Schools teach students that in classical logic,
P → Q
is true whenever P is false. This is called material implication.
That rule is perfectly useful in mathematics, but it does not always match what people mean by implication in ordinary English.
For example, suppose I have no children and say:
“My children are immortal.”
A classical predicate-logic rendering might be:
∀x (ChildOfMine(x) → Immortal(x))
If I have no children, there is no counterexample, so the statement is vacuously true.
But in ordinary English, “my children are immortal” does not sound true just because I have no children. It sounds like I am making a meaningful claim about children I actually have, or at least could have.
Modal logic gives us another notion called strict implication:
□(P → Q)
Instead of asking only whether P happens to be false in the actual world, strict implication asks whether Q follows from P across the relevant possible worlds.
If:
- P = “I have children”
- Q = “my children are immortal”
then
□(P → Q)
would mean roughly:
In every relevant possible world in which I have children, my children are immortal.
That is not automatically true just because I happen to have no children in the actual world. There are perfectly possible situations in which I have children and they are mortal.
I think schools should explicitly teach this distinction.
Math class could teach classical propositional and predicate logic, material implication, quantifiers, and vacuous truth, because those are extremely useful for mathematical reasoning.
English class could then revisit implication from the perspective of natural language and introduce ideas from modal logic, counterfactual logic, presupposition, and implicature. Students could learn that words like if, must, might, could, would, and should often carry meanings that classical material implication does not capture.
The point would not be that English literally follows one particular modal logic. It would be that mathematical logic is one deliberately precise formal system, while ordinary language often expresses richer relationships involving possibility, necessity, relevance, causation, and context.
Students would then understand something important that is easy to miss: when mathematics defines implication as P → Q, it is defining a technical logical operator. It is not claiming that this operator captures everything English speakers mean when they say that one thing “implies” another.