Changes

There is a piece of terminology that is often useful in this connection.  In set-theoretic contexts, either one of the phrases ''X collects Y'' or ''X encases Y'' can be used to mean the same thing as ''Y'' ∈ ''X''.  These formulations can be taken as abbreviated ways of saying that ''X'' enumerates ''Y'' among its cases.  Thus, they express the converse of the membership relation but manage to avoid the ambiguity of the phrase ''X contains Y'', a form that would otherwise have to be qualified on each occasion of its use by specifying whether one means ''contains as an element'' or ''contains as a subset'', as the case may be.

There is a piece of terminology that is often useful in this connection.  In set-theoretic contexts, either one of the phrases ''X collects Y'' or ''X encases Y'' can be used to mean the same thing as ''Y'' ∈ ''X''.  These formulations can be taken as abbreviated ways of saying that ''X'' enumerates ''Y'' among its cases.  Thus, they express the converse of the membership relation but manage to avoid the ambiguity of the phrase ''X contains Y'', a form that would otherwise have to be qualified on each occasion of its use by specifying whether one means ''contains as an element'' or ''contains as a subset'', as the case may be.

To wrap up the development of this reflective project in a single line:  When the mind's original effort to catch itself at work seizes on the inventions of set theory to encapsulate its speculations, the ensuing breed of self reification that comes from mingling an unbridled capacity for self referent expressions with an unchecked propensity for creating abstract objects gives rise to the generation of set theoretic paradoxes.  As a result, it is incumbent on me to show how the concretely limited kinds of constructions that I have in mind can avoid a similar excess and steer clear of the corresponding difficulties.

To wrap up the development of this reflective project in a single line:  When the mind's original effort to catch itself at work seizes on the inventions of set theory to encapsulate its speculations, the ensuing breed of self reification that comes from mingling an unbridled capacity for self referent expressions with an unchecked propensity for creating abstract objects gives rise to the generation of set theoretic paradoxes.  As a result, it is incumbent on me to show how the concretely limited kinds of constructions that I have in mind can avoid a similar excess and steer clear of the corresponding difficulties.

If formalized, a RIF would be an IF that can properly, if only partially, refer to itself as an OF.  Thus, as formalized, a RIF amounts to both a reflexive and a recursive SOI, one that can refer to itself as an object, to the extent that any formal system can.  As a reflexive SOI, a RIF has a sign that refers to itself.  As a recursive SOI, a RIF has a character that can be determined by invoking the record of signs that it uses to refer to simpler versions and earlier developments of itself.

If formalized, a RIF would be an IF that can properly, if only partially, refer to itself as an OF.  Thus, as formalized, a RIF amounts to both a reflexive and a recursive SOI, one that can refer to itself as an object, to the extent that any formal system can.  As a reflexive SOI, a RIF has a sign that refers to itself.  As a recursive SOI, a RIF has a character that can be determined by invoking the record of signs that it uses to refer to simpler versions and earlier developments of itself.

But more than all this, in order to be genuinely reflective a RIF's consideration of itself as a situated form of activity must extend to the consideration of alternative selves.  This means that a RIF must have references to other SOI's, not only those that are continuous with the space of its own potential conduct and correlated to the course of its own form of activity, but also those that are discontinuous from and independent of its own way of being.

But more than all this, in order to be genuinely reflective a RIF's consideration of itself as a situated form of activity must extend to the consideration of alternative selves.  This means that a RIF must have references to other SOIs, not only those that are continuous with the space of its own potential conduct and correlated to the course of its own form of activity, but also those that are discontinuous from and independent of its own way of being.

In keeping with the spirit of a discussion based on concrete examples, the RIF to be improvised here is restrained to the scale of a minimal IF that can reflect on the scene of A and B, in this case, synthesizing a portion of the OF's and IF's suggested by the sign relations A and B into an integrated SOI.  While I do not plan to specify the additional constraints that would be needed to determine this RIF uniquely, even to say whether it is finite or infinite, it forms a convenient reference point for the rest of this section to designate the purported ideal as "the RIF generated by A and B" and to notate it as "RIF (A, B)".

In keeping with the spirit of a discussion based on concrete examples, the RIF to be improvised here is restrained to the scale of a minimal IF that can reflect on the scene of A and B, in this case, synthesizing a portion of the OFs and IFs suggested by the sign relations <math>A\!</math> and <math>B\!</math> into an integrated SOI.  While I do not plan to specify the additional constraints that would be needed to determine this RIF uniquely, even to say whether it is finite or infinite, it forms a convenient reference point for the rest of this section to designate the purported ideal as ''the RIF generated by <math>A\!</math> and <math>B\!</math>'' and to notate it as <math>\operatorname{RIF}(A, B).\!</math>

In accord with the customary figure of speech, a RIF can be personified in the agency of a "reflective interpreter" that possesses the faculties to carry out its actions, and this agent is in turn characterized as the localized representative of a suitably reflective and situated process of interpretation.

In accord with the customary figure of speech, a RIF can be personified in the agency of a "reflective interpreter" that possesses the faculties to carry out its actions, and this agent is in turn characterized as the localized representative of a suitably reflective and situated process of interpretation.