The dual semantics of indefiniteness and its implications for the referentiality of intensional sets (Abstract)

A close look at the semantics of indefiniteness in terms of information raises reasonable doubts about the extent to which predicate logic accurately captures natural language (NL) semantics. The intensional description of a set S involves the statement of some property that the members of S are required to satisfy. In NL, any such property is expressed by means of a predication, which will be here defined as any NL means to increase information. This definition involves the indentification of: (1) where to add information ('the zebra in the zoo'), and (2) what information to add ('is young').

Where to add information - To identify a particular item of information, NLs can use a specific name or, if the name is ambiguous, a disambiguation device based on structures such as

Office Place
------ Paris
------ Bonn (1)

Here, a construct such as Office(x)·Place(Paris) can be used to inambiguously refer to a particular office x. If we denote as T the empty table X(x)·X(x), then we can write

Office(x)·Place(Paris) = Office(x) + T + Place(Paris)

This inambiguous reference to an Office item can be expressed as W r W', where W denotes an ambiguous meaning, and W' denotes some meaning related to W by a topological structure r.

What information to add - The same device can be used to state a predication as an information process W -> W r W'. A relation r that links a general concept C to a particular instance I of that concept (e.g. 'color' to 'green') will be denoted as C(I), and any such C shall be called a category. Summing up, a general syntactic form can be expressed as a combination of:

I = C r I' (specification)
C -> C r I' (predication)

where I denotes an instance of C and I' denotes an instance of a category C' related to C by r.

Ambiguity is heavily dependent on the context, which some NLs use as a 'circumstancial' category X containing the symbol to be disambiguated. But the context is not always available. Let i in C(i) be a particular instance i the name of which is not known. As such, the symbol i is 'empty' and, therefore, is not related to any particular category, i.e. we can write W(i), W'(i), ... This meaning can be associated to the English indefinite 'a'. It would denote that a choice has been made, though no name has been assigned to the item chosen. The reference is incomplete, but inambiguous, and the name can be filled at a further stage of the information process.

Structures such as (1) can be combined to build more complex structures S(C(i)·C'(i)·C''(i)···). As we shall not be concerned with the specific (topological) relations that make up such structures, they will be simply denoted as '·'. A complex structure can combine with an item C(i) if any of its component items binds to it: jump(···X(x)···) + frog(i) = jump(···frog(i)···).

However, the indefinite 'a' has an alternative meaning, strongly reminiscent of the quantifier 'any', and implied in expressions such as

[what to do if] "a frog jumps" [into the office]

Clearly, this sentence is not about a specific frog(i), but rather about an indefinite frog(x)—the opposite meaning, in terms of scope. And yet both meanings are associated to the same word 'a'. This reflects the fact that not all of the components in a complex structure may be concurrently definite. Thus, if S is the structure S(C(x)·C'(x)·C''(x)···) then S(i) = S(C(i)·C'(i)·C''(i)···) is merely the most definite form for S, but other forms are also possible which may or may not be categorised as indefinite. If we define F(x) [resp. H(x)] as the category of all possible frog shapes [resp. places], then frog(i) may account for four different structural forms:

(a) F(x)·H(x), i.e. the abstract concept of 'frog'.
(b) F(i)·H(i), i.e. a frog of a particular shape, at a particular place.
(c) F(x)·H(i), i.e. a particular frog whatever its shape.
(d) F(i)·H(x), i.e. a frog of a particular shape, whatever its place, which is a case interpretation (something that would happen to be a frog).

The case (a) could be related to the NL expression 'frog', while (b) to (d) could be related to the expression 'a frog', even if this expression encompasses three essentially different meanings.

If a property P is described as the information r W' added by a predication W -> W r W', e.g.

S(C(John)·C'(i)·C''(i)···) => John satisfies P = S(C(x)·C'(i)·C''(i)···)

then the expressions (b) to (d) above imply the property P1 = F(x)·H(x), but (b) implies also P2 = F(x)·H(i) and P3 = F(x)·H(i), and both (c) and (d) imply the same property P1.

If we now define a set S as Set·Member(S, x), where Set·Member(x, x) describes the semantics of 'include', then any Member(i) could be identified by means of a property P and would be represented by a complex form, so it might denote different kinds of indefiniteness. The expression 'a set S that is a member of itself' would be formulated as:

Set(x)·Member(Set(x)·Member(Set(x)···))

which is not a finite structure of the kind described here. Actually, this structure is a self-referent topological configuration, that may be called selfinclude(S), in the same way that a loop is not a segment even if can be described as such. This suggests an inconsistent translation of NL concepts into set theory. The concept of a set that is member of itself is valid when referring to a real-world set that includes itself. However, abstractly formulated, the expression cannot refer to an open choice x as if it were a specific item.

Conclusion. By way of their features, topological structures and their combinations may be worth considering as absolute semantic primitives. A structure is an objective semantic referent that can be handled by means of symbols and rules. This paper suggests that both NL semantics and the basic axioms of logic might be reformulated in terms of the essential features of all possible topological structures—which are provided by nature.