The inverted Greek letter iota ‘’ is used by Peano and by Whitehead and Russell (in Principia Mathematica), always followed by a variable, to begin a definite description.
is read as
2. the x such that x is φ.
or more simply, as
3. the φ.
Grice's example: 'the king'.
Such expressions may occur in subject position, as in
5. the φ is ψ’
6. The king of France is bald.
The formal part of Whitehead's and Russell's famous ‘theory of definite descriptions’ consists of a definition of all formulas
in which a description occurs.
To distinguish the portion
from the rest of a larger sentence (indicated by the ellipses above) in which the expression
occurs, the Griceian scope of the description is indicated by repeating the definite description within brackets:
[(x)φx] . ψ(x)φxThe notion of scope is meant to explain a distinction which Russell famously discusses in “On Denoting”.
Russell says that the sentence
The present King of France is NOT bald.
is ambiguous between two readings.
(1) the reading where it says of the present King of France that he is not bald, and
(2) the reading on which denies that the present King of France is bald.
The former reading requires that there be a unique King of France on the list of things that are not bald, whereas the latter simply says that there is not a unique King of France that appears on the list of bald things.
Russell, like Grice,, but unlike Strawson, says the latter, but not the former, can be true in a circumstance in which there is no King of France.
Russell and Grice analyze this difference as a matter of the scope of the _definite description+, though as we shall see, some modern logicians tend to think of this situation as a matter of the scope of the negation sign.
Thus, Whitehead and Russell introduce a method for indicating the scope of the definite description.
To see how Whitehead's and Russell's method of scope works for this case, we must understand the definition which introduces definite descriptions (i.e., the inverted iota operator).
Whitehead and Russell define:
*14·01 [(x)φx] . ψ(x)φx . = : (∃b) : φx . ≡x . x=b : ψb DfThis kind of definition is called a contextual definition, which are to be contrasted with explicit definitions.
An explicit definition of the definition description would have to look something like the following:
(x)(φx) = : … Dfwhich would allow the definite description to be replaced in any context by whichever defining expression fills in the ellipsis.
By contrast, *14·01 shows how a sentence, in which there is occurrence of a description (x)(φx) in a context ψ, can be replaced by some other sentence (involving φ and ψ) which is equivalent.
To develop an instance of this definition, start with the following example:
The present King of France is bald.
Using PKFx to represent the propositional function of being a present King of France and B to represent the propositional function of being bald, Whitehead and Russell would represent the above claim as:
[(x)(PKFx)]. B(x)(PKFx)which by *14·01 means:
(∃b) : PKFx . ≡x . x=b : BbIn words, there is one and only one b which is a present King of France and which is bald.
In modern symbols, using b non-standardly, as a variable, this becomes:
(∃b)[∀x(Kx ≡ x=b) & Bb]Now we return to the example which shows how the scope of the description makes a difference:
There are two options for representing this sentence.
The present King of France is not bald.
[(x)(Kx)] . ~B(x)(Kx)and
~[(x)(Kx)] . B(x)(Kx)In the first, the description has “wide” scope, and in the second, the description has “narrow” scope.
Russell says that the description has “primary occurrence” in the former, and “secondary occurrence” in the latter.
Given the definition *14·01, the two PM formulas immediately above become expanded into primitive notation as:
(∃b) : PKFx ≡x x=b : ~BbIn modern notation these become:
~(∃b) : PKFx ≡x x=b : Bb
∃x[∀y(Ky ≡ y=x) & ~Bx]The former says that there is one and only one object which is a present King of France and which is not bald; i.e., there is exactly one present King of France and he is not bald.
~∃x[∀y(Ky ≡ y=x) & Bx]
This reading is false, given that there is no present King of France.
The latter says it is not the case that there is exactly one present King of France which is bald.
This reading is true.
Although Whitehead and Russell take the descriptions in these examples to be the expressions which have scope, the above readings in both expanded PM notation and in modern notation suggest why some modern logicians take the difference in readings here to be a matter of the scope of the negation sign.