Avoiding a potential problem with break statements

The common way to understand a for statement is that it creates a for all condition. At the end of the statement, we can assert that, for all items in a collection, some processing has been done.

This isn't the only meaning for a for statement. When we introduce the break statement inside the body of a for, we change the semantics to there exists. When the break statement leaves the for (or while) statement, we can assert only that there exists at least one item that caused the statement to end.

There's a side issue here. What if the loop ends without executing the break? We are forced to assert that there does not exist even one item that triggered the break. DeMorgan's Law tells us that a not exists condition can be restated as a for all condition: ¬∃_xB(x) ≡ ∀_x ¬B(x). In this formula, B(x) is the condition on the if statement that includes the break. If we never found B(x), then for all items ¬B(x) was true. This shows some of the symmetry...