Peano and Grice on "the": the implicatures

Speranza

Singular propositions, especially those in which definite descriptions occur, have been termed "Russellian propositions", so called because of their designation by Bertrand Russell in terms of the "iota" quantifier or iota operator, employing an inverted iota to be read "the individual x".

Thus, e.g. (x)(x).

In Principia Mathematica [Whitehead and Russell 1910, 54], Russell writes

"!x"

for the first-order function of an indi-vidual, that is, for any value for any variable which involves only individuals.

Thus, for example, we might write !(Socrates) for

"Socrates is a man".

In the section on "Descriptions" of Principia [Whitehead and Russell 1910, 180], the iota operator replaces the notation "!x" for singulars with "(x)(Φx)"

so that one can deal with definite descriptions in as well as names of individuals.

This is a great simplification of the lengthy and convoluted explanation that Russell undertook in his Principles of Mathematics [Russell 1903, 77–81, §§76–79], in order to summarize the point that "Socrates is a-man" expresses identity between Socrates and one of the terms denoted by a man’.

‘"Socrates is one among men" is a proposition which raises difficulties owing to the plurality of men.

Whitehead and Russell do this in order to distinguish a particular individual which is an element of a class having more than one member from a unary class, i.e. a class having only one member.