This Sentence No Verb Rich Cochrane's Blog

It’s a bit late for Christmas puzzles but these occurred to me this afternoon and I thought they were worth jotting down.

The answers are at the bottom of the post. In each case a “solution” tells us which of the sentences in question is true and which is false. In some cases you may not be able to tell (a sentence could be either true or false) and some may be paradoxical (assigning either truth or falsehood leads to a contradiction). The best approach is to assign a truth value (True or False) and see if it works.

There’s just one thing you need to know: by definition, the sentence “if x then y” is true in all cases unless x is true and y is false. So “If it has a tail then it’s a cat” is true if it has no tail, whether it’s a cat or not, and true if it has a tail and is in fact a cat. But if it has a tail and isn’t a cat the sentence must be false.

Here are the sentences; in each case determine whether it’s true, false, could be either or cannot be either, assuming the whole set of sentences must be consistent:

A: If this sentence is true then it’s false

B: If this sentence is false then it’s true

C: If sentence D is true then it’s false

D: If sentence C is false then it’s true

E: If sentence F is true then it’s false

F: If sentence E is true then it’s false

G: If sentence H is false then it’s true

H: If sentence G is false then it’s true

Solutions

Of course these are similar to the better-known variations on the “liar” paradox that create the same conditions. They have their own particular deviousness, though, due to the asymmetric nature of the material conditional, and they’ll catch some of your students out for sure even if you breezed through them (in which case, clever you).