Changes

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.

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 <math>(o, s, i)\!</math> 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 &ldquo;tantamount to&rdquo; 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.

In particular, it is not absolutely necessary for a sign relation to contain a HU sign in order for it to contain a description of itself or another sign relation.  As long the sign relation is "content" to maintain its reference to the object sign relation in the form of a constant name, then it suffices to use a HI sign that makes a PIR to all of its triples.

In particular, it is not absolutely necessary for a sign relation to contain a HU sign in order for it to contain a description of itself or another sign relation.  As long the sign relation is &ldquo;content&rdquo; to maintain its reference to the object sign relation in the form of a constant name, then it suffices to use a HI sign that makes a PIR to all of its triples.

In the theory of sign relations, as in formal language theory, one tends to spend a lot of the time talking about signs as objects.  Doing this requires one to have signs for denoting signs and ways of telling when a sign is being used as a sign or is just being mentioned as an object.  Generally speaking, reflection on the usage of an established order of signs recruits another order of signs to denote them, and then another, and another, until a limit on one's powers of reflection is ultimately reached, and finally one is forced to conduct one's meaning in forms of interpretive practice that fail to be fully reflective in one critical respect or another.  In the last resort one resigns oneself to letting the recourse of signs be guided by casually intuited inklings of their potential senses.

In the theory of sign relations, as in formal language theory, one tends to spend a lot of the time talking about signs as objects.  Doing this requires one to have signs for denoting signs and ways of telling when a sign is being used as a sign or is just being mentioned as an object.  Generally speaking, reflection on the usage of an established order of signs recruits another order of signs to denote them, and then another, and another, until a limit on one's powers of reflection is ultimately reached, and finally one is forced to conduct one's meaning in forms of interpretive practice that fail to be fully reflective in one critical respect or another.  In the last resort one resigns oneself to letting the recourse of signs be guided by casually intuited inklings of their potential senses.

In this text a number of linguistic devices are used to assist the faculty of reflection, hopefully forestalling the relegation of its powers to its own natural resources for a long enough spell to observe its action.  Two of the most frequently used strategies toward this purpose can be described as follows:

In this text a number of linguistic devices are used to assist the faculty of reflection, hopefully forestalling the relegation of its powers to its own natural resources for a long enough spell to observe its action.  Two of the most frequently used strategies toward this purpose can be described as follows: