A MODEL OF ENTAILMENT MESHES
DRAFT V 0.8 2001-10-30 This text is a working document.
a> Topics are represented as words or phrases.
For example, "circle", "compass",
"sky", "blue", "freedom."
a> If 2 or more topics are connected by either SIMILARITIES or DIFFERENCES, then an analogy exists between them.
b> For example, (plane · table -> flat), (plane Æ table -> geometric figure, physical object).
c> The connection can be extremely weak or non-specific in detail, for example, "what I am pointing at now."
Analogies are the weakest form of relation in
a> Some topics can be represented as derived from others.
b> For example, one can say that the topic "circle" can be derived from the topics "compass" and "plane" as follows: (compass + plane) => circle
Informally, this means that the topics compass and plane
are the necessary and sufficient topics to explain circle: how
"circle" comes about, is derived, is made or produced from, the other
topics. The consideration of what must, or need not be, included in the
derivation is subjective.
a> If all possible derivations exist among a set of 3 or more topics, then a coherence is said to exist, made up of those topics.
b> For example, if (compass + circle => plane) AND (compass + plane => circle) AND (plane + circle => compass) THEN (compass plane circle) is a coherence.
c> In practice, coherences may consist of 3, 4 or more topics. Three is the minimum because with only 2 topics, it is impossible to unambiguously create one topic from the other, and this productive component is the essence of what "coherence" means. (See CONTRADICTION below.)
d> Formally, a coherence comprises both descriptive procedures AND prescriptive procedures.
(a) A descriptive procedure explains the relationship among the topics: "A compass is an instrument that may be used to create the geometric figure of a circle on a geometric surface called a plane." This indicates the WHY of the procedure being described in terms of the roles of the topics. Note that this does not indicate HOW the operation of creating the circle takes place with these topics/objects.
A prescriptive procedure explains how specifically
the topics are combined in order to achieve the result claimed by the
descriptive procedure: "Place the point of the compass into the plane and
keep it fixed. Place the marking side of the compass onto the plane, and,
without lifting either the marking end or the pointed end of the compass from
the plane, rotate the marking end around the pivot made by the pointing end,
until a complete rotation has taken place. The resulting mark on the plane is a
Note: The specific distinction of "descriptive" and "prescriptive" is evocative and useful, but it is not the only way to express the complementarity of the two types of procedure that comprise coherences.
a> A possible contradiction exists when a set of topics do not unambiguously produce/re-produce a further, single topic. This is said to be contradictory because the basis of coherence is that of productive and re-productive processes that are stable over time. If contradictions exist, the system is not stable.
b> For example, if (circle compass plane) AND (circle compass table) exist, the combination of circle + compass may produce plane OR it may produce table.
c> The situation is called a possible contradiction because it may be a flaw in the representation, or a flaw in the underlying model. Detecting the potential contradiction is a "call for information" to the author to resolve the situation.
d> Formally, given 2 coherences, a full contradiction exists when all but 2 topics are in common to both coherences, where each of the 2 non-overlapping topics is present in each of the coherences, respectively [same as in the example just above with (circle compass plane) and (circle compass table)].
coherences may be judged in partial contradiction as follows:
(circle compass plane) AND (circle compass sphere 3D). It is a stylistic choice to decide that these are to be allowed or banned in a given representation.
f> Other, partial contradiction cases exist. For example, subset or superset do not allow for unambiguous production/re-production of the topics involved. Again, it is useful to consider each of these cases as a "call for information" to the author, to resolve the situation.
Identical coherences are not distinct and are not
considered to exist independently (a cognitive repertoire or model either has
or does not have a given coherent relation). Detection of two identical
coherences in an entailment mesh is a representation error and calls for the
deletion of one of the instances.
a> List the names of distinct topics.
redundancies where similar phrases name the same topic, but maintain valid differences
by modifiers to phrases or different words/phrases altogether.
a> Propose connections among topics that appear to relate to each other. At any point, add or remove topics to the proposed relation; however, after doing so, repeat the required confirmation in each step below.
b> Indicate via notational convention whether the relation is a derivation, coherence, or analogy.
c> When asserting a derivation, ensure that the derivation is valid by confirming that the source topics are the necessary and sufficient set of topics for producing/re-producing the derived topic.
d> When asserting a coherence, ensure that all necessary derivations are valid. It is necessary to step through each one, rehearsing an explanation that shows that the topics involved are the necessary and sufficient set to express the production of the derived topic. This may seem laborious, but it is necessary and ultimately rewarding.
asserting an analogy, ensure that the analogy is valid by checking its similarities
CHECK FOR CONTRADICTIONS ("Rule of Genoa")
a> For every coherence that overlaps with any other, calculate whether contradiction exists as per the definition of contradiction above. If so, then execute one or more of the following steps.
b> (Combine topics) Propose that ambiguous topics are actually a false distinction; that is, table and plane are unnecessarily differentiated, and a single topic would suffice for the purposes of the representation.
c> (Split topics) Propose that a single topic is really insufficient to represent the intention of the author; namely, circle should really be split into 2 topics, that of the abstract geometric figure, and that of the actual, pencil-drawn circle on this tabletop (hopefully on a piece of paper).
topics) Propose that additional distinctions (topics) are appropriate to add to
one or more of the coherences; that is, the context of the relation including
plane is that of geometric constructions, while the other is not.
PROPOSE ADDITIONAL COHERENCES ("saturation")
a> Propose a new neighborhood of topics as a potential coherence.
b> Check for contradiction with existing coherences; if none, continue.
to install the new neighborhood as a coherence; that is, ensure that all
derivations are valid. If they are, install as a new coherence.
on the above procedures until the mesh is considered stable or complete for a
checking — stimulates the author to change or add distinctions (topics)
in order to increase the coherence of the whole. This is calculated based only
on the structural relationships among the topics (distinctions), not on any
semantic component. Therefore it can be calculated via digital, symbolic
— proposed addition of new coherences up to, but not including,
of memory — redundant interconnections across coherences in an entailment
mesh create highly robust, failure-resistant structures (similar to, but not
identical to, the metaphor of holographic memories).
of "knowing" — the ability to interpret the non-directional
coherence into a directional derivation for the purpose of action or
explanation. Enables extraction of specific aspects of individual topics that leads
to application in new, specific circumstances, that is, innovation.
a> Entailment: To have, impose, or require as a necessary accompaniment or consequence. From Middle English, into + tail.
b> Coherence: A sticking or cleaving together; union of parts of the same body; cohesion. From Latin, same meaning.
c> Contradiction: An assertion of the contrary to what has been said or affirmed. From Latin, answer, objection.
d> Catalytic, from catalysis: Acceleration of a chemical reaction induced by the presence of material that is chemically unchanged at the end of the reaction. From Greek, to dissolve, to loosen.