BFO 2 0 Axiomatization Modularization and Semantic Verification

BFO 2. 0: Axiomatization, Modularization, and Semantic Verification John Beverley 1

Big Picture for Today • The Basic Formal Ontology, and Axiomatization in First Order Logic • Modularization of the Axioms • Semantic Verification of the Axioms • Summary and Future Work 2

The Basic Formal Ontology (BFO) Everything you think you know about BFO is…. 3

The Basic Formal Ontology Everything you think you know about BFO is…. Probably about right, but here’s a refresher. 4

The BFO 2. 0 Taxonomy 5

Some Important Distinctions • Continuant Entities vs. Occurrent Entities • Dependency vs. Independency • Types vs. Instances 6

Continuant vs. Occurrent • Continuants are enduring entities, wholly present whenever they are, and which lack temporal parts Continuant thing, quality … 7

Continuant vs. Occurrent • Occurrents are perduring entities, not wholly present (usually) at a single instants, and which have temporal parts Continuant Occurrent thing, quality … process, event 8

Dependence vs. Independence Continuant Independent Continuant Dependent Continuant thing quality Occurrent process, event e. g. temperature depends on a bearer 9

Dependence vs. Independence Continuant Independent Continuant Dependent Continuant thing quality, … Occurrent process, event e. g. an event depends on some participant 10

Types vs. Instances Continuant Independent Continuant Dependent Continuant thing quality . . Occurrent process, event . . . . 11

Types vs. Instances types Continuant Independent Continuant Dependent Continuant thing quality . . Occurrent process, event . . . . 12

Types vs. Instances types Continuant Independent Continuant Dependent Continuant thing quality . . instances Occurrent process, event . . . . 13

Dependence vs. Independence Again Continuant Occurrent process Independent Continuant Dependent Continuant thing quality . . temperature depends on bearer . . . . 14

Instance of Independent Continuant Occurrent process Independent Continuant Dependent Continuant thing quality . . John temperature depends on bearer . . . . 15

Instance of Quality Dependent on… Occurrent Continuant process Independent Continuant Dependent Continuant thing quality . . John temperature depends on bearer John’s 101 Degree Temperature . . . . 16

Instance of Process Dependent on… Occurrent Continuant process Independent Continuant Dependent Continuant thing quality . . John temperature depends on bearer John’s 101 Degree Temperature . . . . John’s running process 17

Example Binary Relations • Type-Type Relations • human is_a mammal • human heart part_of human • Instance-Type Relations • John instance_of the type human • Sally allergic_to the type codeine • Instance-Instance Relations • John’s heart part_of John • John’s aorta connected_to John’s heart 18

More Basic Formal Ontology Details • Basic Formal Ontology History • Developed by Smith/Grenon • Based, in part, on Theory of Granular Partitions of Smith/Bittner/Donnelly • Designed as upper level, domain neutral, ontology • Computationally tractable fundamental ontology • Upper level ontologies are response to data silo problems • Also allow easy computational inferencing due to class/subclass structure 19

BFO Versioning • Developing BFO is a community effort, see here: https: //raw. githubusercontent. com/BFOontology/BFO/master/releases/2. 0/bfo. owl • BFO has undergone significant changes since introduced • Recent version, 2. 0, codified in a reference document, and MIT Press published “Building Ontologies with the Basic Formal Ontology” 20

BFO Implementations • A BFO implementation is the realization of a codified BFO technical specification (e. g. reference manual) as a program or as a software component • Implementations: • OWL – Used in web development/Protégé • FOL – Commonly used; theorem prover friendly • CLIF – Common logical language (FOL with abstract semantic language) 21

BFO 2. 0 Implementation • New versions require either new implementations, or updating old implementations • Given BFO 2. 0 changes, implementations are: • OWL – Description Logic Base • FOL – First Order Logic • CLIF – Common Logic 22

BFO 2. 0 Implementation • New versions require either new implementations, or updating old implementations • Given BFO 2. 0 changes, implementations are: • OWL – Description Logic Base (up-to-date!) • FOL – First Order Logic • CLIF – Common Logic 23

BFO 2. 0 Implementation • New versions require either new implementations, or updating old implementations • Given BFO 2. 0 changes, implementations are: • OWL – Description Logic Base (up-to-date!) • FOL – First Order Logic (up-to-date!) • CLIF – Common Logic 24

BFO 2. 0 Implementation • New versions require either new implementations, or updating old implementations • Given BFO 2. 0 changes, implementations are: • OWL – Description Logic Base (up-to-date!) • FOL – First Order Logic (up-to-date!) • CLIF – Common Logic (given FOL implementation, easily generated) 25

BFO 2. 0 Implementation • New versions require either new implementations, or updating old implementations • Given BFO 2. 0 changes, implementations are: • OWL – Description Logic Base (up-to-date!) • FOL – First Order Logic (up-to-date!) • CLIF – Common Logic (given FOL implementation, easily generated) • The work I’ll discuss today concerns the FOL implementation… 26

Axiom Alignment • The FOL implementation is generated by perusing BFO source material, and representing philosophical claims in first order logic • To my knowledge, all claims in the literature concerning BFO have been axiomatized, i. e. represented as axioms in FOL • But let’s look at an example of what I mean… 27

Example: Continuant Axiom • Perusing the reference manual, you’ll find claims like the following: • “Every continuant is an entity. ” • Which can be represented in FOL as: 28

Theorem Alignment • Additionally, theorems are often stated (in the source material) to follow from given philosophical claims, and these must be proven • To my knowledge, all claimed theorems have been proven…by me and some relatively new, and very powerful, software: 29

Theorem Alignment • Additionally, theorems are often stated (in the source material) to follow from given philosophical claims, and these must be proven • To my knowledge, all claimed theorems have been proven…by me and some relatively new, and very powerful, software: 30

Theorem Alignment • Additionally, theorems are often stated (in the source material) to follow from given philosophical claims, and these must be proven • To my knowledge, all claimed theorems have been proven…by me and some relatively new, and very powerful, software • Prover 9 is useful in automating theorem proofs, and it comes bundled with Mace 4 which is useful for finding models of subsets of axioms 31

Prover 9: What? • FOL axioms can be used to prove theorems about BFO 2. 0 within Prover 9 • Prover 9’s user interface 32

Toy Example: Sentential Logic • Sentential logic input illustrated with MP: Set Goal(s): 33

Toy Example: Sentential Logic Sentential logic MP ‘resolution’ proof: 34

Resolution Proof? • Common automated prover technique. • Claims ‘clausified’ in conjunctive normal form • Goal is negated (for reductio ad absurdam) • Contradictions sought and resolved 35

Toy Example: FOL • First Order Logic input illustrating DS: • Set Goal(s): 36

Toy Example: FOL • Disjunctive Syllogism resolution proof: 37

Time to Put Away the Toys: BFO 2. 0 • BFO 2. 0 modularized txt files can be uploaded to prover 9: 38

Time to Put Away the Toys: BFO 2. 0 • Generating axioms (and associated Reference Manual citations + comments) 39

Time to Put Away the Toys: BFO 2. 0 • Look! Here’s our example from earlier, all continuants are entities… 40

BFO 2. 0 -FOL Theorem: S-Depends. On. At • But let’s not waste time with nostalgia; let’s get provin’ • ‘Specifically Depends On At’ is a BFO 2. 0 relation • Holds between a, b, t such that b and a share no parts; if b exists c must exist; and b is neither a boundary nor Site of c • Similar to “existential dependence”, e. g. pain on organism; gait on walking, etc. 41

BFO 2. 0 -FOL Theorem: S-Depends. On. At • Intuitively, this relation is irreflexive, i. e. it is not the case for any x, x s -depends-on x • This is because BFO accepts parthood is reflexive; everything is a (maximal) part of itself • Hence, every continuant shares a part with itself; but s-depends-on requires relata share no parts; we should be able to prove irreflexivity then for this relation (in the presence of parthood axioms) 42

BFO 2. 0 -FOL Theorem: S-Depends. On. At • S-depends-on-at irreflexive proof: 43

BFO 2. 0 -FOL Theorem: Material Entity • If an Entity has a continuant part that is a material entity, then the Entity in question is a Continuant • Entities are…everything…but not everything is a continuant…(e. g. there are occurrents) • Import the source file Material Entity Axioms and Continuant Axioms… 44

BFO 2. 0 -FOL Theorem: Material Entity 45

BFO 2. 0 -FOL Theorem: Material Entity • Voila, theorem is proved: 46

Limitations of Theorem Proving • Proving theorems is useful for uncovering relationships among axioms and axiom commitments • But capturing the philosophical underpinning of BFO requires a large number of axioms; proving theorems here and there just isn’t going to help us understand BFO very well… • We need to prove meta-theoretic results about BFO; we accomplish this by working with the logical structure of BFO directly, i. e. the relations each class bears to other classes 47

BFO 2. 0 Logical Structure 48

Displaying Logical Dependencies • BFO-FOL axioms have been organized to display logical dependencies 49

Displaying Logical Dependencies • BFO-FOL axioms have been organized to display logical dependencies • Example: Object class logically depends on Continuant class 50

Displaying Logical Dependencies • BFO-FOL axioms have been organized to display logical dependencies • Example: Object class logically depends on Continuant class 51

The BFO 2. 0 Hierarchy 52

The BFO 2. 0 Hierarchy 53

Displaying Logical Dependencies • BFO-FOL axioms have been organized to display logical dependencies • Example: Object class logically depends on Continuant class 54

Displaying Logical Dependencies • BFO-FOL axioms have been organized to display logical dependencies • Example: Object class logically depends on Continuant class 55

The BFO 2. 0 Hierarchy 56

The BFO 2. 0 Hierarchy 57

Displaying Logical Dependencies • BFO-FOL axioms have been organized to display logical dependencies • Example: Object class logically depends on Continuant class 58

Displaying Logical Dependencies • BFO-FOL axioms have been organized to display logical dependencies • Example: Object class logically depends on Continuant class 59

The BFO 2. 0 Hierarchy 60

The BFO 2. 0 Hierarchy 61

Why Logical Dependencies? • Focusing on logical dependencies helps with: 62

Why Logical Dependencies? • Focusing on logical dependencies helps with: • Spotting axioms used by all/most classes/relations in BFO 2. 0 63

Why Logical Dependencies? • Focusing on logical dependencies helps with: • Spotting axioms used by all/most classes/relations in BFO 2. 0 • Spotting redundant axioms 64

Why Logical Dependencies? • Focusing on logical dependencies helps with: • Spotting axioms used by all/most classes/relations in BFO 2. 0 • Spotting redundant axioms • Spotting useful modularizations of the BFO 2. 0 axioms 65

Why Logical Dependencies? • Focusing on logical dependencies helps with: • Spotting axioms used by all/most classes/relations in BFO 2. 0 • Spotting redundant axioms • Spotting useful modularizations of the BFO 2. 0 axioms • What is, and why should I care about, modularization, I hear you cry… 66

Modularization: What? • Modules are extracted subsets of axioms from BFO-FOL 2. 0 • Example: Proper subset of BFO-FOL 2. 0 axioms compose the “Taxonomy Module”, i. e. structural axioms governing all entities (the examples I just gave!) 67

Taxonomy Module • Subset of axioms governing taxonomy for BFO; sample axioms: 68

Modularization: What? • Modules: Extracted subsets of axioms from BFO-FOL 2. 0 • Example: Proper subset of BFO-FOL 2. 0 axioms compose the “Taxonomy Module”, i. e. structural axioms governing all entities 69

Modularization: What? • Modules: Extracted subsets of axioms from BFO-FOL 2. 0 • Example: Proper subset of BFO-FOL 2. 0 axioms compose the “Taxonomy Module”, i. e. structural axioms governing all entities • Example: Proper subset of BFO 2. 0 axioms compose the “Temporal Region Mereology Module”, i. e. axioms governing temporal region mereology (a parthood relation governing regions of time) 70

Temporal Region Mereology Module • Subset of axioms applying to temporal regions; sample axioms: 71

Modularization: What? • Modules: Extracted subsets of axioms from BFO-FOL 2. 0 • Example: Proper subset of BFO-FOL 2. 0 axioms compose the “Taxonomy Module”, i. e. structural axioms governing all entities • Example: Proper subset of BFO 2. 0 axioms compose the “Temporal Region Mereology Module”, i. e. axioms governing temporal region mereology 72

Modularization: What? • Modules: Extracted subsets of axioms from BFO-FOL 2. 0 • Example: Proper subset of BFO-FOL 2. 0 axioms compose the “Taxonomy Module”, i. e. structural axioms governing all entities • Example: Proper subset of BFO 2. 0 axioms compose the “Temporal Region Mereology Module”, i. e. axioms governing temporal region mereology • Example: Proper subset of BFO-FOL 2. 0 axioms compose the “Exists At Module”, i. e. axioms governing the Exists At relation (an existence at a time relation ranging over all entities in BFO) 73

Exists At Module • Subset of axioms governing the exists at relation; sample axioms: 74

Modularization: How? • Start with the most general subset of axioms; proceed to next most general, and so on… • Generality is a heuristic characterized in terms of logical dependence; the proper subset of axioms on which all other subsets of axioms are logically dependent is the most general • The proper subset(s) of axioms on which no others depend, are the least general 75

Modularization: How? • Hypothesis: The BFO Taxonomy Module is the most general, followed in order of (loosely) decreasing generality: • Taxonomy Module • Exists At Module • Temporal Region Mereology Module… • As mentioned, modularization is a “divide and conquer” approach to proving meta-theoretic properties about BFO 76

Modularization: Why? • Proving meta-theoretic properties concerning the BFO axioms is easier if we use a ‘divide-and-conquer’ approach when dealing with the axioms; ultimately we are trying to answer: • What are BFO’s ontological and relational commitments? • This is too tough to answer directly, since BFO is so big, but if we divide the axioms up, we can approach this question in steps… • Still, a natural follow-up: Why do I want to find such commitments? 77

Modularization: Why? • Given two theories, if they have different commitments which satisfy them, then they cannot be computationally integrated • Because one theory might say “X” is true under one interpretation, while the second says “~X” is true under the same interpretation; if we attempt to integrate these theories together in a computing environment, disaster will result • We want to be able to predict disaster, so we can avoid it…finding commitments allows us to make predictions; we find them by a method of semantic verification, which uses the modules we construct… 78

Semantic Verification: What? • Logical theories are satisfied or not; if they are, the axioms representing theory has a true interpretation, we say the interpretation is a model • Theories have intended models, or standard interpretations • E. g. The intended models of Peano Arithmetic include just the natural numbers as the domain and arithmetic operations as relations • Semantically verification consists in (1) finding, for a given theory, all the models which satisfy that theory, and (2) making sure the intended models of theory line up with the models that satisfy it 79

Semantic Verification: What? • But we accomplish this in pieces, by using proper modules of BFO… • Claim: Intended models of BFO 2. 0 are captured with the BFO-FOL 2. 0 implementation • To verify this claim for BFO-FOL 2. 0, we must characterize the models of the axioms (i. e. find the ontological and relational commitments) • Then check to see if they line up with BFO 2. 0 (i. e. the reference manual, Barry’s intuitions, etc. ) 80

Semantic Verification: How? • This is no easy task… • It requires tedious sorting through well-understood mathematical theories to uncover axiom sets that constrain specific models 81

Semantic Verification: How? • This is no easy task… • It requires tedious sorting through well-understood mathematical theories to uncover axiom sets that constrain specific models • There are…so many well-understood mathematical theories…and trust me, staring at, and thinking hard about, how axioms relate to one another is not as productive as you might first think. 82

Semantic Verification: How? • This is no easy task… • It requires tedious sorting through well-understood mathematical theories to uncover axiom sets that constrain specific models • There are…so many well-understood mathematical theories…and trust me, staring at, and thinking hard about, how axioms relate to one another is not as productive as you might first think. • So, I turned to a recent development in Toronto, an axiom repository 83

COLO(REpository) and Hierarchies • COLORE is an open repository of ontologies represented as sets of axioms in Common Logic • COLORE consists of hierarches, or partially ordered finite set of theories (modules closed under entailment) in same language: 84

COLO(REpository) and Hierarchies • COLORE is an open repository of ontologies represented as sets of axioms in Common Logic • COLORE consists of hierarches, or partially ordered finite set of theories (modules closed under entailment) in same language: • Distinct theories in the same language, may be compared based on conservative/non-conservative extension properties 85

COLO(REpository) and Hierarchies • COLORE is an open repository of ontologies represented as sets of axioms in Common Logic • COLORE consists of hierarches, or partially ordered finite set of theories (modules closed under entailment) in same language: • Distinct theories in the same language, may be compared based on conservative/non-conservative extension properties • Distinct theories in distinct languages, may be compared by definable equivalence for specific theories, and reducibility for sets of theories 86

Definitions • A Conservative extension to an axiom set, is an extension that ‘doesn’t get you anything new’, but makes stuff easier to prove • E. g. defining the binary relation ‘is a member of’ in naïve set theory results in a conservative extension • A non-conservative extension ‘gets you new stuff’ • E. g. adding a new non-equivalent axiom to an existing axiom set, trivially gets you something new, and is a non-conservative extension 87

Hierarchy Properties (same Language) • Ordering Hierarchy houses theories concerning orderings of points. 88

Ordering Hierarchy 89

Hierarchy Properties (same Language) • Ordering Hierarchy houses theories concerning orderings of points. 90

Hierarchy Properties (same Language) • Ordering Hierarchy houses theories concerning orderings of points; includes (among others): • Theory of Linear Ordering (theory of linear order over points) 91

Ordering Hierarchy 92

Ordering Hierarchy 93

Hierarchy Properties (same Language) • Ordering Hierarchy houses theories concerning orderings of points; includes (among others): • Theory of Linear Ordering (theory of linear order over points) 94

Hierarchy Properties (same Language) • Ordering Hierarchy houses theories concerning orderings of points; includes (among others): • Theory of Linear Ordering (theory of linear order over points) • Theory of Dense Linear Ordering (theory of dense linear order over points) 95

Ordering Hierarchy 96

Hierarchy Properties (same Language) • Ordering Hierarchy houses theories concerning orderings of points; includes (among others): • Theory of Linear Ordering (theory of linear order over points) • Theory of Dense Linear Ordering (theory of dense linear order over points) 97

Hierarchy Properties (same Language) • Ordering Hierarchy houses theories concerning orderings of points; includes (among others): • Theory of Linear Ordering (theory of linear order over points) • Theory of Dense Linear Ordering (theory of dense linear order over points) • Dense Linear Point is a non-conservative extension of Linear Point (Linear Point is ‘agnostic’ about density; Dense Linear requires it) 98

Hierarchy Properties (different Languages) • Relationships can be examined between theories in distinct hierarchies, via definable equivalence: • We start with one theory in one language, make a ‘translation manual’ that allows this theory to speak the language of other theories, then see how much of what theories ‘say’ is the same • Toy example: When I use the word ‘parthood’ how much of what you mean by ‘parthood’ matches up with what I mean? 99

Semantic Verification: How? • I’ve used the COLORE repository’s extensive catalogue of wellunderstood mathematical theories (such as Dense Linear Point), to semantically verify BFO • Mathematical theories in the repository are linked together based on language, conservativity, definable equivalence, etc. • My task: Construct a BFO hierarchy in COLORE, built out of modules, and determine how each module relates to other theories in COLORE 100

BFO/COLORE Observations • BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others): 101

BFO/COLORE Observations • BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others): • The BFO Taxonomy constrains the same models as a COLORE “Theory of Lines” 102

BFO: Taxonomy and COLORE: Theory of Lines 103

BFO/COLORE Observations • BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others): • The BFO Taxonomy constrains the same models as a COLORE “Theory of Lines” 104

BFO/COLORE Observations • BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others): • The BFO Taxonomy constrains the same models as a COLORE “Theory of Lines” • The Temporal Region Mereology constrains the same models as the COLORE “Minimal Extensional Mereology” 105

Minimal Extensional Mereology (MEM) • Concerns parthood relation, minimal in that theory only requires: • Reflexive (every x is part of itself) • Antisymmetric (if x part of y & y part of x then x=y) • Transitive (if x part of y & y part of z then x part of z)… • Applied here, the result is every temporal region is part of itself, putatively symmetrical parts of temporal regions are identical, and, of course, transitivity… 106

COLORE: Mereology Hierarchy 107

COLORE: Mereology Hierarchy 108

Minimal Extensional Mereology (MEM) • Concerns parthood relation, minimal in that theory only requires: • Reflexive (every x is part of itself) • Antisymmetric (if x part of y & y part of x then x=y) • Transitive (if x part of y & y part of z then x part of z)… • Applied here, the result is every temporal region is part of itself, putatively symmetrical parts of temporal regions are identical, and, of course, transitivity… 109

Minimal Extensional Mereology (MEM) • Concerns parthood relation, minimal in that theory only requires: • Reflexive (every x is part of itself) • Antisymmetric (if x part of y & y part of x then x=y) • Transitive (if x part of y & y part of z then x part of z)… • Applied here, the result is every temporal region is part of itself, putatively symmetrical parts of temporal regions are identical, and, of course, transitivity… • MEM contrasts with stronger mereologies, which may require the existence of unrestricted composite objects (BFO doesn’t!) 110

COLORE: Mereology Hierarchy 111

COLORE: Mereology Hierarchy 112

BFO/COLORE Observations • BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others): • The BFO Taxonomy constrains the same models as a COLORE “Theory of Lines” • The Temporal Region Mereology constrains the same models as the COLORE “Minimal Extensional Mereology” 113

BFO/COLORE Observations • BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others): • The BFO Taxonomy constrains the same models as a COLORE “Theory of Lines” • The Temporal Region Mereology constrains the same models as the COLORE “Minimal Extensional Mereology” • Exists At constrains the same models of “Lower MEM Weak Mereology” 114

BFO/COLORE Observations • BFO-FOL modules based on generality, when compared to existing COLORE theories, motivate the following hypotheses (among others): • The BFO Taxonomy constrains the same models as a COLORE “Theory of Lines” • The Temporal Region Mereology constrains the same models as the COLORE “Minimal Extensional Mereology” • Exists At constrains the same models of “Lower MEM Weak Mereology” 115

Taxonomy Models: Setup • We show the BFO Taxonomy semantically equivalent to the COLORE: Theory of Lines • We introduce translation defs between Taxonomy and Lines and vice versa (sample): • Entity(x) <-> (x=x) • Continuant(x) <-> L 1… • Restricts the domain to entities, and translates ‘Continuant’ in BFO 2. 0 to ‘Line 1’ in the COLORE Theory of Lines, and so on… 116

Taxonomy Models: Characterized • Given the appropriate two-way translation definitions, derivation of target axioms can be conducted: • In one direction, with the translation definitions from Taxonomy to Lines, we prove each axiom of the COLORE Theory of Lines • In the other direction, with the translation definitions that from Lines to Taxonomy, we prove each axiom of the BFO Taxonomy • Prover 9 semi-automates both tasks, showing definable equivalence between theories 117

Temporal Region Models: Setup • We next show the Temporal Region Mereology (TRM) semantically equivalent to the COLORE: Minimal Extensional Mereology (MEM) • We introduce translation defs between TRM and MEM and vice versa (sample): • Temporal. Region(x) <-> (x=x) • Temporal. Region. Part. Of(x, y) <-> part(x, y)… • Restricts the domain to Temporal Region, and translates ‘Temporal. Region. Part. Of’ in BFO 2. 0 to ‘part’ in the COLORE mereology hierarchy 118

Temporal Region Models: Characterized • Given the appropriate two-way translation definitions, derivation of target axioms can be conducted: • In one direction, with the translation definitions from TRM to MEM, we prove each axiom of COLORE: MEM • In the other direction, with the translation definitions that from MEM to TRM, we prove each axiom of the BFO Temporal Region Mereology • Again, we use Prover 9 to show definable equivalence between TRM and the COLORE theory MEM 119

Exists At Models: Setup • A more complicated example, we want to show Exists At semantically equivalent to a formal theory in COLORE…but which? • Following Gruninger & Chui work on DOLCE’s ‘present’ relation, what might be called the Lower MEM Weak Mereological Geometry seems a good place to start (but let me explain…) • Motivating Intuition: Entity existing at a time can be represented by a point incident to a line 120

Exists At Models: Setup • Lower MEM Weak Mereological Geometry is composed of theories from various hierarchies in COLORE (continued): • Mereological Geometry combines theories from the ordering, mereology, and incidence geometry hierarchies • Incidence relation concerns lines/points and is reflexive and symmetric • MEM is the underlying mereology (partial order, unique product, etc. ) • ‘Lower’ indicates parthood preserves incidence (e. g. if Entity x Exists At at Temporal Region t’, and Temporal Region t’’ is part of t’, then x Exists At t’’) 121

Exists At Models: Characterized • We show Exists At semantically equivalent to Lower MEM Weak Mereological Geometry (LWMG) • We introduce translation defs from LWMG to Exists at, and vice versa (sample): • Point(x) <-> (Continuant(x) v Occurrent(x)) • Line(x) <-> Temporal Region(x)… • Definable equivalence shown as before with Prover 9 and Mace 4 122

Verification So Far… • The following definable equivalencies have been carried out (BFO modules on left side; COLORE on right): • • • BFO Taxonomy – Theory of Lines Temporal Region – Minimal Extensional Mereology (MEM) Exists At – Lower MEM Weak Mereology* Occurrent Mereology – MEM Continuant Mereology – Lower MEM Foliation* *Indicates proposed theory extensions to be introduced to COLORE 123

Future Work • The remainder of BFO-FOL 2. 0 needs to be modularized and verified…in works are Theory of Time and Theory of Specific Dependence • BFO theories should also be reduced, if possible, to existing theories in COLORE • Modularization and Verification of BFO-FOL 2. 0 will allow for direct comparisons of semantic properties with other ontologies in COLORE (such as DOLCE and SUMO, which have been partially modularized and verified) 124

Future Work • COLORE + Hierarchies approach fleshes out the logical space of axioms, and allows information gleaned from well-known subsets of axioms to clarify other subsets of axioms • Hence, Modularization and Verification will also, likely, lead to clarification of classes of models for various subsets of axioms in general • And in particular, full verification will likely lead to a better understanding of BFO-FOL 2. 0’s overall models 125