Changes

==3.  Instrumental Focus==

==3.  Instrumental Focus==

===3.1.  Propositional Calculus===

3.1.  Propositional Calculus

A symbolic calculus is needed to assist our reasoning and computation in the realm of propositions.  With an eye toward efficiency of computing and ease of human use, while preserving both functional and declarative properties of propositions, I have implemented an interpreter and assorted utilities for one such calculus.  The original form of this particular calculus goes back to the logician C.S. Peirce, who is my personal favorite candidate for the grand-uncle of AI.  Among other things, Peirce discovered the logical importance of NAND/NNOR operators (CP 4.12 ff, 4.264 f), (NE 4, ch. 5), inspired early ideas about logic machines (Peirce, 1883), is credited with "the first known effort to apply Boolean algebra to the design of switching circuits" (M. Gardner, p. 116 n), and even speculated on the nature of abstract interpreters and other "Quasi-Minds" (Peirce, CP 4.536, 4.550 ff).

A symbolic calculus is needed to assist our reasoning and computation in the realm of propositions.  With an eye toward efficiency of computing and ease of human use, while preserving both functional and declarative properties of propositions, I have implemented an interpreter and assorted utilities for one such calculus.  The original form of this particular calculus goes back to the logician C.S. Peirce, who is my personal favorite candidate for the grand-uncle of AI.  Among other things, Peirce discovered the logical importance of NAND/NNOR operators (CP 4.12 ff, 4.264 f), (NE 4, ch. 5), inspired early ideas about logic machines (Peirce, 1883), is credited with "the first known effort to apply Boolean algebra to the design of switching circuits" (M. Gardner, p. 116 n), and even speculated on the nature of abstract interpreters and other "Quasi-Minds" (Peirce, CP 4.536, 4.550 ff).

Thought is not necessarily connected with a brain.  It appears in the work of bees, of crystals, and throughout the purely physical world;  and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there.  (CP 4.551).

Thought is not necessarily connected with a brain.  It appears in the work of bees, of crystals, and throughout the purely physical world;  and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there.  (CP 4.551).

One could hardly invent a better anthem for the work being done today in the AI/systems hybrid areas of cellular automata (Burks, 1970), (Ulam, ch. 12), (Nicolis & Prigogine, 1989), emergent computation (Forrest, 1991), and "society of mind" theories (Minsky, 1986).  I hope it will emerge that these workers achieve the same grade of well-honed insight regarding the mind's apical functions that Peirce was able to inspire, having once acquired a taste for it in the higher combines of logic's hive.

One could hardly invent a better anthem for the work being done today in the AI/systems hybrid areas of cellular automata (Burks, 1970), (Ulam, ch. 12), (Nicolis & Prigogine, 1989), emergent computation (Forrest, 1991), and "society of mind" theories (Minsky, 1986).  I hope it will emerge that these workers achieve the same grade of well-honed insight regarding the mind's apical functions that Peirce was able to inspire, having once acquired a taste for it in the higher combines of logic's hive.

All of these issues that occupied Peirce would be encountered again later in the 20th century when computer scientists, linguists, communication engineers, media theorists, and others would be forced to deal with them in their daily practice and would perforce discover many workable answers.  These are the topics that have come to be recognized as the reality of information and uncertainty, the physicality of symbol systems, the independent dimension of syntax, the complexity of semantics and evaluation, the pragmatic metes and bounds of interactive communication and interpretive control.  All in all, as acutely discovered in AI systems engineering, these factors sum up to the general resistance of matter to being impressed with our minds.

All of these issues that occupied Peirce would be encountered again later in the 20th century when computer scientists, linguists, communication engineers, media theorists, and others would be forced to deal with them in their daily practice and would perforce discover many workable answers.  These are the topics that have come to be recognized as the reality of information and uncertainty, the physicality of symbol systems, the independent dimension of syntax, the complexity of semantics and evaluation, the pragmatic metes and bounds of interactive communication and interpretive control.  All in all, as acutely discovered in AI systems engineering, these factors sum up to the general resistance of matter to being impressed with our minds.

3.1.1.  Peirce's Existential Graphs

====3.1.1.  Peirce's Existential Graphs====

Peirce devised a graphical notation for predicate calculus, or first order logic, that he called the system of "Existential Graphs" (EG).  In its emphasis on relations and its graphic depiction of their logic, EG anticipated many features of present-day semantic networks and conceptual graphs.  Not only does it remain logically more exact than most of these later formulations, but EG had transformation rules that rendered it a literal calculus, with a manifest power for inferring latent facts.  An explicit use of Peirce's EG for knowledge base representation appears in (Sowa, 1984).  A software package that uses EG to teach basic logic is documented in (Ketner, 1990).  The calculus presented below is related in its form and interpretation to the propositional part of Peirce's EG.  A similar calculus, but favoring an alternate interpretation, was developed in (Spencer-Brown, 1969).

Peirce devised a graphical notation for predicate calculus, or first order logic, that he called the system of "Existential Graphs" (EG).  In its emphasis on relations and its graphic depiction of their logic, EG anticipated many features of present-day semantic networks and conceptual graphs.  Not only does it remain logically more exact than most of these later formulations, but EG had transformation rules that rendered it a literal calculus, with a manifest power for inferring latent facts.  An explicit use of Peirce's EG for knowledge base representation appears in (Sowa, 1984).  A software package that uses EG to teach basic logic is documented in (Ketner, 1990).  The calculus presented below is related in its form and interpretation to the propositional part of Peirce's EG.  A similar calculus, but favoring an alternate interpretation, was developed in (Spencer-Brown, 1969).

3.1.1.1.  Blank and Bound Connectives

=====3.1.1.1.  Blank and Bound Connectives=====

Given an alphabet A = {a1,...,an} and a universe U = <A>, we write expressions for the propositions p: U -> B upon the following basis.  The ai: U -> B are interpreted as coordinate functions.  For each natural number k we have two k-ary operations, called the blank or unmarked connective and the bound or marked connective.

Given an alphabet A = {a1,&nbsp;…,&nbsp;an} and a universe U = <A>, we write expressions for the propositions p&nbsp;:&nbsp;U&nbsp;&rarr;&nbsp;B upon the following basis.  The ai&nbsp;:&nbsp;U&nbsp;&rarr;&nbsp;B are interpreted as coordinate functions.  For each natural number k we have two k-ary operations, called the blank or unmarked connective and the bound or marked connective.

The blank connectives are written as concatenations of k expressions and interpreted as k-ary conjunctions.  Thus,

The blank connectives are written as concatenations of k expressions and interpreted as k-ary conjunctions.  Thus,

e1 e2 e3 means "e1 and e2 and e3".

: e1 e2 e3 means "e1 and e2 and e3".

The bound connectives are written as lists of k expressions (e1,...,ek), where the parentheses and commas are considered to be parts of the connective notation.  In text presentations the parentheses will be superscripted, as (e1,...,ek), to avoid confusion with other uses.  The bound connective is interpreted to mean that just one of the k listed expressions is false.  That is, (e1,...,ek) is true if and only if exactly one of the expressions e1,...,ek is false.  In particular, for k = 1 and 2:

The bound connectives are written as lists of k expressions (e1,&nbsp;…,&nbsp;ek), where the parentheses and commas are considered to be parts of the connective notation.  In text presentations the parentheses will be superscripted, as (e1,&nbsp;…,&nbsp;ek), to avoid confusion with other uses.  The bound connective is interpreted to mean that just one of the k listed expressions is false.  That is, (e1,&nbsp;…,&nbsp;ek) is true if and only if exactly one of the expressions e1,&nbsp;…,&nbsp;ek is false.  In particular, for k = 1 and 2:

(e1) means "not e1".

(e1) means "not e1".

  or "e1 neq e2",  "e1 =/= e2".

  or "e1 neq e2",  "e1 =/= e2".

A remaining sample of typical expressions will finish illustrating the use of this calculus.

A remaining sample of typical expressions will finish illustrating the use of this calculus.

(e1 (e2)) means "e1 => e2",

(e1 (e2)) means "e1 => e2",

(e1,(e2,e3)) means "e1 + e2 + e3".

(e1,(e2,e3)) means "e1 + e2 + e3".

3.1.1.2  Partitions:  Genus and Species

=====3.1.1.2  Partitions:  Genus and Species=====

Especially useful is the facility this notation provides for expressing partition constraints, or relations of mutual exclusion and exhaustion among logical features.  For example,

Especially useful is the facility this notation provides for expressing partition constraints, or relations of mutual exclusion and exhaustion among logical features.  For example,

((p1),(p2),(p3))

: ((p1),(p2),(p3))

says that the universe is partitioned among the three properties p1, p2, p3.  Finally,

says that the universe is partitioned among the three properties p1, p2, p3.  Finally,

(g,(s1),(s2),(s3))

: (g,(s1),(s2),(s3))

says that the genus g is partitioned into the three species s1, s2, s3.  Its Venn diagram looks like a pie chart.  This style of expression is also useful in representing the behavior of devices, for example:  finite state machines, which must occupy exactly one state at a time;  and Turing machines, whose tape head must engage just one tape cell at a time.

says that the genus g is partitioned into the three species s1, s2, s3.  Its Venn diagram looks like a pie chart.  This style of expression is also useful in representing the behavior of devices, for example:  finite state machines, which must occupy exactly one state at a time;  and Turing machines, whose tape head must engage just one tape cell at a time.

3.1.1.3  Vacuous Connectives and Constant Values

=====3.1.1.3  Vacuous Connectives and Constant Values=====

As a consistent downward extension, the nullary (or 0-ary) connectives can be identified with logical constants.  That is, blank expressions " " are taken for the value "true" (silence assents), and empty bounds "()" are taken for the value "false".  By composing operations, negation and binary conjunction are enough in themselves to obtain all the other boolean functions, but the use of these k-ary connectives lends itself to a flexible and powerful representation as graph-theoretical data-structures in the computer.

As a consistent downward extension, the nullary (or 0-ary) connectives can be identified with logical constants.  That is, blank expressions " " are taken for the value "true" (silence assents), and empty bounds "()" are taken for the value "false".  By composing operations, negation and binary conjunction are enough in themselves to obtain all the other boolean functions, but the use of these k-ary connectives lends itself to a flexible and powerful representation as graph-theoretical data-structures in the computer.

3.1.2.  Implementation Details

====3.1.2.  Implementation Details====

The interpreter that has been implemented for EG employs advanced data-structures for the reprsentation of both lexical terms and logical expressions.

The interpreter that has been implemented for EG employs advanced data-structures for the reprsentation of both lexical terms and logical expressions.

Mathematically, all this verbiage is just a way of talking about two topics:  (1) functions and relations from structured objects to sets of features, and (2) equivalence relations (for example, orbits under symmetry group actions) on these structured objects.  But the visual metaphors seem to assist thought, most of the time, and are in any case a part of the popular iconography.

Mathematically, all this verbiage is just a way of talking about two topics:  (1) functions and relations from structured objects to sets of features, and (2) equivalence relations (for example, orbits under symmetry group actions) on these structured objects.  But the visual metaphors seem to assist thought, most of the time, and are in any case a part of the popular iconography.

3.1.2.1.  Painted Cacti

=====3.1.2.1.  Painted Cacti=====

Viewing a propositional expression in EG as a "cactus", the bound connectives ( , , , ) constitute its "lobes" (edges or lines) and the positive literals ai are tantamount to "colors" (paints or tints) on its "points" (vertices or nodes).  One of the chief tasks of processing logical expressions is their systematic clarification.  This involves transforming arbitrary expressions into logically equivalent expressions whose latent meaning is manifest, their "canonical" or "normal" forms.  The normalization process implemented for EG, in the graphical language just given, takes an arbitrary tinted cactus and turns it into a special sort of painted cactus.

Viewing a propositional expression in EG as a "cactus", the bound connectives (&nbsp;,&nbsp;,&nbsp;,&nbsp;) constitute its "lobes" (edges or lines) and the positive literals ai are tantamount to "colors" (paints or tints) on its "points" (vertices or nodes).  One of the chief tasks of processing logical expressions is their systematic clarification.  This involves transforming arbitrary expressions into logically equivalent expressions whose latent meaning is manifest, their "canonical" or "normal" forms.  The normalization process implemented for EG, in the graphical language just given, takes an arbitrary tinted cactus and turns it into a special sort of painted cactus.

3.1.2.2.  Concept and Purpose

=====3.1.2.2.  Concept and Purpose=====

What good is this?  What conceivable purpose is there for these inductive and deductive capacities, that enable the personal computer to learn formal languages and to turn propositional calculi into painted cacti?  By developing these abilities for inductive learning and accurate inference, aided by a facility for integrating their alternate "takes" on the world, I hope that AI software will gain a new savvy, one that helps it be both friendly to people and faithful to truth, both politic and correct.  To do this demands a form of artificial intelligence that can do both, without the kinds of trade-off that make it a travesty to both.

What good is this?  What conceivable purpose is there for these inductive and deductive capacities, that enable the personal computer to learn formal languages and to turn propositional calculi into painted cacti?  By developing these abilities for inductive learning and accurate inference, aided by a facility for integrating their alternate "takes" on the world, I hope that AI software will gain a new savvy, one that helps it be both friendly to people and faithful to truth, both politic and correct.  To do this demands a form of artificial intelligence that can do both, without the kinds of trade-off that make it a travesty to both.

3.1.3.  Applications

====3.1.3.  Applications====

The current implementation of this calculus is efficient enough to have played a meaningful part in realistically complex investigations, both practical and theoretical.  For example, it has been used in qualitative research to represent observational protocols of event sequences as propositional data bases.  It has also been used to analyze the behavior of finite state machines and space-time limited Turing machines, exploiting a coding that is similar to but more succinct than the one used in Cook's theorem (on the NP-completeness of propositional calculus satisfiability).  See (Garey & Johnson, 1979) and (Wilf, 1986).

The current implementation of this calculus is efficient enough to have played a meaningful part in realistically complex investigations, both practical and theoretical.  For example, it has been used in qualitative research to represent observational protocols of event sequences as propositional data bases.  It has also been used to analyze the behavior of finite state machines and space-time limited Turing machines, exploiting a coding that is similar to but more succinct than the one used in Cook's theorem (on the NP-completeness of propositional calculus satisfiability).  See (Garey & Johnson, 1979) and (Wilf, 1986).

3.2.  Differential Extensions of Propositional Calculi

===3.2.  Differential Extensions of Propositional Calculi===

In order to define a differential extension of a propositional universe of discourse U, the alphabet A of U's defining features must be extended to include a set of symbols for differential features, or elementary "changes" in the universe of discourse.  Intuitively, these symbols may be construed as denoting primitive features of change, or propositions about how things or points in U change with respect to the features noted in the original alphabet A.  Hence, let dA = {da1,...,dan} and dU = <dA> = <da1,...,dan>.  As before, we may express dU concretely as a product of distinct factors:

In order to define a differential extension of a propositional universe of discourse U, the alphabet A of U's defining features must be extended to include a set of symbols for differential features, or elementary "changes" in the universe of discourse.  Intuitively, these symbols may be construed as denoting primitive features of change, or propositions about how things or points in U change with respect to the features noted in the original alphabet A.  Hence, let dA = {da1,&nbsp;…,&nbsp;dan} and dU = <dA> = <da1,&nbsp;…,&nbsp;dan>.  As before, we may express dU concretely as a product of distinct factors:

dU = Xi dAi = dA1 x ... x dAn.

: dU = Xi dAi = dA1 x ... x dAn.

Here, dAi is an alphabet of two symbols, dAi = {(dai), dai}, where (dai) is a symbol with the logical value of "not dai".  Each dAi has the type B, under the ordered correspondence {(dai), dai} = {0, 1}.  However, clarity is often served by acknowledging this differential usage with a distinct type D:

Here, dAi is an alphabet of two symbols, dAi = {(dai), dai}, where (dai) is a symbol with the logical value of "not dai".  Each dAi has the type B, under the ordered correspondence {(dai), dai} = {0, 1}.  However, clarity is often served by acknowledging this differential usage with a distinct type D:

D = {(dx), dx} = {same, different} = {stay, change}.

: D = {(dx), dx} = {same, different} = {stay, change}.

Finally, let U' = U x dU = <A'> = <A + dA> = <a1,...,an, da1,...,dan>, giving U' the type Bn x Dn.

Finally, let U' = U x dU = <A'> = <A + dA> = <a1,&nbsp;…,&nbsp;an,&nbsp;da1,&nbsp;…,&nbsp;dan>, giving U' the type Bn x Dn.

All propositions of U have natural (and usually tacit) extensions to U', with p: U = Bn -> B becoming p: U' = Bn x Dn -> B.  It is convenient to approach the study of the differential extension U' from a globally democratic perspective, viewing all the differential propositions p: U' -> B as equal citizens.  Devolving from this standpoint, the various grades of differential forms are then defined by their placement in U' with regard to the basis A'.  Extending previous usage, we say that p is singular in U' if it has just one satisfying interpretation in U'.  A proposition p: U' -> B is called singular in U if its projection to U is singular in U, that is, if all its interpretations in U' share the same cell in U.

All propositions of U have natural (and usually tacit) extensions to U', with p: U = Bn -> B becoming p: U' = Bn x Dn -> B.  It is convenient to approach the study of the differential extension U' from a globally democratic perspective, viewing all the differential propositions p: U' -> B as equal citizens.  Devolving from this standpoint, the various grades of differential forms are then defined by their placement in U' with regard to the basis A'.  Extending previous usage, we say that p is singular in U' if it has just one satisfying interpretation in U'.  A proposition p: U' -> B is called singular in U if its projection to U is singular in U, that is, if all its interpretations in U' share the same cell in U.

Using the isomorphism between function spaces:

Using the isomorphism between function spaces:

(Bn x Dn -> B) <==> (Bn -> (Dn -> B)),

: (Bn x Dn -> B) <==> (Bn -> (Dn -> B)),

each p: U' -> B has a unique decomposition into a p': Bn -> (Dn -> B) and a set of p": Dn -> B such that:

each p: U' -> B has a unique decomposition into a p': Bn -> (Dn -> B) and a set of p": Dn -> B such that:

p: Bn x Dn -> B <==> p': Bn -> p": (Dn -> B).

: p : Bn x Dn -> B <==> p' : Bn -> p": (Dn -> B).

For the sake of the visual intuition we may imagine that each cell x in the diagram of U has springing from it the diagram of the proposition p'(x) = p" in dU.

For the sake of the visual intuition we may imagine that each cell x in the diagram of U has springing from it the diagram of the proposition p'(x) = p" in dU.

To attempt a clarification let us now make one more pass.  Let x and y be variables ranging over U and dU, respectively.  Then each p: U' = U x dU -> B has a unique decomposition into a p': U -> B and a set of p(x)': dU -> B such that

To attempt a clarification let us now make one more pass.  Let x and y be variables ranging over U and dU, respectively.  Then each p: U' = U x dU -> B has a unique decomposition into a p': U -> B and a set of p(x)': dU -> B such that

p(x,y) = p'(x)(y) = p(x)'(y).

: p(x,y) = p'(x)(y) = p(x)'(y).

The "x" in p(x)'(y) would ordinarily be subscripted as a parameter in the form px', but this does not explain the difference between a parameter and a variable.  Here the difference is marked by the position of the prime ('), which serves as a kind of "run-time marker".  The prime locates the point of inflexion in a piece of notation that is the boundary between local and global responsibilities of interpretation.  It tells the intended division between individual identity of functions (a name and a local habitation) and "socially" defined roles (signs falling to the duty of a global interpreter).  In the phrase p'(x) the p' names the function while the parenthetical (x) is part of the function notation, to be understood by a global interpreter.  In p(x)' the parenthetical (x) figures into the name of an individual function, having a local significance but only when x is specified.

The "x" in p(x)'(y) would ordinarily be subscripted as a parameter in the form px', but this does not explain the difference between a parameter and a variable.  Here the difference is marked by the position of the prime ('), which serves as a kind of "run-time marker".  The prime locates the point of inflexion in a piece of notation that is the boundary between local and global responsibilities of interpretation.  It tells the intended division between individual identity of functions (a name and a local habitation) and "socially" defined roles (signs falling to the duty of a global interpreter).  In the phrase p'(x) the p' names the function while the parenthetical (x) is part of the function notation, to be understood by a global interpreter.  In p(x)' the parenthetical (x) figures into the name of an individual function, having a local significance but only when x is specified.

I am not yet happy with my understanding of these issues.  The most general form of the question at hand appears to be bound up with the need to achieve mechanisms of functional abstraction and application for propositions.  It seems further that implementing moderate and practical forms of this functionality would have to be a major goal of the research projected here.  On the one hand Curry's paradox warns that this is a non-trivial problem, that only approximate and temporizing forms of success can reasonably be expected.  See (Lambek & Scott, 1986), but (Smullyan, chapt. 14) is probably more suitable for summer reading.  On the other hand the work of (Spencer-Brown, 1969), (Aczel, 1988), and (Barwise & Etchemendy, 1989) seems to suggest that logic can be extended to include "fixed points of negation" without disastrous results.  I can only hope that time and work will untie the mystery.

I am not yet happy with my understanding of these issues.  The most general form of the question at hand appears to be bound up with the need to achieve mechanisms of functional abstraction and application for propositions.  It seems further that implementing moderate and practical forms of this functionality would have to be a major goal of the research projected here.  On the one hand Curry's paradox warns that this is a non-trivial problem, that only approximate and temporizing forms of success can reasonably be expected.  See (Lambek & Scott, 1986), but (Smullyan, chapt. 14) is probably more suitable for summer reading.  On the other hand the work of (Spencer-Brown, 1969), (Aczel, 1988), and (Barwise & Etchemendy, 1989) seems to suggest that logic can be extended to include "fixed points of negation" without disastrous results.  I can only hope that time and work will untie the mystery.

Work Area

==Work Area==

Logical Tangent Vectors

Logical Tangent Vectors