NLP: Definite Descriptions

Channeling and extending Russell

Regarding is: there are four different uses (predication, subtype, identity, existence) and we represent these using the logical notations (P x), (subset x y), (= x y), x. 

It was Russell's belief that by using the new logic of his day, philosophers would be able to exhibit the underlying logical form of natural-language statements. A statement's logical form, in turn, would help philosophers resolve problems of reference associated with the ambiguity and vagueness of natural language.

denoting phrases such as descriptions and proper names. for example, Scott, the author of Waverley, the number two, the golden mountain In, On Denoting (1905)

When interpreting natural language we distinguish

1) logically proper names (including words such as this or that which refer to sensations of which an agent is immediately aware) have referents associated with them.

2) descriptive phrases (such as the smallest number less than pi) should be represented by staments of first order logic using quantification and predicates (such as x is a number), ie they are not to be viewed as referring terms but, rather, he called them "incomplete symbols". They cold take on meaning within appropriate contexts, but that are meaningless in isolation.

Consider 

    (1) The present King of France is bald,

the definite description 'the present King of France' plays a role quite different from that of a proper name such as Scott in the sentence

    (2) Scott is bald.

Letting K abbreviate the predicate 'is a present King of France' and B abbreviate the predicate 'is bald', Russell assigns sentence (1) the logical form, in the notation of the predicate calculus,

    (1) (exists x (and (K x) (forall y (implies (K y) (= y x))) (B x))

In contrast, by allowing s to abbreviate the name Scott, Russell assigns sentence (2) the very different logical form

    (2) (B s)

This distinction allowed Russell to explain three important puzzles.

1) Law of Excluded Middle and how this law relates to denoting terms.
   By the Law of Excluded Middle
     it must be the case that either 
       "The present King of France is bald." is true or 
       "The present King of France is not bald is true
   But if so, both sentences appear to entail the existence of a present King of France!!!!
   Russell's analysis shows how this conclusion can be avoided. 
     By (1), it follows that there is a way to deny (1) without being committed to the existence of a present King of France, namely by accepting that It is not the case that there exists a present King of France who is bald is true.

2) Law of Identity as it operates in (so-called) opaque contexts.
   Even though Scott is the author of Waverley is true
    it does not follow that the two referring terms Scott and the author of Waverley
      need be interchangeable in every situation.
    Thus, although 
      George IV wanted to know whether Scott was the the author of Waverley is true
      George IV wanted to know whether Scott was Scott is, presumably, false
'
let
  s abbreviate the name Scott
  w abbreviate Waverley
  A abbreviate the two-place predicate is 'the author of'

then

    (3) s=s

is not equivalent

    (4) (exists x (and (A x w) (forall y (implies (A y w) (= y x)) (= x s)))) 

Sentence (3) is a necessary truth, while sentence (4) is not.

3) true negative existential claims
   For example, "The golden mountain does not exist". 
   We want a speaker be able to belive a negative existential 
     without also beliving that the subject term has reference.

So the claim that Scott does not exist is false since

    (5) (not (exist x (= x s)))

is self-contradictory, eg, (= s s)

More startling, the claim that 'a golden mountain does not exist'  may be true
  
let
  G abbreviate the predicate 'is golden'
  M abbreviate the predicate 'is a mountain'

    (6) (not (exists x (and (G x) (M x))))

is not contradictory.