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Using these notions, the customary methods for disentangling a many-to-many relation can be explained as follows:

Using these notions, the customary methods for disentangling a many-to-many relation can be explained as follows:

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In the logic of the ancients, the many-to-one relation of things to general names ...

In the logic of the ancients, the many-to-one relation of things to general names ...

In early approaches to mathematical logic, from Leibniz to Peirce and Frege, one ordinarily spoke of the extensions and intensions of concepts.

In early approaches to mathematical logic, from Leibniz to Peirce and Frege, one ordinarily spoke of the extensions and intensions of concepts.

Typically, one starts a work of bridge building by casting a thin line across the intervening gap, using this expediency to conduct a slightly more substantial linkage over the rift, and then proceeding through a train of successors to draw increasingly stronger connections between the opposing shores until a load bearing framework can be established.  There is an analogue of this operation that fits the current situation, and this is something I can do this by taking up the sign relations A and B, already introduced in extensional terms, and redescribing the abstract features of their structures in intensional terms.

Typically, one starts a work of bridge-building by casting a thin line across the intervening gap, using this expediency to conduct a slightly more substantial linkage over the rift, and then proceeding through a train of successors to draw increasingly stronger connections between the opposing shores until a load-bearing framework can be established.  There is an analogue of this operation that fits the current situation, and this is something I can do this by taking up the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> already introduced in extensional terms, and re-describing the abstract features of their structures in intensional terms.

This would be the ideal plan.  But bridging the "tensions", ex and in , that subsist within the forms of representation is not as easy as that.  In order to convey the importance of the task and provide a motivation for carrying it out, I will plot a chain of relationships that stretches from signs, names, and concepts to properties, sets, and objects.

This would be the ideal plan.  But bridging the &ldquo;tensions&rdquo;, &ldquo;ex-&rdquo; and &ldquo;in-&rdquo;, that subsist within the forms of representation is not as easy as that.  In order to convey the importance of the task and provide a motivation for carrying it out, I will plot a chain of relationships that stretches from signs, names, and concepts to properties, sets, and objects.

As a way of resolving the discriminated "tensions", posed here to fall into "ex " and "in " kinds, the strategy just described affords a way of approaching the problem that is less like a bridge than a pole vault, taking its pivot on a fixed set of narrowly circumscribed sign relations to make a transit from extensional to intensional outlooks on their form.  With time and reflection, the logical depth of the supposed distinction, the "pretension" of maintaining a couple of separate but equal tensions in isolation from each other, does not withstand a persistent probing.  Accordingly, the gulf between the two realms can always be fathomed by a finitely informed creature, in fact, by the very form of interpreter that created the fault in the first place.  Consequently, converting the form of a transient vault into the substance of a usable bridge requires in adjunction only that initially pliable and ultimately tensile sorts of connecting lines be conducted along the tracery of the vault until the work of castling the gap can begin.

As a way of resolving the discerned &ldquo;tensions&rdquo;, posed here to fall into &ldquo;ex-&rdquo; and &ldquo;in-&rdquo; kinds, the strategy just described affords a way of approaching the problem that is less like a bridge than a pole vault, taking its pivot on a fixed set of narrowly circumscribed sign relations to make a transit from extensional to intensional outlooks on their form.  With time and reflection, the logical depth of the supposed distinction, the &ldquo;pretension&rdquo; of maintaining a couple of separate but equal tensions in isolation from each other, does not withstand a persistent probing.  Accordingly, the gulf between the two realms can always be fathomed by a finitely informed creature, in fact, by the very form of interpreter that created the fault in the first place.  Consequently, converting the form of a transient vault into the substance of a usable bridge requires in adjunction only that initially pliable and ultimately tensile sorts of connecting lines be conducted along the tracery of the vault until the work of castling the gap can begin.

In the pragmatic theory of signs, the word "representation" is a technical term that is synonymous with the word "sign", in other words, it applies to an entity in the most general category of things that can enter into sign relations in the roles of signs and interpretants.  Thus, in this usage the scope of the term "representation" includes all sorts of syntactic, descriptive, and conceptual entities, a range of options I will frequently find it convenient to suggest by drawing on a pair of stock phrases:  "terms and concepts" (TACs) in a conjuctive context, versus "terms or concepts" (TOCs) in a disjunctive context.

In the pragmatic theory of signs, the word "representation" is a technical term that is synonymous with the word "sign", in other words, it applies to an entity in the most general category of things that can enter into sign relations in the roles of signs and interpretants.  Thus, in this usage the scope of the term "representation" includes all sorts of syntactic, descriptive, and conceptual entities, a range of options I will frequently find it convenient to suggest by drawing on a pair of stock phrases:  "terms and concepts" (TACs) in a conjuctive context, versus "terms or concepts" (TOCs) in a disjunctive context.