Three statements...

An interesting challenge set in HCJ this week; to find out what is important about three individual statements and blog about them.

'The evening star is the same thing as the morning star'

This relates to the work of a German logician, mathematician and philosopher called Gottlob Frege, who wanted to give an example of the difference between 'reference' and 'sense' in logic.

Please consider the statements below:

1.  The morning star = the morning star.

2.  The morning star = the evening star.

As you may or may not know, both the 'morning star' and the 'evening star' refer to the planet Venus.  However, the first statement appears superfluous, while the second appears to be giving additional information to the reader.  Frege argues that both statements are equally true, but because the referenced object in each component expression of both statements is identical (the planet Venus) the sense of the component expressions are what determines the informativeness of each statement.  Thus statement 1 seems distinctly unhelpful, while statement 2 provides the reader with additional information.

'The present king of France is not bald'

This was a logical problem set by Bertrand Russell who intended to build upon the work done by Frege.

Western logic is binary in nature, that is, it only allows for two states:  'true' or 'false'.

Russell argues that if a statement such as 'The present king of France is bald.' is examined, it can be split into three separate assertions:

1.  That there is an x, where x is a present king of France
2.  For every x that is a present king of France, and every y that is a present king of France, x = y (i.e. there can only be one present King of France at most)
3.  For every x that is a present king of France, x is bald.

Given that France is a republic, and as such does not have a king, the statement is provably false, as the first assertion is clearly false.  Now consider the following statement:

'The present king of France is not bald'

This second statement is a negation of the first statement, but logically it presents some problems.

This is due to the ambiguity present in the negation.  If you read this statement as saying 'the present king of France is not bald, because there is no present king of France', the statement is true.

However, if you were mistakenly under the impression that France currently had a king, then the negation could, in logical terms, mean 'there is something that is the present king of France, and it is the only present king of France, and that present king of France is not bald.'

Read in the latter manner, the statement is logically false.  One of the primary laws of logic are that a statement can only be 'true' or 'false'.  Yet 'The present king of France is not bald' appears to be both....


There was nobody on the road

I couldn't find this precise phrase discussed directly, but am choosing to believe (possibly incorrectly) that Chris was paraphrasing from Chapter 7 of Lewis Carroll's famous work 'Through the Looking Glass'.

Specifically:

`I see nobody on the road,' said Alice.
`I only wish I had such eyes,' the King remarked in a fretful tone. `To be able to see Nobody! And at that distance, too! Why, it's as much as I can do to see real people, by this light!' 

There is a logical absurdity in the idea of being able to see nobody, which is hopefully visible to all!