=Paper=
{{Paper
|id=None
|storemode=property
|title=Using quantified epistemic logic as a modeling tool in cognitive neuropsychology
|pdfUrl=https://ceur-ws.org/Vol-883/paper5.pdf
|volume=Vol-883
|dblpUrl=https://dblp.org/rec/conf/esslli/Rendsvig12
}}
==Using quantified epistemic logic as a modeling tool in cognitive neuropsychology==
https://ceur-ws.org/Vol-883/paper5.pdf
Using Quantified Epistemic Logic as a Modeling
Tool in Cognitive Neuropsychology
Rasmus K. Rendsvig
Philosophy and Science Studies, Roskilde University
1 Introduction
The classic modus operandi for model construction in cognitive neuropsychology
(CN) is by use of box-and-arrow diagrams1 to capture the functional architecture
of the mental information-processing system under consideration. In such dia-
grams, of which Figure 1 is an example, boxes represent particular components
of the system and arrows represent pathways of information flow. Along with a
suitable interpretation, box-and-arrow diagrams represent typical CN theories,
consisting of statements regarding what specific modules are included in the sys-
tem as well as statements regarding how information may flow between these
components.
Recently, it has been argued that this methodology should be augmented
by the use of computational models, allowing for the realization of CN theories
in the form of executable computer programs, structurally isomorphic to the
box-and-arrow version the theory, [3, p. 166]:
This way of doing cognitive psychology is called computational cognitive psy-
chology, and its virtues are sufficiently extensive that one might argue that all
theorizing in cognitive psychology should be accompanied by computational
modeling.
Though epistemic logics are not by themselves executable programs, as a mod-
eling tool they do however possess many of the features and virtues required
by [3]. It is the purpose of this paper to discuss the viability of using epistemic
logics as a modeling tool in cognitive neuropsychology.
Benefits of epistemic logic. There are three fields that could benefit from
epistemic logical modeling of theories from cognitive neuropsychology. First, a
motivation for constructing formal logical models aimed at cognitive neuropsy-
chology is that precision and logical entailment can provide explanatory force
and working hypotheses. Further, epistemic logics can be used to express higher
cognitive functions such as knowledge and belief in straightforward languages.
This further allows the formulation of derived principles, such as object recogni-
tion. That epistemic logics can straightforwardly express such functions further
makes it easy to read off predictions from logical theories, allowing for simple
comparisons with empirical observations.
Second, logicians interested in modeling information flow and communication
acts could gain more realistic models of the internal parts of these processes, if
they took to modeling the functional architecture underlying our abilities to
perform such actions.
1
The “universal notation in modern cognitive neuropsychology”, according to [4].
49
Finally, if logicians and cognitive neuropsychologists merged formal tools and
empirical insight, philosophers would stand to gain. Having empirical theories
couched in flexible modal logical frameworks would allow precise analyses of
philosophical problems using tools with which many philosophers are already
familiar. To exemplify, the model introduced below has been used in [14] to
give novel analyses of Frege’s Puzzle about Identity, based on formal notions of
semantic competence.
Plan of attack. As argued in [3], a proper modeling of a CN theory should
be true to that theory in an isomorphic sense: the formal model should include
exactly the modules and pathways of the CN theory, while maintaining their
separate and joint functions. However, while many CN theories include a seman-
tic system module, it is far from clear where such a collection of concepts can
be found in, e.g., a quantified S5 logic. To overcome this difficulty, we here take
the perspective that a logical model of a CN theory is composed of both a for-
mal logic as well as semantics for this logic. More specifically, then the modules
of a functional architecture is represented by model-theoretic structures over
which agents’ capabilities can then be expressed using formal logical syntax. A
complete logic for the model-theoretic structure may then be seen as a theory
detailing the capabilities of an agent with such a mental makeup.
The present approach differs from the computational cognitive neuropsy-
chological (CCN) way outlined in [3] in an important aspect. The CCN approach
there discussed construct a model of normal behavior, which can the be ‘lesioned’
to simulate brain damage, and thereby compare to experimental observations.
In the present, we reverse this approach. Instead of constructing first a stronger
logic for normal behavior from which we can then remove, we construct a weaker
logic for the completely damaged, to which abilities can then be added. As such,
the logic is generic: in order to produce a subject-specific logic, further assump-
tions regarding specific knowledge and abilities must be made.
Evaluating logical models. Given that a motivation for logical modeling of
CN theories was a promise of working hypotheses, one would expect such hy-
potheses to be empirically testable. As the observables in CN tests are subjects’
performance based on given input, and these abilities are described by formal
logical statements over the model-theoretic structure modeling the functional
architecture, it is thus natural to take such hypotheses as constituted by the
formulas satisfied in the actual world of the semantic model. As will be shown
below, this is a feasible way of comparing formal model with empirical observa-
tions. However, there is no hope that any normal epistemic logical model can
produce hypotheses all of which are consistent with made observations. Qua the
problem of logical omniscience, any modeled agent will always know all logical
consequences of her knowledge [7], which no subject can do. This results in a
problem for the present approach in relation to evaluating, rejecting and refining
the formal models, for every hypothesis may now be wrong for two reasons: it
may be the case that the modeled functional architecture does not represent
reality, or it may be that the chosen hypothesis requires reasoning skills beyond
normal, human capabilities. In the latter case, this will result in a non-answer,
50
and the danger is that the correct functional architecture is rejected. We offer
no solution here, but note that this is a problem for which one needs to control
when performing theory evaluation.
Structure. The structure of the paper is as follows. In the ensuing section,
we introduce a simplified version of a CN theory of the structure of lexical,
semantic competence (SLC) from [12]. This is to act as a toy CN theory, which
will be modeled using a two-sorted quantified epistemic logic (QEL) introduced
in section 3. Most weight is on section 4, where the connections between the
QEL and the SLC are drawn. It is first argued that the modules of the SLC are
represented in the two-sorted model-theory. Secondly, it is argued that the three
distinct competence types of the SLC are expressible in the formal language,
and that their dissociations are preserved in the two-sorted logic.2 Finally, we
consider studies which show the shortcomings of the present model, and thereby
falsify the presented model. We then conclude.
2 The Structure of Lexical Competence
In [12], a box-and-arrow theory of the structure of semantic, l exical competence
(SLC) is constructed on the basis of studies from cognitive neuropsychology.3
The elements of the SLC consist of three competence types defined over four on-
tologies, two of these being mental modules, see Figure 1. The three competence
types are inferential competence and two types of referential competence, being
naming and application. The four ontologies include one of external objects, one
of external words (e.g., spoken or written words) and two mental modules: a
word lexicon and the semantic system. This structure is illustrated in Figure 1.
Application
Inference
Naming
Words Word Lexicon Semantic System Objects
Fig. 1. A simplified illustration of the SLC. Elements in the WL are not connected,
only elements in the SS are. Inferential competence requires connecting two items
from the word lexicon through the semantic lexicon.
Word Lexicon, Semantic Lexicon and Inferential Competence. Infer-
ential competence lies between three ontologies: external word, word lexicon and
the semantic system. On input, the external word is first analyzed and related
to an mental representation from the word lexicon (WL). In [12], two word lexica
are included for different input, a phonetic and a graphical. Here, attention is
restricted to a simplified structure with only one arbitrary such, consisting only
2
Due to space restrictions, proof theory will not be considered, but a complete ax-
iomatization can easily be constructed based on the general completeness result for
many-sorted modal logics from [13].
3
For the review of these studies, arguments for the structure and references to relevant
literature, the reader is referred to [12]. The presentation here differs slightly, but
in-essentially, from that.
51
of proper names,4 as illustrated in Figure 1. Using a graphical lexicon as an
example, the word lexicon consists of the words an agent is able to recognize in
writing. Secondly, the mental representation of the word is related to a mental
concept in the semantic system (SS). The SS is a collection of non-linguistic,
mental concepts possessed by the agent, distinct from the WL. The semantic sys-
tem reflects the agent’s mental model of the world, and the items in this system
stand in various relations to one another.5 In contrast, in the WL connections
between the various items do not exist. Such only exist via the SS. The third
step is exactly a connection between two entries in the SS. Finally, the latter
of these are connected to an entry in the WL and output can be performed.6
Inferential competence is the ability to correctly connect lexical items via the
SS, e.g., connecting ‘dog’ to ‘animal’. This ability underlies performance such as
stating definitions, paraphrasing and finding synonyms.
Referential Competence and External Objects. Referential competence is
“the ability to map lexical items onto the world” [12, p. 60]. This is an ability
involving all four ontologies, the last being external objects. It consists of two
distinct subsystems. The first is naming. This is the act of retrieving a lexical
item from the WL when presented with an object. It is a two-step process, where
first the external object is connected to an suitable concept in the SS, which
is then connected to a WL item for output. The second subsystem is that of
application. Application is the act of identifying an object when presented with
a word. Again, this is a two-stage process, where first the WL item is connected
to an SS item, which is then connected to an external object. A naming or
application deficit can occur if either stage is affected: if, e.g., either an object
is not mapped to a suitable concept due to lack of recognition, or a suitable
concept is not mapped to the correct (or any) word, then a naming procedure
will not be successfully completed.
Empirical Backing for Multiple Modules and Competence Types. The
SLC may seem overly complex. It may be questioned why one should distin-
guish between word and semantic type modules, or why referential competence
is composed of two separate competence types, instead of one bi-directional. The
reasons for these distinctions are based on empirical studies from cognitive neu-
ropsychology where reviews of subjects with various brain-injuries indicate that
these modules of human cognition are separate (see [12] for references).
The distinction between WL and SS is further supported in [8] by cases where
patients are able to recognize various objects, but are unable to name them (they
cannot access the WL from the SS). In the opposite direction, cases are reported
where patients are able to reason about objects and their relations when shown
4
To only include proper names is technically motivated, as the modeling would oth-
erwise require second-order expressivity. This is returned to below.
5
Marconi uses the term ‘semantic lexicon’, but to keep this presentation in line with
standard cognitive neuropsychological terminology, ‘semantic system’ is used instead.
6
For simplicity, a distinction between input and output lexica will not be made. See
[15] for discussion.
52
objects, yet unable to do the same when prompted by their names (i.e., the
patients cannot access the SS from the WL). The latter indicates that reasoning
is done with elements from the SS, rather than with items the WL.
Regarding competence types, it is stressed in [12] that inferential and refer-
ential competence are distinct abilities. Specifically, it is argued that the ability
to name objects does not imply inferential competence with the used name, and,
vice versa, that inferential knowledge about a name does not imply the ability
to use it in naming tasks. No conclusions are drawn with respect to the rela-
tionship between inferential competence and application. Further, application is
dissociated from naming, in the sense that application can be preserved while
naming is lost. No evidence is presented for the opposite dissociation, i.e. that
application can be lost, but naming maintained.
In the following section, a model will be constructed which include the men-
tioned ontologies over which the competence types can be defined, and in which
these are appropriately dissociated.
3 Modeling the Structure of Lexical Competence
To construct a toy model of the SLC, a two-sorted first-order epistemic logic will
be used. A very limited syntax is used, though the syntax and semantics could
easily be extended to include more agents, sorts, function- and relation symbols,
cf. [13].
A two-sorted language is used to ensure that the model respects the disso-
ciation of word lexicon and semantic system. The first sort, σOBJ , is used to
represent external objects and the semantic system entries. As such, these are
non-linguistic in nature. The second sort, σLEX , is used to represent the exter-
nal words from the agent’s language and entries in the word lexicon. Had terms
been used to represent both simultaneously, the model would be in contradiction
with empirical evidence.
The choice of quantified epistemic logic (QEL) fits well with the SLC, if one as-
sumes the competence types to be (perhaps implicitly) knowledge-based.7 The
notions of object identification required for application is well-understood as
modeled in the quantified S5 framework, cf. [5]. The ‘knowing who/what’ con-
structions using de re-constructions in QEL from [10] captures nicely the knowl-
edge required for object identification by the subjects reviewed in [12]. This is
returned to in the following section.
Syntax. Define language L to include two sorts, σOBJ and σLEX . For sort
σOBJ , include 1) a countable set of object constant symbols, OBJ = {a, b, c, ...},
and 2) a countably infinite set of object variables V AR = {x1 , x2 , ...}. The set
of terms of sort σOBJ is T EROBJ = OBJ ∪ V AR. For sort σLEX , include 1) a
countable set of name constant symbols, LEX = {n1 , n2 , ...}, and 2) a countably
infinite set of name variables, V ARLEX = {ẋ1 , ẋ2 , ...}. The set of terms of sort
σLEX is T ERLEX = LEX ∪ V ARLEX .
7
In [3], visual recognition tasks are explicitly referred to in terms of knowledge (see,
e.g., p. 149). Which type of knowledge is however not discussed.
53
Include further in L a unary function symbol, µ, of arity T ERLEX −→
T EROBJ . The set of all terms, T ER, of L are OBJ ∪V AR ∪LEX ∪V ARLEX ∪
{µ(t)}, for all t ∈ LEX ∪ V ARLEX . Finally, include the binary relation symbol
for identity, =. The well-formed formulas of L are given by
ϕ ::= (t1 = t2 ) | ¬ϕ | ϕ ∧ ψ | ∀xϕ | Ki ϕ
The definitions of the remaining boolean connectives, the dual operator of Ki ,
K̂i , the existential quantifier and free/bound variables and sentences are all
defined as usual. Though a mono-agent system, the operators are indexed by i
to allow third-person reference to agent i.
Semantics. Define a 2QEL model to be a quadruple M = hW, ∼, Dom, Ii where
1. W = {w, w1 , w2 , ...} is a set of epistemic alternatives to actual world w.
2. ∼ is an indistinguishability (equivalence) relation on W × W .
3. Dom = Obj ∪ N am is the (constant) domain of quantification, where Obj =
{d1 , d2 , ...} is a non-empty set of objects, and N am = {ṅ1 , ṅ2 , ..., ṅk } is a
finite, non-empty set of names.
4. I is an interpretation function such that
I : OBJ × W −→ Obj | I : LEX −→ N am | I : {µ} × W −→ Obj N am
To assign values to variables, define a valuation function, v, by
v : V AR −→ Obj | v : V ARLEX −→ N am
and a x-variant of v as a valuation v 0 such that v 0 (y) = v(y) for all y ∈
V AR(LEX) /{x}.
Truth conditions for formulas of L are now defined as follows:
v (ti ) if ti ∈ V AR ∪ V ARLEX
M, w |=v (t1 = t2 ) iff d1 = d2 , where di = I (w, ti ) if ti ∈ OBJ
I (ti ) if ti ∈ LEX
M, w |=v ϕ ∧ ψ iff M, w |=v ϕ and M, w |=v ψ
M, w |=v ¬ϕ iff not M, w |=v ϕ
M, w |=v Ki ϕ iff for all w0 such that w ∼ w0 , M, w0 |=v ϕ
M, w |=v ∀xϕ (x) iff for all x-variants v 0 of v, M, w |=v0 ϕ (x)
Comments on the semantics are postponed to the ensuing section.
Logic. A sound and complete two-sorted logic for the presented semantics can
be found in [13]. The logic is here denoted QS5(σLEX ,σOBJ ) .
4 Model Validation
As mentioned above, the modules of the functional architecture of the SLC is
represented by model-theoretic structures, over which the agent’s capabilities
can then be expressed using the formal logical syntax. So far, the structures
introduced bear little resemblance to the SLC, and it will be a first task to extract
this hidden structure. Secondly, it is shown that the logical model can express
the three competence types and that the dissociation properties are preserved
in the logic.
54
4.1 Ontologies
The two sets of external objects and external words are easy to identify in the
model-theoretic structure. The external objects constitute the sub-domain Obj,
and are denoted in the syntax by the terms T EROBJ , when these occur outside
the scope of an operator. External words (proper names) constitute the sub-
domain N am denoted by the terms T ERLEX , when occurring outside the scope
of an operator.
The word lexicon and the semantic system are harder to identify. The strategy
is to extract a suitable notion from the already defined semantic structure. To
bite the bullet, we commence with the more complicated semantic system.
Semantic System. In order to include a befitting, albeit very simple notion,
define an object indistinguishability relation ∼aw ⊆ Obj × Obj:
d ∼aw d0 iff ∃w0 ∼ w : I (a, w) = d and I (a, w0 ) = d0 .
a
and from this define the agent’s individual concept class for a at w by Cw (d) =
0 a,w 0
{d : d ∼i d }. The semantic system of agent i may then be defined as the
a a
collection of non-empty concept classes: SSi = {Cw (d) : Cw (d) 6= ∅}.
a
The set Cw (d) consists of the objects indistinguishable to the agent by a ∈
OBJ from object d ∈ Obj in the part of the given model connected to w by ∼.
As an example, consider a scenario with two cups (d and d0 from Obj) upside
down on a table, where one cup conceals a ball. Let a denote the cup containing
the ball, say d, so I (a, w) = d. If the agent is not informed of which of the two
cups contains the ball, i.e. which is a, there will be an alternative w0 to w such
that I (a, w0 ) = d0 . Hence, d ∼aw d0 so d0 ∈ Cw a
(d). The interpretation is that the
0
agent cannot tell cups d and d apart with respect to which conceals the ball.8
Important properties of individual concepts can be expressed in L (see [13]).
For present purposes, most importantly we have that
a
|Cw (d)| = 1 iff M, w |=v ∃xKi (x = a), (1)
i.e. the agents has a singleton concept of a in w iff it is the case that the agent
knows which object a is, in the readings of [9,5]. The intuition behind this reading
is that the satisfaction of ∃xKi (x = a) requires that the interpretation of a is
constant across i’s epistemic alternatives. Hence, there is no uncertainty for
i with respect to which object possesses feature a – i knows which object a
is. Using a contingent identity system for objects, i.e. giving these a non-rigid
interpretation as done in the semantics above, results in the invalidity of both
(a = b) → Ki (a = b) and (a = b) → ∃xKi (x = b). Hence, agent i does not by
default know whether objects are identical when identified by different features,
and neither is the agent able to identify objects by default – as in the example
above. This is a good example of how the present is a weak, generic model:
subject specific abilities regarding identificatory abilities needs to be made as
further assumptions on a per subject basis.
Word Lexicon. A suitable representation of the word lexicon is simpler to
extract than for the SS. This is due to the non-world relative interpretation of
8
Though the agent may be able to tell them apart with respect to other features, like
their color or position.
55
name constants n ∈ LEX, which so far has gone without comment. The inter-
pretation function I of the name constants is defined constant in order ensure
that the agent is syntactically competent. From the definition of I, it follows
that (n1 = n2 ) → Ki (n1 = n2 ) is valid on this class of models. This corresponds
formally to the incontingent identity system used in [11]. The interpretation is
that whenever the agent is presented by two name tokens of the same type of
name, the agent knows that these are tokens of the same name type. The as-
sumptions is adopted as the patients reviewed in [12] are able to recognize the
words utilized.
Notice that identity statements such as (n1 = n2 ) do not convey any infor-
mation regarding the meaning of the names. Rather, they express identity of the
two signs. Hence, the identity ‘London = London’ is true, where as the identity
‘London = Londres’ is false – as the two first occurrences of ‘London’ are two
tokens (e.g. n1 , n2 ∈ LEX) of the same type (the type being ṅ ∈ N am), whereas
the ‘London’ and ‘Londres’ are occurrences of two different name types, albeit
with the same meaning.
Due to the simpler definition of I for name constants, we can define i’s name
class for n directly. Where ṅ ∈ N am and n ∈ LEX this is the set Cin (ṅ) =
{ṅ0 : I (n) = ṅ0 }. The word lexicon of i is then the collection of such sets:
WLi = {Cin (ṅ) : n ∈ LEX}. Each name class is a singleton equivalence class,
and WLi is a partition of N am. Further, (1) holds for name classes if suitably
modified, and the construction of WLi therefore fits nicely fit the assumption of
syntactic competence.
4.2 Interlude: Word Meanings
In order to model knowledge of the meanings of word tokens n ∈ LEX, these
must first be assigned a meaning. In the clinical trials reviewed in [12], applying
a name to it’s meaning is done by extension identification. Therefore, a simple
purely extensional theory of meaning have been embedded in the framework:
the function symbol µ of arity T ERLEX −→ T EROBJ . A meaning function
rather than a relation is used as only proper names are included in the agent’s
language, and for these to have unambiguous meanings, the function requirement
is natural. Given it’s defined arity, µ assigns an element of T EROBJ to each
name in T ERLEX . From the viewpoints of the agents, µ hence assigns an object
(meaning) to each name.
On the semantic level, M, w |=v (µ(n) = a) is taken to state that the meaning
of name n is the object a in the actual world w. The reference map µ is defined
world relatively, i.e. the value µ (n) for n ∈ LEX, can change from world to world.
This is the result of the world relative interpretation of µ given in the semantics
above. Hence, names are assigned values relative to epistemic alternatives.
w1 w2
n1 a n1 a Fig. 2. The meaning function µ is defined
µ
µ world relatively, so meaning of a name
n2 b n2 b may shift across epistemic alternatives.
The motivation for a world relative meaning function is the generic nature of
the model. No agents will by default have knowledge of the meaning of words
56
from their language, but further assumptions to that effect can be assumed in
specific cases.
The simplifying assumption that the WL should include only proper names
was technically motivated by the inclusion of µ. Had verbs been introduced in
the agent’s language, then µ should have assigned them relation symbols as
meanings, and a second-order logic would be required.
4.3 Competence Types
Inferential Competence. Due to the restriction to proper names, the model
is extremely limited in the features expressible regarding inferential competence.
The expressible instances of inferential competence are limited to knowing re-
lations between referring names and not inferential knowledge regarding names
and verbs. As an example, one cannot express that the agent knows the true
sentence ‘name is verb’ as the word lexicon does not contain a ‘verb’ entry. We
are however able to express knowledge of co-reference:
Ki (µ(n) = µ(n0 )) (2)
(2) states that i knows that n and n0 co-refer, i.e. knows the two names to be
synonyms. Based on (2), we may define that agent i is generally inferentially
competent with respect to n by
M, w |=v ∀ẋ((µ(n) = µ(ẋ)) → Ki (µ(n) = µ(ẋ))) (3)
where ẋ ∈ V ARLEX . If (3) is satisfied for all n, agent i will have full ‘encyclo-
pedic’ knowledge of the singular terms of her language. This may however be
‘Chinese Room style’ knowledge, as it does not imply that any names can be
applied nor that any objects can be named.
Referential Competence. Referential competence compromises two distinct
relations between names and objects, relating these through the semantic system,
namely application and naming. An agent can apply a name if when presented
with a name, the agent can identify the appropriate referent. This ability can be
expressed of agent i with respect to name n in w by
M, w |=v ∃xKi (µ(n) = x) (4)
i.e. there is an object which the agent can identify as being the referent of n.
Given the assumption of syntactical competence, there is no uncertainty regard-
ing which name is presented. Since the existential quantifier has scope over the
knowledge operator, the interpretation of µ(n) is fixed across epistemic alterna-
tives, and i thus knows which object n refers to.
To be able to name an object, the agent is required to be able to produce
a correct name when presented with an object, say a. For this purpose, the de
re formula ∃ẋKi (µ(ẋ) = a) is insufficient as µ(ẋ) and a may simply co-vary
across states. This means that i will be guessing about which object is to be
named, and may therefore answer incorrectly. Since there may in this way be
uncertainty regarding presented objects, naming must include a requirement that
i can identify a, as well as know a name for a. This is captured by
M, w |=v ∃x∃ẋKi ((x = a) ∧ (µ(ẋ) = a)). (5)
Here, the quantification and first conjunction ensures that i can identify the
presented object a and the second conjunct ensures that the name refers to a in
all epistemic alternatives.
57
Dissociations. As mentioned, inferential competence and naming are dissoci-
ated. This is preserved in the model in that neither (2) nor (3) alone imply (5).
Nor does (5) alone imply either of the two. The dissociation of application from
naming is also preserved, as (4) does not alone entail (5). That application does
not imply naming is illustrated in Figure 3.
w1 w2
µ(n) d1 µ(n) Fig. 3. Application and naming are not correla-
d1
I
I ted. In actual world w1 , n refers to a and i can
a d2 a d2
correctly apply n, but cannot name a using n:
w1 |=v (µ(n) = a) ∧ ∃xKi (µ(n) = x), but w1 |=v ¬∃x∃ẋKi ((x = a) ∧ (µ(ẋ) = a)).
Here, i cannot name a due to an ambiguous concept. a may be either of d1 or d2 ,
and can therefore not be identified precisely enough to ensure a correct answer.
Whether application entails inferential competence, and whether naming en-
tails application is not discussed in [12]. In the present model, however, these are
modeled as dissociated in the sense that (4) does not entail, nor is entailed by,
either (3) or (5). However, the modeled dissociations are single instances of the
various abilities. Once more instances are regarded simultaneously, implicational
relationships arise, as will be discussed below.
5 Hypotheses and Explanations
A motivation for constructing formal logical models is that precision and logical
entailment can provide explanatory force and working hypotheses. One testable
hypothesis of the present model predicts lack of dissociation between multiple
application instantiations and inferential competence. Specifically, the model
entails that subjects capable of applying two co-referring names will be knowl-
edgeable of their co-reference:
If M, w |=v (µ(n) = µ(n0 )) then
(6)
M, w |=v ∃xKi (µ(n) = x) ∧ ∃yKi (µ(n0 ) = y) → Ki (µ(n) = µ(n0 ))
From this, an explanation why none of the studies reviewed in [12] show dissoci-
ation between application and inferential competence can be conjectured: simple
inferential competence can come about as a bi-product of application, memory
and deductive skill and may therefore require much damage before being severely
impaired.
The formalizations of application and naming also suggests a reason why no
cases where naming was intact, but application broken, was reported in [12]: the
ability to name is very close to entailing the ability to apply a name. In fact,
once (5) is instantiated with a specific name, it implies (4) for the same name.
For application, the chosen object is identified by the subject via the mental
representation ‘the referent of n’, whereas for naming, the presented object must
first be identified by some other feature, e.g., a visual trademark. In case the
mental representation in the SS of this feature is then identical to that of ‘the
referent of n’, then the subject will be able to name a. Hence, one of the necessary
conditions for naming almost implies the necessary and sufficient condition for
application, why the latter will be observed accompanying the former.
58
Implicational relationships as the mentioned should allow for the refutation
of the model. If subjects are found who possess abilities represented by the
antecedents, but lack those of the consequents, the model can be regarded as
refuted, though exactly what the problem is may not be obvious, qua the previ-
ously mentioned issue with logical omniscience.
As mentioned above, both the difference between orthographic and and phono-
logical lexica as well as the difference between input and output lexica was
ignored in this presentation. Since the logical model therefore is based on an ar-
guably wrong functional architecture, it should be possible to find inconsistencies
between model and observed subject behavior. This is indeed the case. For, if
the presented model was correct, then the word lexicon entries should play both
orthographic and phonological roles for words the agent knows in both speech
and writing. Given such a word, an agent able to name with the word should
always be able to do so both orally and in writing. That is, the hypothesis that
agent i is able to name a, i.e. ∃x∃ẋKi ((x = a) ∧ (µ(ẋ) = a)) ((5) from above),
requires that the subject can produce name(s) ẋ both orally and in writing.
This, however, is not the case, as is illustrated, e.g., by the case of RCM, an
82-year old woman, reported on in [8, p. 191]. When prompted with a picture of
a turtle, RCM was able to correctly name it orally using ‘turtle’, but named it
incorrectly in writing, using ‘snake’. As RCM repeatedly made similar mistakes
with respect to written word tasks but not with oral naming tasks, this case can
be taken to show that damage to the orthographic (output) lexicon does not
imply damage to the phonological (output) lexicon. This is not possible in the
presented model, why the model is refuted.9
6 Conclusions and Further Perspectives
In the present paper, we have looked at the possibilities of using epistemic logic
as a modeling tool for cognitive neuropsychological theories by a toy model con-
struction. It has been shown how the functional architecture can be represented
using a combination of model theory and formal syntax. It was further shown
that the constructed model respected important dissociations, but also how the
model could be refuted by suitable empirical evidence contradicting an hypoth-
esis of the model. In conclusion, though the model is incorrect and simplistic, a
serious epistemic logical approach to modeling functional architecture theories
from cognitive science could possibly be of value. An attempt at making a proper
model of a full CN theory would be an obvious next step.
A clear limitation of the presented model is that the formal semantic system
lacks content. The limitation to objects only should be lifted as to include various
properties and relations as well, and moreover, the representation of objects are
black boxed behind constants without a precise interpretation. Before a clear
picture such concepts’ role can be given, an explicit theory of object recognition
must be incorporated. It would be interesting to see the effects of formalizing,
e.g., geon theory of [2] and ‘plug it in’ in the place of the object constants.
9
This can easily be remedied by distinguishing between phonological and orthographic
word terms, i.e. by the addition of a further word sort.
59
More structure could also be provided by attempting to incorporate elements
from conceptual spaces theory [6]. Finally, knowledge operators are too strong in
some cases – RCM from above being a case in point. In situations where subjects
answer consistently but wrongly, belief operators would be better suited. A range
of competence levels could be captured using the various operators from [1].
The style of modeling semantic competence presented here differs from the
way the conceptual theory of [12] and other cited literature tend to regard these
matters. Here, competence types was defined relative to specific words, and
competence judged on a case-by-case basis. Many studies from cognitive neu-
ropsychology base their conclusions on percent-wise correct performance over
one or more test batteries and therefore focus on impaired connections between
modules. In order to facilitate comparison of formal models and empirical re-
search, the case-by-case methodology must be reconciled with this more general
approach.
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