Changes

A further reduction in the number of different kinds of signs to worry about can be achieved by means of a special technique — some may call it an “artful dodge” — for referring indifferently to the elements of a set without referring to the set itself.  Under the designation of a ''plural indefinite reference'' (PIR) is included all the various ways of dealing with denominations, multiple denotations, collective references, or objective multitudes that avail themselves of this trick.

A further reduction in the number of different kinds of signs to worry about can be achieved by means of a special technique — some may call it an “artful dodge” — for referring indifferently to the elements of a set without referring to the set itself.  Under the designation of a ''plural indefinite reference'' (PIR) is included all the various ways of dealing with denominations, multiple denotations, collective references, or objective multitudes that avail themselves of this trick.

By way of definition, a sign <math>q\!</math> in a sign relation <math>L \subseteq O \times S \times I\!</math> is said to be, to constitute, or to make a '''plural indefinite reference''' ('''PIR''') to (every element in) a set of objects, <math>X \subseteq O,\!</math> if and only if <math>q\!</math> denotes every element of <math>X.\!</math> This relationship can be expressed in a succinct formula by making use of one additional definition.

By way of definition, a sign q in a sign relation R c OxSxI is said to be, to constitute, or to make a PIR to (every element in) a set of objects, X c O, if and only if q denotes every element of X.  This relationship can be expressed in a succinct formula by making use of one additional definition.

The "denotation of q in R", written as "De(q, R)", is defined as follows:

The '''denotation''' of <math>q\!</math> in <math>L,\!</math> written <math>\operatorname{De}(q, L),\!</math> is defined as follows:

De(q, R) = Den(R).q = ROS.q = {o C O : <o, q, i> C R, for some i C I}.

| <math>\operatorname{De}(q, L) ~=~ \operatorname{Den}(L) \cdot q ~=~ L_{OS} \cdot q ~=~ \{ o \in O : (o, q, i) \in L, ~\text{for some}~ i \in I \}.</math>

Then q makes a PIR to X in R if and only if X c De(q, R).  Of course, this includes the limiting case where X is a singleton, say X = {o}.  When this happens then the reference is neither plural nor indefinite, properly speaking, but q denotes o uniquely.

Then <math>q\!</math> makes a PIR to <math>X\!</math> in <math>L\!</math> if and only if <math>X \subseteq \operatorname{De}(q, L).\!</math> Of course, this includes the limiting case where <math>X\!</math> is a singleton, say <math>X = \{ o \}.\!</math> In this case the reference is neither plural nor indefinite, properly speaking, but <math>q\!</math> denotes <math>o\!</math> uniquely.

The proper exploitation of PIRs in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes.  This is accomplished by connecting the sides of the distinction in two directions.  First, a HI sign that makes a PIR to many triples of the form <o, s, i> can be taken as tantamount to a HU sign that denotes the corresponding sign relation.  Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple.  The relation of one sign being "tantamount to" another is not exactly a full fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.

The proper exploitation of PIRs in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes.  This is accomplished by connecting the sides of the distinction in two directions.  First, a HI sign that makes a PIR to many triples of the form <o, s, i> can be taken as tantamount to a HU sign that denotes the corresponding sign relation.  Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple.  The relation of one sign being "tantamount to" another is not exactly a full fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.