[FOM] A question about the history of a notation

[Neil Tennant]

Can any fom-er throw some light on the history of a certain notation?

I'll use 'E' for the existential quantifier.

The notation that interests me is 'E!' often pronouned as 'E shriek'.
The earliest use that I know of is in Russell, where it is used in
contexts such as 

	E! iota x F(x)

which means "The F exists", or

	E! alpha

which means "(The set) alpha is non-empty".

(See his 'On Denoting', in ed. Marsh, Logic and Knowledge, at p.93 and
p.91 respectively.)

It is clear that, for Russell, 'E!' is a predicate, in that it is
completed by a name (or, more generally, a singular term) to form a
sentence. Note that in the context 'E! iota x F(x)', it is 'iota', not
'E!', which binds the variable x. The expression 'iota x F(x)' is (of the
category of) a singular term.

The usage established by Russell continues in the work of
contemporary free logicians, who either take E! as a primitive existence
predicate, or define E!t to mean Ex(x=t). Once again, on this definition
'E!' turns out to be a predicate. The variable-binding is accomplished by
the existential quantifier 'E' in the definiens, not by 'E!' in the
definiendum.

But there is a modern usage according to which 'E!' is a quantifier, and
directly binds variabls; more specifically, it is the uniqueness
quantifier. On this reading,

	E! x F(x)

means "There is exactly one x such that F(x)". So here 'E!' is completed
by a predicate to form a sentence, as indicated by the variable-binding.

What I would like to know is the provenance of this modern usage. Can
anyone on this list point me to the earliest source for 'E!' as a
variable-binding quantifier? (I am assuming that, in such a role, it has
only ever been the uniqueness quantifier; but I may be mistaken in this
assumption.)

My own preference in rendering the uniqueness quantifier is to use the
numerical subscript 1. This makes it a special instance of the numerically
definite quantifiers

	E_0 x F(x)	=df 	There are no Fs
	E_1 x F(x)	=df	There is exactly one F
	E_2 x F(x)	=df	There are exactly two Fs
	:
	E_n x F(x)	=df	There are exactly n Fs
	:

I would also like to get a sense, also, of the terminological preferences
of members of this list. Which do you prefer for the uniqueness
quantifier:

	E! x F(x) ;
or	
	E_1 x F(x) ?


___________________________________________________________________
Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science

http://www.cohums.ohio-state.edu/philo/people/tennant.html

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