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Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1 (view source)
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, 12:56, 15 September 2010→1.3.4.3. Semiotic Equivalence Relations
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To discuss these types of situations further, I introduce the square bracket notation "[''x'']<sub>''E''</sub>" for "the equivalence class of the element ''x'' under the equivalence relation ''E''". A statement that the elements ''x'' and ''y'' are equivalent under ''E'' is called an ''equation'', and can be written in either one of two ways, as [''x'']<sub>''E''</sub> = [''y'']<sub>''E''</sub> or as ''x'' =<sub>''E''</sub> ''y''.
To discuss these types of situations further, I introduce the square bracket notation <math>[x]_E</math> for ''the equivalence class of the element <math>x</math> under the equivalence relation <math>E</math>''. A statement that the elements <math>x</math> and <math>y</math> are equivalent under <math>E</math> is called an ''equation'', and can be written in either one of two ways, as <math>[x]_E = [y]_E</math> or as <math>x =_E y</math>.
In the application to sign relations I extend this notation in the following ways. When ''L'' is a sign relation whose ''syntactic projection'' or connotative component ''L''<sub>''SI''</sub> is an equivalence relation on ''S'', I write "[''s'']<sub>''L''</sub>" for "the equivalence class of ''s'' under ''L''<sub>''SI''</sub>". A statement that the signs ''x'' and ''y'' are synonymous under a semiotic equivalence relation ''L''<sub>''SI''</sub> is called a ''semiotic equation'' (SEQ), and can be written in either of the forms: [''x'']<sub>''L''</sub> = [''y'']<sub>''L''</sub> or as ''x'' =<sub>''L''</sub> ''y''.
In the application to sign relations I extend this notation in the following ways. When <math>L</math> is a sign relation whose ''syntactic projection'' or connotative component <math>L_{SI}</math> is an equivalence relation on <math>S</math>, I write <math>[s]_L</math> for ''the equivalence class of <math>s</math> under <math>L_{SI}</math>''. A statement that the signs <math>x</math> and <math>y</math> are synonymous under a semiotic equivalence relation <math>L_{SI}</math> is called a ''semiotic equation'' (SEQ), and can be written in either of the forms: <math>[x]_L = [y]_L</math> or <math>x =_L y</math>.
In many situations there is one further adaptation of the square bracket notation that can be useful. Namely, when there is known to exist a particular triple ‹o, s, i› ∈ ''L'', it is permissible to use "[''o'']<sub>''L''</sub>" to mean the same thing as "[''s'']<sub>''L''</sub>". These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.
In many situations there is one further adaptation of the square bracket notation that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i) \in L</math>, it is permissible to use <math>[o]_L</math> to mean the same thing as <math>[s]_L</math>. These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.
In these terms, the SER for interpreter ''A'' yields the semiotic equations:
In these terms, the SER for interpreter <math>\text{A}</math> yields the semiotic equations:
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{| cellpadding=4 style="text-align:center"