Russell’s theory:
I take the notion of the variable as fundamental; I use ‘C(x)’ to mean a proposition in which x is a constituent, where x, the variable, is essentially and wholly undetermined. Then we can consider the two notions ‘C(x) is always true’ and ‘C(x) is sometimes true.’ Then everything and nothing and something are to be interpreted as follows:
C(everything) means ‘C(x) is always true’;
C(nothing) means ‘“C(x) is false” is always true’;
C(something) means ‘It is false that “C(x) is false” is always true’.
Today we do it differently:
C (everything) means ∀x C(x)
C (nothing) means ∀x ~C(x)
C (something) means ~∀x ~C(x) (= ∃x C(x))
Descriptions: The
“The father of Charles II was executed” becomes: ‘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’.
In other words, ‘The father of Charles II was executed’ becomes
(i) Someone fathered Charles II (∃x FCx &)
(ii) At most one person fathered Charles II (∀y ∀z (FCy & FCz ⊃ y = z) &)
(iii) Everyone who fathered Charles II was executed (∀y (FCy ⊃ EXy))
Three Puzzles
(1) Informativeness of Identity: Scott=Waverley
(2) Non-Denoting Terms: The present King of France
(3) Non-existence Claims: the difference between A and B does not subsist.
(KR Notes: basically if a denoting phrase’s function is to denote a real object in the real world, then the above puzzles are difficult to solve. Russell’s genius is to to understand denoting phrase as an abbreviation of a more complicated quantificational structure. Thus, denoting phrase does not denote to anything particular, but to be understood as a set of constraints/requirements to hook singular individuals that answer to them (semantic referent).)
Problems of Meaning and Denotation
We say that when C occurs it is the denotation that we are speaking about; but when “C” occurs, it is the meaning, and the logical relation involved between meaning and denotation is that the meaning denotes the denotation. The difficulty which confronts us is that we cannot succeed in both preserving the connection of meaning and denotation and prevent them from being the same thing.
KR notes: the key to understand this paragraph is to understand C as a phrase/complex was to have both meaning and denotation, and meaning, or the proposition/complex itself, denotes the denotation. However, if such is the case, then “whenever C occurs without inverted commas, what is said is not true of the meaning, but only of the denotation….Thus to speak of C itself, to make a proposition about the meaning, our subject must not be C, but something that denotes C. Thus ‘C’, which is what we use when we want to speak of the meaning, must not be the meaning, but something which denotes the meaning.” Moreover, in such a view, “C is only the denotation, the meaning wholly relegated to ‘C’.” However, it is not the case that only denotation occurs when C is in a proposition.
Solution to the Three Puzzles
Key Point: a denoting phrase ( such as “the author of Waverley”) is essentially part of a sentence, and does not, like most single words, have any significance on its own account.
Primary occurrence v. Secondary occurrence: When the description is given wide scope, Russell calls this a primary occurrence. When it is given narrow scope, it’s a secondary occurrence. Basically, a wide scope means the denoting phrase in a clause is substituted and interpreted in the primary sentence while a narrow scope means the denoting phrase in a clause is substituted and interpreted within the clause. Or more formally, “a secondary occurrence of a denoting phrase may be defined as one in which the phrase occurs in a proposition p which is a mere constituent of the proposition we are considering, and the substitution for the denoting phrase is to be effected in p, not in the whole proposition considered.
There are two ways of negating ‘The F is G.’ (two ways of interpreting “it is not the case that the F is G”)
(i) ∃x (Fx & ∀y (Fy ⊃ y=x) & ~Gx). There is a unique F and it is NOT G. [wide scope]
(ii) ~∃x (Fx & ∀y (Fy ⊃ y=x) & Gx). NOT: there is a unique F and it is G. [narrow scope]
Russell would say the description has a “primary occurrence” in (i) and a “secondary occurrence” in (ii). Nowadays we say the description has wide scope in (i) and narrow scope in (ii).
On Acquaintance
One interesting result of the above theory of denoting is this: when there is anything with which we do not have immediate acquaintance, but only definition by denoting phrases, then the propositions in which this thing is introduced by means of a denoting phrase do not really contain this thing as a constituent, but contain instead the constituents expressed by the several words of the denoting phrase.
