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Revision as of 17:58, 4 February 2009
, 17:58, 4 February 2009→Derived Equivalence Relations: mathematical markup
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The ''denotation of <math>x\!</math> in <math>L,\!</math>'' written <math>\operatorname{Den}(L, x),</math> is defined as follows:
The "denotation of x in R", written "Den(R, x)", is defined as follows:
Den(R, x) = {o C O : <o, x> C Den(R)}.
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Den}(L, x) ~=~ \{ o \in O ~:~ (o, x) \in \operatorname{Den}(L) \}.</math>
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In other words:
In other words:
Den(R, x) = {o C O : <o, x, i> C R for some i C I}.
{| align="center" cellpadding="8" width="90%"
Equivalent expressions for this concept are recorded in Definition 12.
| <math>\operatorname{Den}(L, x) ~=~ \{ o \in O ~:~ (o, x, i) \in L ~\text{for some}~ i \in I \}.</math>
|}
Equivalent expressions for this concept are recorded in Definition 12.
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Definition 12
Definition 12
D12h. {o C O : <o, x, i> C R for some i C I}
D12h. {o C O : <o, x, i> C R for some i C I}
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Signs are ''equiferent'' if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be ''denotatively equivalent'' or ''referentially equivalent'', but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms.
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To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain.
To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain.
