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Context, Perspective, and Generalities in a Knowledge Graph

AI3:::Adaptive Information (Mike Bergman) - Mon, 12/12/2016 - 17:40
Recent Slideshow Posted on KBpedia

Last week I gave a presentation to the Ontolog Forum on Cognonto, with specific reference to the KBpedia knowledge structure. I am pleased to now post this presentation online:

Context, Perspective, and Generalities in a Knowledge Ontology from Mike Bergman

As a standard method, the Forum routinely records the audio of its presentations. So, if you desire, you can listen to my 60 min presentation (followed by a 30 min Q & A) by clicking this audio link. The slide numbers are noted as the audio presentation proceeds so you can synchronize the audio with the slides.

We’re very pleased to have been giving this opportunity by the Forum. We invite you to hear this introduction to Cognonto and KBpedia.

KBpedia Upper Ontology Gets a Refresh

AI3:::Adaptive Information (Mike Bergman) - Mon, 12/05/2016 - 14:40
Part of Our Ongoing Efforts to Better Represent Knowledge

Cognonto today announced the release of version 1.20 of KBpedia, its knowledge structure that integrates six major knowledge bases (Wikipedia, Wikidata, OpenCyc, GeoNames, DBpedia and UMBEL) and 20 subsidiary ones under the KBpedia Knowledge Ontology (KKO). KBpedia’s explicit purpose is to provide a foundation for knowledge-based artificial intelligence (KBAI) by supporting the (nearly) automatic creation of training corpuses and positive and negative training sets and feature sets for deep, unsupervised and supervised machine learning.

The changes in this new release are solely related to KKO, the knowledge graph portion of KBpedia. There are two major drivers for this update to the KBpedia upper ontology. First, internal development efforts are now focusing on the modeling of predicates and time and action. This effort affects the definitions, splits and boundaries between attributes, relations, events and activities. Revisions in this area have been derived from a much closer reading of Charles Sanders Peirce‘s writings, based on our view that CSP has the most logical and sophisticated understanding of knowledge representation yet expressed. Second, where appropriate, we have relied on Peircean terminology to capture specific concepts. We are doing this to make KKO more amenable to review by Peircean scholars. At the same time, we have tried to reduce the use of obscure or difficult Peircean terms where they might be a barrier to understanding.

These changes solely affect two of the three main branches in KKO. The most affected branch is Monads, the branch representing Firstness (see below), reflecting the basic concepts or building blocks used in KKO. The Particulars branch, which captures the representation of individuals or instances, also was modified to capture those changes in the Monads branch. The Generals branch, the main portion for classes and types, was not affected by these changes.

The resulting KKO upper structure now has this form with about 165 key concepts, all organized according to Peirce’s universal categories of Firstness (1ns), Secondness (2ns) and Thirdness (3ns) (I earlier provided a broad overview for the basis of this triadic design):

level 1 level 2 level 3 level 4 level 5 level 6 level 7 Monads [1ns] FirstMonads [1ns] Suchness [1ns] Accidental [1ns] Inherent [2ns] Relational [3ns] Thisness [2ns] Chance [1ns] Being [2ns] Form [3ns] Pluralness [3ns] Absolute [1ns] Inclusive [1ns] Exclusive [2ns] Difference [3ns] SimpleRelative [2ns] Conjugative [3ns] DyadicMonads [2ns] Attributives [1ns] Oneness [1ns] Identity [1ns] Real [2ns] Matter [1ns] SubstantialForm [2ns] AccidentalForm [3ns] Fictional [3ns] Otherness [2ns] Inherence [3ns] Quality [1ns] Negation [2ns] Intrinsic [3ns] Relatives [2ns] Concurrents [1ns] Opponents [2ns] Conjunctives [3ns] Quantity [1ns] Values [1ns] Numbers [1ns] Multitudes [2ns] Magnitudes [3ns] Discrete [2ns] Continuous [3ns] Subsumption [2ns] Connective [3ns] Unary [1ns] Binary [2ns] Conditional [3ns] Indicatives [3ns] Iconic [1ns] Indexical [2ns] Associative [3ns] Denotative [1ns] Similarity [2ns] Contiguity [3ns] TriadicMonads [3ns] Representation [1ns] Icon [1ns] Index [2ns] Symbol [3ns] Mediation [2ns] Mentation [3ns] Particulars [2ns] MonadicDyads [1ns] MonoidalDyad [1ns] EssentialDyad [2ns] InherentialDyad [3ns] Events {2ns] Action [1ns] Change [1ns] Exertion [2ns] Perception [3ns] Reaction [2ns] State [1ns] Volition [2ns] Thought [3ns] Continuous [3ns] Space [1ns] Points [1ns] Areas [2ns] 2D Dimensions SpaceRegions [3ns] 3D Dimensions Time [2ns] Instants [1ns] Intervals [2ns] Eternal [3ns] Duratives [3ns] Situations [1ns] Activities [2ns] Processes [3ns] Entities [3ns] SingleEntities [1ns] Phenomenal [1ns] Ideal [2ns] Conceptual [3ns] PartOfEntities [2ns] Members [1ns] Parts [2ns] FunctionalComponents [3ns] ComplexEntities [3ns] CollectiveStuff [1ns] MixedStuff [2ns] CompoundEntities [3ns] Generals [3ns] (== SuperTypes) SignElements [1ns] AttributeTypes [1ns] RelationTypes [2ns] SituationTypes Symbols [3ns] Primitives [1ns] Structures [2ns] Conventions [3ns] Constituents [2ns] NaturalPhenomena [1ns] SpaceTypes [2ns] Shapes [1ns] Places [2ns] LocationPlace AreaRegion Forms [3ns] TimeTypes [3ns] Times [1ns] EventTypes [2ns] ActivityTypes [3ns] Manifestations [3ns] NaturalMatter [1ns] AtomsElements [1ns] NaturalSubstances [2ns] Chemistry [3ns] OrganicMatter [2ns] OrganicChemistry [1ns] BiologicalProcesses LivingThings [2ns] Prokaryotes [1ns] Eukaryotes [2ns] ProtistsFungus [1ns] Plants [2ns] Animals [3ns] Diseases [3ns] Agents [3ns] Persons [1ns] Organizations [2ns] Geopolitical [3ns] Symbolic [3ns] Information [1ns] AVInfo [1ns] VisualInfo AudioInfo WrittenInfo [2ns] StructuredInfo [3ns] Artifacts [2ns] FoodDrink Drugs Products Facilities Systems [3ns] MentalProcesses [1ns] Concepts [1ns] TopicsCategories [2ns] LearningProcesses [3ns] SocialProcesses [2ns] FinanceEconomy Society Science [3ns]

It is useful to re-cap the three constituents of Peirce’s trichotomy, what he called simply the Three Categories, or the universal categories, as follows:

  • Firstness [1ns] — these are possibilities or potentials, the basic forces or qualities that combine together or interact in various ways to enable the real things we perceive in the world, such as matter, life and ideas. These are the unrealized building blocks, or elements, the essences or attributes or possible juxtapositions. They are not divisible, what Peirce called indecomposables, since they are integral qualities or ideas in themselves;
  • Secondness [2ns] — these are the particular realized things or concepts in the world, what we can perceive, point to and describe. A particular is also known as an event, entity, instance or individual; and
  • Thirdness [3ns] — these are the laws, habits, regularities and continuities that may be generalized from particulars. All generals — what are also known as classes, kinds or types — belong to this category. The process of finding and deriving these generalities also leads to new insights or emergent properties, which continue to fuel knowledge discovery. Insights arising from Thirdness enable us to further explore and understand things, and is a driving force for further categorization.

Note that the three main branches and most of the sub-branches to KKO conform to this triadic structure. The basis for this structure was discussed in an earlier article.

In future posts I will delve and explain further each of the main branches of KKO. It is also likely the changes and refinements to this upper structure may continue for some time. Cognonto has open sourced KKO both for use by others and as a means for Peircean scholars and students to make critical commentary and suggestions. Because of this desire for review, we have also annotated the KKO structure more extensively in this release, including references to specific passages from Peirce’s writings.

Remember, KBpedia and KKO are the first complete attempt to capture Charles S. Peirce’s views of the logical organization of knowledge and the theory of signs into a working computer ontology (knowledge graph). As with Peirce’s views of ‘truth‘ as a limit function that can be approached but never fully attained, we will continue to strive to improve our understanding of how best to model knowledge for artificial intelligence purposes. The good news is we are already realizing significant KBAI benefits from KBpedia in its current form. We expect those benefits to continue to grow with further refinements to KKO and its typologies.

The open source KBpedia Knowledge Ontology (KKO) may be downloaded and inspected from here. We welcome any and all critical commentary.

Upcoming Dec. 7 Webinar on Cognonto

AI3:::Adaptive Information (Mike Bergman) - Thu, 12/01/2016 - 11:49
Public Ontolog Forum Talk to Emphasize KBpedia

The Ontolog Forum, the online community of practice for ontologists, has just announced a public online meeting on Cognonto. I will be giving the roughly 60 min presentation, to be followed by 30 min of open discussion, on December 7 at 9:30am Pacific / 12:30pm Eastern / 6:30pm CEST / 5:30pm GMT / 1730 UTC.  The public is invited.

According to the announcement:

KBpedia is a recently announced knowledge structure that integrates six major knowledge bases (OpenCyc, Wikipedia, Wikidata, GeoNames, DBpedia, UMBEL) under the KBpedia Knowledge Ontology (KKO). KBpedia’s explicit purpose is to provide a foundation for knowledge-based artificial intelligence by supporting the (nearly) automatic creation of training corpuses and positive and negative training sets and feature sets for deep, unsupervised and supervised machine learning. KKO is the upper ontology for KBpedia, and is guided by the universal categories (Firstness, Secondness, Thirdness) of Charles S. Peirce. [This talk] will discuss what KBpedia is, how it is organized and constructed, and why KKO offers some new approaches to vexing metaphysical questions in ontology design related to the knowledge representation of entities, relations, attributes, concepts, and natural kinds. The discussion period will hopefully highlight next potentials and important open questions.

Instructions and dial-in numbers are provided on the meeting announcement site. Please note that Ontolog has an open intellectual property rights policy.

I hope to see you there!

Nice Interview, Excellent Summary of Cognonto

AI3:::Adaptive Information (Mike Bergman) - Mon, 11/28/2016 - 15:46

Dataversity has just published an article on Cognonto based on an interview of me and review of our online materials. The article, Cognonto Takes On Knowledge-Based Artificial Intelligence,” does an excellent job of summarizing the venture. The writer, Jennifer Zaino, did a fantastic job capturing our discussions and framing the Cognonto story. Thanks!

I especially like that the article begins with the simple words, knowledge-based artificial intelligence, which is the quintessential description of what Cognonto is about. Jennifer then goes on to explain the genesis of the venture, the central role of its knowledge structure KBpedia, and the basis of this knowledge graph grounded in the triadic logic of the 19th century philosopher, Charles Sanders Peirce.

We will use this article for months to come in helping others understand what we are doing and why. It is always refreshing to see intelligent, well-written journalism. Thanks, Jenny.

Two New KBpedia Use Cases Published

AI3:::Adaptive Information (Mike Bergman) - Tue, 11/22/2016 - 15:20
Six Diverse Use Cases Now Available

Cognonto today published two new use cases on how to further leverage KBpedia, its knowledge structure that integrates six major knowledge bases (Wikipedia, Wikidata, OpenCyc, GeoNames, DBpedia and UMBEL), plus mappings to another 20 leading knowledge vocabularies. KBpedia provides a foundation for knowledge-based artificial intelligence (KBAI) by supporting the (nearly) automatic creation of training corpuses and positive and negative training sets and feature sets for deep, unsupervised and supervised machine learning.

The two new use cases are in: 1) dynamic tests and refinements of machine learners enabled by KBpedia’s fast creation of training sets and corpuses and reference (‘gold’) standards; and 2) KBpedia’s unique aspects that provide context for various entity types. With these two additions, Cognonto has now published six diverse use cases:

Each use case is summarized according to the problem and our approach to solving it and the benefits that result. The use cases themselves present general workflows and code snippets for how the use case was tackled.

We will continue to publish use cases using Cognonto’s technologies and KBpedia as they arise.

 

New ‘Computable Intelligence’ Article Posted on ODBMS.org

AI3:::Adaptive Information (Mike Bergman) - Thu, 11/17/2016 - 15:18
Invited Article Provides Good KBAI Summary

I am pleased to point to a new invited article, “Wrestling Knowledge into Computable Intelligence,” published today on ODBMS.org. My article provides a high-level summary of recent trends in knowledge-based artificial intelligence and the mindsets and designs necessary to move KBAI forward. I think the article provides a pretty good summary (if I say so myself!) of the approach we take at Cognonto.

I’d like to thank Roberto Zicari for the invite to write in a style and brevity not typical of my normal articles.   Enjoy!

New KBpedia Version Released

AI3:::Adaptive Information (Mike Bergman) - Tue, 11/15/2016 - 20:07
Upper Structure, Typologies Updated

Cognonto today released version 1.10 of its KBpedia knowledge structure. KBpedia integrates six major knowledge bases (Wikipedia, Wikidata, OpenCyc, GeoNames, DBpedia and UMBEL), plus mappings to another 20 leading knowledge vocabularies, under the KBpedia Knowledge Ontology (KKO). KBpedia’s explicit purpose is to provide a foundation for knowledge-based artificial intelligence (KBAI) by supporting the (nearly) automatic creation of training corpuses and positive and negative training sets and feature sets for deep, unsupervised and supervised machine learning.

This new release focused on two major updates. First, certain aspects of the upper structure of the KKO were streamlined. And, second, KBpedia’s core typologies, which capture the overwhelming majority of reference concepts that are classified as entity types, were further organized to create tighter taxonomic structures.

The upper portion of the KBpedia knowledge graph required cleanup because it was still using some of the abstract-tangible distinctions used in Cyc. These distinctions were no longer used with the adoption of the universal categories of Charles S. Peirce (see my earlier article for more on this architectural design). This cleanup resulted in removing nearly 25% of the upper level links from the prior version (which were superfluous to the disjoint design of KBpedia). The typology organizations are part of an ongoing effort to streamline and tighten these structures.

Last week Cognonto’s CTO, Fred Giasson, described the general build processes we have in place for KBpedia. This release is another example of that process in action.

KBpedia contains nearly 40,000 reference concepts (RCs) and about 20 million entities. The combination of these and KBpedia’s structure results in over 6 billion logical connections across the system, as these KBpedia statistics show:

Measure Value No KBpedia reference concepts (RCs) 39,052 No. mapped vocabularies 27 Core knowledge bases 6 Extended vocabularies 21 No. mapped classes 138,987 Core knowledge bases 137,322 Extended vocabularies 1,665 No. typologies (SuperTypes) 63 Core entity types 33 Other core types 5 Extended 25 Typology assignments 372,967 No. of “triples” in KBpedia ontology 1,347,818 No. aspects 80 Direct entity assignments 68,026,551 Inferred entity aspects 204,704,905 No. unique entities 19,643,718 Inferred no of entity mappings 2,541,684,526 Total no. of “triples” 3,689,849,183 Total no. of inferred and direct assertions 6,251,177,427 KBpedia v. 1.10 Statistics

This release of KBpedia is part of an ongoing series of releases to improve and extend the knowledge structure, as well as to increase its mappings to still additional external vocabularies. You can inspect the upper portions of the KBpedia knowledge graph on the Cognonto Web site. Also, if you have an ontology editor, you can download and inspect the open source KKO directly.

About Cognonto

The insight behind Cognonto is that existing knowledge bases can be staged to automate much of the tedium and reduce the costs now required to set up and train machine learners for knowledge purposes. Cognonto’s mission is to make knowledge-based artificial intelligence (KBAI) cheaper, repeatable, and applicable to enterprise needs.

Cognonto (a portmanteau of ‘cognition’ and ‘ontology’) exploits large-scale knowledge bases and semantic technologies for machine learning, data interoperability and mapping, and fact and entity extraction and tagging. Cognonto puts its insight into practice through a knowledge structure, KBpedia, designed to support AI, and a management framework, the Cognonto Platform, for integrating enterprise and external data to gain the advantage of KBpedia’s structure.

Cognonto automates away much of the tedium and reduces costs in many areas. Cognonto offers a number of use cases for how the Cognonto Platform and KBpedia in combination with enterprise information assets may be applied.

Keeping KBpedia Current and Coherent

AI3:::Adaptive Information (Mike Bergman) - Tue, 11/08/2016 - 16:38
Build and Testing Processes Are the Third Leg of Cognonto’s Capabilities

Knowledge is inherently dynamic and constantly changing. We learn new things; make connections between things that were previously hidden; revise our understandings in light of new discoveries; and embrace new domain relationships and facts. In the case of Cognonto‘s knowledge graph, KBpedia, and its six major contributing knowledge bases (KBs) and mappings to a further 20 ontologies, this dynamism takes place at warp speed. This dynamism is evident by simply noting the thousands of changes daily in each of Wikipedia and Wikidata, two of KBpedia’s major KBs.

Cognonto’s services are based on three capabilities. The first is KBpedia, which we have discussed elsewhere. The second is the Cognonto Platform, the means for accessing and using KBpedia in conjunction with enterprise or domain information. And the third are building and testing routines, scripts, logs and processes. It is the latter by which we keep KBpedia current, and is an essential infrastructure to our entire suite of services.

Cognonto’s CTO, Frédérick Giasson, has today published on LinkedIn an overview article on the principal components within this build and testing infrastructure. This is significant, but largely hidden, work. We have honed this infrastructure over a period of years, and are continuously adding to our roster of scripts and procedures.

What is remarkable about this infrastructure is the speed with which we can completely rebuild KBpedia from scratch (less than two hours) and the shortness of the entire cycle of producing a new major version of the system (less than two weeks). This infrastructure is all the more impressive when one considers that KBpedia has and maps to hundreds of thousands of concepts, millions of entities, and billions of assertions. Yet, despite this complexity, each new build of KBpedia is logically consistent, satisfiable, and coherent. Our build and testing scripts are what help ensure this quality.

Fred’s article explains this infrastructure in greater detail. Our build and testing infrastructure brings essential stability to Cognonto’s overall offerings. Great work, Fred!

KBpedia Knowledge Graph Gets Expanded Search

AI3:::Adaptive Information (Mike Bergman) - Mon, 10/24/2016 - 13:45
Search Both Reference Concepts and Entities

We have expanded the search function for Cognonto’s KBpedia knowledge graph. When first released last month, the KBpedia search was limited to reference concepts (RCs) only. With today’s upgrade, all of KBpedia’s 20 million entities can now be searched.

Though you can start investigating the Knowledge Graph (KG) simply by clicking links, to really discover items of specific interest you will need to do a search. These search functions for the Knowledge Graph are described below.

Two Search Options

The Knowledge Graph (KG) may be searched for either reference concepts (RCs) or entities.

On the main KG page (and for all other pages in the KG section except for actual results pages), the search box has a dropdown list function next to the search button. Via this dropdown, you may select to search either Reference Concepts or Entities. Depending on your selection, that choice also shows in the title to the search box. Whatever choice you make in the dropdown list is retained until you select a different option. The default option when you first encounter the Knowledge Graph is to search Reference Concepts.

Once your search specification is entered in the search box, you must click the Search button to invoke the actual search.

Reference Concepts (RC) Search

If you pick the Reference Concepts main search option, there are two different behaviors available to you.

With Autocompletion

Note our search box has Reference Concepts as its title. As you type in the search box, the system’s autocompletion will return RC candidates in a dropdown list. Only preferred labels (the canonical “name”) and terminal URI strings will match for autocomplete. Semsets and terms in the RCs description will not match.

Each newly entered character in the search box narrows the possible matching results. If your query string ceases prompting with RC candidates, there are no matches for the current substring query on preferred labels or URI fragments in the system.

There are some styling and naming conventions used to assign RC preferred labels and URI fragments you may observe over time that may make finding the right RC query more effective. Realize, however, that there are multiple ways RCs can be named and referenced. If autocomplete does not match what you think the RC name might be, try next the search without autocompletion (see next).

If you do get matches to the query string, you will be presented with one or more live link options to specific RCs in the dropdown list box. Pick the option you are seeking to go to the RC Record for your specific reference concept of interest.

Without Autocompletion

The search without autocompletion is a broader one, and is often useful when you have been unable to formulate an effective query for the preferred label of an RC of interest. To conduct a search without autocompletion, simply provide a query string in the search box without picking one of the autocompletion dropdown prompts (should they occur). Clicking the Search Concepts button will take you to a standard search of the concepts in KBpedia. Doing so brings you to a paginated set of search results pages, with 20 results per page.

Once you are on a results listing page, your search options change. Again you are given a dropdown list box, whereby you may restrict the actual concept search to one of these fields:

  • Preferred label — this is the title or standard name for the reference concept
  • Alternative labels — these are semset labels for synonyms for the RC
  • Description — this is the definition or description field for the reference concept; most RCs have this field
  • URI — this is the actual Web identifier for the RC; search is based on a sub-string search of the URI, or
  • All content — all four fields above are included in the search.

When you pick a result from the search list, you are taken to the RC Record report (see the How to Use the Knowledge Graph page).

Entities Search

The standard search box gives you a dropdown where you can choose to conduct an Entities or Reference Concepts search (see first figure above). If you choose Entities, that is so indicated in the label to the search button.

When the Entities search is chosen, you have a choice to restrict the actual search to one of these fields:

  • Preferred label — this is the title or standard name for the entity
  • Description — this is the definition or description field for the entity; not all entities have this field
  • URI — this is the actual Web identifier for the entity; search is based on a sub-string search of the URI, or
  • All content — all three fields above are included in the search.

Whichever of these four choices you make, the selection appears as the title to the search box and remains in effect until you change the dropdown option. The default when you first invoke the Entities search is ‘All’.

To actually conduct the search, you need to click on the Search Entities button.

Depending on which optional field you selected, the results count will vary. Obviously, the largest number of results arise from the ‘All’ field choice.

Entities Search Results

Search results for entities generally presents fewer details in the results than for RCs, as this figure shows:

 

Some results are limited to a title (prefLabel). Others may include altLabels or descriptions, depending on the nature of the record. Each result has an icon to the upper right indicating the source of the entity record. Clicking that icon (background highlighted text) presents the standard entity results listing as described on the How to Use the Knowledge Graph page.

Enjoy!

Cognonto Publishes Use Cases

AI3:::Adaptive Information (Mike Bergman) - Mon, 10/17/2016 - 15:26
Extending KBpedia with Domain and Enterprise Data is a Key Theme

Cognonto, our recently announced venture in knowledge-based artificial intelligence (KBAI), has just published three use cases. Two of these use cases are based on extending KBpedia with enterprise or domain data. KBpedia is the KBAI knowledge structure at the heart of Cognonto.

The Cognonto Web site contains longer descriptions of these use cases, with statistics and results where appropriate. We intend to continue to publish more use cases. Notable ones will be broadly announced.

Use Case #1: Word Embedding Corpuses

word2vec is an artificial intelligence ‘word embedding’ model that can establish similarities between terms. These similarities can be used to cluster or classify documents by topic, or to characterize them by sentiment, or for recommendations. The rich structure and entity types within Cognonto’s KBpedia knowledge structure can be used, with one or two simple queries, to create relevant domain “slices” of tens of thousands of documents and entities upon which to train word2vec models. This approach eliminates the majority of effort normally associated with word2vec for domain purposes, enabling available effort to be spent on refining the parameters of the model for superior results.

Some key findings are:

  • Domain-specific training corpuses work better with less ambiguity than general corpuses for these problems
  • Cognonto (through KBpedia) speeds and eases the creation of domain-specific training corpuses for word2vec (and other corpus-based models)
  • Other public and private text sources may be readily added to the KBpedia baseline in order to obtain still further domain-relevant models
  • Such domain-specific training corpuses can be used to establish similarity between local text documents or HTML web pages
  • This method can also be combined with Cognonto’s topics analyzer to first tag text documents using KBpedia reference concepts, and then inform or augment these domain-specific training corpuses, and
  • These capabilities enable rapid testing and refinement of different combinations of “seed” concepts to obtain better desired results.
Use Case #2: Integrating Private Data

KBpedia provides a rich set of 20 million entities in its standard configuration. However, by including relevant entity lists, which may already be in the possession of the enterprise or from specialty domain datasets, significant improvements can be achieved across all of the standard metrics used for entity recognition and tagging. Here is an example of the standard metrics applied by Cognonto in its efforts:

Cognonto’s standard methodology also includes the creation of reference, or “gold standards”, for measuring the benefits of adding more data or performing other tweaks on the entity extraction algorithms.

Some key findings from this use case in adding private data to KBpedia include:

  • In the example used, adding private enterprise data results in more than a doubling of accuracy (108%) over the standard, baseline KBpedia for identifying the publishing organization of a Web page
  • Some datasets may have a more significant impact than others, but overall, each dataset contributes to the overall improvements of the predictions. Generally adding more data improves results across all measured metrics
  • “Gold standards” are an essential component for testing the value of adding specific datasets or refining machine learning parameters
  • Approx. 500 training instances are sufficient to build a useful “gold standard” for entity tagging; negative training examples are also advisable
  • Even if all specific entities are not identified, flagging a potential “unknown” entity is an important means for targeted next efforts of adding to the current knowledge base
  • KBpedia is a very useful structure and starting point for an entity tagging effort, but that adding domain data is probably essential to gain the overall accuracy desired for enterprise requirements, and
  • This use case is broadly applicable to any entity recognition and tagging initiative.
Use Case #3: The Cognonto Mapper

The Cognonto Mapper includes standard baseline capabilities found in other mappers such as string and label comparisons, attribute comparisons, and the like. But, unlike conventional mappers, the Cognonto Mapper is able to leverage both the internal knowledge graph structure and its use of typologies (most of which do not overlap with one another) to add structural comparators as well. These capabilities lead to more automation at the front end of generating good, likely mapping candidates, leading to faster acceptance by analysts of the final mappings. This approach is in keeping with Cognonto’s philosophy to emphasize “semi-automatic” mappings that combine fast final assignments with the highest quality. Maintaining mapping quality is the sine qua non of knowledge-based artificial intelligence.

Some key findings from this use case are:

  • A capable mapper, including structural considerations, is essential for quickly generating mapping candidates for entities, concepts, vocabularies, schema and ontologies
  • A capable mapper, including structural considerations, is essential for accurate mapping candidates for entities, concepts, vocabularies, schema and ontologies
  • Quality mappings require manual vetting for final acceptance
  • The Cognonto Mapper generates quick and accurate candidates for linking entities, concepts, vocabularies, schema or ontologies
  • The Cognonto Mapper can generate “gold standards” in 15% of the time of standard approaches
  • The Cognonto Mapper scoring can help point to likely mappings even where the candidates are ambiguous
  • The Cognonto Mapper can identity likely, but unknown, entities for inspection and commitment to the knowledge base, and
  • Other structural aspects of KBpedia, such as aspects, relations or attributes, can inform other comparators and mappers.

See the original use case links for further details, code examples, and results and statistics. As noted, we will announce additional use cases as they are published.

Threes All of the Way Down to Typologies

AI3:::Adaptive Information (Mike Bergman) - Mon, 10/03/2016 - 15:30

Download as PDF

Further Details on the Design of KBpedia’s Knowledge Ontology (KKO)

Every knowledge structure used for knowledge representation (KR) or knowledge-based artificial intelligence (KBAI) needs to be governed by some form of conceptual schema. In the semantic Web space, such schema are known as “ontologies”, since they attempt to capture the nature or being (Greek ὄντως, or ontós) of the knowledge domain at hand. Because the word ‘ontology’ is a bit intimidating, a better variant has proven to be the knowledge graph (because all semantic ontologies take the structural form of a graph). In Cognonto‘s KBAI efforts, we tend to use the terms ontology and knowledge graph interchangeably.

Cognonto uses the KBpedia Knowledge Ontology (KKO) as its upper conceptual schema. KKO is the structure by which Cognoto’s hundreds of thousands of reference concepts and millions of entities are organized [1]. This article presents an overview and rationale for this KKO structure. Subsequent articles will delve into specific aspects of KKO where warranted.

A Grounding in Peirce’s Triadic Logic

The upper structure of the KBpedia Knowledge Ontology (KKO) is informed by the triadic logic and basic categories of Charles Sanders Peirce. If the relation of this triadic design to knowledge representation appears a bit opaque, please refer to the introduction of this series in the prior article, The Irreducible Truth of Threes. The key point is that ‘threes’ are the fewest by which to model context and perspective, essential to capture the nature of knowledge.

Peirce’s triadic logic, or trichotomy, is also the basis for his views on semiosis (or the nature of signs). The three constituents of Peirce’s trichotomy, what he called simply the Three Categories, were in his view the most primitive or reduced manner by which to understand and categorize things, concepts and ideas. Peirce’s Three Categories  can be roughly summarized as:

  • Firstness [1ns] — these are potentials, the basic forces or qualities that combine together or interact in various ways to enable the real things we perceive in the world, such as matter, life and ideas. These are the unrealized building blocks, or primitives, the essences or attributes or possible juxtapositions
  • Secondness [2ns] — these are the particular realized things or concepts in the world, what we can perceive, point to and describe. A particular is also known as an entity, instance or individual
  • Thirdness [3ns] — these are the laws, habits, regularities and continuities that may be generalized from particulars. All generals — what are also known as classes, kinds or types — belong to this category. The process of finding and deriving these generalities also leads to new insights or emergent properties, which continue to fuel knowledge discovery. Insights arising from Thirdness enable us to further explore and understand things, and is a driving force for further categorization.

Understanding, inquiry and knowledge require this irreducible structure; connections, meaning and communication depend on all three components, standing in relation to one another and subject to interpretation by multiple agents. (Traditional classification schemes have a dyadic or dichotomous nature, which does not support the richer views of context and interpretation inherent in the Peircean view.)

Peirce argues persuasively that how we perceive and communicate things requires this irreducible triadic structure. The symbolic nature of Thirdness means that communication and understanding is a continuous process of refinement, getting us closer to the truth, but never fully achieving it. Thirdness is a social and imprecise mode of communication and discovery, conducted by us and other agents separate from the things and phenomena being observed. Though it is a fallibilistic process, it is one that also lends itself to rigor and methods. The scientific method is a premier example of Thirdness in action.

What constitutes the potentials, realized particulars, and generalizations that may be drawn from a query or investigation is contextual in nature. That is why the mindset of Peirce’s triadic logic is a powerful guide to how to think about and organize the things and ideas in our world (that is, knowledge representation). Peirce’s triadic logic and views on categorization are fractal in nature. We can apply this triadic logic to any level of information granularity.

Thus, KKO applies this mindset to organizing its knowledge graph. At each level in the KKO upper structure, we strive to organize each category according to the ideas of Firstness (1ns), Secondness (2ns) and Thirdness (3ns), as shown in the upper KKO structure below.

Basic Structural Considerations

Now armed with a basic conceptual and logical grounding, what are the main kinds of distinctions we want to capture in our knowledge structure? Since our purpose is to provide a means for integrating knowledge bases (KBs) of use to artificial intelligence (AI), or KBAI, the answer to this question resides in: 1) what conceptual distinctions are captured by the constituent KBs; and 2) what kinds of work (AI) we want to do with the structure.

The answers to these questions help us to define the basic vocabulary of our knowledge base, what Peirce called its speculative grammar [2]. This base vocabulary of KKO is thus:

  • Attributes are the ways to characterize the entities or things within the knowledge base; while the attribute values and options may be quite complex, the relationship is monadic to the subject at hand. These are intensional properties of the subject
  • Relations are the way we describe connections between two or more things; relations are external-facing, between the subject and another entity or concept; relations set the extensional structure of the knowledge graph
  • Entities are the basic, real things in our domain of interest; they are nameable things or ideas that have identity, are defined in some manner, can be referenced, and should be related to types; entities are the bulk of the overall knowledge base
  • Events are nameable sequences of time, are described in some manner, can be referenced, and may be related to other time sequences or types
  • Activities are sustained actions over durations of time; activities may be organized into natural classes
  • Types are the hierarchical classification of natural kinds within all of the terms above
  • The Typology structure is not only a natural organization of natural classes, but it enables flexible interaction points with inferencing across its ‘accordion-like’ design
  • Base concepts are the vocabulary to the grammar and top-level concepts in the knowledge graph, organized according to Peircean-informed categories
  • Annotations are indexes and the metadata of the KB; these can not be inferenced over. But, they can be searched and language features can be processed in other ways.

We will be talking about specific vocabulary items above in subsequent articles. One important distinction to draw for now, however, is the split between attributes and relations. In standard RDF and OWL ontologies these are lumped together as properties. In OWL, there is the further distinction of datatype and object properties, but these do not quite capture the difference we desire. In KBpedia, attributes are the descriptions or characteristics of a given entity (or its type); relations are the roles, connections or subsumptions between objects [3].

How these vocabulary terms relate to one another and the overall KBpedia knowledge structure is shown by this diagram:

A Knowledge Base Grammar

Note that the three columns of this figure correspond to the three categories of potentials (1ns, left column), particulars (2ns, middle column) and generals (3ns, right column) described above. In terms of KBpedia, all of the instances of the knowledge structure, now numbering over 20 million in the standard version, are affiliated with the middle column (particulars). The classification aspects of KBpedia reside in the right column (generals). The reasoning aspects of KBpedia largely reside in the left and right columns (though reasoning work is also done on selecting and aggregating instances in the middle column) [4].

Below the Upper Structure are Typologies

More than 85% of the classification structure of KBpedia resides in the generals, or types, in the rightmost column. These, in turn, are organized according to a set of typologies, or natural classification structures. Unlike the KKO upper structure, each typology is not necessarily organized according to Peirce’s triadic logic. That is because once we come to organize and classify the real things in the world, we are dealing with objects of a more-or-less uniform character (such as animals or products or atomic elements).

There are about 80 such typologies in the KBpedia structure, about 30 of which are deemed “core”, meaning they capture the bulk of the classificatory system. Another document presents these 30 “core” typologies in more detail.

I have written elsewhere [5] about the basis for “natural” classification systems. These approaches, too, are drawn from Peirce’s writings. Natural classifications may apply to truly “natural” things, like organisms and matter, but also to man-made objects and social movements and ideas. The key argument is that shared attributes, including a defining kind of “essence” (Aristotle) or “final cause” (Peirce) help define the specific class or type to which an object may belong. For Peirce, what science has to tell us, or what social consensus settles upon, holds sway.

If accomplished well, natural classification systems lend themselves to hierarchical structures that may be reasoned over. Further, if the splits between typologies are also done well, then it is also possible to establish non-overlapping (“disjoint”) relationships between typologies that provide powerful restriction and selection capabilities across the knowledge structure. We believe KBpedia already achieves these objectives, though we continue to refine the structure based on our mappings to other external systems and other logical tests.

The KKO Upper Structure

We now have the pieces in hand to construct the full KKO upper structure. Here is the upper structure of KKO with its 144 concepts:

Predica [1ns] Ontics [1ns] Qualities [1ns] Physical [2ns] Being [1ns] One == Haeccity [1ns] True [2ns] Good [3ns] Form [2ns] Structure [3ns] Conceptual [3ns] Absolute [1ns] SimpleRelative [2ns] Conjugative [3ns] Attribuo [2ns] Identity [1ns] Haeccity == One [1ns] Nature [2ns] Beingness [1ns] Real [2ns] Matter [1ns] SubstantialForm [2ns] AccidentalForm [3ns] Fictional [3ns] Quiddity [3ns] Intensional [2ns] Conjunctive [3ns] Quantity [1ns] Values [1ns] Numbers [1ns] Multitudes [2ns] Magnitudes [3ns] Discrete [2ns] Continuous [3ns] Roles [2ns] Typical [3ns] Relatio [3ns] Subsumption [1ns] Similar [2ns] LogicalConnection [3ns] Unary [1ns] Binary [2ns] Conditional [3ns] Particulars [2ns] Entities [1ns] SingleEntities [1ns] Objects [1ns] States [2ns] Events [3ns] PartOfEntities [2ns] Members [1ns] Parts [2ns] FunctionalComponents [3ns] ComplexEntities [3ns] CollectiveStuff [1ns] MixedStuff [2ns] CompoundEntities [3ns] Indices [2ns] Indicators [1ns] Associations [2ns] Annotations [3ns] Selectional [1ns] Referential [2ns] Directional [3ns] Continua [3ns] Space [1ns] Points [1ns] Areas [2ns] 2D Dimensions SpaceRegions [3ns] 3D Dimensions Time [2ns] Instants [1ns] Intervals [2ns] Events [3ns] Duratives [3ns] Situations [1ns] Activities [2ns] Processes [3ns] Generals [3ns] (== SuperTypes) SignElements [1ns] AttributeTypes [1ns] RelationTypes [2ns] SituationTypes Symbols [3ns] Primitives [1ns] Structures [2ns] Conventions [3ns] Constituents [2ns] NaturalPhenomena [1ns] SpaceTypes [2ns] Shapes [1ns] Places [2ns] LocationPlace AreaRegion Forms [3ns] TimeTypes [3ns] Times [1ns] EventTypes [2ns] ActivityTypes [3ns] Manifestations [3ns] NaturalMatter [1ns] AtomsElements [1ns] NaturalSubstances [2ns] Chemistry [3ns] OrganicMatter [2ns] OrganicChemistry [1ns] BiologicalProcesses LivingThings [2ns] Prokaryotes [1ns] Eukaryotes [2ns] ProtistsFungus [1ns] Plants [2ns] Animals [3ns] Diseases [3ns] Agents [3ns] Persons [1ns] Organizations [2ns] Geopolitical [3ns] Symbolic [3ns] Information [1ns] AVInfo [1ns] VisualInfo AudioInfo WrittenInfo [2ns] StructuredInfo [3ns] Artifacts [2ns] FoodDrink Drugs Products Facilities Systems [3ns] MentalProcesses [1ns] Concepts [1ns] TopicsCategories [2ns] LearningProcesses [3ns] SocialProcesses [2ns] FinanceEconomy Society Methodeutic [3ns] InquiryMethods [1ns] KnowledgeDomains [2ns] EmergentKnowledge [3ns]

Where appropriate, each entry is also labeled with one of the Three Categories (1ns, 2ns, 3ns). Note that all of the typologies are shown under the main Generals (3ns) category. The main “core” typologies are shown in orange. (Note that TopicsCategories and KnowledgeDomains are big typologies, but are not disjoint in any way. Shapes is also a big typology, but about half of all entities have that type.)

A Graph View of the Structure

We can differently view this structure through a graph view of the KKO structure, also showing the “core” typologies in orange. As might be expected, the KKO “core” ontologies tend to occur at the periphery of this graph:

Upper Knowledge Graph with ‘Core’ Typologies Highlighted For Further Information

Of course, the purpose of all of this design is to provide a coherent, consistent, logical structure over which Cognonto may reason and link external data and schema. We will be discussing elsewhere the specific use cases and applications of this structure. For now, we wanted to set the stage for the design basis for KKO. This article will be a common reference to those subsequent discussions.

The KKO upper structure may be downloaded and inspected in greater detail.

[1] For an introduction to Cognonto and KBpedia, its knowledge structure, see M.K. Bergman, 2016. “Cognonto is on the Hunt for Big AI Game“, AI3:::Adaptive Information blog, September 20, 2016. [2] See M.K. Bergman, 2016. “A Speculative Grammar for Knowledge Bases“, AI3:::Adaptive Information blog, June 20, 2016. [3] Attributes, Relations and Annotations comprise OWL properties. In general, Attributes correspond to the OWL datatypes property; Relations to the OWL object property; and Annotations to the OWL annotation property. These specific OWL terms are not used in our speculative grammar, however, because some attributes may be drawn from controlled vocabularies, such as colors or shapes, that can be represented as one of a list of attribute choices. In these cases, such attributes are defined as object properties. Nonetheless, the mappings of our speculative grammar to existing OWL properties is quite close. [4] As I earlier wrote, “Description logics and their semantics traditionally split concepts and their relationships from the different treatment of instances and their attributes and roles, expressed as fact assertions. The concept split is known as the TBox (for terminological knowledge, the basis for T in TBox) and represents the schema or taxonomy of the domain at hand. The TBox is the structural and intensional component of conceptual relationships. The second split of instances is known as the ABox (for assertions, the basis for A in ABox) and describes the attributes of instances (and individuals), the roles between instances, and other assertions about instances regarding their class membership with the TBox concepts.” In the diagram above, the middle column represents particulars, or the ABox components. The definition of items and the first and third columns represent TBox components. [5] See M.K. Bergman, 2015. “‘Natural Classes’ in the Knowledge Web“, AI3:::Adaptive Information blog, July 13, 2015.

The Irreducible Truth of Threes

AI3:::Adaptive Information (Mike Bergman) - Tue, 09/27/2016 - 17:15
KR Models Need to Represent Knowledge via Context and Perspective

What does the idea of knowledge mean to you?

The entry for knowledge on Wikipedia says:

“Knowledge is a familiarity, awareness or understanding of someone or something, such as facts, information, descriptions, or skills, which is acquired through experience or education by perceiving, discovering, or learning. Knowledge can refer to a theoretical or practical understanding of a subject. It can be implicit (as with practical skill or expertise) or explicit (as with the theoretical understanding of a subject); it can be more or less formal or systematic. [1]

OK, that’s a lot of words. Rather than parse its specifics, let’s look at the basis of “knowledge” using a variety of simple examples.

Perplexing Perspectives and Confusing Contexts

Let’s take for an example the statement, the sky is blue. We can accept this as a factual statement (thus, assumed knowledge). But, we also know that the sky might be dark or black, if it is night. Or the sky may be gray if it is cloudy. Indeed, when we hear the statement that the sky is blue, if we believe the source or can see the sky for ourselves, then we can readily infer that the observation is occurring during daylight, under a clear sky. Our acceptance of an assertion as factual or being true carries with it the implications of its related contexts. On the other hand, were the simple statement to be le ciel est bleu, and if we did not know French, we would not know what to make of the statement, true or false, with context or not, even if all of the assertions are still correct.

This simple example carries with it two profound observations. First, context helps to determine whether we believe or not a given statement, and if we believe it, what the related context implied by the statement might be. Second, how this information is conveyed to us is via symbols — in this case, the English language, but applicable to all human and artificial and formal notations like mathematics as well — which we may or may not be able to interpret correctly. If I am monolingual in English and I see French statements, I do not know what the symbols mean.

Knowledge is that which is “true” in a given context or perspective. Is it a duck, or is it a rabbit? Knowledge may reside solely in our own minds, and not be part of “common knowledge”. But, ultimately, even personal beliefs not held by others only become “knowledge” that we can rely upon in our discourse once others have “acknowledged” the truth. Forward-looking thinkers like Copernicus or Galileo or Einstein may have understood something in their own minds not yet shared by others, but we do not “acknowledge” those understandings as knowledge until we can share and discuss the insight. (That is, what scientists would call independent verification.) In this manner, knowledge, like language and symbol-creation, is inherently a social phenomena. If I coin a new word, but no one else understands what I am saying, that is not part of knowledge; that is gibberish.

OK, let’s take another example. This time we’ll take the simple case of a flower. What this panel of images shows is how a composite flower may be seen by first humans, then bees and then butterflies:

How Humans, Bees and Butterflies See A Daisy

In this example [2], we are highlighting the fact that different insects see objects with different wavelengths of light than we (humans) do. Bees see much more in the ultraviolet spectrum. The daisy flower knows how to attract this most important pollinator.

In a different example focused on human perception alone, look at these two panels on the left:

Color Blind, on left; What is ‘bank’?, on right

The middle picture shows us how the color-blind person “sees”, with reds and yellows washed out [3].

Or, let’s take another example, this case the black-and-white word for ‘bank’. We can see this word, and if we speak English, even recognize it, but what does this symbol mean? A financial institution? The shore of a river? Turning an airplane? A kind of pool shot? Tending a fire for the evening?

In all of these examples, there is an actual object that is the focus of attention. But what we “know” about this object depends on what we perceive or understand and who or what is doing the perceiving and the understanding. We can never fully “know” the object because we can never encompass all perspectives and interpretations.

KR Models Need to Represent Knowledge via Context and Perspective

Every knowledge structure used for knowledge representation (KR) or knowledge-based artificial intelligence (KBAI) needs to be governed by some form of conceptual schema. In the semantic Web space, such schema are known as ontologies, since they attempt to capture the nature or being (Greek ὄντως, or ontós) of the knowledge domain at hand. Because the word ‘ontology’ is a bit intimidating, a better variant has proven to be the knowledge graph (because all semantic ontologies take the structural form of a graph). In Cognonto‘s KBAI efforts, we tend to use the terms ontology and knowledge graph interchangeably.

In general knowledge domains, such schema are also known as upper ontologies. However, one of the first things we see with existing ontologies is that they tend to be organized around a single, dyadic dimension, even though guided by a diversity of conceptual approaches. In the venerable Cyc knowledge structure, one of the major divisions is between what is tangible and what is intangible. In BFO, the Basic Formal Ontology, the split is between a “snapshot” view of the world (continuant) and its entities versus a “spanning” view that is explicit about changes in things over time (occurrent). Other upper ontologies have different dyadic splits, such as abstract v. physical, perduant v. endurant, dependent v. independent, particulars v. universals, or determinate v. indeterminate [4]. I’m sure there are others.

Ontologies are designed for specific purposes, and the bases for these splits in other ontologies have their rationales and uses. But in Cognonto’s case of needing to design an ontology whose specific purpose is knowledge representation, we need to explicitly model the nature of knowledge. Knowledge is not black and white, nor is it shades of gray along a single dimension. Knowledge is an incredibly rich construct intimately related to context and perspective. The minimum cardinality that can provide such perspective is three.

If we return to the examples that began this article, we begin to see the interaction of three separate things. We have the actual thing itself, be it an object or a phenomenon. It is what is is. Then, we have a way that that thing is conveyed or represented. It might be an image, a sound, a perception, a finger pointing at it, or a symbol (or combination of symbols such as a description) of it. Then we have how that representation is perceived. It is in the interplay of these three separate things that something is “understood” or becomes “knowledge” (that is, a sign). This triadic view of the world was first articulated by Charles Sanders Peirce (1839-1914) (pronounced “purse”), the great American logician, philosopher and polymath.

Peirce’s logic of signs in fact is a taxonomy of sign relations, in which signs get reified and expanded via still further signs, ultimately leading to communication, understanding and an approximation of “canonical” truth. Peirce saw the scientific method as itself an example of this process [5].

A given sign is a representation amongst the triad of the sign itself (which Peirce called a representamen, the actual signifying item that stands in a well-defined kind of relation to the two other things), its object and its interpretant. The object is the actual thing itself. The interpretant is how the agent or the perceiver of the sign understands and interprets the sign. Depending on the context and use, a sign (or representamen) may be either an icon (a likeness), an indicator or index (a pointer or physical linkage to the object) or a symbol (understood convention that represents the object, such as a word or other meaningful signifier).

An interpretant in its barest form is a sign’s meaning, implication, or ramification. For a sign to be effective, it must represent an object in such a way that it is understood and used again. This makes the assignment and use of signs a community process of understanding and acceptance [6], as well as a truth-verifying exercise of testing and confirming accepted associations (such as the meanings or words or symbols).

The key aspect of signs for Peirce, though, is the ongoing process of interpretation and reference to further signs, a process he called semiosis. A sign of an object leads to interpretants, which, as signs, then lead to further interpretants.

Relation to Cognonto’s KBpedia Knowledge Ontology

The essence of knowledge is that it is ever-growing and expandable. New insights bring new relations and new truths. The structures we use to represent this knowledge must themselves adapt and reflect the best of our current, testable understandings. Peirce saw the trichotomous parts of his sign logic as the fewest “decomposable” needed to model the real world; we would call these “primitives” in modern terminology. Robert Burch has called Peirce’s ideas of “indecomposability” the ‘Reduction Thesis’ [7]. The basic thesis is that ternary relations suffice to construct any and all arbitrary relations, but that not all relations can be constructed from unary and binary relations alone. Threes are irreducible to capture the basis of knowledge.

With its express purpose to provide a sound basis for modeling knowledge, essential to knowledge-based artificial intelligence, Cognonto’s governing schema, the KBpedia Knowledge Ontology (KKO), is the first knowledge graph to explicitly embrace this triadic logic. Later articles will discuss KKO in much greater detail. Peirce’s logic of semiosis and his three universal categories provide the missing perspective of classing and categorizing the world around us. The irreducible truth of ‘threes’ is the essential foundation for representing knowledge and language.

[1] Similar senses are conveyed by the Wiktionary definition of knowledge. [2] From Photography of the Invisible World (http://photographyoftheinvisibleworld.blogspot.com/2013/11/today-more-about-my-last-surviving.html). [3] See http://allday.com/post/3919-this-is-what-its-actually-like-to-be-colorblind/ color_blind.png. [4] See, for example, Ludger Jansen, 2008. “Categories: The Top-level Ontology,” Applied ontology: An introduction (2008): 173-196, and Nicola Guarino, 1997. “Some Organizing Principles For A Unified Top-Level Ontology,” National Research Council, LADSEB-CNR Int. Report, V3.0, August 1997. [5] M.K. Bergman, 2012. “Give Me a Sign: What Do Things Mean on the Semantic Web?“, AI3:::Adaptive Information blog, January 24, 2012. [6] See further Catherine Legg, 2010. “Pragmaticsm on the Semantic Web,” in Bergman, M., Paavola, S., Pietarinen, A.-V., & Rydenfelt, H. eds., Ideas in Action: Proceedings of the Applying Peirce Conference, pp. 173–188. Nordic Studies in Pragmatism 1. Helsinki: Nordic Pragmatism Network. [7] See Robert Burch, 1991. A Peircean Reduction Thesis: The Foundations of Topological Logic, Texas Tech University Press, Lubbock, TX. Peirce’s reduction thesis is never stated explicitly by Peirce, but is alluded to in numerous snippets.
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