The upside down world paradox

By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)

As most kids (I suspect), my daughters sometimes play ‘upside down world’, especially when I ask them something to which they should say ‘yes’, but instead they say ‘no’ and immediately regret it: ‘Upside down world!’ The upside down world game basically functions as a truth-value flipping operator: if you say yes, you mean no, and if you say no, you mean yes.

My younger daughter recently came across the upside down world paradox: if someone asks you ‘are you playing upside down world?’, all kinds of weird things happen to each of the answers you may give. If you are not playing upside down world, you will say no; but if you are playing upside down world you will also say no. So the ‘no’ answer underdetermines its truth-value, a bit like the no-no paradox. Now for the ‘yes’ answer: if you are playing upside down world and say ‘yes’, then that means ‘no’, and so you are not playing the game after all if you are speaking truthfully. But then your ‘yes’ was a genuine yes in the first place, and so you are playing the game and said yes, which takes us back to the beginning. (In other words, 'no' is the only coherent answer, but it still doesn't say anything about whether you are actually playing the game or not.)

I do not think the upside down paradox is of particular theoretical interest, but what struck me is that it arose in a fairly mundane situation, and was viewed as paradoxical by a 7-year old (who is admittedly the daughter of a philosopher of logic, fair enough…). She didn’t call it a paradox at first; she just said that this was a really difficult question to answer (‘are you playing upside down world?’); whatever you said, strange things happened. So this may well be a modest example of how Liar-like paradoxes may emerge even in everyday situations. (Hum, maybe I should write a paper with her, following the example of Veronique and Peter Eldridge-Smith on Pinocchio’s paradox.)