This may add to the detailed discussions and explanations above.
I had problems with mathematical implication until I read Keith Devlin's "Introduction to Mathematical Thinking". Hopefully what follows is a faithful summary of some of the ideas in the book.
- Implication is not the conditional
- Truth tables can help
1 We learn and experience that most things happen because of causality, or our interpretation of it. $A$ happens then $B$ happens. This gets expressed in different ways: $A$ happens so $B$ will happen, if $A$ happens then $B$ will happen, etc. These different expressions all get conflated and some people learn or are taught they all are represented by the expressions:
"if $A$ then $B$"
$A \implies B$
All of the above are wrapped up in the the idea of implication.
We all feel comfortable, or we internalise that when $A$ happens, $B$ will certainly happen because $A$ caused $B$. This is implication.
However, we are left with an uncertainty about what happens when $A$ doesn't happen.
To deal with this uncertainty in mathematics we need to abandon the idea of causality and fall back to a reduced idea of the conditional.
In Devlin's terms
"implication is the conditional with causality"
or the equivalent
"the conditional is implication without causality"
Well we can choose to define the conditional for every possible (4 ways) of combining $A$ and $B$ being True or False. I will break off point 1 here then come back.
For point 2 I understand these things in truth table terms, not linguistic terms.
- In parallel, if we think of simple logic (I am not sure of the exact terminology - propositional logic?) we can combine two statements $A$ and $B$ with a variety of operations: and, or. We learn (usually passively) even if we can't express the ideas formally, about the meaning of compound statements $A$ OR $B$, $A$ AND $B$ whose truth value depends on the truth value of $A$ and $B$ and we can define these in truth tables. In more concrete terms, most people are comfortable with TRUE OR FALSE = TRUE and TRUE AND FALSE = FALSE and so on for all possible combinations.
Again, so what?
Well, $A \implies B$ is also a compound statement whose truth value depends upon the truth value of $A$ and $B$. We can now define the truth table values for False $\implies$ False to be TRUE. That's weird and uncomfortable.
There are two ways to deal with this first uncomfortable weirdness.
Either, claim that
$A \implies B$
is equivalent to
$\neg A \lor B$
derive the truth table values and demonstrate no contradictions,
alternatively (and this is how I deal with the situation - Devlin does not write this), we do not think about the result (truth value) of a compound statement for various "input" values as being True or False, but just ask whether the logical argument is valid (in place of true) or invalid (in place of false).
For example if A: x = 3, is False and B: x < 7 is False
$A \implies B$ in this case is a valid argument. In this case:
if x $\neq$ 3 then x $\geq$ 7 is valid and hence true.
Rather than the truth table values being some arbitrary definition of the "meaning" of $A \implies B$, the conditional contains and is consistent with our hazier understanding of implication or if ... then, where A "causing" B is wrapped up in our understanding.
So back to 1. My understanding is this
$A \implies B$ is the conditional without causality, and all written or spoken expressions such as "implies" or "so" or "when" or "sufficent" or "necessary" I translate into the form $A \implies B$ ($A \to B$ ???) and work from there.
I would recommend Devlin's book (I have no commercial interests or connection) and the free Stanford/Coursera course by Devlin that covers the same ground.