Fundamentals¹

A proposition is the meaning of a clause minus its basic illocution. Derivatively, it is also a semantic representation of such an entity. Some of the properties mentioned below relate rather to this latter concept of proposition.

A proposition is based on a predication whose predicate takes one or more arguments. With the exception of certain equations, a predicate is based on a predicable notion. There are various ways in which an argument may be constituted. In a fully specified proposition, all of the argument positions are occupied by constants or bound variables. An open proposition is a proposition one (or more) of whose argument positions is not occupied by a constant or variable bound in the same proposition. Such a proposition has an empty place (s. Dahl 1974). In , #a is a (closed) proposition, while #b and #c are open propositions.

In a formal representation, an open position = empty place is indicated by an unbound variable. In linguistic description, open propositions appear, inter alia, in the representation of the valency of a verb, e.g.
‘xsbj buys ydir.obj’.

The most transparent way of coding an open proposition is by an open clause, i.e. a clause which lacks the syntactic component that would occupy the open position. Maybe the extrafocal clause of a cleft-sentence, as in , is the clearest example.

Given that in a cleft-sentence, the focus constituent is extracted from its clause, it leaves an empty place there, indicated by y in .

Since they are syntactically incomplete, open clauses are dependent clauses. Of course, there are possibilities to leave some reference open in an independent sentence, too, as in someone bought a dress yesterday. Here, however, the variable in question is bound in the same clause. Apart from these minimal conditions, there is no biunique mapping of the semantic configuration of ‘open proposition’ onto some grammatical construction. Three linguistic properties of relevant constructions are discussed in the following sections.

External binding

The position that is empty in an open clause may or may not be interpreted as occupied by a referent identified in the context. In , it is clear that the position indicated by y is interpreted as occupied by ‘a dress’. In this sense, the variable y is bound outside the open clause. This is compatible with the notion of an open clause. Another construction involving an open clause whose empty place is interpreted as occupied by a referent mentioned outside the clause is the external-head relative construction illustrated by .

The “gap” strategy of relative clause formation simply leaves the relativized position empty; and just as in other relativization strategies, it is interpreted as occupied by the nucleus. On the other hand, there are constructions that allow one to leave in the dark what would otherwise be an argument of their predicate, as in .

The subject position of the subordinate construction in is open, and it would be easy to supply a context compatible with the speaker's intention to leave the referent of x open, too.

Finiteness

As already suggested by the foregoing examples, an open clause may be finite, as in and , or non-finite, as in . In the latter example, occupancy of the open position is left in the dark. The very same non-finite construction in has its open position semantically occupied by the referent of the subject of the matrix clause.

Designating an open proposition is a typical function of non-finite constructions.

Orientation

As explained in the dedicated section, both a closed and an open clause may be oriented towards one of its syntactic positions. Among the open clauses of the above example series, those of , and are not oriented, while the subordinate clauses of are oriented towards their object and subject position, resp.

¹ An elementary introduction to this topic is elsewhere.