the beginning of the course was not true."

So, we don't have basis for assigning the value T for the last two cases.

"I don't know where you got the idea that we "randomly" choose anything. I think of it rather as "innocent until proven guilty"."
I should have been more precise. I read that it's a matter of "convenience", which means that it would still work if we have chosen different values, and hence I said "randomly".

If it's "innocent until proven guilty", that means there is considerable amount of thoerems not being correct for this reason alone, is that right?

 

[Deveno]

a lot of times a proof will begin like so:

if A is a schwartzenabber, then by the Godenkindicker Theorem, we know that A is propertizalbe. Therefore...

the thing is, it may well be the case that A is not a schwartzenabber. in fact, there might not BE any schwartzenabbers at all. in those cases, the theorem could truthfully go on and quote the entirety of tolstoy's war and peace as justification, because it's irrelevant.

why? because we're interested in schwartzenabbers. and by golly, schwartzenabbers better be propertizable, or we have a real problem (or at least Goderkindicker does).

 

[jaja1990]

a lot of times a proof will begin like so:

if A is a schwartzenabber, then by the Godenkindicker Theorem, we know that A is propertizalbe. Therefore...

the thing is, it may well be the case that A is not a schwartzenabber. in fact, there might not BE any schwartzenabbers at all. in those cases, the theorem could truthfully go on and quote the entirety of tolstoy's war and peace as justification, because it's irrelevant.

why? because we're interested in schwartzenabbers. and by golly, schwartzenabbers better be propertizable, or we have a real problem (or at least Goderkindicker does).

I think I understood the point. And if I did understand correctly, this is another issue that is related to theorems.

I'm asking if the randomness of the (last two) values given to "implication" lead to theorems being possibly/potentially incorrect. Aside from the fact that the prepositions themselves could be wrong.

 

[SteveL27]

I think I understood the point. And if I did understand correctly, this is another issue that is related to theorems.

I'm asking if the randomness of the (last two) values given to "implication" lead to theorems being possibly/potentially incorrect. Aside from the fact that the prepositions themselves could be wrong.

I'm a little confused by your question about randomness. Is your concern that "False => anything" is true? In other words, if 1+1=3 then I'm the King of Siam. That's a logically true statement. Is that your concern?

I am not following the part about randomly assigning truth values.

 

[jaja1990]

I am not following the part about randomly assigning truth values.

As HallsOfIvy puts it "innocent until proven guilty". "innocent until proven guilty" means, in reality, we don't know if the truth-value of FT and FF are T and T respictively. What we do is "consider" them True. Why did we chose them to be True? (We choose them to be True randomly)

Clarification: by randomly I mean, there is a 50/50 chance of our choices being True.

The question is, then,
"I'm asking if the randomness of the (last two) values given to "implication" lead to theorems being possibly/potentially incorrect. Aside from the fact that the prepositions themselves could be wrong. "

 

[Stephen Tashi]

"I'm asking if the randomness of the (last two) values given to "implication" lead to theorems being possibly/potentially incorrect. Aside from the fact that the prepositions themselves could be wrong. "

How much more wrong could a theorem be than to have the propositions it stated be wrong?

For example, suppose you decide that "A implies B" is false when A is false and B is true. Someone states the theorem "If a > 0 and b > 0 then ab > 0". The theorem would disproven by the example a = -1 and b = -6.

The conventions on the truth table are the only ones that make sense when you consider the fact that most mathematical theorems involve (explicitly or implicitly) quantifiers - such as "for each" or "there exists". So rather than a specific implication like "if ( 3 > 0 and 2 > 0 ) then (3)(2) > 0", a theorem is usually a quantified statement like "For each real number a, and for each real number b, if a > 0 and b > 0 then ab > 0. To be true, the theorem must be true for all real numbers a and b, even when a = -2 and b = 4. You can think of the fact that the implication is true for a = -2 and b = 4 as a demonstration that the if-part of the statement correctly filters out -2 and 4 from serious consideration.

 

[jaja1990]

How much more wrong could a theorem be than to have the propositions it stated be wrong?


For example, suppose you decide that "A implies B" is false when A is false and B is true. Someone states the theorem "If a > 0 and b > 0 then ab > 0". The theorem would disproven by the example a = -1 and b = -6.

A theorem can be proven according to axioms, and axioms can be wrong. Regardless of that, I just don't want to take this in consideration (or discuss it), that's why I said "Aside from the fact that ...".

"I'm asking if the randomness of the (last two) values given to "implication" lead to theorems being possibly/potentially incorrect. Aside from the fact that the propositions themselves could be wrong. "

Edit: a mistake removed.

 

[Stephen Tashi]

"I'm asking if the randomness of the (last two) values given to "implication" lead to theorems being possibly/potentially incorrect. Aside from the fact that the propositions themselves could be wrong. "

You've already been told there is no "randomness" about the truth table for implication.

A theorem is a example of a proposition. What property of a theorem do you call being "incorrect" that is not covered by it being a false proposition?

 

[jaja1990]

You've already been told there is no "randomness" about the truth table for implication.

A theorem is a example of a proposition. What property of a theorem do you call being "incorrect" that is not covered by it being a false proposition?

If it's not random, then how are the last two truth values chosen?

If I tell someone "you are innocent until proven guilty", that doesn't mean he really isn't guilty. He could be guilty, but "considered" not guilty. And he still could be innocent. I can't conclude what he really is.

 

[Stephen Tashi]

If it's not random, then how are the last two truth values chosen?

They are chosen so that an example which does not satisfy the if-part of the implication cannot be used to prove the implication false.

If I tell someone "you are innocent until proven guilty", that doesn't mean he really isn't guilty. He could be guilty, but "considered" not guilty. And he still could be innocent. I can't conclude what he really is.

I have no idea what that is supposed to illustrate. Are you saying the statement is "incorrect" because it does not allow you to make a conclusion about the persons guilt? How would changing the truth table for implication alter that property of the statement?

 

[chiro]

The truth table for an implication follows a specific truth table that HallsOfIvy has pointed out: it is not random at all.

Also HallsOfIvy like other posters have pointed out have provided why the implication is the way it is with the 'innocent until proven guilty' example. It would helpful for you to re-read the example again if you need to.

 

[Deveno]

As HallsOfIvy puts it "innocent until proven guilty". "innocent until proven guilty" means, in reality, we don't know if the truth-value of FT and FF are T and T respictively. What we do is "consider" them True. Why did we chose them to be True? (We choose them to be True randomly)

Clarification: by randomly I mean, there is a 50/50 chance of our choices being True.

The question is, then,
"I'm asking if the randomness of the (last two) values given to "implication" lead to theorems being possibly/potentially incorrect. Aside from the fact that the prepositions themselves could be wrong. "

perhaps the situation would be a bit clearer, if we had another truth-value: U for "unknown".

we could make a truth table for this like so:

p q p→q

T T T
T F F
T U U
F T T
F F T
F U T
U T T
U F U
U U U

so that's one approach.

but i think the best way to look at it is this: p→q being true/false, is an expression of the validity of the chain of reasoning. often, we know p, and we want to know whether or not q is true. so we go:

p→q
p
therefore, q

this form of reasoning is called "modus ponens" (or: by way of affirming).

note that if p is false, even though p→q is technically true, we still know nothing about q. that's the way we want it, if arguing p→q reveals nothing about q, we need another p to "start from".

to put it another way, often the statement p is of the form:

p is true under certain conditions. for example, a typical p might be:

k is an integer.

expanded, this is something like: k exists, and sets exist, and one of the sets that exists is a set called "the integers" and k is one of the elements of that set. and each of THOSE statements could be broken down into even more assertions we are claiming all are "true".

one of these statements (buried way,way down there) is something like:

"it makes sense to talk about sets because the definition of sets is contradiction-free". and it could well be, that indeed THAT statement is false. the truth is, no one knows. would that invalidate all of mathematics? no. we would just need "a better p".

what I'm trying to get at, is that p→q true when p is false, gives us 0 information. if p is false, p→q and p→(~q) both make "equal amounts of sense". that's not what we use implication FOR. we use it to restrict the possibilities, and it only works when starting from "a solid starting point" (or what are often called "first principles", or sometimes "axioms"). the situations where p is false, just aren't interesting, because they don't help in "narrowing down the possibilities".

you have to start somewhere. logic, language, thought itself, doesn't "come out of nothing". in talking about, and exploring the world we live in, we share a collection of undefined shared experiences, called (for won't of a better name) "common knowledge". truths we do hold to be "self-evident", things like: the sun shines in the sky, food alleviates hunger, rain is wet like water. if we develop these shared ideas to a certain level of complexity, we can talk about these ideas abstractly, to the point where we can wonder if the ways we think about the experiences we have are actually faithful to the experiences we have. as far as i know, that is undecided, but we act "as if" it were true, and so far that hasn't caused any major problems.

 

[jaja1990]

what I'm trying to get at, is that p→q true when p is false, gives us 0 information. if p is false, p→q and p→(~q) both make "equal amounts of sense". that's not what we use implication FOR. we use it to restrict the possibilities, and it only works when starting from "a solid starting point" (or what are often called "first principles", or sometimes "axioms"). the situations where p is false, just aren't interesting, because they don't help in "narrowing down the possibilities".

That's what I was trying to understand.

Saying "innocent until guilty" doesn't help. It still means that there is a 50% chance it is "not innocent"; as you put it, the implication alone when 'p' is false gives us 0 information, it has to be accompanied with more information. I still need to read about axioms, and how implication works well with axioms, which was the second half of my question in the first place.

 

[friend]

That's what I was trying to understand.

Saying "innocent until guilty" doesn't help. It still means that there is a 50% chance it is "not innocent"; as you put it, the implication alone when 'p' is false gives us 0 information, it has to be accompanied with more information. I still need to read about axioms, and how implication works well with axioms, which was the second half of my question in the first place.

The answer is easy. The relationship of material implication, p→q, can be expressed in terms of other connectives like the AND symbol, ^. All that implication means is that it cannot be true that you have p=true and q=false. Or in symbols p→q=~(p^~q). where ~ is the symbol for negation. And since you don't have a problem with NOT and AND, then you should not have a problem with implication.