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One aspect of semantics is concerned with the reference that a sign has to its object, which is called its ''denotation''.  For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed.  Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects.  In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

One aspect of semantics is concerned with the reference that a sign has to its object, which is called its ''denotation''.  For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed.  Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects.  In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

The dyadic relation that constitutes the ''denotative component'' of a sign relation <math>L</math> is written <math>\operatorname{Den}(L)</math>.  Information about the denotative component of semantics can be derived from <math>L</math> by taking its ''dyadic projection'' on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, <math>\operatorname{proj}_{OS} L</math>, <math>L_{OS}</math>, or <math>L_{12}</math>, and defined as follows:

The dyadic relation that constitutes the ''denotative component'' of a sign relation <math>L</math> is denoted <math>\operatorname{Den}(L)</math>.  Information about the denotative component of semantics can be derived from <math>L</math> by taking its ''dyadic projection'' on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, <math>\operatorname{proj}_{OS} L</math>, <math>L_{OS}</math>, or <math>L_{12}</math>, and defined as follows:

: <math>\operatorname{Den}(L) = \operatorname{proj}_{OS} L = L_{OS} = \{ (o, s) \in O \times S : (o, s, i) \in L ~\text{for some}~ i \in I \}</math>.

: <math>\operatorname{Den}(L) = \operatorname{proj}_{OS} L = L_{OS} = \{ (o, s) \in O \times S : (o, s, i) \in L ~\text{for some}~ i \in I \}</math>.

The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object.  As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations.

The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object.  As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations.

The connection that a sign makes to an interpretant is called its ''connotation''. In the general theory of sign relations, this aspect of semantics includes the references that a sign has to affects, concepts, impressions, intentions, mental ideas, and to the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct.  This complex ecosystem of references is unlikely ever to be mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the ''connotative'' import of language.  Given a particular sign relation ''L'', the dyadic relation that constitutes the ''connotative component'' of ''L'' is denoted ''Con''(''L'').

The connection that a sign makes to an interpretant is called its ''connotation''. In the general theory of sign relations, this aspect of semantics includes the references that a sign has to affects, concepts, impressions, intentions, mental ideas, and to the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct.  This complex ecosystem of references is unlikely ever to be mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the ''connotative'' import of language.  Given a particular sign relation <math>L</math>, the dyadic relation that constitutes the ''connotative component'' of <math>L</math> is denoted <math>\operatorname{Con}(L)</math>.

The bearing that an interpretant has toward a common object of its sign and itself has no standard name.  If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this omits the mediational character of the whole transaction.

The bearing that an interpretant has toward a common object of its sign and itself has no standard name.  If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this omits the mediational character of the whole transaction.

Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory glosses on objective scenes and their descriptive texts, it is easy to regard them as ''annotations'' both of objects and of signs, but this function points in the opposite direction to what is needed in this connection.  What does one call the inverse of the annotation function?  More generally asked, what is the converse of the annotation relation?

Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory glosses on objective scenes and their descriptive texts, it is easy to regard them as ''annotations'' both of objects and of signs, but this function points in the opposite direction to what is needed in this connection.  What does one call the inverse of the annotation function?  More generally asked, what is the converse of the annotation relation?

In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned dimension of semantics.  On a trial basis, I refer to it as the ''ideational'', the ''intentional'', or the ''canonical'' component of the sign relation, and I provisionally refer to the reference of an interpretant sign to its object as its ''ideation'', its ''intention'', or its ''conation''.  Given a particular sign relation ''L'', the dyadic relation that constitutes the ''intentional component'' of ''L'' is denoted ''Int''(''L'').

In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned dimension of semantics.  On a trial basis, I refer to it as the ''ideational'', the ''intentional'', or the ''canonical'' component of the sign relation, and I provisionally refer to the reference of an interpretant sign to its object as its ''ideation'', its ''intention'', or its ''conation''.  Given a particular sign relation <math>L</math>, the dyadic relation that constitutes the ''intentional component'' of <math>L</math> is denoted <math>\operatorname{Int}(L)</math>.

A full consideration of the connotative and intentional aspects of semantics would force a return to difficult questions about the true nature of the interpretant sign in the general theory of sign relations.  It is best to defer these issues to a later discussion.  Fortunately, omission of this material does not interfere with understanding the purely formal aspects of the present example.

A full consideration of the connotative and intentional aspects of semantics would force a return to difficult questions about the true nature of the interpretant sign in the general theory of sign relations.  It is best to defer these issues to a later discussion.  Fortunately, omission of this material does not interfere with understanding the purely formal aspects of the present example.