I can't understand logical implication

I just started studying logic (high school) anyway...for the truth table of logical implication

If sentence $A$ is true and $B$ is true then $A\implies B$ is true.

does that mean if $A$ and $B$ are both true then there is a way to prove $B$ is true from $A$, always?

the same for if $A$ is false can you get anything either True or false proved from this $A$?

Solution 1:

As a logical proposition, the material conditional $A \implies B$ is a very weak one: as you've noticed, it's very easy to satisfy it just by accident. In fact, this happens whenever $A$ is false, or whenever $B$ is true. Thus, merely observing that $A \implies B$, for some specific $A$ and $B$, says very little.

Instead, the usefulness of implication lies in the fact that, precisely because of its weakness, it is often possible to assert $A \implies B$ as a universal statement (either an axiom or a provable theorem) that holds for any valuation of any free variables mentioned in the propositions $A$ and $B$.

For example, consider the statement: $$x > 2 \;\land\; x \text{ is prime} \implies x \text{ is odd}.$$ Merely observing that this statement holds for some $x$ says very little — there are plenty of numbers for which it is trivially true, either because they are odd, or because they are not primes greater than 2. What makes this statement useful is that we can prove that it holds for all $x$ — there isn't a single number which would be greater than 2 and prime, but not odd.

Solution 2:

One way to understand implication is to remember that $A\Rightarrow B$ is equivalent to $\neg A \lor B$. If you understand negation ($\neg$) and disjunction ($\lor$), then you understand implication.

Solution 3:

Look at $A$ and $B$ as something that is either false or true. For example let $A$ be the event that tomorrow is Tuesday and let $B$ be the event that the day after tomorrow is Wednesday.

Look at $$ A\implies B $$

as a promise - if $A$ is true then so is $B$.

In our example, if $A$ is true then indeed so is $B$ and so the implication $A\implies B$ is true.

However, now consider $C$ as the statement that tomorrow is Friday, and I state $$ C\implies B $$

that is - I promise you that if $C$ will happen so will $B$.

Tomorrow is not Friday (at the time of writing), and so $C$ is false, regardless of if $B$ is false or true - my promise was kept.

Now regarding the terminology of proofing $B$ is both $A,B$ are true. Note that statements like $$ \text{My cat walks on four}\implies1+1=2 $$

is true, since both are true, but what would it mean to prove $B$ from $A$ ?

Solution 4:

Maybe it's more clear if we separate the logical operator meaning of implication from its logical statement meaning.

When we use it as a logical operator, we conceive it simply as an entity, that given two logical values (thus true or false), produces a third logical value, using a common defined rule (its truth table).
So it makes perfect sense to say $A \implies B = true$ if $A = B = true$, and we don't concern about what actually proposition $A$ and $B$ means, we care exclusively about their logical values.

It's different when we use implication as a logical statement.
In this case we really say something about the meaning of the propositions involved in our statement. So, while proposition $A = My\;cat\;is\;black$ is true, and proposition $B = I\;am\;hungry$ is true as well, $A \implies B$ is not a valid statement.
Such meaning is linked to set theory and formal logic. Using implication in this context means that you can infer $B$ from $A$, in a way called modus ponens.

Solution 5:

I would say that $A$ being true and $B$ being true does not mean you can always prove (deduce) $B$ from $A$.

Here's an example. A: Alice lives in Atlanta. B: Bob lives in Boston. Even if these are both true, there is no (apparent) relationship. So you can't logically deduce $B$ from $A$ even though $A\Rightarrow B$ is true in this case.

I guess this means that if you can logically deduce statement $Q$ from statement $P$, then $P\Rightarrow Q$ is true; but knowing $P\Rightarrow Q$ is true does not guarantee the existence of a deduction of $Q$ from the assumption of $P$.

Logical implication is a defined logical connective, so as long as $P$ and $Q$ have truth values (true or false), so does $P\Rightarrow Q$.