## On logical paradoxes and talking sheep

It’s time for a bit of serious concentration again – here’s another fun paradox to get your head around…

Today is the birthday of ** Haskell Brooks Curry** (12 Sep 1900 – 1 Sep 1982), an American mathematician and a pioneer of mathematical logic. He specialised in combinatorial logic, and some of his work found application in the development of modern computer programming languages.

While working on a strand of logic called *‘naïve logic’*, he came up with a logical construct that became known as *Curry’s paradox*.

The paradox is based on the idea of a *‘conditional claim’*, or *(If A, then B)*. Consider the following conditional claim:

*“If this sentence is true, then sheep can speak English.” *

Even though the second part of the sentence is false *(last time I checked)*, there’s nothing stopping us from analysing the truth of the sentence.

The quoted sentence is of the form *(If A then B)* where *(A)* refers to the sentence itself and *(B)* refers to the claim *“sheep can speak English”*. Within the context of Curry’s naïve logic, the way to prove a conditional sentence is to assume that the hypothesis *(A)* is true, and then to prove, based on that assumption, that the conclusion *(B)* is true.

So, lets start with the assumption *(A)* is true. Because *(A)* refers to the overall sentence, therefore assuming *(A)* is true implies that the statement *(If A then B)* is also true. So, because *(A)* is true, *(B)* must be true. Assuming the truth of *(A)* is therefore sufficient to guarantee that *(B)* is true, regardless of the actual truth of statement *(B)*. Which of course results in a paradox if *(B)* is, in fact, false.

Phew….

We can even show Curry’s paradox occurring in formal symbolic logic. Assuming there is a formal sentence (X → Y), where X itself is equivalent to (X → Y), then a formal proof can be given for Y:

1. X → X

*(rule of assumption, also called restatement of premise or of hypothesis)*

2. X → (X → Y)

*(substitute right side of 1, since X is equivalent to X → Y by assumption)*

3. X → Y

*(from 2 by contraction)*

4. X

*(substitute 3, since X = X → Y)*

5. Y

*(from 4 and 3 by rule of inference)*

There you have it – convincing mathematical proof that sheep *CAN* speak English! 🙂