Changes

In general, a good way to specify the meaning of a new notation is by means of a semantic equation, or a system of semantic equations, that expresses the function of the new signs in terms of familiar operations.  If it is merely a matter of introducing new signs for old meanings, then this method is sufficient.  In this vein, the intention and use of the “supercilious notation” for reflecting on signs could have its definition approximated in the following way.

In general, a good way to specify the meaning of a new notation is by means of a semantic equation, or a system of semantic equations, that expresses the function of the new signs in terms of familiar operations.  If it is merely a matter of introducing new signs for old meanings, then this method is sufficient.  In this vein, the intention and use of the “supercilious notation” for reflecting on signs could have its definition approximated in the following way.

<pre>

Let <math>{}^{\langle} x {}^{\rangle} ~=~ {}^{\backprime\backprime} x {}^{\prime\prime}</math> as signs for the object <math>x,\!</math> and let <math>{}^{\langle\langle} x {}^{\rangle\rangle} ~=~ {}^{\langle\backprime\backprime} x {}^{\prime\prime\rangle} ~=~ {}^{\backprime\backprime\langle} x {}^{\rangle\prime\prime}</math> as signs for the object <math>{}^{\backprime\backprime} x {}^{\prime\prime},</math> an object that incidentally happens to be sign.  An alternative way of putting this is to say that the members of the set <math>\{ {}^{\langle} x {}^{\rangle}, {}^{\backprime\backprime} x {}^{\prime\prime} \}</math> are equivalent as signs for the object <math>x,\!</math> while the members of the set <math>\{ {}^{\langle\langle} x {}^{\rangle\rangle}, {}^{\langle\backprime\backprime} x {}^{\prime\prime\rangle}, {}^{\backprime\backprime\langle} x {}^{\rangle\prime\prime} \}</math> are equivalent as signs for the sign <math>{}^{\backprime\backprime} x {}^{\prime\prime}.</math>

Let <x> = "x" as signs for the object x, and let <<x>> = <"x"> = "<x>" as signs for the object "x", an object that incidentally happens to be sign.  An alternative way of putting this is to say that the members of the set {<x>, "x"} are equivalent as signs for the object x, while the members of the set {<<x>>, <"x">, "<x>"} are equivalent as signs for the sign "x".

</pre>

===6.10. Higher Order Sign Relations : Examples===

===6.10. Higher Order Sign Relations : Examples===