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On reflection, the observation that appeared just before these last questions arose can be seen to make a very broad claim about a certain class of properties affecting expressions, namely, all those properties that can be analogous to the ordered measures of expressive quality.  For future reference, let me call this the ''monotone assumption'' (MA).  This generatrix of so many future and specious assumptions takes for granted a sweeping claim about the ways that an order of analysis of expressions translates into an order of comparison of their measures under one of these properties.  But this entire and previously unstated assumption is itself just another manner of working hypothesis for the mental procedure or the process of inquiry that makes use of it, and its proper understanding is perhaps better served if it is rephrased as a question:  Can the <math>X</math> of a claim or a concept be greater than the <math>X</math> of the subordinate claims and concepts that it calls on, where <math>{}^{\backprime\backprime} X {}^{\prime\prime}</math> stands for ''certainty'', ''clarity'', or any one of the corresponding class of measures, orders, properties, qualities, or virtues?

On reflection, the observation that appeared just before these last questions arose can be seen to make a very broad claim about a certain class of properties affecting expressions, namely, all those properties that can be analogous to the ordered measures of expressive quality.  For future reference, let me call this the "monotone assumption" (MA).  This generatrix of so many future and specious assumptions takes for granted a sweeping claim about the ways that an order of analysis of expressions translates into an order of comparison of their measures under one of these properties.  But this entire and previously unstated assumption is itself just another manner of working hypothesis for the mental procedure or the process of inquiry that makes use of it, and its proper understanding is perhaps better served if it is rephrased as a question:  Can the X of a claim or a concept be greater than the X of the subordinate claims and concepts that it calls on, where "X" stands for "certainty", "clarity", or any one of the corresponding class of measures, orders, properties, qualities, or virtues?

Rather than taking this claim for granted, suppose I go looking for any properties, that might be similar to certainty or clarity, for which the measure of a whole expression is capable of exceeding the measure of its parts.  Is there an order property that is dependent on the constitution of the whole expression and a function of its analytic constituents but not necessarily tied down to monotonely conservative relationships like the sum, the average, or the lowest common denominator of the measures affecting its syntactic elements?  Once I take the trouble to formulate the question in explicit terms, any number of familiar examples are free to come to mind as fitting its requirements.  Indeed, since the notions of dependency and independence that accompany the use of mathematical functions and mathematical forms of decomposition do not by themselves implicate the more constrained types of dependency and the more radical types of independence that arise in relation and in reaction to the MA, it is rather easy to think of many that will do.

Rather than taking this claim for granted, suppose I go looking for any properties, that might be similar to certainty or clarity, for which the measure of a whole expression is capable of exceeding the measure of its parts.  Is there an order property that is dependent on the constitution of the whole expression and a function of its analytic constituents but not necessarily tied down to monotonically conservative relationships like the sum, the average, or the lowest common denominator of the measures affecting its syntactic elements?  Once I take the trouble to formulate the question in explicit terms, any number of familiar examples are free to come to mind as fitting its requirements.  Indeed, since the notions of dependency and independence that accompany the use of mathematical functions and mathematical forms of decomposition do not by themselves implicate the more constrained types of dependency and the more radical types of independence that arise in relation and in reaction to the MA, it is rather easy to think of many that will do.

=====5.2.11.10. Problems and Methods=====

=====5.2.11.10. Problems and Methods=====