If $2=2$, then "some apples are green"

Question

I don't understand why the implication below is defined to be true...

If $2=2$, then "some apples are green".

There is no bound between the two sentences, so how can we say that the last one is implied by the first? There is not a deductive process...

Answer 1

Material implication is not causation. There needs to be no "bound" between the two statements at all, as long as the implication (i.e. the truth value) is true.
And here, it true: "2 = 2" is true and "Some apples are green" is true, so $1 \to 1$ is, by definition of the operator, true.
Implication is nothing more than that. It is precisely defined by the corresponding truth table, and does not make any claims abuot causation or any other kind of semantic relation between the two statements other than their truth values.

Answer 2

When you want to prove/study $P\Rightarrow Q$, you need to check if when $P$ is true, then $Q$ is also true.

So in your case, $2=2$ is always true, so you need to check if the rest is always true too.

And "some apples are green" is true too, so you do have

$$2=2\Rightarrow \text{ some apples are green}.$$

You just have in your case

$$\text{True }\Rightarrow \text{ True}$$

which is always true.

Answer 3

When one finds an implication $p \to q$, it is wise to replace it with the disjunction $\neg p \lor q$. Hence,

$$\begin{array}{rl} (2 = 2) \to \text{some apples are green} &\equiv \overbrace{(2 \neq 2)}^{\equiv \text{False}} \lor \text{some apples are green}\\ &\equiv \text{False} \lor \text{ some apples are green}\\ &\equiv \text{some apples are green}\end{array}$$

Thus, if some apples are indeed green, then the implication is true.