Denoting phrases and definite descriptions

In On denoting, Russell explains what he called denotative phrases and gives examples of them: «a man, some man, any man, every man, all men, the present King of England, the present King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth».

And groups them into three cases:

Without reference: the present King of France
With a particular reference: the center of mass of the solar system for ...
With ambiguous reference: a man

Such kind of prases can be find within sentences. As he considers 'all', 'nothing' and 'something' the primitive denotative phrases, he proceeds to analyze them schematically. According to Russell, the notion 'C(x) is always true' (corresponding to 'all', ie 'all x is C') is fundamental and indefinable, other notions are defined through it.


C(everything) = "C (x) is always true"
C(nothing) = "«C(x) is false» is always true"
C(something) = "It is false that «C(x) is false» is always true"

He also defines the notion "C(a man)" using the example:

"I met a man", where C(x) is the property "I met x", and "C(a man)" would be:
"«I met x, and x is human» is not always false."

We also define the following expressions (according to that indicated above):

C(all men) = "«if x is human, then C(x) is true» is always true"
C(no men) = "«if x is human, then C(x) is false» is always true"
C(some men) = C(a man) = "It is false that «C(x) and x is human» is always false"

C(every man) = C(all men)

Then there are the cases with a definite article such as "the". An example might be: "The father of Charles II was executed," which can be interpreted as "C(the father of Charles II)" (where: C(x) = "x was executed"). So we obtain:


"It is not always false of x that x begat Charles II and that x was executed and that «if y begat Charles II, y is identical with x» is always true of y".

To simplify all this, that may seem confusing, here is its formal representation.

C(all) = ∀x Cx
C(nothing) = ∀x ¬Cx
C(something) = ¬∀x ¬Cx

C(all men) = ∀x (Hx ⊃ Cx)
C(no men) = ∀x (Hx ⊃ ¬Cx)
C(a man) = ¬∀x ¬(Hx ∧ Cx)

C(the father of Charles II) = ¬∀x ¬[Px ∧ Cx ∧ ∀y (Py ⊃ y = x)]


The foregoing allows a reduction of all propositions in which denotative phrases occur to some where they not. That is, instead of expressions that denote objects in the world as subjects of sentences, we will have sentences that will denote either the truth or falsehood.

According to Strawson, this view was still widely accepted among logicians when he wrote a critical review in On Referring.

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