Dear Jason

I think you may be right. In the applied use of logic [in which words are mixed with symbols] we often seem to use free (‘real’) variables as an indefinite mode of reference, and this may have been the usage Russell had in mind. Imagine a group of people that we want to keep anonymous [eg in a court of law] then we may agree to call them ‘Mr X’, ‘Mr Y’, etc. In each case the variable denotes a particular person without identifying that person; we may call this ‘indefinite reference’. Thus in Russell’s example, ‘I met x’ there would be a unique person denoted by ‘x’ which, for reasons of concealment or poor memory the speaker has decided not to identify by his singular term. Of course it is only the act of reference that is indefinite, not the person; the person is a definite individual.

Thus the real meaning of ‘I met x’ would seem to be

(1) I met the person referred to by ‘x’ .

The variable in ‘I met x’ is used here to indicate that we want to refer to a person only as being the referent of the symbol ‘x’. It is intended by this means to create anonymity.

Consequently, ‘I met x’ does not mean the same as ‘I met y’ since currently ‘y’ may be used to designate a different person. Free variables used in this way may be combined in a shared narrative: thus I may have been to a party and met a father and son, whom I cannot now identify, in which case I could assert truly,

(2) ‘I met x and I met y and x is the father of y’

Here it is the speaker who decides who to designate as x, and who as y. Isn’t this a formalisation of ‘I met a father and his son’ ?

I believe this provides a theory for what Russell referred to as ‘indefinite descriptions’ and, as Russell required, it does not appear to assert the existence of anything, ie it uses no quantifiers. However, if I am right, since ‘I met x’ means (1) “ I met the person referred to by ‘x’ ” then this translation is itself none other than a definite description, and according to Russell’s own theory of definite descriptions it says,

(3)     There is one and only one person referred to by ‘x’ and I met that person.

Similarly for each of the clauses in the narrative (2).

So it turns out, ironically, that indefinite descriptions seem to require definite descriptions [in the metalanguage] for their analysis. If ‘I met x’ is a propositional function then clearly, with this full expansion (3) the propositional function is indeed true or false as you claimed because, implicitly, the variable itself is being referred to, as well as the object.

I’d be most interested to hear your comments on this theory of indefinite descriptions,

Dave.