Volume 1 (2013)
The Revival of Reliabilism:
A Mathematical Approach to the Generality Problem
Justin Svegliato
Department of Philosophy and Religious Studies, Marist College
Abstract
In this paper, reliabilism, a theory of epistemic justification most clearly articulated by Alvin
Goldman, is amended in order to avert the threat of the generality problem. More particularly, I
argue that the reliability of belief-forming processes can no longer merely be in terms of the
success rate; rather, it must be a function of both the success rate and the corresponding number
of instantiations. Thereafter, I introduce a method to objectively specify belief-forming processes
in terms of all relevant considerations that affect reliability. Finally, potential simplifications to
the proposed theory are discussed as well.
I. Introduction
In “Reliabilism: What is Justified Belief?,” Alvin Goldman offers an externalist account of
epistemic justification where a belief is justified if and only if it is formed via a reliable process.1
Despite its initial attractiveness, the generality problem—namely that that the reliability of a
given process is contingent upon the broadness of its specification—undermines the initial
tenability of reliabilism insofar as one can no longer assign an objective degree of reliability to
belief-forming processes. In this paper, I will proceed by explaining (1) the fundamental features
of the current formulation of reliabilism and (2) the threat of the generality problem. Thereafter, I
will (3) argue for an altered account of reliabilism that effectively handles the generality problem
by defining the reliability of a process in terms of its success rate and corresponding number of
instantiations, (4) demonstrate the revised account ascribing an accurate level of reliability to
1
Alvin Goldman, "Reliabilism: What Is Justified True Belief?," in Arguing about Knowledge, ed. Ram Neta et al.
(New York: Routledge Taylor & Francis Group, 2009), 157-73.
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singularly instantiated processes, (5) provide a methodology for objectively describing beliefforming mechanisms to circumvent the reliance of the success rate, a component of reliability,
upon an arbitrary degree of generality, and finally (6) offer a reductive account of process
specifications to reduce the infinite set of all potential belief-forming mechanisms to a finite
subset.
II. An Explanation of Reliabilism
Before outlining reliabilism, it is important to understand the concept of a belief-forming process
since the justificational status of a belief depends upon the reliability of a process according to
the theory. In a word, a belief-forming process is “a functional operation or procedure, i.e.
something that generates a mapping from certain states—“inputs”—into other states—“outputs.”
The outputs in the present case are states of believing this or that at a given moment.”2 That is to
say, it is a process, particularly perception, reasoning, or perhaps recollection, that yields a
belief.3 For instance, consider Ryan, an agent of average intellectual capacity not suffering from
any physical limitations, who notices a tree outside while sitting at his desk. He consequently
forms a belief that a tree is outside on the basis of what he sees. In this case, the belief-forming
process is visual perception to the extent that it is the functional operation that generates his
belief. Now consider Ryan attempting to solve a simple arithmetic problem, perhaps the addition
of five and six. The belief-forming process is the method by which he arrives at an answer to the
problem; in particular, his use of reasoning is the operation that generates his proposed solution.
2
Alvin Goldman, "Reliabilism: What Is Justified True Belief?,” 164.
The distinction between types and tokens is important in understanding the discourse that will follow. “Types are
generally said to be abstract and unique; whereas tokens are concrete particulars, composed of ink, pixels of light (or
the suitably circumscribed lack thereof) on a computer screen, electronic strings of dots and dashes, smoke signals,
hand signals, sound waves, etc.” Linda Wetzel, “Types and Tokens,” Stanford Encylopedia of Philosophy (Spring
2006 Edition), Edward N. Zalta (ed.), URL= http://plato.stanford.edu/entries/types-tokens/. For instance, a specific
type is the abstract category of a bicycle. However, the road bicycle that Lance Armstrong rides or perhaps the
mountain bicycle that I ride on the weekend are both examples or more accurately tokens of the given type bicycle.
In discourse regarding belief-forming processes, there is a difference between types and tokens. A type may be
considered visual perception whereas tokens of this type would be specific instances of when a belief is formed via
visual perception. More specifically, in the scope of this paper, a process is considered an abstract category, i.e., a
type, whereas the instantiations of this process are considered tokens.
3
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Justin Svegliato/Volume 1 (2013)
More importantly, however, Ryan may or may not arrive at the correct answer to the
problem. He may wrongly conclude that the sum of five and six is fifty-six. Despite arriving at
an incorrect answer, his reasoning generates a belief and is still a belief-forming process. Thus a
belief-forming process is independent of whether the generated belief is true—the mere
production of beliefs is sufficient for a process to be belief-forming. In fact, one may utilize a
process that never produces true beliefs in some cases. Nevertheless, this process is still
classified as belief-forming in virtue of its ability to generate beliefs—without respect to their
truth or falsity—given certain inputs.
Insofar as these processes may yield false beliefs, Goldman claims that a level of
reliability, a quantity contingent upon the likelihood that a generated belief will be true rather
than false, can be assigned to a belief-forming process. He then offers an account of epistemic
justification where one’s doxastic attitude—namely, an attitude that pertains to beliefs and
perhaps others mental states such as desires, judgments, thoughts, and opinions—is justified if
and only if the process that led to the doxastic attitude is reliable, “where (as a first
approximation) reliability consists in the tendency of a process to produce beliefs that are true
rather than false.”4 That is, a belief-forming process must have a high success rate, namely the
likelihood of yielding a true belief must be sufficiently high, for the agent’s doxastic attitude,
specifically his or her belief, to be justified. Hence, reliability is solely a function of the success
rate, particularly the number of tokens that result in true beliefs divided by the total number of
tokens generated by the given belief-forming process.
For instance, suppose Ryan is looking at a tree in a meadow during a clear day.
According to reliabilism, he is justified in believing it is a tree provided that visual perception
under the conditions specified—the weather and location, two variables that significantly affect
sight in part due to the degree of illumination and the distance to the observed object, as well as
various other factors—is generally reliable. In particular, the success rate of the process that
4
Alvin Goldman, "Reliabilism: What Is Justified True Belief?," 163.
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yielded his belief is sufficiently high due to these circumstances, and thus his belief is justified
according to the reliabilist account of epistemic justification. Conversely, suppose that Ryan’s
belief is the result of an unreliable process. Perhaps the conditions under which the sighting
occurs are poor due to factors, such as fog and bad lighting, that hinder his visual perception. As
a result, these circumstances lower the success rate and thereby lower the reliability of the beliefforming process to such an extent that any belief generated by the said process, including the
belief held by Ryan, will be unjustified. In these sorts of prototypical cases, reliabilism captures
the way we think about the attribution of justification to beliefs; the markedly simple and
intuitive manner in which this attribution occurs is an attractive feature that demonstrates the
efficacy of the theory.
III. The Threat of the Generality Problem
Although reliabilism seems to accurately attribute justification to doxastic attitudes, the
generality problem challenges the precision of these ascriptions. In a word, the reliability of a
given belief-forming process is wrongly contingent upon the broadness or narrowness of its
specification. According to Goldman, “input-output relations can be specified very broadly or
very narrowly, and the degree of generality will partly determine the degree of reliability” 5. That
is to say, the justificational status of beliefs will fluctuate with the level of generality used to
specify that process.6
Recall the example of Ryan forming the belief that a tree is outside his window. The
process in this case may be expressed as “looking at a tree while sitting at a desk.” However, the
process may also be described as visual perception or perhaps more broadly as mere perception.
Conversely, the process specification may be narrower, namely “looking at a tree while sitting at
a desk during a cloudy day.” Or the process could be described as “looking at a tree in a field
while sitting at a desk during a cloudy summer day in Australia as night is just about to fall.” The
5
Alvin Goldman, "Reliabilism: What Is Justified True Belief?," 165.
Similar concerns have been raised in John Pollock, “Reliability and Justified Belief,” Canadian Journal of
Philosophy, Vol. 14, No. 1(1984): 103-114. Also see Earl Conee and Richard Feldman, “The Generality Problem for
Reliabilism,” Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 89, No.
1 (1998): 1-29.
6
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description of the process whereby he forms his belief could become considerably if not perhaps
infinitely more specific. Thus, there are countless specifications, all of which are equally viable
according to reliabilism, that could sufficiently describe and encompass all tokens of the given
belief-forming process that Ryan employs. This is to say that the validity of all specifications for
a given process rests not upon the degree of generality but rather upon their ability to encompass
all tokens under consideration. So long as the description covers all tokens in question, the
generality of a given specification has no bearing upon its correctness.
Such a consideration has an impact upon the feasibility of reliabilism. For instance, in
the original specification of the process, namely “looking at a tree while sitting at a desk,” the
reliability is maximal, and thus the agent is justified in forming his belief in virtue of the
conditions of the process. However, when the specification is broader, i.e. visual perception and
perception, the reliability, whatever it may be, may not be of the same reliability as the original
specification. Similarly, in the more specific descriptions of the process, say, “looking at a tree
while sitting at a desk during a cloudy day,” the reliability decreases provided that the
augmentation adds a condition that undermines perception. With this narrower specification,
Ryan is most likely still justified in his belief that he is looking at a tree; however, the reliability
of the belief-forming process changes. Therefore, in certain scenarios the justificational status of
a belief may vary if the reliability deviates too greatly between two satisfactory albeit distinct
specifications of the process.
Thus, which of the specifications of the belief-forming process in the example of Ryan is
correct? Which should determine whether his doxastic attitude is justified? Without an answer to
these questions, the theory cannot determine the justificational status of a belief since the
reliability of processes supervenes upon an arbitrarily chosen degree of generality of their
process specifications. That is, the justificational status of a doxastic attitude can change by
manipulating how the process that yielded that belief is specified. Consequently, reliabilism
appears to be an untenable theory because it fails to ascribe an objective justificational status to a
doxastic attitude given its inability to offer a sole level of reliability.
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It is important to consider one last consequence of the generality problem. When a beliefforming process is narrowly specified, there is only one token upon which the reliability of that
process, a function in terms of the success rate, is contingent. That is to say, the reliability will
rest upon a single instantiation of the process given a narrow specification. For instance, suppose
an agent who is nearly blind, Sally, notices a “Route 912” sign on a foggy winter night while
driving her car at 130 miles per hour on an expressway. Naturally, most would agree that this
process is unreliable. Suppose, however, she fortuitously forms a belief that the sign reads
“Route 912.” As a result of this detailed specification, Sally’s belief formation in this rare
scenario is the only instantiation—i.e., the only token—of the process in question. Hence, this
intuitively unreliable process is reliable on the standard account of reliabilism because the only
instantiation of the process generated a true belief. Therefore, given the function of reliability
offered by the standard account—namely a quantity only in terms of the success rate that is
further reducible to the number of instantiations that yield true beliefs divided by the total
number of instantiations—the reliability of the process will be maximal. This conclusion
necessitates that her doxastic attitude is fully justified, a result that is inconsistent with the way
we intuitively think about justification.
IV. The Mathematical Definition of Reliability
Resolution of the singularly-instantiated process aspect of the generality problem demands that
the reliability of a belief-forming process cannot only be in terms of the success rate. In
particular, reliability cannot merely rest upon the tendency of a given process to produce true
rather than false beliefs, as this quantity is partly determined by the degree of generality of the
specification. Hence we must modify the traditional definition of reliabilism to consider another
quantity in addition to the success rate. However, discovering this other quantity involves
analyzing the process by which reliabilism determines the justificational status of a belief
generated by a singularly-instantiated process, particularly the scenario of Sally forming her
belief that the sign reads “Route 912.”
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In this case, we established that the reliability of the belief-forming process in question is
maximal, and thus her belief is justified despite that such a conclusion is inconsistent given the
undesirable conditions of the process. Her forming the belief that the sign reads “Route 912”
should—at least to some extent—contribute to the level of reliability; yet, it should not confirm
nor deny that the reliability of the process is maximal merely in virtue of a single instantiation. In
other words, we are not confident in whether this mere instantiation is an accurate representation
of the process in totality, and it is therefore conceivable that the likelihood that the success rate is
accurate increases with number of instantiations. In essence, if there are many instantiations for a
given process, we are more confident that the success rate correctly portrays every instantiation
whereas singularly instantiated mechanisms intuitively lack this sort of empirical support.
Hence the reliability of a singularly instantiated process should be rather low to the extent
that the token may have generated a true belief on the basis of luck. Consider, however, a
multiply instantiated process of, say, eighty instantiations, and suppose that that the success rate
is roughly eighty-five percent. Insofar as we are confident that the success rate is accurate given
a sufficient number of instantiations, the reliability of the belief-forming process will
approximately equal the success rate. Thus the high number of instances of the process ensures
that luck is not the factor generating true beliefs; rather, this process is reliable as the result of
some truth-conducive characteristics or abilities. In virtue of the tendency of the mechanism to
generate true rather than false beliefs on the basis of some property or disposition, we are more
confident that the process is reliable for good reasons. Necessarily, the likelihood that the success
rate is accurate relies upon the total number of tokens because a high quantity of instantiations
suggests that there exists some truth-conducive disposition possessed by the mechanism.
Therefore, to confirm that the reliability of a process is accurate and not the result of
fortuity, reliability must be a function of the success rate as well as the total number of
instantiations of the process requiring that the reliability of a given belief-forming mechanism is
defined by the function
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(
where s is the success rate and
)
7
is the total number of instantiations. However, the success rate
may be further reduced to the equation
,
where
is the number of instantiations that yield true beliefs. So the reliability of a process may
be more simply expressed as the function
(
)
8
It is important to note that the function will never generate a reliability exceeding the success rate.
That is to say, the reliability will get infinitely close to but never equal the success rate as the
number of instantiations increases. In other words, the asymptote of the function is always the
specified success rate. This result is intuitive since the reliability of a process is partly the
measure of how confident we are in believing that the success rate is accurate. For instance, if a
high number of instantiations of a particular process generate true beliefs then it is likely that the
success rate of that process is accurately represented; namely, the success rate is roughly
equivalent to the reliability. Following a similar line of reasoning, the reliability will be
significantly lower than the success rate given a low number of instantiations. Therefore, the
reliability of the particular process logarithmically increases with respect to the total number of
instantiations until the process reliability equals the asymptotic success rate.
7
See the appendix for the derivations of all equations used throughout the paper.
Throughout the paper, I correctly state that reliability is a function of both the number of instantiations and the
success rate of the process. However, in order to simplify the number of variables used in the reliability formula, I
restate the success rate in terms of its constituent parts—namely the number of tokens that generate true beliefs and
the total number of tokens—and then substitute that proportion in for every occurrence of the success rate in the first
statement of the reliability formula. I initially express reliability in terms of the success rate and the total number of
instantiations for pedagogical concerns; however, it is important to understand that reliability can be viewed as a
function of both the success rate and the total number of tokens as well as a function of the total number of tokens
and the number of tokens that generate true beliefs.
8
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V. The Correct Specificity of Belief-Forming Processes
While the reliability of belief-forming processes is now in terms of both the success rate and the
total number of instantiations, the generality problem still challenges the validity of the theory
because the degree of generality of a process specification partly determines the degree of
reliability. For example, recall the example of Ryan looking at a tree in a field. We determined
earlier in the paper that the success rate of a belief-forming process is partly contingent upon an
arbitrary degree of generality thereby resulting in a subjective ascription of reliability. Therefore,
the revised account of reliabilism—despite its success in assigning accurate reliabilities to
singularly instantiated processes—cannot yet handle an important consequence of the generality
problem, namely its inability to assign an objective level of reliability to belief-forming
mechanisms.
There is, however, a clear resolution to this problem as a result of revising the calculation
of reliability. When the reliability of a process was merely a function of the success rate,
processes were specified broadly to prevent reliability from resting upon a single instantiation.
Fortunately, however, a broad specification of belief-forming processes is no longer
indispensable to the theory given the increased accuracy of reliability ascriptions to infrequently
instantiated processes, that is, mechanisms that have but a few tokens. Instead the modified
theory conversely necessitates narrow process specifications to account for all conditions upon
which the efficacy of a particular process depends; viz., a specification must contain only those
details affecting the reliability of a process. Any superfluous factors that have no bearing upon
the reliability of a process ought not to be contained within its specification. As a result, a
process is sufficiently described if and only if all relevant factors—particularly, those that affect
the success rate of the process—are included in the specification. For instance, consider the case
of Ryan noticing a tree in a field. Various details, such as the color of his shirt or jeans, are not
relevant considerations with respect to the reliability of the process and therefore ought not to be
contained in the specification. On the other hand, details, such as the location, weather, or
perhaps physical or mental defects of the observer, are necessary constituents of the specification
since each affects his capacity for visual perception.
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Justin Svegliato/Volume 1 (2013)
It is again important to note that specifications may be unavoidably narrow necessitating
that there may exist only one instantiation for certain processes; however, the modified account
of reliabilism that I have proposed can adequately handle these situations. On the contrary, if the
process specification does not contain all relevant details affecting the instantiation’s efficacy,
the reliability of that process will not be accurate to the extent that a token that ought to be
categorized under one specification may be perhaps categorized under another specification. For
instance, consider two situations where the descriptions of the processes are sufficient. First,
suppose that James is eavesdropping on a conversation in a crowded mall. Further, suppose that
Ryan, shopping at the mall as well, is eavesdropping on the same conversation while also
wearing hearing aids. They both then form beliefs about the topic of the conversation and are
both one of many instantiations of each corresponding process specification. However, suppose
that the specification of the process of the latter case was insufficient: it did not include that
Ryan is wearing hearing aids. As a result, the situation of Ryan, an instantiation of a process,
would be inappropriately grouped with an instantiation of a much different process. Thus a token
that should have been categorized under one process is now inaccurately identified under another
process. Hence the reliability of each process will not represent the entire set of instantiations
that belong under each corresponding specification. By incorporating all relevant factors that
affect reliability, we are enforcing an objective level of generality, and so the reliability of
process will not be contingent upon an arbitrary degree of generality.
VI. The Simplification of Process-Type Specifications
Assume the process specification of some set of tokens, say looking at a tree in a field while
wearing glasses, is sufficient and is expressed in terms of all factors affecting reliability.
Conceivably, our current specification of processes is colloquial and informal, for the expression
of relevant considerations relies upon everyday adjectives and language. This further complicates
the calculation of reliability, as the informality of process specifications causes many to falsely
claim that two identical processes are distinct. For instance, again recall the case of Ryan
noticing a tree while sitting at his desk and furthermore consider a scenario whereby Molly is
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looking at a tree through a window. In both scenarios, both Ryan and Molly are using visual
perception but the relevant considerations or factors, namely the location of the agent in question,
are distinct according to language and thus both examples represent two disjoint processes.
However, are these processes truly distinct? Are the mere locational details of each instantiation
enough to identify the two tokens as two distinct processes?
It is rational to consider the case of Ryan and Molly as instantiations of two distinct
processes insofar as the difference in location affects reliability. The location, however, is only a
relevant consideration in virtue of its affect on the efficacy of the sight of Ryan and Molly. What
if the locations of each agent affected visual perception in an identical way? That is, we can
perceive the location of visual perception as important only with respect to its affect on the
degree of illumination in the proximity of the observer and the observed. Therefore, the
processes of Ryan and Molly—rather than being in terms of mere location—may be expressed in
terms of the degree of illumination at that particular location. For example, the process in
Molly’s scenario can be better described as “observing a tree with a degree of illumination x”
whereas the process-type in Ryan’s situation can be expressed as “observing a tree with a degree
of illumination y.” In these cases, each process is not specified in terms of the location; rather, it
is in terms of a more simplistic and informative descriptor, namely the degree of illumination.
Hence, if the degree of illumination is identical—i.e. x is equal to y in reference to the cases of
Ryan and Molly—in both locations, then the processes are surprisingly identical and there would
be no significant distinction between the two instantiations despite a difference in environment.
Thus our original conception of process specifications must change. Particularly, instead
of representing instantiations with informal descriptions consisting of commonplace modifiers
concerning environmental factors, the specification must contain only those attributes, which are
derivations of more colloquial language, directly affecting the mechanisms in question. This
reductive formulation yields a simplified set of processes insofar as many previously distinct
mechanisms are now considered equivalent thereby transforming the once infinite set of
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processes into a more fundamental subset.9 In a word, eliminating unnecessarily specific
environmental details and instead incorporating the conditions affecting reliability, such as the
degree of illumination, will reduce many previously distinct processes into one mechanism.
Using this sort of reasoning, other nontrivial details, such as the time, weather, and even
psychological and physical deficiencies of the process instantiator, are reducible to more
fundamental factors affecting the process. As a result, process specifications must always be
expressed in terms of the most fundamental qualities to yield a simplified finite set of
mechanisms.
VII. The Categorization of Belief-Forming Processes
At present, I have offered a modified account of reliability that handles all facets of the
generality problem; that is, the theory now yields objective specifications for processes since
reliability no longer relies upon an arbitrary degree of generality while also assigning accurate
reliabilities to singularly instantiated processes. However, in order to provide a more practical
account of reliabilism, I will describe a method by which the total reliability of primary
processes, particularly the most general mechanisms that generate beliefs, can be calculated. All
mechanisms can be essentially interpreted as either a process of cognition or perception.
Intuitively, perception is composed of smaller yet more specific subcategories, i.e., sight, hearing,
taste, smell, and touch, whereas cognition consists of merely reasoning and recollection. The
below method, which is used to calculate the reliability of a primary process, namely perception
and cognition, may be applied to the aforementioned subcategories or any arbitrarily chosen
category as well.
9
Since there are infinitely many scenarios where each of which has distinct quantity of illumination, it may be
argued that there are still infinitely many processes. Insofar as an organism’s sense of sight is limited by biological
considerations, it cannot perceive infinitesimally small changes in light. Thus there is a discrete quantity, a function
of the effectiveness of the organs that handle visual perception, required for an organism to perceive changes in
illumination. That is, if there are two locations that are infinitesimally close to each other in terms of the degree of
illumination, the two locations are effectively identical with respect to the senses of the organism. Moreover, since
there is intuitively a threshold for which an organism can no longer visually perceive any objects in an environment
where the degree of light is too low or high, i.e., absolute darkness or blindingly bright, there are still finitely many
processes.
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In a word, the reliability of primary processes is also defined by the same function that is
used to calculate the reliability of any other process. That is, the reliability of a primary process
(or any subcategory of the primary process) is also defined by the function
(
In this case, however,
)
is the total number of tokens for that given primary process, whereas
is the number of tokens that yield true beliefs.
Though it is important to note that the reliability of a primary process is not the reliability
of any specific process; that is, it cannot be used to determine the justificational status of a belief.
To ascertain the status of a belief from the reliability of a process, the specification of the process
must be sufficient; particularly, it must contain all relevant details affecting the success rate.
Hence there is a significant difference between the reliability of primary processes and the
reliability of any specific subprocess. Namely, the reliability of a primary process only serves as
an estimate or indicator of the reliability of more specific process whereas the reliability of the
specific process that generated a given belief is the precise quantity needed for the calculation of
the justificational status of that belief.
VIII. Conclusion
Despite the initial tenability of Alvin Goldman’s account of epistemic justification, the generality
problem undeniably challenges the ability of reliabilism to attribute justification to a doxastic
attitude merely based upon the tendency of a belief-forming process to generate true rather than
false beliefs. However, by redefining the reliability of belief-forming mechanisms in terms of
this tendency as well as their corresponding number of instantiations, not only can accurate
reliabilities be ascribed to singularly instantiated processes but the methodology by which we
specify processes can now require the inclusion of only those details affecting the efficacy of the
process in order to circumvent the influence of an arbitrary degree of generality. Hence the
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Justin Svegliato/Volume 1 (2013)
generality problem—a once inescapable deterioration of reliabilism—is no longer a threat to an
intuitive account of epistemic justification.
Appendix: The Derivation of the Reliability Equation
It is important to note that the result of this derivation does not result in a concrete formula—it is
merely a family of functions that includes an arbitrary constant. We determine the value of that
constant by substituting our initial condition for the appropriate variables and solving. Further
note that
is the reliability growth rate parameter with respect to the number of occurrences and
our choice of this variable is arbitrary. Thus we need to choose a
that will accurately represent
the growth of the function.
However, before solving for the arbitrary constant and choosing , we must first solve
(
the ordinary differential equation
) by the method of separation of variables since
it is both linear and autonomous. Dividing and multiplying both sides by
respectively yields
have that
(
(
) and
. Subsequently, by integrating both sides of the equation, we
)
(
))
(
because
)
(
)
(
)
(
By multiplying both sides by
(
and simplifying we have that
exponentiating both sides yields
dividing by
(
as well as distributing
(
)
(
which can be rewritten as
yields
31
))
) and then
. Next,
and then also
Justin Svegliato/Volume 1 (2013)
from both sides and simplifying yields (
subtracting
have that
when both sides are divided by
Furthermore, let
and
rate of the process and
)
. Finally, we
.
where is the dependent variable representing success
is the independent variable signifying the total number of tokens. Thus
the reliability of a given belief-forming process can be modeled by
(
)
Yet the above model still contains an underdetermined constant —essentially selecting
the initial point of the equation—that can be determined using an arbitrary initial condition. The
reliability of any when
should conceivably be ; however, due to the nature of the
equation, the minimum reliability must be infinitesimally larger than 0 in order to retain the
characteristics of the equation in question. Hence we will assume that a reliability of
equal to . Next, we will solve for
using the initial condition for any when
is
and
. Observe that
(
)
and so
Cross-multiplying yields
us
. Subtracting
or
(
and
). By dividing
we have that
32
from both sides gives
from both sides
Justin Svegliato/Volume 1 (2013)
Substituting the right hand side of the equation for
(
in the reliability formula above yields
)
or
(
Now
)
must be arbitrarily chosen in order to specify the slope of the graph. We let
. This will cause the slope of the graph to be more gradual; that is, the rate at which the
reliability increases for each
will decrease and thus
(
)
However, , the success rate, may be further reduced to the number of tokens that yield true
beliefs
divided by the total number of tokens . Thus
reliability formula yields
(
)
or
(
)
33
. By substituting
in for in the