It might be helpful to think of this by considering each of the disjuncts of a disjunctive statement as containing unspoken implications. Consider, for example, the following disjunction:
Either no storm will come, or the boat will sink.
For each of the disjuncts, we can assert the following implications:
- If no storm comes, no storm comes.
- If a storm comes, the boat will sink.
It's intuitively true that either a storm will come or not. Therefore, we can put all that together to derive the following, which is a logical consequence to the first statement:
If no storm comes, no storm comes; or if a storm does come, the boat will sink.
The first part is redundant, so this can be simplified to:
Either no storm comes; or if a storm comes, the boat will sink.
From this it can be seen that disjunctive statements can be thought of as having conditional statements hidden within their elements.
Vacuity
It might be argued that it is vacuous to insert an antecedent into the disjunction in that way. For example, it could be argued that any antecedent could be inserted, such as:
Either no storm comes; or if the moon is made of cheese, the boat will sink.
Although it is true that that logically follows from the premise, there's an important difference to be noted. The antecedent "if a storm comes" complements the other disjunct, whereas "if the moon is made of cheese" does not. This involves the important difference that the simpler conditional follows logically from the disjunctive statement:
- ~A ∨ (A → B) implies A → B
- But, ~A ∨ (C → B) does not imply C → B
In other words, it cannot be concluded that if the moon is made of cheese, the boat will sink; but, it can be concluded that if a storm does come, the boat will sink.