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, 20:16, 3 May 2012→6.10. Higher Order Sign Relations : Examples: markup
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Finally, the reflectively extended naming function <math>\operatorname{Nom}' : O' \to S'\!</math> is defined as <math>\operatorname{Nom}' = \operatorname{Nom} \cup Nom_1.\!</math>
Finally, the reflectively extended naming function <math>\operatorname{Nom}' : O' \to S'\!</math> is defined as <math>\operatorname{Nom}' = \operatorname{Nom} \cup \operatorname{Nom}_1.\!</math>
A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.
A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.
In the present context an application of the arch notation, for example, <math>{}^{\langle} x {}^{\rangle},\!</math> is read on analogy with the use of any other functional notation, for example, <math>f(x),\!</math> where <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is the name of a function <math>f,\!</math> <math>{}^{\backprime\backprime} f(~) {}^{\prime\prime}\!</math> is the context of its application, <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is the name of an argument <math>x,\!</math> and where the functional abstraction <math>{}^{\backprime\backprime} x \mapsto f(x) {}^{\prime\prime}\!</math> is just another name for the function <math>f.\!</math>
In the present context an application of the arch notation, for example, <math>{}^{\langle} x {}^{\rangle},\!</math> is read on analogy with the use of any other functional notation, for example, <math>f(x),\!</math> where <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is the name of a function <math>f,\!</math> <math>{}^{\backprime\backprime} f(~) {}^{\prime\prime}\!</math> is the context of its application, <math>{}^{\backprime\backprime} x {}^{\prime\prime}\!</math> is the name of an argument <math>x,\!</math> and where the functional abstraction <math>{}^{\backprime\backprime} x \mapsto f(x) {}^{\prime\prime}\!</math> is just another name for the function <math>f.\!</math>
It is clear that some form of functional abstraction is being invoked in the above definition of <math>\operatorname{Nom}_1.\!</math> Otherwise, the expression <math>x \mapsto {}^{\langle} x {}^{\rangle}\!</math> would indicate a constant function, one that maps every <math>x\!</math> in its domain to the same code or sign for the letter <math>{}^{\backprime\backprime} x {}^{\prime\prime}.\!</math> But if this is allowed, then it seems either to invoke a more powerful concept, ''lambda abstraction'', than the one being defined or else to attempt an improper definition of the naming function in terms of itself.
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Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in actuality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of HO signs.
Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in actuality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of HO signs.
