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§ 15.   There is an order of logical reasoning that is typically described as ''propositional'' or ''sentential'' and represented in a type of formal system that is commonly known as a ''propositional calculus'' or a ''sentential logic'' (SL).  Any one of these calculi forms an interesting example of a formal language, one that can be used to illustrate all of the preceding issues of style and technique, but one that can also serve this inquiry in a more instrumental fashion.  This section presents the elements of a calculus for propositional logic that I described in earlier work (Awbrey, 1989 and 1994).  The imminent use of this calculus is to construct and analyze logical representations of sign relations, and the treatment here focuses on the concepts and notation that are most relevant to this task.

§ 15.   There is an order of logical reasoning that is typically described as ''propositional'' or ''sentential'' and represented in a type of formal system that is commonly known as a ''propositional calculus'' or a ''sentential logic'' (SL).  Any one of these calculi forms an interesting example of a formal language, one that can be used to illustrate all of the preceding issues of style and technique, but one that can also serve this inquiry in a more instrumental fashion.  This section presents the elements of a calculus for propositional logic that I described in earlier work (Awbrey, 1989 and 1994).  The imminent use of this calculus is to construct and analyze logical representations of sign relations, and the treatment here focuses on the concepts and notation that are most relevant to this task.

The next four sections (§§ 16–19) treat the theme of self-reference that is invoked in the overture to a RIF.  To inspire confidence in the feasibility and the utility of well chosen reflective constructions and to allay a suspicion of self reference in general, it is useful to survey the varieties of self reference that arise in this work and to distinguish the forms of circular referrals that are likely to vitiate consistent reasoning from those that are relatively innocuous and even beneficial.

The next four sections (§§ 16–19) treat the theme of self-reference that is invoked in the overture to a RIF.  To inspire confidence in the feasibility and the utility of well chosen reflective constructions and to allay a suspicion of self-reference in general, it is useful to survey the varieties of self-reference that arise in this work and to distinguish the forms of circular referrals that are likely to vitiate consistent reasoning from those that are relatively innocuous and even beneficial.

§ 16.   Recursive Aspects

§ 16.   Recursive Aspects

§ 17.   Patterns of Self Reference

§ 17.   Patterns of Self-Reference

§ 18.   Practical Intuitions

§ 18.   Practical Intuitions

§ 19.   Examples of Self Reference

§ 19.   Examples of Self-Reference

The intertwined themes of logic and time will occupy center stage for the next eight sections (§§ 20–27).

The intertwined themes of logic and time will occupy center stage for the next eight sections (§§ 20–27).

Robert Burns, A Sonnet Upon Sonnets

Robert Burns, A Sonnet Upon Sonnets

One of the main problems that the present TOI has to address is how a TOI can address the problems of self reference that an inquiry into inquiry involves.  If a sonnet can say something true about sonnets, then a TOI, far less limited in the number and measure of its lines, ought to be able to say something true about TOIs in general, unless the removal of these limitations takes away the only things whereof and whereby it has to speak, the ends and means of its own form of speech.

One of the main problems that the present TOI has to address is how a TOI can address the problems of self-reference that an inquiry into inquiry involves.  If a sonnet can say something true about sonnets, then a TOI, far less limited in the number and measure of its lines, ought to be able to say something true about TOIs in general, unless the removal of these limitations takes away the only things whereof and whereby it has to speak, the ends and means of its own form of speech.

Using the pragmatic theory of signs, the forms of self reference that have to be addressed in this project can be divided into two kinds, or classified in accord with two dimensions of referential involvement.  Roughly speaking, reference in the broader sense can suggest either a denotative reference to an object or a connotative reference to a sense.  Therefore, a projected self reference can be classified according to the ways that its components of reference propose to recur on themselves:  how much pretends to be a self description along denotative lines and how much purports to be a self address in the connotative direction.

Using the pragmatic theory of signs, the forms of self-reference that have to be addressed in this project can be divided into two kinds, or classified in accord with two dimensions of referential involvement.  Roughly speaking, reference in the broader sense can suggest either a denotative reference to an object or a connotative reference to a sense.  Therefore, a projected self-reference can be classified according to the ways that its components of reference propose to recur on themselves:  how much pretends to be a self description along denotative lines and how much purports to be a self address in the connotative direction.

Under suitably liberalized conditions of interpretation, then, what is meant by "a self referent text", whether one that denotatively describes or connotatively addresses itself?  Apparently, it can mean a text that addresses, describes, refers to, or speaks to either one of two issues:  (1) the outwardly passing features of its own succession of signs, or (2) the inwardly relied on properties of its own regenerative sources.

Under suitably liberalized conditions of interpretation, then, what is meant by "a self referent text", whether one that denotatively describes or connotatively addresses itself?  Apparently, it can mean a text that addresses, describes, refers to, or speaks to either one of two issues:  (1) the outwardly passing features of its own succession of signs, or (2) the inwardly relied on properties of its own regenerative sources.

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In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far.  There are a couple of catches that save this form of representation from falling into a "vicious circle", that is, into a pattern of self reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action.  First, the regular representations do not represent that a group element is literally "equal to" a set of ordered pairs involving that very same group element, but only that it is "mapped to" something like this set.  Second, careful usage would dictate that the "something like" that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements xj themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form "xj".

In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far.  There are a couple of catches that save this form of representation from falling into a "vicious circle", that is, into a pattern of self-reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action.  First, the regular representations do not represent that a group element is literally "equal to" a set of ordered pairs involving that very same group element, but only that it is "mapped to" something like this set.  Second, careful usage would dictate that the "something like" that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements xj themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form "xj".

These reservations are crucial to understanding the form of explanation that a regular representation provides, that is, what it explains and what it does not.  If one is seeking an ontological explanation of what a group and its elements "are", then one would have reason to object that it does no good to represent a group and its elements in terms of their actions on the group elements themselves, since one still does not know what the latter entities "are".  Notice that the form of this objection is reminiscent of a dilemma that is often thought to obstruct the beginning of an inquiry into inquiry.  A similar pattern of knots occurs when one tries to explain the process of formalization in terms of its effects on the term "formalization".  In each case, the resolution of the difficulty turns on recognizing a distinction between the active and passive modes of existence that go with each nameable objective.

These reservations are crucial to understanding the form of explanation that a regular representation provides, that is, what it explains and what it does not.  If one is seeking an ontological explanation of what a group and its elements "are", then one would have reason to object that it does no good to represent a group and its elements in terms of their actions on the group elements themselves, since one still does not know what the latter entities "are".  Notice that the form of this objection is reminiscent of a dilemma that is often thought to obstruct the beginning of an inquiry into inquiry.  A similar pattern of knots occurs when one tries to explain the process of formalization in terms of its effects on the term "formalization".  In each case, the resolution of the difficulty turns on recognizing a distinction between the active and passive modes of existence that go with each nameable objective.

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In setting out the plan of a full scale RIF there is an unspoken promise to justify eventually the thematic motives that experimentally tolerate its indulgence in self reference, and it seems that this implicit hope for a full atonement in time is a key to the tensions of the work being borne.  This section, in order to inspire confidence in the prospects of a RIF being achievable, and by way of allaying widespread suspicions about all types of self reference, examines several forms of circular referral and notes that not all contemplation of self reference is incurably vicious.

In setting out the plan of a full scale RIF there is an unspoken promise to justify eventually the thematic motives that experimentally tolerate its indulgence in self-reference, and it seems that this implicit hope for a full atonement in time is a key to the tensions of the work being borne.  This section, in order to inspire confidence in the prospects of a RIF being achievable, and by way of allaying widespread suspicions about all types of self-reference, examines several forms of circular referral and notes that not all contemplation of self-reference is incurably vicious.

In this section I consider signs, expressions, sign relations, and systems of interpretation (SOIs) that involve forms of self reference.  Because it is the abstract forms of self reference that constitute the chief interest of this study, I collect this whole subject matter under the heading "patterns of self reference" (POSRs).  With respect to this domain I entertain the classification of POSRs in two different ways.

In this section I consider signs, expressions, sign relations, and systems of interpretation (SOIs) that involve forms of self-reference.  Because it is the abstract forms of self-reference that constitute the chief interest of this study, I collect this whole subject matter under the heading "patterns of self-reference" (POSRs).  With respect to this domain I entertain the classification of POSRs in two different ways.

In this section I take notice of a broad family of formal structures that I refer to as "patterns of self reference" (POSRs), because they seem to have in common the proposed description of a formal object by means of recursive or circular references.  In their basic characters, POSRs range from the familiar to the strange, from the obvious to the problematic, and from the legitimate to the spurious.  Often a POSR is best understood as a formal object in its own right, or as a formal sign that foreshadows a definite object, but occasionally a POSR can only be interpreted as something in the character of a syntactic pattern, one that goes into the making of a questionable specification and represents merely a dubious attempt to indicate or describe an object.  All in all, POSRs range from the kinds of functions and objects, or programs and data structures, that are successfully defined by recursion to the sorts of vitiating circles that doom every attempt to define an unknown term in terms of itself.

In this section I take notice of a broad family of formal structures that I refer to as "patterns of self-reference" (POSRs), because they seem to have in common the proposed description of a formal object by means of recursive or circular references.  In their basic characters, POSRs range from the familiar to the strange, from the obvious to the problematic, and from the legitimate to the spurious.  Often a POSR is best understood as a formal object in its own right, or as a formal sign that foreshadows a definite object, but occasionally a POSR can only be interpreted as something in the character of a syntactic pattern, one that goes into the making of a questionable specification and represents merely a dubious attempt to indicate or describe an object.  All in all, POSRs range from the kinds of functions and objects, or programs and data structures, that are successfully defined by recursion to the sorts of vitiating circles that doom every attempt to define an unknown term in terms of itself.

Because POSRs span the spectrum from the moderately straightforward to the deliberately misleading, there is a need for ways to tell them apart, at least, before pursuing their consequences too far.  Of course, if one cannot rest without having all computable functions at one's command, then no program can tell all the good and bad programs apart.  But if one can be satisfied with a somewhat more modest domain, then there is hope for a way, an experimental, fallible, and incremental way, but a way nonetheless, that eventually leads one to know the good and ultimately keeps one away from the bad.

Because POSRs span the spectrum from the moderately straightforward to the deliberately misleading, there is a need for ways to tell them apart, at least, before pursuing their consequences too far.  Of course, if one cannot rest without having all computable functions at one's command, then no program can tell all the good and bad programs apart.  But if one can be satisfied with a somewhat more modest domain, then there is hope for a way, an experimental, fallible, and incremental way, but a way nonetheless, that eventually leads one to know the good and ultimately keeps one away from the bad.

When it comes to their propriety, POSRs are found on empirical grounds to fall into two varieties:  the "exculpable" and the "indictable" kinds.  Thus, it is reasonable to attempt an empirical distinction, proposing to let experience mark each POSR as an "excusable self reference" (ESR) or an "improper self reference" (ISR), as the case may be.  But empirical grounds can be a hard basis to fall back on, since a recourse to actual experience with POSRs can risk an agent's participation in pretended sign relations and promissory representations that amount in the end to nothing more than forms of interpretive futility.  Therefore, one seeks an arrangement of methods in general or an ordering of options in these special cases that makes the empirical trial a court of last resort and that avoids resorting to the actual experience of interpretation as a routine matter of course.

When it comes to their propriety, POSRs are found on empirical grounds to fall into two varieties:  the "exculpable" and the "indictable" kinds.  Thus, it is reasonable to attempt an empirical distinction, proposing to let experience mark each POSR as an "excusable self-reference" (ESR) or an "improper self reference" (ISR), as the case may be.  But empirical grounds can be a hard basis to fall back on, since a recourse to actual experience with POSRs can risk an agent's participation in pretended sign relations and promissory representations that amount in the end to nothing more than forms of interpretive futility.  Therefore, one seeks an arrangement of methods in general or an ordering of options in these special cases that makes the empirical trial a court of last resort and that avoids resorting to the actual experience of interpretation as a routine matter of course.

First, I recognize an "empirical distinction" that seems to exist between the less problematic and the more problematic varieties of self reference, allowing POSRs to be sorted according to the consequential features that they have in actual experience.  There are the "good" sorts, those cleared up to the limits of accumulated experience as innocuous usages and even as probable utilities, and then there are the "bad" sorts, those marked by hard experience as definitely problematic.

First, I recognize an "empirical distinction" that seems to exist between the less problematic and the more problematic varieties of self reference, allowing POSRs to be sorted according to the consequential features that they have in actual experience.  There are the "good" sorts, those cleared up to the limits of accumulated experience as innocuous usages and even as probable utilities, and then there are the "bad" sorts, those marked by hard experience as definitely problematic.