## 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! 🙂

## John Macarthur and the birth of the Australian wool industry (not just another Aussie sheep joke!)

Today we celebrate the birthday of John Macarthur (3 Sep 1767 – 11 Apr 1834) the English-born Aussie who is recognised as the pioneer of the wool industry that boomed in Australia in the early 19th century, and has since been one of the country’s agricultural trademarks.

Macarthur was born in Plymouth, Devon in the UK. He began his career in the army, and after various assignments and activities became part of the New South Wales corps in 1789 and was posted to faraway Sydney, Australia. A fiery character, his life story reads like a historical romance novel, with way too many saucy details (battles with authorities, involvement in a military coup, land battles and much more) to get into on this forum.

Suffice to say, after settling in Australia, Macarthur got involved in rearing sheep for mutton, purchasing his first flock in 1795. He also purchased a small flock of Spanish Merino, imported from the Cape Colony (part of the later South Africa) in 1797. The merino is an excellent wool breed, and it didn’t take long for Macarthur to recognise the economic potential of wool production for export, as opposed to simply rearing sheep for the local meat-market. What made wool a potential export hit was the fact that it was a non-perishable commodity (a necessary feature, given Australia’s distance from the markets of the UK and Europe) and offered a high value per unit of weight.

On a trip back to London he lobbied for more land, and succeeded in being granted 5000 acres of the best pasture land in New South Wales. He became the largest sheep rearer in the colony, and made a fortune exporting merino wool to the UK, who were at the time cut off from their traditional wool supplier, Spain, as a result of the Napoleonic Wars. He also gained recognition for producing wool of the finest quality, which further upped the prices at which he was able to sell his produce.

Macarthur’s ventures opened the door for others to follow, and Australia’s wool export market started to boom in the early 19th century. It remains a key export commodity, with Australia remaining the world’s largest producer of the wool, mainly from merino sheep. New Zealand is in second place, and China in third. The wool produced in Australia and New Zealand is considered to be of the finest international quality – the best Aussie and Kiwi merino wool is known as grade 1PP, and is the industry benchmark of excellence for merino wool.

Natural wool is one of nature’s super-products. It is technically superior to synthetic materials in various ways – it has better heat insulation and superior hydryphilic properties, it is naturally flame-retardant, resistant to static electricity, and hypoallergenic.  Researchers at the Royal Melbourne Institute of Technology have developed a material blending wool and kevlar (the material often used in body armour) and it was found that the blend was lighter and cheaper, and outperformed kevlar in damp conditions.

What’s more, wool is also environmentally preferable to materials like nylon or polypropylene. According to the latest research on the role of wool in the natural  carbon cycle, it has been suggested that under the correct circumstances, wool production can potentially be carbon neutral.

So while the Aussies and Kiwis may suffer endless jokes relating to their sheep, the product being produced is something very special.  And John Macarthur deserves a tip of the hat as the bloke who kicked it all off more than 200 years ago.