Changes

===6.28. The Bridge : From Obstruction to Opportunity===

===6.28. The Bridge : From Obstruction to Opportunity===

There are many reasons for using intensional representations to describe formal objects, especially as the size and complexity of these objects grows beyond the bounds of finite information capacities to represent them in practical terms.  This is extremely pertinent to the progress of the present discussion.  As often happens, when a top-down investigation of complex families of formal objects actually succeeds in arriving at examples that are simple enough to contemplate in extensional terms, it can be difficult to see the relation of such impoverished examples to the cases of original interest, all of them typically having infinite cardinality and indefinite complexity.  In short, once a discussion is brought down to the level of its smallest cases it can be nearly impossible to bring it back up to the level of its intended application.  Without invoking intensional representations of sign relations there is little hope that this discussion can rise far beyond its present level, eternally elaborating the subtleties of cases as elementary as <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math>

There are many reasons for using IRs to describe formal objects, especially as the size and complexity of these objects grows beyond the bounds of finite information capacities to represent in practical terms.  This is extremely pertinent to the progress of the present discussion.  As often happens, when a top down investigation of complex families of formal objects actually succeeds in arriving at examples that are simple enough to contemplate in extensional terms, it can be difficult to see the relation of such impoverished examples to the cases of original interest, all of them typically having infinite cardinality and indefinite complexity.  In short, once a discussion is brought down to the level of its smallest cases it can be nearly impossible to bring it back up to the level of its intended application.  Without invoking IRs of sign relations there is little hope that this discussion can rise far beyond its present level, eternally elaborating the subtleties of cases as elementary as A and B.

There are many obstacles to building this bridge, but if these forms of obstruction are understood in the proper fashion, it is possible to use them as stepping stones, to capitalize on their redoubtable structures, and to convert their recalcitrant materials into a formal calculus that can serve the aims and means of instruction.

There are many obstacles to building this bridge, but if these forms of obstruction are understood in the proper fashion, it is possible to use them as stepping stones, to capitalize on their redoubtable structures, and to convert their recalcitrant materials into a formal calculus that can serve the aims and means of instruction.

This approach requires me to consider a chain of relationships that connects signs, names, concepts, properties, sets, and objects, along with various ways that these classes of entities have been viewed at different periods in the development of mathematical logic.

This approach requires me to consider a chain of relationships that connects signs, names, concepts, properties, sets, and objects, along with various ways that these classes of entities have been viewed at different periods in the development of mathematical logic.

I would like to begin by giving an "impressionistic capsule history" of the relevant developments in mathematical logic, admittedly as viewed from a certain perspective, but hoping to allow room for alternative persepectives to have their way/ present themselves in their own best light.

I would like to begin by giving an &ldquo;impressionistic capsule history&rdquo; of the relevant developments in mathematical logic, admittedly as viewed from a certain perspective, but hoping to allow room for alternative perspectives to have their way and present themselves in their own best light.

The human mind, boggling at the many to many relation between objects and signs that it finds in the world as soon as it begins to reflect on its own reasoning process, hits upon the strategy of interposing a realm of intermediate nodes between objects and signs, and looking through this medium for ways to factor the original relation into simpler components.

Variant 1.  The human mind, boggling at the many to many relation between objects and signs that it finds in the world as soon as it begins to reflect on its own reasoning process, hits upon the strategy of interposing a realm of intermediate nodes between objects and signs, and looking through this medium for ways to factor the original relation into simpler components.

At the beginning of logic, the human mind, as soon as it begins to reflect on its own reasoning process, boggles at the many to many relation between objects and signs that it finds itself conducting through the world.

Variant 2.  At the beginning of logic, the human mind, as soon as it begins to reflect on its own reasoning process, boggles at the many to many relation between objects and signs that it finds itself conducting through the world.

There are two methods for attempting to disentangle this confusion that are generally tried, the first more rarely, the second quite frequently, though apparently in opposite proportion to their respective chances of actual success.  In order to describe the rationales of these methods I need to introduce a number of technical concepts.

There are two methods for attempting to disentangle this confusion that are generally tried, the first more rarely, the second quite frequently, though apparently in opposite proportion to their respective chances of actual success.  In order to describe the rationales of these methods I need to introduce a number of technical concepts.

Suppose that P and Q are dyadic relations, with P c XxY and Q c YxZ.  Then the "contension" of P and Q is a triadic relation R c XxYxZ that is notated as R = P&Q and defined as follows:

Suppose that P and Q are dyadic relations, with P c XxY and Q c YxZ.  Then the "contension" of P and Q is a triadic relation R c XxYxZ that is notated as R = P&Q and defined as follows: