PUZZLE:
Larry is married but Nick isn’t. Larry is looking at Mary, and Mary is looking at Nick.
QUESTION:
Is someone married looking at someone not married?
CHOICES:
A. Yes
B. No
C. Not enough information to answer
This puzzle was posed to me by a student (thanks Richard!) after class one morning. A cursory google suggests 80% choose incorrectly. I am skeptical; I’ve found no empirical evidence supporting this claim (I’d be interested if anyone else has). Be warned, searching for the puzzle will likely turn up solutions, so if you’d like to solve it, best settle here and reflect for a bit. I’ll wait. Once you are finished, check your answer by clicking it above. Afterwards, scroll down for a solution and some discussion.
SOLUTION:
The hallmark of a logical solution to a puzzle is that once presented with the solution, nearly everyone agrees it is correct. The Wason Selection Test is an example. Many choose incorrectly. Bentham reports a psychologist once confessed to him that nearly everyone accepts the standard solution as correct once it is explained [1]. The puzzle under discussion strikes me as logical in this sense. I’m curious if you agree.
I’ve translated the puzzle into a standard classical first-order language, where the predicate and relations symbols are obvious. It is straightforward to prove the solution (I’m using Hardegree’s natural deduction system in Symbolic Logic: A First Course). In symbols:
1. Ml & ~Mn Premise
2. Llm & Lmn Premise
3. SHOW∃x∃y(Mx & ~My & Lxy) ID
4. ~∃x∃y(Mx & ~My & Lxy) AID
5. ∀x∀y(~Mx v My v ~Lxy) 4, Substitution
6. ~Ml v Mm v ~Llm 5, Universal Instantiation
7. Mm 1,2,6 DE
8. ~Mm v Mn v ~Lmn 5, Universal Instantiation
9. ~Mm 1,2,6, DE
10. ! 7,9 Contradiction
I’ve assumed as premises the information provided in the puzzle. On the SHOW line you’ll find a symbolization of the follow-up question. On line 4, I assume the negation of the SHOW line. On line 5, I substitute negated existential quantifiers for universal quantifiers trailed by negation, which is then distributed via De Morgan application. Since both variables in line 5 are under universal scope, I instantiate without restriction; in particular, to the constants denoting Larry and Mary. The result is line 6, which says Larry is unmarried, Mary is, and Larry is not looking at Mary. However, on line 1 we assumed Larry was married. Similarly, on line 2 we assumed Larry was looking at Mary. Hence, by disjunctive syllogisms, we infer Mary must be married on line 7. Instantiating line 5 once more, this time with Mary unmarried, Nick married, and Mary not looking at Nick, results by similar reasoning in Mary not being married on line 9. Since we already inferred that Mary was married, we now find ourselves in a contradiction. Hence, we infer someone is looking at someone not married.
Another way to think about the solution is to observe (or suppose what is plausible) that Mary is either married or not. If married, then since Mary is looking at Nick who is unmarried, someone is looking at someone who is unmarried. If unmarried, then since married Larry is looking at Mary, someone married is looking at someone unmarried. Either way, the answer is "Yes."
DISCUSSION:
Several of my students, when initially posed with the puzzle, claim there is “Not enough information to answer”. I take this to suggest students have trouble thinking to make certain plausible suppositions. To be clear, I do not think students have trouble making plausible suppositions and reasoning from them in general. They do, after all, deliberate about the future. Rather, it just doesn’t naturally occur to them to make even plausible suppositions in certain contexts, such as the context of puzzles and the contexts of proofs.
Works Cited
[1] Bentham, J. (2008). Logic and Reasoning: Do the Facts Matter? Studia Logica. 88:67-84