Peano and Grice on "the"

Speranza

The inverted iota was understood as a function

mapping the propositional function it takes as argument onto the sole argument to

that propositional function that yields a true proposition if there is such an argument,

  and a chosen object, otherwise.

Russell borrowed the inverted iota from Peano (1897, 1906), who uses it for a function mapping a singleton class onto its single

  

member.

Russell’s understanding of the inverted iota also owes a lot to Frege’s sign ‘\’ from Grundgesetze §11, which

stands for a function that has as value, if its argument is the value-range of a concept true of ONLY ONE object (Frege's example: "God") the sole

object that falls under the concept, and a chosen object otherwise.

Using this device, in a certain sense, Russell maintains

that propositional functions are to be taken as fundamental.

  

Russell goes on to  analyse

‘y=the father of x’

by beginning with the relation

R, a function whose value for every argument x is a propositional function that itself

yields a true proposition only for fathers of x as argument.

He would then analyse

‘THE father of x’ as

i‘(R‘x)





Another denoting function Russell takes as
primitive is the function that maps a propositional function onto the class of entities

satisfying that function.

However, Russell thinks sets

can be done away with entirely. Russell considers this function to

be in fact definable in a similar way to the "the father" case using the inverted iota and

what Whitehead calls relation Kl .

However, Russell was never happy

with the function i, and it bothered him to have only a single function that worked

so differently than the others he had on the table.

For a discussion of Russell's dislike of this

use of the symbol I see Klement). Of course, he finally managed to purge

his logic completely of denoting functions in 1905 when he came across the theory of

‘incomplete symbols’ of ‘On Denoting’ and the possibility of defining away both descriptive

phrases and class-terms in context.


And then came Grice.