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The dyadic relation that constitutes the ''denotative component'' of a sign relation ''L'' is denoted ''Den''(''L'').  Information about the denotative component of semantics can be derived from ''L'' 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, ''Proj''_''OS'' ''L'', ''L''_''OS'' , or ''L''₁₂ , and defined as follows:

The dyadic relation that constitutes the ''denotative component'' of a sign relation ''L'' is denoted ''Den''(''L'').  Information about the denotative component of semantics can be derived from ''L'' 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, ''Proj''_''OS'' ''L'', ''L''_''OS'' , or ''L''₁₂ , and defined as follows:

: ''Den''(''L'') = ''Proj''_''OS'' ''L'' = ''L''_''OS'' = {‹''o'', ''s''› ∈ ''O'' × ''S'' : ‹''o'', ''s'', ''i''› ∈ ''L'' for some ''i'' ∈ ''I''}.

: ''Den''(''L'') = ''Proj''_''OS'' ''L'' = ''L''_''OS'' = {‹''o'', ''s''› ∈ ''O'' × ''S'' : ‹''o'', ''s'', ''i''› ∈ ''L'' for some ''i'' ∈ ''I''}.

Looking to the denotative aspects of the present example, various rows of the Tables specify that ''A'' uses "i" to denote ''A'' and "u" to denote ''B'', whereas ''B'' uses "i" to denote ''B'' and "u" to denote ''A''.  It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.

Looking to the denotative aspects of the present example, various rows of the Tables specify that ''A'' uses "i" to denote ''A'' and "u" to denote ''B'', whereas ''B'' uses "i" to denote ''B'' and "u" to denote ''A''.  It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.

The connotative component of a sign relation ''L'' can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

The connotative component of a sign relation ''L'' can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

: ''Con''(''L'') = ''Proj''_''SI'' ''L'' = ''L''_''SI'' = {‹''s'', i› ∈ ''S'' × ''I'' : ‹''o'', ''s'', ''i''› ∈ ''L'' for some ''o'' ∈ ''O''}.

: ''Con''(''L'') = ''Proj''_''SI'' ''L'' = ''L''_''SI'' = {‹''s'', ''i''› ∈ ''S'' × ''I'' : ‹''o'', ''s'', ''i''› ∈ ''L'' for some ''o'' ∈ ''O''}.

The intentional component of semantics for a sign relation ''L'', or its ''second moment of denotation'', is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

The intentional component of semantics for a sign relation ''L'', or its ''second moment of denotation'', is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

: ''Int''(''L'') = ''Proj''_''OI'' ''L'' = ''L''_''OI'' = {‹o, i› ∈ ''O'' × ''I'' : ‹o, s, i› ∈ ''L'' for some ''s'' ∈ ''S''}.

: ''Int''(''L'') = ''Proj''_''OI'' ''L'' = ''L''_''OI'' = {‹''o'', ''i''› ∈ ''O'' × ''I'' : ‹''o'', ''s'', ''i''› ∈ ''L'' for some ''s'' ∈ ''S''}.

As it happens, the sign relations ''L''_''A'' and ''L''_''B'' in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of (''L''_''A'')_''OS''  and (''L''_''B'')_''OS''  is merely echoed in (''L''_''A'')_''OI''  and (''L''_''B'')_''OI'' , respectively.

As it happens, the sign relations ''L''_''A'' and ''L''_''B'' in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of (''L''_''A'')_''OS''  and (''L''_''B'')_''OS''  is merely echoed in (''L''_''A'')_''OI''  and (''L''_''B'')_''OI'' , respectively.