The Lover’s Fallacy

by Carl Pierer

Often, people don't do a particular thing. Even if they are supposed to. Tamino didn't talk to Pamina, Kant didn't leave Königsberg and Peter Singer doesn't donate all his money (to the point of marginal utility) to charity. We like to take this not acting as evidence for something more. Tamino's silence for unrequited love. Someone never leaving their hometown as a conservative old bore. Singer's “selfishness” as falsifying his philosophy. While intuitively plausible, this reasoning is flawed. It is the lover's fallacy.

In Act 2, Scene IV, of the magic flute, Pamina hears Tamino playing on his flute and hurries to talk to him. But he, undergoing the second ordeal, is bound to remain silent:

“You're here, Tamino? I heard your flute and ran towards the sound. – But you are sad? Will you not say a word to your Pamina?”
Tamino motions her to go away.
“What? I am to keep away from you? Do you love me no more? Oh, this is worse than an offence – worse than death.”

Pamina, deeply disappointed, reasons: “There are two possibilities, either Tamino doesn't talk to me or he loves me. So, if he loves me, then he talks to me. He doesn't talk, so he doesn't love.”

The fallacy is based off a close link between or and if-then sentences, or disjunctions and conditionals. A truth table will readily illustrate this:

~P is the negation of P, so whenever P is true ~P is false and vice versa. We see that for a conditional to be false we need the antecedent (~P) to be true, while the consequent (Q) is false. Here, this is the case when P is false and Q is false. It is important to notice that if the antecedent is false, the conditional is necessarily true. Now, a disjunction is true as long as at least one of the disjuncts (P or Q) is true. Since the truth table entries for P ∨ Q and ~P → Q are the same, we say they are (logically) equivalent.

Suppose we symbolise the sentences in Tamino's case the following way:

P: Tamino talks to Pamina

Q: Tamino loves Pamina

Formalising Pamina's steps of reasoning then gives:

~P

∴ ~P ∨ Q

Tamino didn't talk to Pamino, so either he doesn't talk to her or he loves her. While this is a less intuitive principle, we can add anything to a given fact in a disjunctions. The disjunction is true as long as at least one of the disjuncts is true. We know one disjunct to be true, so the truth value of the other does not influence whether or not the disjunction as a whole is true.

~P ∨ Q

∴ Q → P

Either Tamino doesn't talk to Pamina or he loves her, therefore if he loves her, then he talks to her. This seems at least intuitively plausible. The next step is a perfectly fine modus tollens:

Q → P

~P

∴ ~Q

If Tamino loves Pamina, then he talks to her. He didn't talk to her, so he doesn't love her.

This should be a bit worrisome. Not only for Pamina. Q was chosen to symbolise Tamino loves Pamina, but Q and P could symbolise anything we'd like. For instance, we could have:

P: Kant leaves Königsberg at least once

Q: 1 + 1 = 2

The argument then becomes: Kant never left Königsberg. So either Kant never left Königsberg or 1 + 1 = 2. Thus, if 1 + 1 = 2, then Kant left Königsberg at least once. Kant never left Königsberg. Therefore, 1 + 1 ≠ 2.

The dodgy move is from the disjunction to the conditional. The disjunction, at least in classical logic, is inclusive. This means, it is true even if both disjuncts are true. The sentence “either Kant never left Königsberg or 1 + 1 = 2”
(~P ∨ Q) is still true when both ~P and Q are the case. However, the conditional “if 1 + 1 = 2, then Kant left Königsberg at least once” (Q → P) is false if both ~P and Q. A truth table comes in handy:

It is evident from the truth table that ~P ∨ Q is not equivalent to Q → P, because their truth tables are different. The correct equivalent conditional to ~P ∨ Q is: ~Q → ~P. Quite a difference. Our (causally influenced) intuition tells us that there is some relation between these two conditionals, but there isn't. Compare the two statements: i) If I push the billiard ball, then it will move and ii) If I don't push the billiard ball, then it won't move. ii) means that unless I push the billiard ball, it won't move. There can be no other cause. i) just says that I may be the cause of the ball's movement (if I push it), but the ball could as well move without my help.

The lover's fallacy is related to affirming the consequent:

P

Q → P

∴ Q

Pamina could also commit this fallacy, then she would reason: Tamino talks to me. If he loves me, then he talks to me. Therefore, he loves me. This would be very handy indeed! But it should be evident that this is false. Quite a lot of people will talk to Pamina, yet not all of them love her.

In contrast to affirming the consequent, the given fact in the lover fallacy is used to establish the conditional. Instead of deriving faultily from a given conditional, the lover's fallacy occurs at a dodgy transition from a disjunction to a conditional. It plays fast and loose with our ideas about “or”. In ordinary language, “or” can be either exclusive or inclusive. An example of an inclusive or would be: “It's hot today, let's go swimming or eat ice cream.” The exclusive use would be: “Tamino loves me or he doesn't”. In the exclusive sense, just one of the disjuncts can be true. So the same fallacy occurs if we reason the following way:

~P

~P ∨ Q

∴ ~Q

Because “v” is chosen to symbolise the inclusive use of “or”. The lover's fallacy is a bit more sophisticated, since it covers up the evident mistake by an intuitively plausible transition to a conditional.

This same reasoning is applied by some of Peter Singer's critics, especially in early ethics courses at university. Freshers feel incredibly smug and confident with the following argument: i) Peter Singer, a rather successful and hence rather wealthy philosopher, does not give all his money to charity. ii) According to his moral philosophy, this would be the only right thing to do. iii) Either he does not give all his money to charity or his theories are right. iv) If his theory was right, then he would give all his money to charity. v) So his theory must be wrong.

As seen before, if we symbolise the sentences (~P: Peter Singer doesn't give all his money to charity; Q: Singer's theory is right) the argument follows the very pattern of the lover's fallacy:

i) ~P

ii) (this premise does not add anything to the argument, it is just mentioned for clarifying purposes)

iii) ~P ∨ Q

iv) Q → P

v) ∴ ~Q

This is not to defend Singer's philosophy, it is very well possible that he is wrong (why this might be the case, look here). But the simple fact that he doesn't follow his own rules, doesn't imply that they are wrong. There's an extra argument needed for that, because people often don't do a particular thing. Even if they are supposed to.