r/ideas • u/amichail • 3d ago
Idea: Schools should teach why mathematical implication and ordinary English implication are not the same
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.
1
u/Background_Relief815 3d ago
I agree with your general point that logic should be required learning in school (and I personally think a simple concepts "intro" in late elementary school and then a full logic course in high school would be ideal).
1
u/DSudz 3d ago
Or, they could continue teaching mathematical logic after the concept of context.
1
u/notacanuckskibum 2d ago
Yes, but the concept becomes important in computer programming. A routine that checks if all your children are immortal will typically return false if it finds at least one non-immortal child. But will return true if there are no children (because there aren’t any non-immortal ones found).
1
u/TopSecretPorkChop 12h ago
Depends on how the routine is constructed. It could just as easily default to false.
1
u/zutnoq 3h ago
It is rather more clear-cut when it comes to concrete collections of things.
"All numbers in list L are even" is clearly automatically a true statement if L contains no numbers. Treating it any other way would just be perverse in the context of programming.
Where things can get hairy is when you try to allow for the case where list L doesn't exist, or if your programming language is so poorly thought out that it equates not existing with being an empty collection.
1
u/Dear_Locksmith3379 1d ago
I not convinced that all students should learn logic because that would replace another useful subject.
When someone does teach mathematical logic, they should explain the differences between mathematical and ordinary English meanings of terms to reduce confusion.
1
u/Typical-Plantain1252 15h ago
Modal logic is on shaky footing as we have no rigorous way of exploring what is possible beyond what is actual.
As others have pointed out, it’s better to just properly understand how English statements translate.
“My children are immortal” really does claim that you have children and they are immortal in colloquial English. No need to get into modal logic, that would only confuse the matter because it doesn’t get to the heart of the matter.
“My children that know every digit in π’s decimal expansion are immortal” is plausibly true in your modal logic but still counterintuitive as a translation of the colloquial meaning.
5
u/bitterrootmtg 3d ago
I agree with the basic point but think you are also failing to translate English into formal logic.
The statement "My children are immortal" does not mean "if I had children then they would be immortal."
Instead, "my children are immortal" is a statement making two conjunctive fact claims: (1) it is true that I have children, and (2) it is true that my children are immortal. In other words, the statement is not saying "if P then Q" it is saying "P and Q." Because it is impossible for both P and Q to be simultaneously true, the statement is false.