It’s well known that the material implication → of propositional logic does not do a perfect job of capturing what we mean when we make “if… then…” statements in English. The usual examples of failure rest on the fact that any material conditional with a false antecedent is vacuously true (so “if 2 is odd then 2 is even” turns out to be true). But over time, philosophers have come up with a whole lot of different ways in which → can catch us by surprise.
Here’s a list of some such cases. In each case, I will present an argument using if…then… statements that is clearly invalid, but which is actually valid in propositional logic if the if…then… statements are translated as the material conditional!
1. Harper
If I put sugar into my coffee, it will taste fine.
Therefore, if I put sugar and motor oil into my coffee, it will taste fine.
S → F
(S ∧ M) → F
2. Distributivity
If I pull both switch A and switch B, the engine will start.
Therefore, either the engine will start if I pull switch A or the engine will start if I pull switch B.
(A ∧ B) → S
(A → S) ∨ (B → S)
3. Transitivity
If Biden dies before the election, Trump will win.
If Trump wins the election, Biden will retire to his home.
Therefore, if Biden dies before the election, Biden will retire to his home.
B → T
T → R
B → R
4. Principle of Explosion
Either zombies will rise from the ground if I bury a chicken head in my backyard, or zombies will rise from the ground if I don’t bury a chicken head in my backyard.
(B → D) ∨ (¬B → D) is a tautology
5. Contraposition
If I buy a car, I won’t buy a Pontiac.
Therefore, if I buy a Pontiac, I won’t buy a car.
C → ¬P
P → ¬C
6. Simplification
If John is in London then he’s in England, and if he’s in Paris then he’s in France.
Therefore, either (1) if John’s in London he’s in France or (2) if John’s in Paris then he’s in England.
(L → E) ∧ (P → F)
(L → F) ∨ (P → E)
7. False Antecedent
It’s not the case that if God exists then human life is a product of random chance.
Therefore, God exists.
¬(G → C)
G
8. True Consequent
If I will have eternal life if I believe in God, then God must exist.
I do not believe in God.
Therefore, God exists.
(B → E) → G
~B
G
You can check for yourself that each of these is logically valid! Can you figure out what’s going wrong in each case?
1 and 5, Harper and Contraposition, don’t rely on vacuous truth, you could produce those paradoxes without teaching someone about vacuousness, and without using axioms motivated by considering vacuousness. 3,4, and 6-8, Transitivity, Principle of Explosion, Simplification, False Antecedent, and True Consequent, rely on vacuous truth for their result. I’m not sure where Distributivity comes from, my guess is that comes from vacuousness. I remember some paradox that I forgot the details of where if several things work together for a result, one of them is sufficient to bring about that result. What stayed with me was that a logician argued to defend that result of a logic, even though, by my lights anyways, it was a clear failure of the theory.
I was wrong about 3, Transitivity, it doesn’t depend on vacuous truth.
I want to try and describe how vacuous truth is implicated in the surprising implications of material implication. A material conditional is vacuously true when the antecedent is false. In 8, True Consequent, we are surprised that we committed to saying that an antecedent conditional would imply some consequent EVEN if the conditional is only true because it is vacuously true. In 4, Principle of Explosion, we are surprised that either (B->D) or (-B->D) for any (B,D). The reason is that one of those conditionals is vacuously true because one of their respective antecedents is false. In 7, False Antecedent, we are surprised that the negation of a material conditional is dependent on the antecedent of the conditional being true. The reason is that the conditional would be vacuously true if the antecedent was false. It’s difficult to detail a deep connection between vacuous truth and 6, simplification, but I’m pretty sure there is one.