Scope and Truth value gaps

Here is a bit of clarification of what goes on in relation to scoping of definite descriptions in Principia and in Grice's formal work, and where truth value gaps do or don't appear.

Three philosophers appear here, Russell, Grice and Church, and three logical systems, with an aside to Strawson on truth value gaps.

Lets get the latter out of the way first.
Strawson was not so fond of Russell's theory of descriptions.
He thought that statements involving definite descriptions were meaningless if the definite description was not uniquely satisfied, and hence that they lacked a truth value.
To realise that maxim in a formal logic you would typically use a "non-classical" logic with more than two truth values (and you translate "no truth value" as "neither true nor false").

The three systems discussed here (Russell's PM, Church's STT, and Grice's system Q) are all "classical", i.e. two valued, and hence there can be no truth value gap in definite descriptions in any of these systems, every sentence is either true or false (under every interpretation), so if there is a problem (as Strawson might insist) it is just the lack of a truth value gap or a third truth value.

Passing on to curiosities of scope, the main issue arises in PM, persists in Q but is addressed in a different manner, but is absent from STT.

It arises from Russell's Theory of Descriptions, which treats descriptions as "incomplete symbols" involving contextual definitions.
For the full low-down on this see the SEP article on the notation of Principia Mathematica, this is the shortened bowdlerized version.
Russell's incomplete symbols are defined only in specific contexts and the elimination is sensitive to the context chosen.
This gives rise to potential ambiguities of scope, to resolve which Russell adopts a rather cumbersome notation, for which Grice offers an alternative in his System Q (for Quine) in the form of subscripted quantifiers.

The simple example offered by Speranza involves the infamous King of France, and the ambiguity which arises when a negation is added to the allegation of baldness.

 "The King of France is not bald"
 

can be translated in two different ways depending on whether the negation is considered to be in the context for elimination of the description or outside it.


There are two options for representing this sentence according to Russell's theory (from SEP):

[(x)(Kx)] . ~B(x)(Kx)

and

~[(x)(Kx)] . B(x)(Kx)

where the placement of the notation "[(x)(Kx)]" serves to identify the relevant context for the elimination of the definite description, and we can see that in the first case the negation is inside that context and in the second case it is not.

When these are eliminated we get the two sentences previously quoted by Speranza:

∃x[∀y(PKFy ≡ y=x) & ~Bx]

~∃x[∀y(PKFy ≡ y=x) & Bx]

(the SEP article has slipped from using "K" for "King of France" to using "PKF"!)

So long as there is a King of France these two sentences will have the same truth value, according as he is or is not Bald, but if there is no King of France then the first sentence will be false and the second true.

The method used by Church is quite different, and no special scoping problems arise in that system.  Church does not use incomplete symbols and instead may be understood as using an explicit non-contextual definition of the definite description operator (though in fact the symbol is a primitive and so is defined by an axiom, an implicit definition in Hilbert's terminology).
Church here is more interested a solution which is technically satisfactory than one which correctly renders definite description as it is used in natural languages.  The result is clearer and easier to work with, and though definitely not the same as ordinary language, it is probably no worse than Russell's rendition.

A few years back I did a bit of formal analysis of Grice's system Q using ProofPower, a proof tool for HOL a language based on Church's STT.  Because the treatment of descriptions in STT does not follow Russell's Theory of Descriptions the aspects of system Q providing an alternative scoping mechanism (which were not the main thrust of Grice's work) were not treated, as no scoping issues arise when system Q is rendered in Church's system.