=Paper=
{{Paper
|id=Vol-1144/paper3
|storemode=property
|title=A Formal Framework for Identity in Cyber and Other Universes
|pdfUrl=https://ceur-ws.org/Vol-1144/paper3.pdf
|volume=Vol-1144
|dblpUrl=https://dblp.org/rec/conf/maics/EsterlineD14
}}
==A Formal Framework for Identity in Cyber and Other Universes==
https://ceur-ws.org/Vol-1144/paper3.pdf
A Formal Framework for Identity in Cyber and Other Universes
Albert Esterline1 and Jules Chenou2
1. Department of Computer Science
2. Department of Electrical and Computer Engineering
College of Engineering
North Carolina Agricultural and technical State University
Greensboro, North Carolina, 27411
{esterlin, jchenou}@ncat.edu
Abstract expression as the relation between situations, viz., between
We present preliminary work on a framework for an utterance (situation) and a described situation. Channel
constructing identities that takes as its point of departure theory and situation semantics were two parts of a coherent
Barwise’s channel theory, founded on category theory, and program, and situation theory was developed as
allied work on situation semantics. The framework takes
mathematical support for situation semantics.
actions relating to identity as fundamental. Actions and
events are situations, although situations can include larger The framework sketched here takes actions relating to
context. Identity-related situations are classified under identity, both judgments and assertions, as fundamental.
(identity) types, which can be fused. Identity-related Actions are events (with an agent), and events are basically
activity is partitioned according to the ‘form of life’ situations, although situations can include larger regions of
involved into (identity) environments, similar to Barwise’s
space and time and generally accommodate context.
classifications. An identity-related action is often based on
another identity-related action as a resource, and all these Identity-related situations are classified under (identity)
actions may reference events in an individual’s life. The types, which can be fused. Identity-related activity is
result is a layered picture of identity-related situations. partitioned according to the ‘form of life’ involved into
Identities retain references to the situations from which they environments, similar to Barwise’s classifications. An
are constructed, so provenance, context, and narrative detail
identity-related action is often based on another identity-
are retained. The category-theoretical constructs of
Barwise’s channel theory, however, must be relaxed when related action as a resource, and all these actions may
identity in multiple environments is considered. reference events in an individual’s life. The result is a
layered picture of identity-related situations, and this
layering was part of the motivation for situation semantics.
1 Introduction Identities in this framework retain references to the
This paper presents preliminary work on a framework for situations from which they are constructed, so provenance,
constructing identities that takes its point of departure work context, and narrative detail are retained. Multiple
by Barwise and his associates on the “flow of information” identities are easily accommodated yet fusion results in
and situations. Barwise’s channel theory is an application constraints that converge to real-life individuals behind the
of an area of abstract mathematics known as category perspectives that arise in the environment, somewhat as a
theory, which addresses the form of mathematical theories genotype is behind phenotypes.
and interrelates their structures. Channel theory gives an It turns out that the category-theoretical constructs of
account of how x being P carries the information that y is Barwise’s channel theory must be relaxed when identity in
Q, where x and y are “tokens” and P and Q are “types,” x multiple environments is considered. We explain the
and P being in one classification system, y and Q being in reasons for this and sketch what needs to be done to
another. Barwise and his associates also developed recover the category-theoretic foundation.
situation semantics, which accounts for the meaning of an The next two sections provide brief introductions to
category theory and channel theory. Section 4 introduces
the technical notion of a situation. Section 5 introduces the
notion of identity as a type in a classification (system), that
is, an environment, which involves a coherent set of
practices and standards used to identify and characterize computer scientists; (Fiadeiro 2005) addresses category
individuals. Different environments provide different theory in the context of software engineering.
perspectives on a given person, but note that the same id-
relevant situation (or id-situation) may be classified under
several types since it may involve several individuals. 3 Channel Theory
Section 5 concludes with a sketch of three environments. The point of departure for channel theory is the notion of a
Section 6 reconciles identification-relevant actions (id- classification, which consists of a set of individuals
actions) that are first person assertions with third-person (“tokens”), a set of types, and a binary relation indicating
identifications and characterizations in terms of the notion that a given token is of a given type. Barwise and Seligman
of theory of mind from developmental psychology. Section (Barwise and Seligman 1997) presented a framework for
7 introduces resource situations as meta-representations in the “flow of information” in (generally implicit) category-
an id-situation, and it introduces referenced events as real- theoretic terms. They address the question, “How is it that
world facts that are the background content of the id- information about any component of a system carries
situations. The next section analyzes examples where information about other components of the system?” They
identities (as types in environments) are fused. Finally, define a classification A to be a structure with non-empty
Section 9 considers how this framework can be expressed sets 𝑡𝑦𝑝(𝑨) of types and 𝑡𝑜𝑘(𝑨) of tokens as well as a
in category-theoretic terms similar to channel theory. binary relation ⊨A between 𝑡𝑜𝑘(𝑨) and 𝑡𝑦𝑝(𝑨) such that,
for 𝑎 ∈ 𝑡𝑜𝑘(𝑨) and 𝛼 ∈ 𝑡𝑦𝑝(𝑨), 𝑎 ⊨𝐴 𝛼 indicates that a
is of type α. The theory does not limit what a or α might
2 Category Theory
be (as long as it makes sense for a to be of type α). It
A category 𝒞 consists of a class of objects and a class of could be that a is an object and α a property (monadic first-
morphisms (or arrows or maps) between the objects. Each order relation), or a might be a situation and α a type of
morphism f has a unique source object 𝑎 and target object situation; often, different tokens of a classification amount
b; we write 𝑓: 𝑎 → 𝑏. The composition of 𝑓: 𝑎 → 𝑏 and to the same physical system across different time points
𝑔: 𝑏 → 𝑐 is written as 𝑔 ∘ 𝑓 and is required to be and types are instantaneous partial state descriptions of the
associative: if in addition ℎ: 𝑐 → 𝑑, then ℎ ∘ (𝑔 ∘ 𝑓) = system.
(ℎ ∘ 𝑔) ∘ 𝑓. It is also required that, for every object x, For classifications A and C, an infomorphism f from A
there exists a morphism 1𝑥 : 𝑥 → 𝑥 (the identity morphism to C is a pair of functions (𝑓 ∧ , 𝑓 ∨ ), 𝑓 ∧ : 𝑡𝑦𝑝(𝑨) → 𝑡𝑦𝑝(𝑪),
for x) such that, for every morphism 𝑓: 𝑎 → 𝑏, we have and 𝑓 ∨ : 𝑡𝑜𝑘(𝑪) → 𝑡𝑜𝑘(𝑨) satisfying, for all tokens
1𝑏 ∘ 𝑓 = 𝑓 = 𝑓 ∘ 1𝑎 . It follows from these properties 𝑐 ∈ 𝑡𝑜𝑘(𝑪) and all types 𝛼 ∈ 𝑡𝑦𝑝(𝑨),
that there is exactly one identity morphism for every 𝑓 ∨ (𝑐) ⊨𝐴 𝛼 iff 𝑐 ⊨𝐶 𝑓 ∧ (𝛼)
object. A functor from one category to another is a That is, the image of token c (under 𝑓 ∨ ) is classified to be
structure-preserving mapping, preserving the identity and of type α if and only if c is classified to be of the type that
composition of morphisms. More exactly, if 𝒞 and 𝒟 are is of the image of α (under 𝑓 ∨ ). Intuitively, an
categories, then a functor F from 𝒞 to 𝒟 is a mapping that infomorphism is a part-to-whole, A-to-C, informational
associates with each object 𝑥 ∈ 𝒞 an object, 𝐹(𝑥) ∈ 𝒟 relationship. Figure 1.a shows function pairs (𝑓 ∧ , 𝑓 ∨ ) and
and, with each morphism :𝑥 →𝑦 ∈ 𝒞 , a (𝑔∧ , 𝑔∨ ), each depicted in Figure 1.b as a single
morphism 𝐹(𝑓): 𝐹(𝑥) → 𝐹(𝑦) ∈ 𝒟 such that 𝐹(𝑖𝑑𝑥 ) = infomorphism with codomain C, while f has domain A and
𝑖𝑑𝐹𝑥 for every object x ∈ C and 𝐹(𝑔 ∘ 𝑓) = 𝐹(𝑔) ∘ 𝐹(𝑓) g domain B.
for all morphisms f: x → y and g: y → z . Information (not the “amount” of information, as per
In category theory, a commutative diagram is a diagram Shannon) is assumed to “flow” among the components of a
of objects (as vertices) and morphisms (arrows between system. Components may, but need not, be distant from
objects) such that all directed paths in the diagram with the one another in time and space, and they may be very
same start and end points lead to the same result by different things. The system is “distributed” in this sense
composition. (not necessarily in the sense in which that term is used in
The classic presentation of category theory is (MacLane computer science). For exemplar, the students,
1998). Two reasonably comprehensive and rigorous texts classrooms, scheduling system, and attendance records at a
that are not too sophisticated for someone with a strong school together form a distributed system.
undergraduate math background are (Simmons 2011) and An information channel is a family of infomorphisms
(Awodey 2010). A light introduction is provided by with a common codomain, called the core. Essentially, a
(Lawvere and Schanuel 2009). (Pierce 1991) and (Barr channel consists of a set {A1,...,An} of classifications that
and Wells 1990) are texts addressed specifically to represent the parts of the distributed system, a
classification C (the core) that represents the system as a
whole, and a set of infomorphisms {f1,...,fn} from each of Basically, all classifications in D are “informational parts”
the parts onto C. Tokens in C are the connections of the of the core whose channel covers D.
system: a given token c in C connects the tokens it is Turning to regularities in a classification’s type, let A be
related to by means of {f1,...,fn}. Parts {A1,...,An} carry a classification and Γ and ∆ be sets of types in A. A token
information about each other as long as they all are part of a of A satisfies the “sequent” 〈Γ,∆〉 provided that, if a is of
C. In category-theoretic terms, the core is the sum (or every type in Γ, then it is of some type in ∆. If every token
colimit) of the parts. Figure 2 shows the sense in which the of A satisfies 〈Γ,∆〉, then Γ is said to entail ∆ and 〈Γ,∆〉 is
core is a universal construction, making it unique. called a constraint supported by A. The set of all
constraints supported by A is called the complete theory of
A, denoted by Th(A). These constraints are system
regularities, and it is by virtue of regularities among
connections that information about some components of a
distributed system carries information about other
components. These regularities are relative to the analysis
of the distributed system in terms of information channels.
Barwise and Seligman’s summary statement of their
analysis of information flow, restricted to the simple case
of a systems with two components, a and b, is as follows.
Figure 1. (a) shows infomorphisms f (with domain A)
Suppose that the token a is of type α. Then a’s being of
and g (domain B) with common comdain C as
type α carries the information that b is of type β, relative
function pairs. (b) is the single-arrow view.
to channel C, if a and b are connected in C and if the
translation α′ of α entails the translation β′ of β in the
theory Th(C), where C is the core of 𝐶. (Barwise and
Seligman 1997, p.35)
4 Situations
Channel theory is neutral about what tokens are, for
example, objects (or object states) or system states. One
alternative, which is quite common and natural, is to take
tokens to be situations. Situation semantics, articulated in
Barwise and Perry’s Situations and Attitudes (Barwise and
Perry 1984), attempted to provide a solid theoretical
foundation for reasoning about commonsense and real
world situations and was complemented by Barwise’s
other work in channel theory. Situation theory provides the
Figure 2: A core (universal cocone) (𝒇𝒊 ∶ 𝓐𝒊
mathematical foundations to situation semantics and was
→ 𝓒)𝒊∈𝑰 is a family of informorphisms with a common
developed by Barwise and others, including Devlin in
codomain 𝓒; if (𝒉𝒊 ∶ 𝓐𝒊 → 𝓓)𝒊∈𝑰 is a another such
Logic and Information (Devlin 1991).
familywith a common codomain 𝓓 then there exists a
According to Devlin, a situation is some part of the
unique informorphism 𝝃: 𝓒 → 𝓓 such that for all
ongoing happenings in the world and allows situation
𝒊 ∈ 𝑰, 𝒉𝒊 = 𝝃 ∘ 𝒇𝒊 .
theory to handle context. Situations include not only
A distributed system D is a collection of elements that connected space-time regions but also spatially
carry information about each other. Formally, D consists of disconnected (e.g., telephone calls) and temporally
an indexed class 𝑐𝑙𝑎(𝐷) of classifications together with a disconnected situations (e.g., multi-day sporting events).
class 𝑖𝑛𝑓(𝐷) of infomorphisms whose domains and Situation theory introduces the notion of an infon as the
codomains are all in 𝑐𝑙𝑎(𝐷). An information channel K basic item of information. We assume as given sets A of
covers distributed system D if and only if 𝑐𝑙𝑎(𝐷) are the individuals, R of properties and relations, L of spatial
classifications of the channel and, for every infomorphism locations, and T of temporal locals. The general form of an
𝑓 in 𝑖𝑛𝑓(𝐷), there are infomorphisms from both the infon, then, is
domain and codomain of 𝑓 to the core of K such that the 〈〈𝑅𝑛 , 𝑎1, … , 𝑎𝑛 , 𝑙, 𝑡, 𝑖〉〉,
diagram formed by these three infomorphisms commutes. where 𝑅𝑛 ∈ 𝑅 is an n-place relation, 𝑎1 , … , 𝑎𝑛 ∈ 𝐴 are
objects appropriate for the corresponding argument places
of 𝑅𝑛 , 𝑙 ∈ 𝐿, 𝑡 ∈ 𝑇, and 𝑖 ∈ {0,1} is the polarity. A characterizing agent. We handle both these perspectives
polarity of 1 indicates that the objects are thus related in l within a given environment.
at t; a polarity of 0 indicates otherwise. Where s is a 5.1 Online Identity
situation and σ an infon, it might be that s supports σ, that Our first example of an identity environment is that in
is, that σ holds in s. which we establish online identities in a rather narrow
A real situation (according to Devlin) is a part of reality sense. Here we can look at the FOAF properties that have
(and considered a single entity) that supports an indefinite Person (or, more broadly, Agent) as domain. This
number of infons, while an abstract situation is a set of includes, for example, mbox (a person’s email address) and
infons. Real situations are tokens for us. An abstract workplaceHomepage. As per FOAF (and the Semantic
situation for us is a type. If abstract situation s is the set Web in general), resources are generally identified with
{𝜎1 , … , 𝜎𝑚 } of infons and we identify the classification as URIs. A type is associated with a pattern that (in the first
I, then, for real situation r, 𝑟 ⊨𝑰 𝑠 if and only if r supports instance) is a URI. As a situation type, such a type is a
all infons in s. context in which the URI is used (e.g., a given email
address is used in the ‘To’ field of an email). Tokens are
situations where someone engages in an online activity or
5 Identity as Classification
reflects on such an activity. Such a situation is classified
We take an identity to be a classification in the sense of as of a given type if the activity involved makes use of the
channel theory, where types are abstract situations and URI in question. The situation could involve an individual
tokens are real situations. The similarity between asserting their identity (as in the ‘From’ field of an email)
situations and events has been frequently noted (cf., e.g., or characterizing an individual (as in the ‘To’ field of an
Kratzer 2011), and actions are a kind of event (where one email or when someone analyzes the content of an email).
can identify an agent). We take situations as they relate to 5.2 Physiological Identity
identity (what we call id-situations) to be those that include Another identity environment is that in which one
identity-relevant actions (what we call id-actions). establishes what we refer to as physiological identity.
An (identity) environment is a coherent set of practices Types here relate to the physical features of a person that
and standards used to identify and to characterize may figure in their identification or characterization, such
individuals. This notion of environment relates to things as fingerprints, iris patterns, photographs of faces,
Wittgenstein’s observation that “to imagine a language and recordings of voices. Thus, for example, we have a
means to imagine a form of life” (Wittgenstein 1991, §19) situation type in which a given fingerprint pattern is used
if we extend this observation to cover not only for identification of an individual. A token here is a
grammatical utterances but any action that has significance situation in which one uses physical evidence for
by virtue of social convention. Below, we sketch some identification or characterization. Perhaps the more
environments by which one establishes what we call online common cases are where one identifies or characterizes an
identity, physiological identity, or social identity. Our individual, but there are notable cases where one asserts
analysis to date is modest, and the example environments one’s identity (e.g., “shows one’s face”). To ground the
are concrete. The ‘forms of life’ involve id-actions that are notion of a pattern and relate it to officially sanctioned
overt and subject to conventions that have evolved to evidence, we distinguish an instance of a pattern that is
sanction (sometimes legally) evidence for identity. preserved by an authority (such as a law-enforcement
Different environments provide different perspectives on agency) and accepted as a standard against which other
a given person. The question of how these perspectives are instances of the pattern may be compared for authenticity.
fused into a single identity is deferred to Section 8. How Examples of such standards include, for example, the
one partitions identity-related activity into environments is images that the FBI keeps of convicted criminals’
somewhat arbitrary, and our example environments could fingerprints. The pattern instantiated by such a standard
easily be partitioned finer. Indeed, we mention fusing may also be instantiated, for example, on a doorknob (in,
identities within each of these environments. say, a crime scene investigation), but it is also instantiated
The same id-situation may be classified under several in the texture of the thumb of an actual person. Note that a
types, as when an action relates to the identities of several standard need not be pictorial. For example, an iris pattern
different individuals. One generally tries to identify id- may be stored as a multitude of properties and distributions
situations that relate to as few distinct identities as possible in computer memory.
although some events inherently involve multiple players.
5.3 Social Identity
Some id-actions amount to asserting an identity (which
Our final example of an identity environment is where we
is a first-person perspective) while others identify or
establish what we call (in a narrow sense) social identities,
characterize an individual distinct from the identifying or
concerned with social aspects of a person that may figure
in an attempt to identify or characterize them. says the puppet will look in the original box. This is
Paradigmatic examples come from historical research. A because only the older child has a notion of false belief,
token here is a situation in which one uses social evidence here that the puppet may falsely believe that the cookie is
for identifying or characterizing an individual. Types in the box where it saw the cookie placed. Other areas of
include such things as the observing a given name in a ToM research include perspective taking, pretend play, and
given military roster or a church record. A name is to be recognizing the referents of words.
understood in terms of its denotation independent of A quite comprehensive coverage of ToM is presented by
language and variants within the same language. Thus, for Alvin Goldman in Simulating Minds (Goldman 2008).
example, ‘Charlemagne’, ‘Carolus Magnus’, and ‘Karl der Goldman uses the term “mindreading” for representing
Große’ all count as the same name. On the other hand, a another as having a certain mental state, and he covers the
phrase structured as a proper name (e.g., “John Smith”) two main accounts of mindreading. According to theory
often (if not usually) denotes different individuals in theory (TT), one uses a naïve psychological theory to guide
different circumstances and may not denote a real one’s assignments of mental states, while according to
individual in any circumstance (e.g., “Lieutenant Kijé”). A simulation theory (ST), one tries to replicate others’ mental
definite description is a way of identifying an individual states and project the resulting pretend states onto them.
based on characteristics that are distinguishing in the The simulation advocated by ST is not like the
context in which they are used. Extended discourse can computational simulation of, say, a weather system, but
provide rich characterization of an individual, who is rather is replication (or duplication).
denoted with various linguistic devices, including proper Regarding identity-relevant actions, what is known of
name, definite descriptions, and anaphoric pronouns. ToM suggests that, in identifying or characterizing an
While examples of actions relevant to social identity that individual distinct from the agent doing the identifying or
come readily to mind generally involve identifying or characterizing, the agent uses the ToM relevant to the
characterizing an individual, there are notable cases of identity environment in question. According to TT, all
asserting social identity (e.g., as per Shelley, “My name is states of the mindreader are meta-representations, none of
Ozymandias, king of kings: Look on my works, ye Mighty, which are attributed to the target, although the contents of
and despair!”). the final state are (Goldman 2008). When an agent
Note that physiological identity lacks the conventional identifies or characterizes another individual, this action
or social nature of the other two. In some standard sense generally involves a meta-representation by the agent of an
of “identity,” it is quite common for people to have identity-relevant action by that individual.
multiple online identities, and FBI Associate Director
Mark Felt was revealed to be the character Deep Throat
only 31 years after Watergate. In contrast, perspectives that 7 Resource Situations and Referenced Events
apparently have a life of their own do not arise with The denotations of the meta-representations invoked by TT
physiological evidence unless the person himself are essentially what Devlin identifies as resource
undergoes radical change. situations. For example, the utterance "The dog I saw
yesterday is back" carries information about the presence
of a dog and does so by reference to a previous situation
6 Identity Environments and Theory of Mind
(in which the same dog was present). Regarding id-
We have maintained that, for a given identity environment, actions, when one inspects an email, for example, and
there are id-actions that are assertions of identity and notes that it is from someone, the id-action is the
others that identify or characterize an individual distinct inspection and the resource situation is the sending of the
from the identifying agent. That the same ‘form of life’ is email. When an id-action asserts identity, no resource
involved in both aspects (when the identity environment is situation is generally needed, but when it identifies or
held constant) is supported by research in developmental characterizes an individual, it generally (but not always)
psychology addressing what is known as theory of mind references a resource situation. An example of the latter
(ToM) (Flavell 2004), the human ability to attribute mental that does not involve a resource situation is taking a picture
states (beliefs, desires, intentions, etc.) to oneself and of an individual. Taking a picture, however, can be a
others and to understand that others have mental states resource situation for any number of id-situations because
different from one’s own. One area covered by ToM that id-action leaves an enduring object.
research is false belief. For example, a child watches as a Another component in many id-situations is what we
puppet sees a cookie put in one of two boxes and leaves. call the referenced event (or, sometimes, the referenced
Someone moves the cookie to the other box. When the state), a real-world fact that is the background content of
puppet then returns, the older child, but not the younger, the id-situation. For example, the content of an email often
makes reference to one or more events, fingerprints on the If we judge that ref is indeed a common referenced event,
doorknob may relate to someone opening the door at a we thereby fuse JohnDoeOL and JohnDoePhy, forming the
certain time, and analyzing a passage in a chronicle usually super-type, say, JohnDoe, and o1 and o2 are then taken as
involves reading about events that happened in the society evidence for ref by way of propositions (1) and (2).
being chronicled. Frequently, a referenced event is what a Now suppose the eleventh was a Saturday, and suppose
resource situation is about. It may also be what an we have the following.
assertion of identity is about (as in, e.g., a confession). An (i′) We observe an email from John.Doe@Acme.com that
id-action often involves a referenced event without a states that he was at Joe’s Bar at 11:00 PM on the fourth
resource situation as when someone describes another's (ii′) We observe a picture of the inside of Joe’s Bar taken
personality, background, or work. When an object is at 11:00 PM on the eighteenth that contains a face that
provided for reference, we take the referenced 'event' to be closely matches the face in a picture labeled ‘John Doe.’
the event producing the object. For example, when a mug Call the situation that involves the id-action in (i′) o1a and
shot is attached to an email, the referenced event is the the situation that involves the id-action in (ii) o2a. Both
subject posing for the mug shot (which includes involve resource situations similar to those in o1 and o2,
provenance and topic information missing in the bare respectively. The referenced event for o1a, call it ref1, is
picture). John being in the bar at 11:00 PM on the fourth, and the
For establishing identity, sometimes we focus on the id- referenced event for o2a, call it ref2, is John being there at
situation itself, sometime on resource situation, and 11:00 on the eighteenth. We assert the following.
sometimes on the referenced event. Times and spatial (1′) o1a |= JohnDowOL
locations are often details that bind together events to form (2′) o2a |= JohnDowPhy
a coherent picture, and these parameters are relevant to all Id-situations (i′) and (ii′) may be further evidence, by way
three of these aspects. These details, however, are often of propositions (1′) and (2′), for fusing JohnDowOL and
lacking for some or all of these aspects. In such cases, JohnDowPhy. Fusing the identity types produces an
sometimes we can impose a non-metric (and possibly only additional correspondence between the referenced event
partial) temporal order on situations and events and ref, ref1, and ref2 in that they involve the same identity
determine relative locations. Spatial and temporal type. These referenced events (and hence, indirectly, id-
inclusion and overlap relations are also often indicated. situations (i), (ii), (i′), and (ii′)) may provide evidence
regarding John Doe’s habit of visiting Joe’s Bar Saturday
evenings. A large number of diverse id-situations relating
8 Fusing Identity Types to a single, fused identity may allow us to build up not just
Often the more interesting issues that arise regarding a history of an individual but even a full-blown identity.
identity relate to fusing identities from several identity As another example, suppose we do the following.
environments. Suppose we do the following. (iii) We determine that the DNA from skeletal remains
(i) We observe an email from John.Doe@Acme.com that found beneath a car park in Leicester match the DNA
states that he was at Joe’s Bar at 11:00 PM on the from the blood on a garment known to have been worn
eleventh by Richard III.
(ii) We observe a picture of the inside of Joe’s Bar taken at (iv) We read in a chronicle that Richard III was hastily
11:00 PM on the eleventh that contains a face that buried at a certain location near medieval Leicester.
closely matches the face in a picture labeled ‘John Doe.’ Call the id-situation that involves the id-action in (iii) o3
Call the situation that involves the id-action in (i) o1 and and the id-situation that involves the id-action in (iv) o4.
the situation that involves the id-action in (ii) o2. Note that Note that o4, but not o3, involves a resource situation,
both involve a resource situation: in o1, it is the action of namely, the situation in which the chronicler wrote about
sending the email, and in o2 it is the action of taking and Richard III’s burial. The id-situation in (iii) is of type, say,
labeling John’s picture. The id-situation in (i) is of type, RichardIIIPhy, while that in (iv) is of type, say,
say, JohnDowOL, a type in the online environment, and we RichardIIISoc, and we assert
assert the identity proposition (3) o3 |= RichardIIIPhy
(1) o1 |= JohnDowOL (4) o4 |= RichardIIISoc
The id-situation in (ii) is of type, say, JohnDowPhy, a type Relevant to what is of interest here, o3 relates to a
in the physiological environment, and we assert the referenced event, call it ref3, to do with the demise of the
identity proposition last monarch of the House of York in a certain time
(2) o2 |= JohnDowPhy interval and his remains being interred at a given location.
Apparently, there is a common referenced event, ref, of Id-situation o4 relates to a referenced event, call it ref4, to
John Doe being at Joe’s Bar at 11:00 PM on the eleventh. do with the same monarch being buried near a certain city
on a particular day in 1485. If we fuse RichardIIIPhy and
RichardIIISoc, giving the super-type, say, RichardIII, we cases, we referred to various combinations of the id-
thereby relate ref3 and ref4, in effect inferring a new, more situations, resource situations, and referenced events. The
specific event, say, ref34, that (unlike ref3) involves a type function can remain a total function 𝑓 ∧ : 𝑡𝑦𝑝(𝐴) →
specific day and (unlike ref4) involves a specific location. 𝑡𝑦𝑝(𝐶) and must be such that the type of an id-situation in
Where e is an event, let prop(e) be a statement in some a given environment is reflected in its type in the core. We
sufficiently well-defined language whose sense is e. Then call such a function-relation pair (𝑓 ∧ , 𝑓 ∨ ) an id-map,
we have prop(ref34) ⇒ prop(ref3) ∧ prop(ref4), where ⇒ denoted by undecorated 𝑓: 𝐴 → 𝐶. Instead of an
and ∧ are the usual truth-functional material implication information channel, we might refer to an identity channel.
and conjunction connectives. Also note that the referenced Each identity proposition c |= γ in the core C can be
event in (iv) helps us rule out such things as a reburial. given a confidence indicating that some identity
proposition in one identity environment is evidence for
some identity proposition in another environment.
9 Category-theoretic View of Identity Fusion Suppose we have environments A and B with id-maps f: A
For a systematic account of the interrelation and fusion of → C and g: B → C. Suppose also that c |= γ in the core has
identities, we would like something like Barwise and a certain confidence p ∈ [0, 1]. Let 𝑎 = 𝑔∨ (𝑐), 𝑏 =
Seligman’s information channels. The “parts” would be the 𝑓 ∨ (𝑐), and 𝑔∧ (𝛼) = 𝑓 ∧ (𝛽) = 𝛾. Then a |= α carries the
identity environments, and an infomorphism f from information that b |= β with confidence p (and vice versa).
classification A to classification B would at least include (Fig. 3 shows a variation with domains Ai, Aj, and Ak,
the type function 𝑓 ∧ : 𝑡𝑦𝑝(𝐴) �⎯⎯� 𝑡𝑦𝑝(𝐵). The token associated identity propositions ai |= αi, aj |= αj, and ak |=
function, however, is problematic, as we shall see. The αk, respectively, and id-maps fi, fj, and fk, respectively.)
core C of such an “identity channel” would compose the Confidence in a core proposition is less than certainty since
types of the various identity environments (not necessarily tokens corresponding to the connection token are
as a sum), and it would support the following. Suppose “missing” in some environments. Note, however, that
that id-situation o0 in environment E0 is of identity type T0 some environments may contribute more tokens, which
and that id-situation o1 in environment E1 is of identity would generally increase confidence.
type T1. If the translation T0′ of T0 entails the translation
T1′ of T1 in the theory Th(C), then o0 being of type T0
carries the information that o1 is type T1.
The problem is that not every token (i.e., id-situation) in
one “part” (i.e., environment) can be related to some id-
situation in some fixed other environment, nor is there a
guarantee that an id-situation in one environment relates to
at most one in the other environment For example,
perhaps I inspect an email from Ed@Acme.com to
Al@Acme.com. There is no reason to expect any id-actions
relating to Ed’s or Al’s physical aspects; on the other hand,
there might be several such id-actions. Resource situations
and referenced events do not help here since they might Fig. 3. An identity channel with corresponding identity
simply be missing from the record for a given environment propositions for the “parts” and the “whole” (core). The
or duplicated with trivial changes. confidence in the core proposition is the confidence that a
We are forced to accept not functions between sets of proposition in a part carries the information expressed by
tokens but rather relations between sets of tokens: for a any of the corresponding propositions in the other parts.
given identity environment (i.e., classification) A, where C
As mentioned, the partition of identity activity into
denotes the core, the relation 𝑓 ∨ : 𝑡𝑜𝑘(𝐶) ⟷ 𝑡𝑜𝑘(𝐴) in
environments is somewhat arbitrary. In particular, there is
general relates some elements in tok(C) to multiple
a general-to-specific hierarchy of environments. Fusion of
elements in tok(A) and relates some to no element in tok(A)
identity types, therefore, is often a multistage process as
(when the id-situations are “missing”). This requires us to
one moves up this hierarchy on multiple branches. Indeed,
relax the sum construction. The “core” is no longer
in our sketches of three environments, we noted fusions
unique, that is, the construction is no longer universal. We
within the environments. Since what are parts in Fig. 3
have, then, a cocone, but not a universal cocone (which is a
could themselves be cores in subsumed identity channels,
sum). The token relations have to be constructed ad hoc for
we would typically associate a confidence less than 1.0
each environment. Some indication of how this is done
with the identity propositions associated with these parts.
was given in the above discussion, where, for various
Fusing types generally reduces confidence in identity identities may be induced from the evidence. Category
propositions since fusion is an inductive process. On the theory is an ideal tool here since it is ultimate mathematics
other hand, fusion generally increases confidence in the for representing and reasoning about structure.
identification of the players in the referenced events and We note that how individuals construct their identities in
asserted identities. For example, fusing the identity types online settings has become an active area of research
relating to the email fom and picture of John Doe in (Poletti and Rak 2014). We also suggest that our
Section 8 results in an identity proposition with confidence framework, being neutral regarding how identities are
less than 1.0 but increases the confidence that we have put discovered or constructed, can be useful across a range of
our finger on a particular person. The theory associated interests, including personal and group identity as
with the core of an identity channel can be thought of as addressed in psychology and the social sciences (Vignoles
merged constraints that pick out a noumenal individual (a et al. 2011).
sort of genotype) behind the id-situations that arise
(collectively forming various phenotypes) from its
interactions with the various environments. As we fuse References
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