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As is well known, we have the least intuitive of basic operations, the 'implication' or '=>'. Consider 'A => B'.

Most beginners get stumped on the vacuous truth, that implication could be true even if A is false.

When I first came across this very recently, I stumped on it too. But yesterday I realized I don't even understand the non-vacuous parts as well. Consider this:

  • A: 'Earth has one moon'
  • B: 'Two plus two equals four'

Both are true (or we know both to be true). In this case, according to logical rules, 'A => B' would be true. In other words, "earth having one moon implies that two plus two equals four" or "two plus two equals four when earth has one moon". That is utterly confusing. How could two unrelated true statements have a connection through implication just because both are true?

I asked this on a forum and the consensus was that I'm not supposed to infer any meaning out of A and B, and that I should just use '=>' as a truth-functional. I assume that means I should just think of '=>' as an operation that has a certain definition without thinking why that definition should make sense or if '=>' of formal logic has any connection with the real-life word "implies".

I googled some more and came across 'some philosophical problems' regarding this but I haven't read them yet:

  • Article link and reddit discussion here
  • Another discussion here

However, today I came across a different way of looking at it and wanted to know if this is the correct way of thinking.

The problem might be that we (sometimes or always?) teach 'A => B' implication as a conclusion of its components A and B. This appears to be the case when we have 'A => B' column at the far right of the 3-column truth table, as if it's the result of A and B truth-values. We shouldn't be doing that? A and B's truth-values don't dictate whether A implies B or not (except in the single case when A is true and B is false then A => B must be false)?

However if we think of 'A => B' as a hypothesis and then look at the values of A and B, it makes more sense. In that case the four cases turn out to be:

  • If 'A => B' is true, and A is true, then B is true.
  • If 'A => B' is true, and A is true, but B is false, that's a logical impossiblity (so 'A => B' must be false)
  • If 'A => B' is true, and A is false, then B could be true or could be false

In other words, I'm assuming 'implication' and then constructing the truth table instead of defining implication in terms of a truth table.

So the question is: Can implication really be a conclusion of its components the way it is depicted as standard in truth tables? Because in that case it really doesn't make sense to me. Because that means that every true statement (in a gigantic set of all true statements) implies every other true statement, whether those statements are related or not.

(P.S.: I guess one problem with my approach is that how can we use (assume) 'implication' without defining it first. And the only way to define it is to state its truth-table without paying attention to why the truth-table should make sense. I'm completely confused).

EDIT 1: This is how implication should be defined (or the way it makes sense to me):

 ---------------------------------
| A | B | A => B                  |
|---|---|-------------------------|
| T | T | could be T, could be F! |
| T | F | F                       |
| F | T | could be T, could be F! |
| F | F | could be T, could be F! |
 ---------------------------------

plus this:

 --------------------------------------
| A => B | A | B                       |
|--------|---|-------------------------|
| T      | T | T                       |
| T      | F | could be T, could be F! |
 --------------------------------------

So another question would be, what is wrong with this way of thinking?

EDIT 2: For the sake of completeness I should add another table that takes A as a conclusion and 'A => B' and 'B' as given:

 --------------------------------------
| A => B | B | A                       |
|--------|---|-------------------------|
| T      | T | could be T, could be F! |
| T      | F | F                       |
 --------------------------------------
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    $\begingroup$ "every true statement implies every other true statement, whether those statements are related or not." - this happens to be the case. In fact, all statements, true or not, imply true statements. Just to make it worse for you. :) $\endgroup$ – GFauxPas Nov 16 '14 at 18:38
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    $\begingroup$ What is your question? Please put it in bold if you've already included it here. $\endgroup$ – Doug Spoonwood Nov 16 '14 at 18:42
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    $\begingroup$ "How could two unrelated true statements have a connection through implication just because both are true?" Are you sure that even the two statements you've put forth "two plus two equals forth" and "earth has one moon" are not related? I mean, isn't it possible that there exist relationships here that you don't know about? $\endgroup$ – Doug Spoonwood Nov 16 '14 at 18:45
  • $\begingroup$ Please refer to math.stackexchange.com/questions/439987/… $\endgroup$ – JMoravitz Nov 16 '14 at 18:45
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    $\begingroup$ You may be interested in Wikipedia's article on "Paradoxes of material implication", one of which is exactly the point that is puzzling you. $\endgroup$ – MJD Nov 17 '14 at 8:14
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How could two unrelated true statements have a connection through implication just because both are true?

And that's your mistake: that they have to have a connection. Syntax (the relevant branch of mathematical logic) isn't about connections; it's about deductive proof. How to infer things. In the context of Peano arithmetic, 2+2=4 is a theorem. You don't need any hypotheses to deduce 2+2=4. Having the additional hypothesis that the Earth has one moon is superfluous: you can still deduce 2+2=4 in its presence.

However, English is about communicating information to other people. For the sake of efficient communication, you have been trained to read between the lines; to add unspoken connections between things said together.

e.g. if 2+2=4 were true, then why would anyone bother saying "If the Earth has one moon, then 2+2=4"? So, given the fact that someone says it, you make an inference that you could not have deduced 2+2=4 on its own, and that you need an additional premise (such as the fact the Earth has one moon) to arrive at that conclusion.

This is a fine habit for every-day communication, aside from the fact that it leaves you vulnerable to propaganda and marketing. But you should drop that habit when reading formal mathematics -- one of the main points of formal logic is that formal language 'means' only what is said, nothing more, nothing less.

All that the truth* of "if A then B" means is that if A also happens to be true*, then you can infer B.

Note that you can deduce the entire truth table from this fact: e.g.

  • If $T \implies F$ were true, then if both $A$ and $A \implies B$ were true, it would still be possible that $B$ is false, and so you can't infer it. Thus, $(T \implies F) = F$
  • If $F \implies F$ were false, then the truth of $A \implies B$ would automatically imply $B$, regardless of whether or not $A$ is true. Since we're not supposed to be able to infer $B$ from $A \implies B$ alone, that means we have to have $(F \implies F) = T$.
  • If $F \implies T$ were false, then the truth of $A \implies B$ would mean that if $B$ were true, then we could conclude $A$ was true. This is backwards; we're only supposed to be able to make implications in the forward direction. Thus, we must have $(F \implies T) = T$ as well.

*: I really shouldn't use the word "true" here either, since that is about semantics, not syntax. But I find it extremely awkward to talk about these things in English without using the word "true", and the distinction is probably too subtle until you are actually start studying these things properly.


Aside: there is a variant of logic called "linear logic" where making inferences consumes the hypotheses; in this logic you cannot infer 2+2=4 from the hypothesis that the Earth has one moon: instead, from that hypothesis you have to infer "The Earth has one moon and 2+2=4". However, linear logic is very weird, and you'll probably find it even less satisfactory. (and I don't really know all that much about it anyways)

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A good way to resolve this would be to think of such a statement in the context of the axioms it comes from. In particular, I think what is confusing about the statement

The earth has one moon implies 2+2=4

is that we intuitively want to read it as

The earth has one moon proves 2+2=4

which isn't really true in any rigorous sense- there's no way you can derive that $2+2=4$ just by knowing that the earth has one moon. These statements, as you have noticed, are quite different from one another and it's confusing if you try to put them together.

A better way to think about it would be something like (informally)

The usual rules of arithmetic prove that "The earth has one moon implies 2+2=4"

which we could write more succinctly, by taking "The earth has one moon" as an axiom as:

The usual rules of arithmetic and the fact that the earth has one moon proves that 2+2=4.

which makes more sense - we can easily see that the earth having one moon is irrelevant here, since the statement

The usual rules of arithmetic prove that 2+2=4

is also true - but having irrelevant axioms seems more okay when you put them next to the relevant ones - and even in the above statement, we don't need to know everything about arithmetic to prove 2+2=4 - we just need to know a little about addition. However, a statement like

The earth has one moon implies the earth has less than two moons

has the consequent being unprovable if we only use arithmetic, thus we actually do need "The earth has one moon" to prove it.

From this, it is clear that a statement like "$A$ implies $B$" ought to be read as "The truth of $B$ follows from $A$, and whatever axioms we're working with". The truth table captures this idea effectively. Suppose we have our usual axioms $X$, and some system $X+A$ where we take those axioms and add $A$. The implication $A\rightarrow B$ states that, in $X+A$, it must be that $B$ holds (since if $A$ is true, the only way to make $A\rightarrow B$ true is to make $B$ true). However, it doesn't care whether $B$ was already true in $X$ - making $A$ irrelevant - or whether, as our intuition suggests, $B$ was independent.

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The point is that material implication just states that $v(\varphi \rightarrow \psi)=1$ when $v(\varphi) \leq v(\phi)$.

It is mathematically defined so because it is useful: it is not supposed to have any connection with our intuitive notion of conditionality.

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The implication can be read as an ordering relationship among the truth values of statements. P -> Q can be understood to mean "The truth value of P is less than or equal to the truth value of Q", or "Q is not less true than P". This is most useful when we want to mark arguments as invalid because they proceed from true assumptions to reach false conclusions. The "could be true, could be false entries" in the original poster's table are conventionally assumed true because they are not logically invalid.

How we know that this relationship holds is irrelevant.

1) It may hold trivially, i.e. when we already know that P is false or Q is true. Any statement Q at all is at least a true as a known falsehood P. A known truth Q is no less true than any other statement P. I call this a trivial sense, because we cannot then use P -> Q to prove anything we don't already know. In this sense, it is correct to assume that P->Q is true for any pair of true statements, but this has nothing to do with any deductive relationship.

2) It may be a hypothesis, or an assumption. We may sometimes assume that this relationship holds, solely for purposes of argument and examination of the consequences and when we would not otherwise accept it. In this case, we typically assume that P -> Q is true, and then with information about P or Q, see what else we can establish. (if P is false, P-> Q tells us nothing about Q. If Q is true, P -> Q tells us nothing about P.)

3) It may be the derived conclusion of an argument: Q follows from P and whatever other assumptions are involved. The major difficulties arise when people try to interpret the trivial sense to mean something on this order. Two two statements may very well both just happen to be true (and therefore have a logical, truth functional relationship) but in every other sense be entirely independent and even unrelated.

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