Changes

One characteristic of Peirce's definition is crucial in supplying a flexible infrastructure that makes the formal and mathematical treatment of sign relations possible.  Namely, this definition allows objects to be characterized in two alternative ways that are substantially different in the domains they involve but roughly equivalent in their information content.  Namely, objects of signs, that may exist in a reality exterior to the sign domain, insofar as they fall  under this definition, allow themselves to be reconstituted nominally or reconstructed rationally as equivalence classes of signs.  This transforms the actual relation of signs to objects, the relation or correspondence that is preserved in passing from initial signs to interpreting signs, into the membership relation that signs bear to their semantic equivalence classes.  This transformation of a relation between signs and the world into a relation interior to the world of signs may be regarded as a kind of representational reduction in dimensions, like the foreshortening and planar projections that are used in perspective drawing.

One characteristic of Peirce's definition is crucial in supplying a flexible infrastructure that makes the formal and mathematical treatment of sign relations possible.  Namely, this definition allows objects to be characterized in two alternative ways that are substantially different in the domains they involve but roughly equivalent in their information content.  Namely, objects of signs, that may exist in a reality exterior to the sign domain, insofar as they fall  under this definition, allow themselves to be reconstituted nominally or reconstructed rationally as equivalence classes of signs.  This transforms the actual relation of signs to objects, the relation or correspondence that is preserved in passing from initial signs to interpreting signs, into the membership relation that signs bear to their semantic equivalence classes.  This transformation of a relation between signs and the world into a relation interior to the world of signs may be regarded as a kind of representational reduction in dimensions, like the foreshortening and planar projections that are used in perspective drawing.

This definition takes as its subject a certain three-place relation, the sign relation proper, envisioned to consist of a certain set of three-tuples.  The pattern of the data in this set of three-tuples, the extension of the sign relation, is expressed here in the form:  ‹Object, Sign, Interpretant›.  As a schematic notation for various sign relations, the letters "s", "o", "i" serve as typical variables ranging over the relational domains of signs, objects, interpretants, respectively.  There are two customary ways of understanding this abstract sign relation as its structure affects concrete systems.

This definition takes as its subject a certain three-place relation, the sign relation proper, envisioned to consist of a certain set of three-tuples.  The pattern of the data in this set of three-tuples, the extension of the sign relation, is expressed here in the form:  ‹Object, Sign, Interpretant›.  As a schematic notation for various sign relations, the letters "''s''", "''o''", "''i''" serve as typical variables ranging over the relational domains of signs, objects, interpretants, respectively.  There are two customary ways of understanding this abstract sign relation as its structure affects concrete systems.

In the first version the agency of a particular interpreter is taken into account as an implicit parameter of the relation.  As used here, the concept of interpreter includes everything about the context of a sign's interpretation that affects its determination.  In this view a specification of the two elements of sign and interpreter is considered to be equivalent information to knowing the interpreting or the interpretant sign, that is, the affect that is produced ''in'' or the effect that is produced ''on'' the interpreting system.  Reference to an object or to an objective, whether it is successful or not, involves an orientation of the interpreting system and is therefore mediated by affects ''in'' and effects ''on'' the interpreter.  Schematically, a lower case "j" can be used to represent the role of a particular interpreter.  Thus, in this first view of the sign relation the fundamental pattern of data that determines the relation can be represented in the form ‹o, s, j› or ‹s, o, j›, as one chooses.

In the first version the agency of a particular interpreter is taken into account as an implicit parameter of the relation.  As used here, the concept of interpreter includes everything about the context of a sign's interpretation that affects its determination.  In this view a specification of the two elements of sign and interpreter is considered to be equivalent information to knowing the interpreting or the interpretant sign, that is, the affect that is produced ''in'' or the effect that is produced ''on'' the interpreting system.  Reference to an object or to an objective, whether it is successful or not, involves an orientation of the interpreting system and is therefore mediated by affects ''in'' and effects ''on'' the interpreter.  Schematically, a lower case "''j''" can be used to represent the role of a particular interpreter.  Thus, in this first view of the sign relation the fundamental pattern of data that determines the relation can be represented in the form ‹''o'', ''s'', ''j''› or ‹''s'', ''o'', ''j''›, as one chooses.

In the second version of the sign relation the interpreter is considered to be a hypostatic abstraction from the actual process of sign transformation.  In other words, the interpreter is regarded as a convenient construct that helps to personify the action but adds nothing informative to what is more simply observed as a process involving successive signs.  An interpretant sign is merely the sign that succeeds another in a continuing sequence.  What keeps this view from falling into sheer nominalism is the relation with objects that is preserved throughout the process of transformation.  Thus, in this view of the sign relation the fundamental pattern of data that constitutes the relationship can be indicated by the optional forms ‹o, s, i› or ‹s, i, o›.

In the second version of the sign relation the interpreter is considered to be a hypostatic abstraction from the actual process of sign transformation.  In other words, the interpreter is regarded as a convenient construct that helps to personify the action but adds nothing informative to what is more simply observed as a process involving successive signs.  An interpretant sign is merely the sign that succeeds another in a continuing sequence.  What keeps this view from falling into sheer nominalism is the relation with objects that is preserved throughout the process of transformation.  Thus, in this view of the sign relation the fundamental pattern of data that constitutes the relationship can be indicated by the optional forms ‹''o'', ''s'', ''i''› or ‹''s'', ''i'', ''o''›.

Viewed as a totality, a complete sign relation would have to consist of all of those conceivable moments — past, present, prospective, or in whatever variety of parallel universes that one may care to admit — when something means something to somebody, in the pattern ‹s, o, j›, or when something means something about something, in the pattern ‹s, i, o›.  But this ultimate sign relation is not often explicitly needed, and it could easily turn out to be logically and set-theoretically ill-defined.  In physics, it is important for theoretical completeness to regard the whole universe as a single physical system, but more common to work with "isolated" subsystems.  Likewise in the theory of signs, only particular and well-bounded subsystems of the ultimate sign relation are likely to be the subjects of sensible discussion.

Viewed as a totality, a complete sign relation would have to consist of all of those conceivable moments — past, present, prospective, or in whatever variety of parallel universes that one may care to admit — when something means something to somebody, in the pattern ‹''s'', ''o'', ''j''›, or when something means something about something, in the pattern ‹''s'', ''i'', ''o''›.  But this ultimate sign relation is not often explicitly needed, and it could easily turn out to be logically and set-theoretically ill-defined.  In physics, it is important for theoretical completeness to regard the whole universe as a single physical system, but more common to work with "isolated" subsystems.  Likewise in the theory of signs, only particular and well-bounded subsystems of the ultimate sign relation are likely to be the subjects of sensible discussion.

It is helpful to view the definition of individual sign relations on analogy with another important class of three-place relations of broad significance in mathematics and far-reaching application in physics:  namely, the binary operations or ternary relations that fall under the definition of abstract groups.  Viewed as a definition of individual groups, the axioms defining a group are what logicians would call highly non-categorical, that is, not every two models are isomorphic (Wilder, p. 36).  But viewing the category of groups as a whole, if indeed it can be said to form a whole (MacLane, 1971), the definition allows a vast number of non-isomorphic objects, namely, the individual groups.

It is helpful to view the definition of individual sign relations on analogy with another important class of three-place relations of broad significance in mathematics and far-reaching application in physics:  namely, the binary operations or ternary relations that fall under the definition of abstract groups.  Viewed as a definition of individual groups, the axioms defining a group are what logicians would call highly non-categorical, that is, not every two models are isomorphic (Wilder, p. 36).  But viewing the category of groups as a whole, if indeed it can be said to form a whole (MacLane, 1971), the definition allows a vast number of non-isomorphic objects, namely, the individual groups.