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Now if a sentence <math>s\!</math> really denotes a proposition <math>q,\!</math> and if the notation <math>^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime}</math> is merely meant to supply another name for the proposition that <math>s\!</math> already denotes, then why is there any need for the additional notation?  It is because the interpretive mind habitually races from the sentence <math>s,\!</math> through the proposition <math>q\!</math> that it denotes, and on to the set <math>Q = q^{-1} (\underline{1})</math>  that the proposition <math>q\!</math> indicates, often jumping to the conclusion that the set <math>Q\!</math> is the only thing that the sentence <math>s\!</math> is intended to denote.  This higher order sign situation and the mind's inclination when placed in its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from <math>s\!</math> to <math>q\!</math> to <math>Q.\!</math>

Now if a sentence S really denotes a proposition P, and if the notation "[S]" is merely meant to supply another name for the proposition that S already denotes, then why is there any need for the additional notation?  It is because the interpretive mind habitually races from the sentence S, through the proposition P that it denotes, and on to the set X = P-1(1) that the proposition P indicates, often jumping to the conclusion that the set X is the only thing that the sentence S is intended to denote.  This HO sign situation and the mind's inclination when placed within its setting calls for a linguistic mechanism or a notational device that is capable of analyzing the compound action and controlling its articulate performance, and this requires a way to interrupt the flow of assertion that typically takes place from S to P to X.

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=====1.3.10.8.  The Cactus Patch=====

=====1.3.10.8.  The Cactus Patch=====