Changes

In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having '''I''' ⊆ '''S'''.  In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''.  In the forthcoming examples, '''S''' and '''I''' are identical as sets, so the very same elements manifest themselves in two different roles of the sign relations in question.  When it is necessary to refer to the whole set of objects and signs in the union of the domains '''O''', '''S''', and '''I''' for a given sign relation '''L''', one may refer to this set as the ''world of '''L''''' and write '''W''' = '''W'''_'''L''' = '''O''' ∪ '''S''' ∪ '''I'''.

In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having '''I''' ⊆ '''S'''.  In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''.  In the forthcoming examples, '''S''' and '''I''' are identical as sets, so the very same elements manifest themselves in two different roles of the sign relations in question.  When it is necessary to refer to the whole set of objects and signs in the union of the domains '''O''', '''S''', and '''I''' for a given sign relation '''L''', one may refer to this set as the ''world of '''L''''' and write '''W''' = '''W'''_'''L''' = '''O''' ∪ '''S''' ∪ '''I'''.

To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible when the examples become more complicated, I introduce the following abbreviations:

To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations:

O = Object Domain,

| align="center" | '''O''' || = || Object Domain

S = Sign Domain,

I = Interpretant Domain.

| align="center" | '''S''' || = || Sign Domain

| align="center" | '''I''' || = || Interpretant Domain

O = { Ann, Bob } = { A, B }.

Introducing a few abbreviations for use in considering the present Example, we have the following data:

S = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}.

| align="center" | '''S'''

| =

| {"Ann", "Bob", "I", "You"}

| =

| {"A", "B", "i", "u"}

In the present Example, '''S''' = '''I''' = Syntactic Domain.

 

In the present examples, ''S'' = ''I'' = Syntactic Domain.

Tables 1 and 2 give the sign relations associated with the interpreters ''A'' and ''B'', respectively, putting them in the form of relational databases.  Thus, the rows of each Table list the ordered triples of the form ‹''o'', ''s'', ''i''› that make up the corresponding sign relations:  ''A'', ''B'' ⊆ ''O''×''S''×''I''.  The issue of using the same names for objects and for relations involving these objects will be taken up later, after the less problematic features of these relations have been treated.

Tables 1 and 2 give the sign relations associated with the interpreters ''A'' and ''B'', respectively, putting them in the form of relational databases.  Thus, the rows of each Table list the ordered triples of the form ‹''o'', ''s'', ''i''› that make up the corresponding sign relations:  ''A'', ''B'' ⊆ ''O''×''S''×''I''.  The issue of using the same names for objects and for relations involving these objects will be taken up later, after the less problematic features of these relations have been treated.