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TYPE Conceptual Analysis
PUBLISHED 30 April 2024
DOI 10.3389/fpsyg.2024.1361580
OPEN ACCESS
EDITED BY
Tomer Fekete,
Ben-Gurion University of the Negev, Israel
REVIEWED BY
Omid Khatin-Zadeh,
University of Electronic Science and
Technology of China, China
Kiran Pala,
University of Eastern Finland, Finland
*CORRESPONDENCE
Steven Phillips
steven.phillips@aist.go.jp
RECEIVED 24 January 2024
ACCEPTED 08 April 2024
PUBLISHED 30 April 2024
CITATION
Phillips S (2024) A category theory perspective
on the Language of Thought: LoT is universal.
Front. Psychol. 15:1361580.
doi: 10.3389/fpsyg.2024.1361580
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A category theory perspective on
the Language of Thought: LoT is
universal
Steven Phillips*
Mathematical Neuroscience Group, Human Informatics and Interaction Research Institute, National
Institute of Advanced Industrial Science and Technology, Tsukuba, Japan
The Language of Thought (LoT) hypothesis proposes that some collections
of mental states and processes are symbol systems to explain language-like
systematic properties of thought. Recent proponents of this hypothesis point to
additional LoT-like properties in non-linguistic domains to claim that LoT remains
the “best game in town” in terms of explanatory coverage. Nonetheless, LoT
assumes but does not explain why/how symbolic representations connect to
other (non-symbolic) formats. The perspective presented here is supposed to
bridge this gap as a duality in a category theoretical sense: (perceptual) data are
projected onto a base (conceptual) space in one direction, and in the opposite
direction, these data are referenced by that space. Accordingly, perception is dual
to conception. These constructions follow from a universal mapping principle
aording an explanation for why/how symbolic and non-symbolic formats are
connected: as the “best” possible transformation between the two forms— so
the slogan, LoT is universal. This view also sheds some light on the apparent
pervasiveness of logic-like capacities across age-groups and species, and these
constructions constitute special types of categories called toposes (topoi), and
every topos has an interpretation in first-order logic.
KEYWORDS
Language of Thought, category theory, topos theory, category, topos, product,
subobject
1 Introduction
The Language of Thought (LoT) hypothesis supposes that (some) collections of mental
states and processes constitute symbol systems having a combinatorial syntax and semantics
that is akin to a language: LoT is a symbol system that represents complex entities by
compositions of symbols representing the entities’ constitutents so that syntactic relations
among constituent symbols capture in a structurally consistent manner the semantic
relations among corresponding constituent entities (Fodor, 1975,2008). Moreover, for
instance, the state of affairs John loves Mary is represented by a composition of symbols
John,Mary and Loves for constituents John and Mary, and their loves relationship that
are composed of a complex symbol Loves(John,Mary) representing John loves Mary. This
situation is distinguished from a different, but related, state of affairs Mary loves John by
the composite symbol Loves(Mary,John) that swaps the positions of John and Mary in
accordance with the exchange of roles. The roles are encoded by the relative position of
the filler symbols, which allows one to infer the lover of the relationship independently of
the particular instance. Symbolic expressions assumedly correspond to physical states in
a consistent way via a physical instantiation mapping (Fodor and Pylyshyn, 1988). Causal
relations between these physical states are supposed to underwrite the inferential relations
afforded by transformation of corresponding symbolic expressions—LoT is realized as a
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Phillips 10.3389/fpsyg.2024.1361580
physical symbol system (Newell, 1980), analogous to the
instantiation of a programming language with a computer,
and situates within a representational/computational theory of mind
(Wilson, 1999).
A motivation for LoT is to explain systematicity properties of
thought: e.g., why having the ability to represent John loves Mary
and infer that John is the lover in that relationship implies having
the ability to represent Mary loves John and infer that Mary is the
lover in this relationship (Fodor and Pylyshyn, 1988). As a simple
illustration, LoT assumes (atomic) symbols for John and Mary,
John and Mary, an operation that combines those symbols into
strings of symbols, (John,Mary)7→ John Mary, and an operation
that transforms the string to recover the first symbol as the lover,
as in John Mary 7→ John. Moreover, having representational and
inferential abilities with regard to John loves Mary is supposed to
imply having representational and inferential abilities with regard
to Mary loves John because both involve the same processes.
However, an explanation for systematicity requires more than
simply positing a collection of (atomic) symbols, processes for
constructing complex symbolic representations from those atoms,
and processes for transforming those representations. LoT assumes
that the construction processes are the “canonical” ones affording
all possibly needed combinations (not just some of them) and that
all possibly needed transformations are always in lock-step with
those constructions. These assumptions have led to conclude that
LoT lacks a complete explanation for systematicity in failing to
explain why these assumptions should hold (Aizawa, 2003), i.e.,
LoT suffers from the same type of problem raised for connectionist
(neural network) theory (Fodor and Pylyshyn, 1988). Others,
however, contend that these assumptions are the core principles of
LoT, not ad hoc assumptions (McLaughlin, 2009).
This contention has become more pressing with evidence
of LoT-like properties in more diverse (non-linguistic) forms of
cognition than previously realized (Quilty-Dunn et al., 2023)—
LoTs appear seemingly everywhere in everyone and by extension
some capacity for logic (Mandelbaum et al., 2022). However,
accounting for the emergence of these properties is still a question
for further study (Quilty-Dunn et al., 2023), which depends on
explaining why/how symbolic representations connect to non-
symbolic formats, given the usual distinction between (System 1)
processes that are fast, non-symbolic, and developmentally early vs.
(System 2) processes that are slow, symbolic, and developmentally
late (Kahneman, 2011;Evans and Stanovich, 2013). Category
theory (Awodey, 2010;Leinster, 2014) was invented to compare
mathematical constructions (Eilenberg and Mac Lane, 1945), e.g.,
across algebra and topology, cf. symbolic and spatial representation.
In this vein, we return to a category theory explanation for
systematicity (Phillips and Wilson, 2010) to account for LoT-
like properties and their relationships to non-symbolic systems.
A categorical explanation says that systematicity follows from
universal construction, i.e., a type of “best” solution for the given
situation. For instance, systematicity for John loves Mary pertains
to a universal construction called a categorical product on the
set of atomic symbols, {John,Mary}, i.e., the set of all pairwise
combinations of symbols together with two projection maps
retrieving the first and second symbols. The purpose of the current
study is to apply this approach to bridge the explanatory gap
between symbolic and non-symbolic properties and, in doing so,
begin to address questions about how LoTs (may) differ within and
between species (Mandelbaum et al., 2022). An outline follows:
The perspective presented here is supposed to bridge this gap
as a duality in a category theoretical sense: (perceptual) data are
projected onto a base (conceptual) space in one direction, and
in the opposite direction, these data are referenced by that space.
Accordingly, perception is dual to conception. For a preview, these
maps pertain to fiber bundles and presheaves, respectively, and
their maps are related by universal mapping properties affording an
explanation for why/how symbolic and non-symbolic formats are
connected: as a best possible transformation between formats. An
upshot of this approach is that each additional LoT-like property
also pertains to a universal construction which constitutes a special
type of category called a topos. Every topos has an interpretation in
first-order logic, hence the apparent pervasiveness of a capacity for
logic across age groups and species. A summary of these LoT-like
properties is given next and a category theory account in the section
that follows. Discussion of this view is given in the final section.
This study provides a novel category theory perspective on the LoT
hypothesis. Category theory may seem abstruse to many cognitive
scientists, so the presentation style is informal rather than technical
to facilitate some conceptual intuitions. More technical details are
found in references and Supplementary material, which provides
the background theory pertaining to formal conceptions of LoT
and some further perspective. The central intuition driving this
category theoretical perspective on LoT is that the relations between
representational states, i.e., the maps (also called morphisms or
arrows) of a category, have “shape”, and shape is given by topology,
which affords the common ground for moving between (LoT-like)
symbolic and non-symbolic representational formats.
1.1 Additional LoT-like properties
LoT is usually considered for System 2 (contra, System 1)
cognition. However, other LoT-like properties are also apparent
in non-linguistic domains, including perceptual, social, infant, and
non-human cognition (Quilty-Dunn et al., 2023). Six additional
properties were educed from an analysis of the literature with
a particular focus on visual cognition in the context of object
files (Kahneman et al., 1992) that supposedly store and update
information on external objects over time. The following is a
summary of the six properties (Quilty-Dunn et al., 2023, with the
quoted text from there). Language-like examples are given for their
familiarity with a more detailed connection to object files given in
the next section.
1. Discrete constituents: representations have “distinct constituents
corresponding to individuals and their separable features”. For
example, the representation, Loves(John,Mary) for John loves
Mary has distinct constituents John and Mary corresponding to
individuals John and Mary.
2. Role-filler independence: “[LoT architectures] combine
constituents in a way that maintains independence between
syntactic roles and the constituents that fill them”: e.g., the
constituents John and Mary represent the same individuals,
i.e., John and Mary independently of their syntactic roles as
subject or object in the loves relation.
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3. Predicate-argument structure: “a predicate is applied to an
argument to yield a truth-evaluable structure”, e.g., it may or
may not be true that John loves Mary.
4. Logical operators: AND, OR, IF, and NOT.
5. Inferential promiscuity: “computational processes that transform
representations with one logical form into representations
with another logical form”. For example, if John is tall and
Mary is tall,John is tall corresponds to transformation from
tall(John)AND tall(Mary) to tall(John).
6. Abstract conceptual content: the capacity to “represent
abstract categories without representing specific details”, e.g.,
representing a concept of chair as an abstraction of specific
chair instances.
Parallel examples in visual cognition involve symbol-like
representations for external objects and their visual features in ways
that exhibit the six LoT-like properties. The authors argue that the
LoT hypothesis is currently the “best game in town” by providing a
broader account of cognitive behavior (Quilty-Dunn et al., 2023),
which aligns with an earlier assessment from the classicist vs.
connectionist debate over properties, such as systematicity (Aizawa,
2003).
Despite the explanatory successes and renewed interest in LoT,
the question of how LoT is supposed to arise remains (Quilty-Dunn
et al., 2023). Note that a probabilistic version of LoT (Goodman
et al., 2015) does not necessarily help here, since the probabilistic
inferences are over LoT-like representations. This situation would
seem to favor deep learning approaches that eschew symbols
and learn directly from data (see Kriegeskorte, 2015;Krizhevsky
et al., 2017;Vaswani et al., 2017). However, there is an ongoing
debate as to whether deep learning models are psychologically
plausible in regard to the training which are needed for human-
level performance and the specific types of predictions and errors
made (Bowers et al., 2022). Our focus here is a categorical
perspective on LoT (which may also help reconcile these views in
the System 1/System 2 sense, already mentioned), not an attempt to
settle this debate.
2 A category theory perspective
Our category theory perspective begins with a brief overview
of basic concepts used to recast LoT in category theoretical terms.
We start with the concept of map, which is also called morphism
or arrow (Awodey, 2010;Leinster, 2014). A map is a “directed
relation” between a pair of objects, written f:A→B, to indicate
that map fgoes from object Ato object B, called the domain and
codomain of f, respectively. The two senses of the word “object”
used throughout the current study are distinguishable from the
context. A pair of maps f : A →Band g:B→C(i.e., where
the codomain of fis the domain of g) compose to form a map:
h=g◦f:A→C, where ◦denotes the composition operation.
A collection of objects and a collection of maps together with a
composition operation satisfying certain rules constitute a category,
C. The archetypal example is the category of sets and functions,
Set, where the composition operation is function composition.
Categories may also be objects in a larger category. In this situation,
the arrows in the larger category are called functors,F:C→
D, which map objects to objects and morphisms to morphisms
in a structurally coherent way: e.g., monotonic functions (order-
cohering maps), f:P→Q, are functors between categories
that are ordered sets, a≤Pbimplies f(a)≤Qf(b). Functors
are also objects in categories whose maps are called natural
transformations,η:F.
→G, which transform the images of the
domain functors, F(A) and F(f), into the images of the codomain
functors, G(A) and G(f), in a coherent way, i.e., the result of a
transformed map is the same as the transformed result of a map,
G(f)◦ηA=ηB◦F(f). Deeper and broader expositions are
found in many introductions to category theory (Awodey, 2010;
Leinster, 2014), along with a conceptual introduction specifically
for cognitive scientists (Phillips, 2022), scientists generally (Spivak,
2014), and a broader audience (Lawvere and Schanuel, 2009).
Here, categories are regarded as collections of symbolic or non-
symbolic states and functors and natural transformations as maps
between them. Our categorical approach is presented in three parts
by, first, introducing two principles guiding our perspective on
LoT (Section 2.1), second, recasting the six LoT-like properties
using these principles (Section 2.2) and, third, observing that these
constructions constitute a special type of category, called a topos,
which also pertain to object files (Section 2.3).
2.1 Universal mapping properties and
principles
There are two category theory principles that guide our
perspective on LoT: the universal mapping principle and the
duality principle. The universal mapping principle (UMP) says
that constructions are determined by a universal mapping property
(UMP) that is given by a unique-existence condition. For example,
the categorical product of sets Aand B(introduced earlier) is
a set Pand two maps πA:P→Aand πB:P→B, called
projections, that together satisfy the following unique-existence
condition: for every set Zand pair of maps f:Z→Aand
g:Z→B, there exists aunique map u:Z→Psuch
that (f,g)=(πA,πB)◦u. The product—pair (P,π), where
π=(πA,πB)—has a universal property in that all maps into
Aand Bfactors through π. This construction is optimal in the
sense that the unique-existence condition is only satisfied with
sets that have at least as many elements as the number of A-
elements multiplied by the number of B-elements (existence) and
at most that many elements (uniqueness). There is generally more
than one universal construction for the given situation (here,
the maps into Aand B), but all such universal constructions
are unique up to a unique isomorphism, meaning that there is
one and only one way to go between two such constructions.
Every construction given by a UMP acts like a global minimum,
i.e., all constructions point to the universal construction as a
structural analog (abstraction) of gradient descent—there are
no “local minima” in this situation. UMP says that universal
constructions are obtained by following this “gradient”. In this
way, the category theory approach (Phillips and Wilson, 2010,
2016) is supposed to address the basic explanatory criterion for
systematicity—i.e., to explain why systematicity necessarily follows
from the core principles of the theory, not just how systematicity
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may be consistent with that theory (Fodor and Pylyshyn, 1988;
Aizawa, 2003)—essentially, as a universal construction, All roads
lead to Rome! This principle is supposed as a driver of cognitive
constructions (Phillips, 2021).
The duality principle pertains to the directionality of maps.
Formally, for every category C, there is an opposite category Cop
that has the objects and the “reversed” arrows of C, i.e., an arrow
f:A→Bin Cis an arrow fop :B→Ain Cop. A construction in
Chas a dual construction in Cop (NB. Copop =C). This principle
also applies to universal constructions. For instance, a product in C
has a dual universal construction called the coproduct, which is the
product in Cop. Coproducts are universal constructions so follow
the same pattern for products with the arrows reversed. To wit,
the coproduct of sets Aand Bis a set Qtogether with two maps
(injections, ιA:A→Qand ιB:B→Q) infusing Aand Binto
Qsuch that for any set Zand maps f:A→Zand g:B→Z
the pair (f,g) factors uniquely through the injections: i.e., there
exists a unique map u:Q→Zsuch that (f,g)=u◦(ιA,ιB). This
construction is also optimal in that the unique-existence condition
is only satisfied with sets that have at least as many elements as the
number of A-elements plus the number of B-elements (existence)
and at most that many elements (uniqueness). A category and its
opposite are related to a functor, Iop :C→Cop. More generally,
a functor of the form F:C→Dop (equivalently, F:Cop →D) is
called contravariant from Cto D(a functor of the form F:C→
Dis called covariant from Cto D). A pair of opposing functors
(possibly contravariant), F:C⇆D:G, that are related by a certain
type of universal construction is called an adjoint situation, i.e.,
the best possible recovery of the objects Cand maps fin Cfrom
their images, F(C) and F(f) in D, by the opposing functor Gas the
objects GF(C) and maps GF(f) in C; equivalently, the best possible
recovery of the objects Dand maps gin Dfrom their images, G(D)
and G(g) in C, by the opposing functor Fas the objects FG(D) and
maps FG(g) in D. Adjoint situations are also regarded as a form
of duality. In this way, the universality and duality principles are
intimately related.
These two principles guide our categorical perspective on
thought as mappings between percepts and concepts representing
entities. We start with a basic, dual-view of a function between
sets f:A→Bas an A-labeling of the elements in Bor the B-
properties of the elements in A(Lawvere and Schanuel, 2009).
In a deck of cards, for example, each card has the property of
suit (i.e., club,spade,diamond, and heart) and rank (i.e., two,
three,..., queen,king,ace). Accordingly, there are maps from the
deck listing these properties, i.e., suit :D→Sand rank :D→R.
We may also think of the deck as a set of (perceptual) instances for
the concepts of suit and rank. Moreover, a map in one direction
(percepts to concepts) is a conceptualization of percepts and a map
in the opposite direction is a perceptual instantiation of concepts.
In general, a function in Set does not have an opposite as a
function in Setop, unless that function has an inverse. The best
one can do is to recover the preimage from every element bin
B, i.e., the set of elements in Awhose image under fis b. For
instance, suppose a set of colored shapes, CS = {♠,♣,♦,♥}, a
set of colors, C= {b,r}, and a set of shapes, S= {s,c,d,h}.
The color function produces a color image for each item, e.g.,
fc:♠ 7→ b, and the preimage function for fcrecovers the subset
of colored shapes with the given color, e.g., f−1
c[C]:b7→ {♠,♣}.
Similarly, for shape, fs:♥ 7→ hand f−1
s[S]:h7→ {♥}. These
two perspectives illustrate a simple case of a more general duality
(adjoint situation) that exists between fiber bundles and presheaves,
where the sets have the structure of a topological space. There are
also maps between bundles and maps between presheaves that are
determined by universal properties (Mac Lane and Moerdijk, 1992;
Awodey, 2010), which can be interpreted as changes in perception
and conception. Simple examples of these situations are used for
our categorical view of LoT-like properties.
2.2 A LoT like a category
Thought is supposed to have (at least) six additional LoT-
like properties. Quilty-Dunn et al. (2023) consider properties in
visuospatial formats pertaining to object files (Kahneman et al.,
1992). An object file is characterized as a representation that
“(i) sustains reference to an external object over time, and
(ii) stores and updates information concerning the properties
of that object” (Green and Quilty-Dunn, 2021). Object-tracking
and visual memory effects, e.g., forgetting/remembering of a
color feature independently of an orientation feature or tracking
an object despite a feature change is observed to support
a propositional/compositional format over formats that are
iconic/wholistic (Green and Quilty-Dunn, 2021;Quilty-Dunn et al.,
2023). Other domains include reasoning and inference over visually
presented information. The purpose here is to show how these
six LoT-like properties pertain to the universality and duality
principles with examples of case situations from Quilty-Dunn et al.
(2023).
2.2.1 Discrete constituents
The discrete constituents property, in the context of object
files, pertains to visual features such as color and shape that are
represented independently of each other as implied by, say, a
recall task where accuracy in recalling color does not change with
shape and accuracy in recalling shape does not change with color.
LoT captures the discreteness property by positing symbols for
the constituent features and a process that combines symbols to
construct complex representations, e.g., rs for a red square and
rt for a red triangle. The symbol for red does not change with
the juxtapositioning of shape symbols capturing discreteness. The
order of the symbols does not matter for this property.
From a categorical (topological) perspective, discreteness is
a universal (mapping) property. A set Xcan be given a spatial
(topological) structure by imposing a topology T consisting of
subsets of X, called open sets, that specify the relative proximity
of elements in X. The inclusions of open sets imply an order. In
this way, an order is imposed on the elements of a set by specifying
a corresponding topology. Every set has two extreme topologies:
the discrete topology where every subset of Xis an open set and the
indiscrete topology where the only open sets are the empty set and X.
In the discrete case, every element x∈Xbelongs to an open set with
just that element, {x}, and corresponds to the discrete constitutent
property (every symbol is “equiproximal” to every other symbol).
Discreteness in this topological sense is a universal construction.
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A space Tis finer than a space T′if every open set of T′is an
open set of T; dually, Tis coarser than T′if every open set of T′
is an open set of T. The discrete topology is the finest space on X,
since every subset of Xis an open set. The indiscrete topology is
the coarsest space, since the only open sets are the empty set and
X, which are required for any topology. Moreover, in the category
of topologies on Xordered by inclusion of open sets, the indiscrete
and discrete spaces are universal constructions called the initial and
terminal objects, respectively: an object is initial (terminal), denoted
0 (1) if for every object Zin the category there exists a unique
map u:0→Z(u:Z→1). Hence, discreteness follows from a
universal construction.
The importance of this topological view is in placing the
notion of symbolic (discreteness) on common ground with a notion
of non-symbolic (non-discreteness). A non-discrete space is an
indiscrete space or a space that is not a discrete space. A continuous
function between topological spaces, f:X→Y, preserves
proximity at possibly courser levels a granularity, meaning that
for every open set Uof Y, the preimage of Uis an open set of
X. The collection of topological spaces on Xand their continuous
functions constitute a category, which affords the common ground
for moving between symbolic and non-symbolic systems.
2.2.2 Role-filler independence
Evidence for role-filler independence arises in a way that is
similar to discrete constitutents in that the representations of roles
are not affected by particular fillers and the representations of
fillers are not affected by particular roles. The canonical example
is the representations of John loves Mary and Mary loves John
where representing the role of lover is the same regardless of
being filled by the symbol for John or Mary (e.g., as always being
encoded as the first position) and the representation for John
is the same regardless of the role (as in always being the same
symbol). In the context of object files, role-filler independence
appears in object tracking tasks where the identity of the object
(role) remains unchanged despite changes in visual features such as
color (filler)—role independence from filler. Conversely, attributes
can maintain their integrity despite swapping attributed objects—
filler independence from role (Quilty-Dunn et al., 2023). In this
situation, roles can also be (explicitly) represented as symbols that
are composed of symbols for fillers to indicate their role.
A corresponding categorical view is to treat the roles and fillers
in terms of presheaves, i.e., set-valued (contravariant) functors on
topological spaces, F:Xop →Set. A presheaf maps open sets
and inclusions, interpreted as roles and role relations, to sets and
restrictions maps, interpreted as fillers and filler relations. For
instance, the part-whole relations among the symbols ⊠,⊞,×,+
are given by a presheaf that sends the inclusion {∗} ⊆ {∗,⋆}to
the restriction map ⊠7→ ×,⊞7→ +. In this way, a presheaf
has spatial and non-spatial properties (more complex relationships
beyond part-whole relations are also given by presheaves on other
topological structures, such as graphs, or complexes). A sheaf is a
universal presheaf, which has the special property that the attached
data are recoverable from just the data attached to the points of
the underlying space, called the stalks of the sheaf. The dual view
is the category of fiber bundles on that space: presheaves map
roles to fillers and bundles map fillers to roles. A perspicuous way
to think about presheaves is as relational database tables, where
table headings correspond to the points of the space and table
rows correspond to the image of the presheaf (Abramsky and
Brandenburger, 2011;Abramsky, 2013). For instance, the loves
relation can be regarded as a presheaf on the discrete topological
space for the two-point set {s,o}, where srepresents the subject
and orepresents the object of the relation, that is mapped to a set of
pairs including (John,Mary) representing the instance John loves
Mary. Two types of maps of presheaves are presheaf morphisms
which are maps between the data attached to the same space (table
rows) and maps between possibly different spaces on which the
data are attached (table headings). Suppose, for instance, John loves
Sue instead of Mary. This change is conveyed by the mapping,
(John,Mary)7→ (John,Sue). The passive form, Mary is loved by
John is obtained from a continuous function mapping subject to
object and object to subject, s7→ o,o7→ s, inducing a functor
that swaps the corresponding columns creating a new table for
the passive form. Accordingly, roles and fillers of a relation are
independently represented by presheaves.
This situation also applies to the type of role-filler independence
that is claimed for object files (Quilty-Dunn et al., 2023). The roles
for object files are points of a (discrete) topological space and their
attributes are the attached data. Change in a color feature, say,
from red to green for a tracked object corresponds to a presheaf
morphism on a one-point space: a mapping between corresponding
one-tuples, e.g., (r)7→ (g). The “swapping” of color features
in multiple object tracking scenarios (Quilty-Dunn et al., 2023)
corresponds to the functor induced by the continuous function
that exchanges the role of each object as points (locations) in a
topological space, l17→ l2,l27→ l1. This situation generalizes
to other scenarios, e.g., the continuous function on a three-point
space, l17→ l2,l27→ l1,l37→ l3leaves the attributes attached to the
third role unchanged. Moreover, object files as presheaves capture
role-filler independence.
Structured (part-whole) relations also appear among objects
in a complex visual scene that are akin to grammatical
structure, which provides further support for a LoT in visual
cognition (Quilty-Dunn et al., 2023). Hierarchical structures are
orders, which have corresponding topologies. Hence, the LoT-like
relational structure of visual scenes is also captured by presheaves
on such topologies.
2.2.3 Predicate-argument structure
Quilty-Dunn et al. (2023) also argue that object files have a
predicate-argument structure akin to language, as in their examples
“That spherical object is red” or “That red object is spherical”. The
object is predicated by a color in the first example and by a shape
in the second example. A map f:A→Bcan be regarded as listing
the B-properties of A-objects (Lawvere and Schanuel, 2009). Every
continuous function f:Y→Xinduces a sheaf of sections on X,
F:Xop →Set, i.e., for each open set Uof X, the set of functions
F(U)= {s:U→Y|f◦s=1U}, called the sections (right-
inverses) of frestricted to the open set U. Moreover, suppose a
set of colored shapes, CS ⊆C×Swhere Cand Sare the sets of
colors and shapes with the discrete topologies. The (trivial) fiber
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bundle on Cconsists of the projection πc:C×S→Cwhich
induces the sheaf of sections on C, i.e., the dual map, F:Cop →
Set sending the color red, for instance, to the set of red objects.
Similarly, the projection πs:C×S→Sinduces the sheaf of section
on S. Thus, the duality between bundles and presheaves captures
predicate-argument structure.
In the sense of (classical) logic, predicate-argument structure
pertains to propositions that either true or false. For relations as
subsets of Cartesian products, evaluation is given by the relation’s
characteristic function: for a (binary) relation R⊆A×B, the
characteristic function is a function χR:A×B→2, where 2 =
{⊥,⊤} is the set of truth values, sending pair (a,b) to ⊤(true)
whenever ais R-related to b, i.e., (a,b)∈R, otherwise ⊥(false).
There is a one-to-one correspondence between relations and their
characteristic functions, which is given by the following “square”
of arrows (see Goldblatt, 2006):
R!
−→ 1
⊆
y
y
t
A×B−→
χR
{⊥,⊤}
(1)
where tpicks out ⊤(true). This unique correspondence follows
from another instance of a universal construction involving the
terminal object, 1 = {∗}, and the pullback of the arrow talong χR
yielding R. Pullbacks are given by a universal mapping property that
is closely related to products. For sets, the pullback of a function
f:A→Calong a function g:B→C(equivalently, galong f) is a
set A×CB= {(a,b)|f(a)=g(b)}and its two projections, which acts
like a product constrained by the maps into C(if Cis the terminal
object, implying no constraints, then the pullback is just a product).
More generally, for a category Cwith a terminal object, an
object in Cis called a subobject classifier for Cif there exists an
arrow τ:1→such that for each monomorphism (cf. injection)
ι:U→Xthere exists a unique arrow χι:X→such that the
following diagram is a pullback (see Goldblatt, 2006):
U!
−→ 1
ι
y
y
τ
X−→
χι
(2)
where χιis called the classifying morphism,τis called the truth
arrow, and Uis called a subobject of X. This abstraction from subset
to subobject affords a richer notion of truth. In the category of
presheaves on X, for instance, the subobject classifier is interpreted
as assigning truth to just that part of the data on the open sets
for which the predicate holds. If the predicate holds nowhere
or everywhere on the space, the empty set or Xis returned,
respectively, corresponding to false or true in the usual sense.
2.2.4 Logical operators
Logical operators AND, OR, IF, and NOT have corresponding
abstractions in category theory. We have already observed that the
two-element set corresponds to the set of Boolean truth values,
2= {⊥,⊤}. This set is constructed from the coproduct of singleton
sets, 1 = {∗} (terminal objects), i.e., 2 =1+1, where +denotes the
coproduct. The four logical operators correspond to maps between
these sets: e.g., AND (∧) is the map ∧:2×2→2;(⊥,⊥)7→
⊥, (⊥,⊤)7→ ⊥, (⊤,⊥)7→ ⊥, (⊤,⊤)7→ ⊤; similarly, for OR and
IF, NOT is the map ¬:2→2; ⊥ 7→ ⊤,⊤ 7→ ⊥. These operators in
coordination with the subobject classifier afford logical operations
over relations. The complement of a relation R⊆A×B, denoted R,
is given by composing the subobject classifier χRwith the map for
NOT, i.e., the subobject classifier χR= ¬ ◦ χR.
The two-cups task is supposed to demonstrate a capacity for
logic, i.e., if an object is NOT under one cup, it must be under the
alternative cup (Quilty-Dunn et al., 2023). Suppose CONTAINS
is the predicate on the set of cups yielding true if the object is
under the cup. Composing with the NOT map yields the subobject
classifier identifying the alternative cup as containing the object.
2.2.5 Inferential promiscuity
Inferential promiscuity in the visual context is supposed
when inferences transcend changes in visual features. LoTs have
the inferential promiscuity property by virtue of computational
processes that transform representations with logical form: e.g., P∧
Q⇒P, or P⇒P∨Q. The categorical analogs of conjunction and
disjunction are the categorical product and coproduct. In a category
with products and coproducts, C, the product functor sends pairs
of objects (and maps) to their products, 5:(A,B)7→ A×B. The
implication P∧Q⇒Pcorresponds to the natural transformation
from the product functor to the functor picking out the first object,
´
5:(A,B)7→ A, i.e., the natural transformation π:5.
→´
5
whose maps are the natural projections πA:A×B7→ A. The
analogous situation applies to disjunction, which corresponds to
the natural injection, ιA:A7→ A+B. Moreover, functors act as
representations, hence natural transformations as representation
transformations that range over the objects Ain C, such as Pranges
over propositions.
2.2.6 Abstract conceptual content
Object-specific preview benefit, i.e., a reaction-time benefit for
test items related to previously viewed stimuli, particularly across
modalities such as observing an image of an apple for basic category
APPLE and testing with the word “apple” suggests that object-
files store abstract conceptual content along with specific instances
of that concept (Quilty-Dunn et al., 2023). LoT accommodates
this situation by assuming that perceptual representations of the
instances are linked to a symbolic representation of the concept.
We have already observed a categorical version of abstract
conceptual content in the deck of cards, for instance, the map from
cards to suits, suit :C→S, which assigns specific cards to their suit,
e.g., suit :2♥ 7→ ♥. This function (projection) constitutes a fiber
bundle and induces a sheaf of sections that sends each suit to the
set of cards with that suit, as in {♥} 7→ {2♥,..., 10♥, J♥,..., A♥}.
Naturally, this situation also extends to individual cards that are
given by different decks, i.e., where cards with different faces, say
for the two-of-hearts, indicate the same card, 2♥. In this way,
perception is dual to conception. The map from percept to concept
(fiber bundle) acts as the conceptual interpretation of the stimuli,
and the map from concept to percept (presheaf) acts as the best
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possible recovery of the stimuli after the percept is no longer
available due to an occlusion event.
2.3 A topos of thought (object files)
The universal and dual constructions pertaining to the six LoT-
like properties constitute a special type of category called a topos
(Mac Lane and Moerdijk, 1992;Lawvere and Rosebrugh, 2003;
Goldblatt, 2006). A topos is a category that has all (finite) limits and
exponential objects. We have already observed examples of limits in
the form of terminal objects, products, and pullbacks and their dual
constructions, called (finite) colimits, in the form of initial objects
and coproducts. (The dual of pullback is pushout.) An exponential
object in Set consists of a collection of functions from a set Ato a
set B, denoted BA, called a function space—the number of functions
in BAis |B||A|, where |A|and |B|are the number of elements in
Aand B, respectively. Note the two special cases where Aand
Bare the initial and terminal objects, i.e., the empty set and the
singleton set, respectively—the number of functions in B0and 1A
is |B|0=1|A|=1, as required by the definitions for initial and
terminal [exponential objects pertain to another type of universal
construction, which have applications in computer science as the
curry-uncurry transform: e.g., addition as an operator whereby a+
b=(+a)b]. Having finite limits and an exponential object implies
having finite colimits and a subobject classifier. These conditions
define an elementary topos, which are easier to state, though more
general, than the conditions for the original Grothendieck topos
(differences between these concepts are not explored here). Object
files were regarded in terms of fiber bundles and presheaves, in
the previous section. Categories of presheaves on a space Xand
categories of fiber bundles over Xare topoi, which evokes another
slogan, LoT is a topos.
A category/topos theory approach arguably affords the most
concise, systematic, and formally precise framework for studying
the structure of thought. The basic ingredients of a topos
are constructions derived from universal mapping properties.
All (finite) limits are constructed from just two special cases,
i.e., pullbacks and terminal objects—equivalently, products and
another type of limit called equalizer (see Leinster, 2014). Moreover,
limits are constructed from adjoint functors which are themselves
obtained from Kan extensions (Mac Lane, 1998). For instance, the
product functor 5:C2→Cis obtained from a Kan extensions of
the identity functor 1Calong the diagonal functor 1:C→C2, as
indicated by the following “triangle” of maps:
C2
1
y
x
5
C−→
1C
C
(3)
This situation is akin to an autoassociative neural network
model that attempts to recover the input from its hidden unit
representations by adjusting parameters to minimize the distance
between the target given by the identity map and the network
response. The best that one can do in this categorical situation
is a natural transformation from the target given by the identity
functor to the system response given by the composition G◦F,
i.e., η:1C
.
→G◦F; dually, ǫ:F◦G.
→1D. A Kan extension
in this situation may be regarded as the best that a (cognitive)
system can do to recover what was lost by taking an action as a
(left/right adjoint) functor F:C→D, i.e., the right/left adjoint
functor G:D→C, thus absorbing the universality and duality
principles as a single universal-duality (adjointness) principle.
This adjointness principle affords a basis for explaining
why/how symbolic representations connect to other, non-symbolic
representational formats. Recall that categories of topological
spaces and continuous functions provide common ground for
connecting symbolic and non-symbolic representations as discrete
and non-discrete spaces, respectively. For instance, there is a
continuous function from the two-element set {∗,⋆}with the
discrete topology to the same set with the indiscrete topology, i.e.,
the map sending each element to itself, thus realizing a transition
between a symbolic to a non-symbolic format. This difference
is analogous to the difference between a multi-slot vs. single-
slot view of object files (Green and Quilty-Dunn, 2021)—each
point (open set) in the discrete space corresponds to a different
object role vs. a single object role (non-empty open set) in the
indiscrete case by their topologies. In the discrete case, each point
belongs to a distinct open set, {∗} and {⋆}. For the indiscrete
case, the only non-empty open set contains both points, {∗,⋆}.
For the corresponding categories of presheaves, the discrete space
affords compositionality, whereas the indiscrete space does not.
For example, the deck of cards represented as a presheaf on the
set representing suit and rank, {s,r}, with the discrete topology
affords recovery of the suit and rank of each card: for instance, the
inclusion {s} ⊆ {s,r}is sent to the restriction map that consists of
the mapping 2♥ 7→ ♥. No such access to constituents is afforded
with the indiscrete topology because the indiscrete topology does
not have any non-trivial inclusion maps.
Some theoretical and empirical implications of this difference
in topology pertain to the respective presheaf morphisms, as
illustrated by a simple example. Suppose the string “CAT” is
associated with the string “DOG”. What should the strings “ACT”
and “TAC” be associated with? If the strings are interpreted as
words and the mapping as a free association, one might respond
with “NOW” and “TIC”, respectively, by ignoring associations
between constituent letters. This situation corresponds to a map
between presheaves on an indiscrete space. However, if the
strings are interpreted as letter sequences and the association
as a mapping of letters, one might respond with “ODG” and
“GOD”, respectively, using the three constituent letter maps for
the map from “CAT” to “DOG”. This situation corresponds
to a map between presheaves on a discrete space. Thus, the
empirical implications pertain to responses on novel strings:
the discrete case affords generalization—correct prediction of
target responses for novel strings; the indiscrete case does
not. These two situations were observed experimentally when
subjects trained on letter-pair to color-shape associations exhibited
both forms, depending on the difficulty of the task, i.e.,
number of associations to be learned (Phillips et al., 2016).
Moreover, the relationship between these two cases is given
by an adjoint situation (Phillips, 2018). An analogous situation
may be used to reconcile multi-slot and single-slot views of
object files, where visual features spontaneously do vs. do not
break apart.
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3 Discussion
What is gained by rendering LoT-like properties in terms
of category/topos theory constructions? The idea of treating
relations between representational states as arrows affords two
gains, which is discussed here. First, arrows have shape (in addition
to direction) that is given by topology in the context of sheaves
and bundles, providing the common ground needed to connect
symbolic and non-symbolic formats via continuous functions. This
idea that the (causal) relations between representational states are
determined by the “shape” of those states which is familiar to the
classical view as syntactic form (Fodor and Pylyshyn, 1988). What
category theory adds is the way that certain shapes (constructions)
are universal—satisfy universal mapping properties—and that
every universal construction follows from the universal mapping
principle: “gradient” descent (ascent) to the corresponding terminal
(initial) object (Phillips and Wilson, 2016;Phillips, 2021). In this
way, category theory affords an extra level of explanation for why
such LoT-like properties arise, beyond the classical account of
stipulating symbolic representations as needed (see Section 1), and
how such constructions connect to other formats, beyond just
assuming that they do.
Second, category/topos theory provides a systematic treatment
of LoTs in terms of a small number of conditions (limits and
exponential objects) and principles (universality and duality). We
already noted that limits and exponential objects are obtained from
universal properties and that all finite limits arise from just two
types, terminals and pullbacks. The conciseness goes even further—
all limits are universal with respect to just one type of functor,
the diagonal functor 1:C→CJwhere Jis the “shape” of the
functor into C, i.e., a category, J, typically consisting of a small
number of objects and arrows, whose identities are unimportant,
used to pick out objects and arrows in the target category, C.
A terminal is a limit with respect to the empty shape and a
pullback with respect to a shape that has three distinct objects
and two distinct arrows, i.e., · → · ← ·. Dually, colimits
derive from just two types, initial objects and pushouts. The shape
of an initial object is also the empty category, 0op =0. The
shape category for pushouts is opposite the one for pullbacks, i.e.,
· ← · → ·. The diversity of limits and colimits stems from the
variety of categories, C, to which the diagonal functor is applied.
Not all categories have all types of limits, but all limits derive
from the same universality principle (UMP), i.e., the principle of
construction from a universal mapping property (UMP). Moreover,
in each case, the limit/colimit functor is right/left adjoint to the
diagonal functor, thus combining the universality and duality
principles into a single adjointness principle, whence the slogans,
Adjoints are everywhere and All concepts are Kan extensions (Mac
Lane, 1998). For the purpose of assessing the differences in LoTs
within and across species (Mandelbaum et al., 2022), one needs
a systematic organizational framework. As a formal framework in
this regard, category/topos theory is arguably the “best game in
town” for concision and formal precision. Topos theory brings
together algebra, topology/geometry, and logic (see Leinster, 2011),
cf. language, perception, and reasoning, which affords an analogous
panoramic view for cognitive science: an integrative view of LoT
(Fodor, 1975) as an algebra of thought, perception as a geometry
of thought (Gärdenfors, 2000), and reasoning as mental models of
thought (Johnson-Laird, 2008).
There are, however, important caveats to this categorical view.
In particular, capacities for inference are not the same across
age-groups and species. For instance, older children but not
younger children around the age of 5 years old have a capacity for
transitive inference (Halford, 1984;Andrews and Halford, 1998):
if aRb and bRc then aRc. How should these differences be treated
within this category/topos theory approach? One may appeal to
cognitive capacity limitations, such as the number of items that
can be concurrently held in working memory (Cowan, 2001)
or the number of variables that can be concurrently processed
(Halford et al., 1998). Presheaves as relational tables have a “two
dimensional” structure in terms of the number of columns and
the number of rows. The number of columns can be interpreted
as relational arity: a one-column table is a unary relation and a
two-column table is a binary relation. Relational complexity theory
says that differences in inferential ability correspond to differences
in the capacity to process relational information (Halford et al.,
1998,2014), which has also been considered in terms of categorical
products/pullbacks (Phillips et al., 2009), which underlie sheaves
(Mac Lane and Moerdijk, 1992). The number of rows can be
interpreted as the number of items that can be stored or referenced.
These two types of cognitive capacities (Halford et al., 2007) are
potentially relatable via sheaf theory.
Another caveat is that not all aspects of a construction derive
from universal properties. For instance, the universal property in
an adjoint situation pertains to the left/right adjoint functor for
the given functor, F, if such an adjoint exists. The given functor,
F, itself is not necessarily obtained from some universal properties
(if Fhas a left/right adjoint, G,Fmay be dually viewed as a universal
construction given G). In these situations, a pair of adjoint functors
act like hypothesis generation and test, i.e., given a hypothesis as the
functor F, test this hypothesis for a universal property as the left or
right adjoint Gto F. There may be more than one adjoint situation
between a pair of categories of which only some are relevant to
the task at hand. In this case, the category/topos theory approach
needs to be augmented with a measure of relevance or fitness akin
to a neural network model. Incorporating category/sheaf theoretic
constructions into such models is another possibility (Clark et al.,
2008;Hansen and Gebhart, 2020;Barbero et al., 2022;Bradley et al.,
2022). Cognitive systems are supposed to be resource-dependent
and goal-driven. A categorical theory of resources (see Coecke
et al., 2016) may also afford an explanation for why a universal
property, hence systematicity, fails to emerge. A LoT is after all a
physical symbol system (as mentioned earlier) embedded in some
environment. If the environment provides insufficient information
to establish the needed construction, the universal construction
may be suboptimal with regard to the construction that would
have been obtained had more information been made available,
i.e., in a larger context. In this case, the universal construction
is from a functor interpreted as the available resources to some
object interpreted as the task goal (see Supplementary material,
remarks 132).
Viewing LoT as a topos leads to other properties beyond the
six canvassed in the previous section. The archetypal topos is the
category of sets and functions, Set, whose classifying object is the
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set of Boolean values (true and false). Thus, Set acts like ordinary
classical logic. A category of presheaves on a space Xis also a
topos but with a different classifying object consisting of the open
sets of Xwhich affords the graduated notion of truth mentioned
earlier (NB. The classifying object is not a “free parameter” but
a universal mapping property of the particular category/context
that one is working in). The law of excluded middle, i.e., ¬¬A=
A, characterizing classical logic, need not hold for some topoi.
For instance, the category of directed graphs is a topos whereby
¬¬A6= A, but ¬¬¬A= ¬A(Lawvere and Schanuel, 2009).
What these additional properties mean for a language of thought
has yet to be worked out. However, a general implication of
this graded notion of truth is that response errors should align
with the open sets (roles), e.g., correctly recalling object color but
not shape, which accords with the view that accounting for the
types of errors that humans make is more important than simply
modeling overall response accuracy, at least in vision (Bowers
et al., 2022). A topos theory approach is also apt for a formal
theory of goals that are organized into subgoals given that the
concept of subobject is a generalization of the concept of subset,
hence dependency order. To continue the analogy, functors are to
subfunctors (cf. subsets) as goals are to subgoals, or partial solutions
to a problem (see Supplementary material, remarks 132). Exploring
the implications of these formal abstractions are topics for future
studies.
The claim that LoT is universal is to be understood in
the context of a particular aspect of cognition, not that all of
cognition is necessarily LoT-like. LoT is not the only game in
town (Quilty-Dunn et al., 2023) and universality is not absolute
with respect to cognition in toto—universal constructions are local
with respect to some category or relative with respect to some
functor. As already discussed, systematic failures point to other
aspects of cognition that are better accounted for by incorporating
other types of theories, such as embodied cognition (Shapiro and
Spaulding, 2024), that realize the requisite universal properties.
Other approaches such as dynamical systems (Port and van Gelder,
1998) and Bayesian probability (Griffiths et al., 2008) also have
categorical connections (Lawvere and Rosebrugh, 2003;Culbertson
and Sturtz, 2014;St. Clere Smithe, 2023). Where category theory
stands apart from other theoretical approaches is the way that
such alternatives are incorporated into a universal way, vis-a-vis,
adjointness. Indeed, topos theory is the quintessential example
for mediation of opposites (Lawvere, 1989,1992—see Rosiak,
2022, for further philosophical discussion). Category theory was
developed as a type of meta-mathematics (Eilenberg and Mac Lane,
1945), and cognitive science was founded as an interdisciplinary
science of mind (Miller, 2003). The pertinent parallel here is that
category/topos theoretical approaches are the natural approach
to explain why/how these different aspects live together. LoT
may well have a good game, but category theory is the better
sport.
Author contributions
SP: Conceptualization, Formal analysis, Writing – original
draft.
Funding
The author(s) declare financial support was received
for the research, authorship, and/or publication of
this article. This study was supported by a Japanese
Society for the Promotion of Science Grant-in-aid
(23K11797).
Acknowledgments
An earlier version of some of this study was presented at
the 45th Annual Conference of the Cognitive Science Society,
CogSci2023 (Phillips, 2023).
Conflict of interest
The author declares that the research was conducted
in the absence of any commercial or financial relationships
that could be construed as a potential conflict
of interest.
Publisher’s note
All claims expressed in this article are solely those
of the authors and do not necessarily represent those of
their affiliated organizations, or those of the publisher,
the editors and the reviewers. Any product that may be
evaluated in this article, or claim that may be made by
its manufacturer, is not guaranteed or endorsed by the
publisher.
Supplementary material
The Supplementary Material for this article can be found
online at: https://www.frontiersin.org/articles/10.3389/fpsyg.2024.
1361580/full#supplementary-material
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