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# In a direction of ''vagueness'', with ''vague'' signs and concepts, one loses a degree of security as to exactly what property the sign or concept implies in the current context, and thus this can be classified as an intensional mode of abstraction.

# In a direction of ''vagueness'', with ''vague'' signs and concepts, one loses a degree of security as to exactly what property the sign or concept implies in the current context, and thus this can be classified as an intensional mode of abstraction.

The first order of business is to draw some distinctions, and at the same time to note some continuities, between the varieties of partiality that remain to be sufficiently clarified and the more mundane brands of partiality that are already familiar enough for present purposes, but lack perhaps only the formality of being recognized under that heading.

The first order of business is to draw some distinctions, and at the same time to note some continuities, between the varieties of partiality that remain to be sufficiently clarified and the more mundane brands of partiality that are already familiar enough for present purposes, but lack perhaps only the formality of being recognized under that heading.

The most familiar illustrations of information theoretic "partiality", "partial indication", or "signs bearing partial information about objects" occur every time one uses a general name, for example, the name of a genus, class, or set.  Almost as commonly, the formula that expresses a logical proposition can be regarded as a partial specification of its logical models or satisfying interpretations.  Just as the name of a genus or class can be taken as a "partially informed reference" or a "plural indefinite reference" (PIR) to one of its species or elements, so the name of an n place relation can be viewed as a PIR to one of its elementary relations or n tuples, and the formula or expression of a proposition can be understood as a PIR to one its models or satisfying interpretations.  For brevity, this variety of referential indetermination can be called the "generic partiality" of signs as information bearers.

The most familiar illustrations of information-theoretic partiality, partial indication, or &ldquo;signs bearing partial information about objects&rdquo; occur every time one uses a general name, for example, the name of a class, genus, or set.  Almost as commonly, the formula that expresses a logical proposition can be regarded as a partial specification of its logical models or satisfying interpretations.  Just as the name of a class or genus can be taken as a ''partially informed reference'' or a ''plural indefinite reference'' (PIR) to one of its elements or species, so the name of an <math>n\!</math>-place relation can be viewed as a PIR to one of its elementary relations or <math>n\!</math>-tuples, and the formula or expression of a proposition can be understood as a PIR to one its models or satisfying interpretations.  For brevity, this variety of referential indetermination can be called the ''generic partiality'' of signs as information bearers.

Note.  In this discussion I will not systematically distinguish between the logical entity typically called a "proposition" or "statement" and the syntactic entity usually called an "expression", "formula", or "sentence".  Instead, I work on the assumption that both types of entity are always involved in everything one proposes and also on the hope that context will determine which aspect of proposing is most apt.  For precision, the abstract category of propositions proper will have to be reconstituted as logical equivalence classes of syntactically diverse expressions.  For the present, I will use the phrase "propositional expression" whenever it is necessary to call particular attention to the syntactic entity.  Likewise, I will not always separate "higher order propositions" (HOPs), that is, propositions about propositions, from their corresponding formulations in the guise of "higher order propositional expressions" (HOPEs).

Note.  In this discussion I will not systematically distinguish between the logical entity typically called a "proposition" or "statement" and the syntactic entity usually called an "expression", "formula", or "sentence".  Instead, I work on the assumption that both types of entity are always involved in everything one proposes and also on the hope that context will determine which aspect of proposing is most apt.  For precision, the abstract category of propositions proper will have to be reconstituted as logical equivalence classes of syntactically diverse expressions.  For the present, I will use the phrase "propositional expression" whenever it is necessary to call particular attention to the syntactic entity.  Likewise, I will not always separate "higher order propositions" (HOPs), that is, propositions about propositions, from their corresponding formulations in the guise of "higher order propositional expressions" (HOPEs).