Shirleen Stibbe http://www.shirleenstibbe.co.uk

Investigation: Sentence first - verdict afterwards Proofs Workshop

From Alice in Wonderland: Trial of the Knave of Hearts, alleged to have stolen the Queen of Hearts' tarts: 'Let the jury consider their verdict,' the King said, for about the twentieth time that day. 'No, no!' said the Red Queen. 'Sentence first - verdict afterwards.'

Draw a conclusion from the following sentences, using all the information given.

a) Babies are illogical;

b) Nobody is despised who can manage a crocodile;

c) Illogical persons are despised.

a) My saucepans are the only things that I have that are made of tin

b) I find all your presents useful

c) None of my saucepans is useful

Lewis Carroll's real name was Charles Lutwidge Dodgson. He was the son of a clergyman and a Mathematics lecturer at Christ Church College, Oxford. As a mathematician he had a liking for symbolic logic. He was also keen on photography, particularly the photographing of children, and the picture shows him cleaning a photographic lens. He published 60 logic puzzles each of which listed some statements and the reader was invited to draw a conclusion from given sentences using all the information given.

Shirleen Stibbe http://www.shirleenstibbe.co.uk

Sentence first - verdict afterwards. Solutions Proofs Workshop Draw a conclusion from the following sentences, using all the information given:

Example 1 Example 2

a) Babies are illogical; b) Nobody is despised who can manage a crocodile; c) Illogical persons are despised.

a) My saucepans are the only things I have that are made of tin; b) I find all your presents useful; c) None of my saucepans is useful.

Solutions:

First we pick out and label each of the component propositions in the question, with the aim of paraphrasing each sentence in formal logic terms. Using x to denote anyone, the components may be denoted as:

Example 1 Example 2

p: x is a baby q: x is illogical r: x is despised s: x can manage a crocodile

p: x is one of my saucepans q: x is made of tin r: x is a present from you s: x is useful

Now it is easy to put the sentences in formal logic terms:

Example 1 Example 2

Paraphrase Logic Paraphrase Logic

a) If x is a baby, then x is illogical p ⇒ q a) If x is made of tin, then x is a saucepan q ⇒ p

b) If x can manage a crocodile, then x is not despised

s ⇒ ¬r b) If x is a present from you, then x is useful r ⇒ s

c) If x is illogical, then x is despised q ⇒ r c) If x is useful, then x is not a saucepan s ⇒ ¬p

To draw a conclusion, we need to find a chain of the form a ⇒ b ⇒ c ⇒ d, from which we may deduce a ⇒ d.

Example 1 Example 2

The contrapositive of b) is r ⇒ ¬s, i.e. If x is despised, then x can’t manage a crocodile, which links to sentence c), so we can say: p ⇒ q ⇒ r ⇒ ¬s, from which we deduce p ⇒ ¬s. The conclusion is therefore: If x is a baby, then x can’t manage a crocodile, or

Babies can’t manage crocodiles.

The contrapositive of a) is ¬p ⇒ ¬q, i.e. If x is not a saucepan, then x is not made of tin, which links to sentence c), so we can say: r ⇒ s ⇒ ¬p ⇒ ¬q, from which we deduce r ⇒ ¬q. The conclusion is therefore: If x is a present from you, then x is not made of tin,, or

None of your presents is made of tin.