The Value of Knowledge
Wolfgang Spohn
Workshop
Full and Partial Belief
TiLPS, Oct. 20-22, 2014
Table of Contents
The Issue
A Brief Word on the Gettier Business
Modal Theories of Knowledge
Nozick’s Sensitivity Analysis (2)
Pritchard’s Safety Analysis
Freitag’s Normality Analysis
A Discontent with Modal Theories (2)
The Epistemic Interpretation of Conditionals
The “Circumstances are Such That” Reading of Conditionals
The Sensitivity Analysis Epistemically Interpreted (2)
The Other Analyses Epistemically Interpreted
Returning to the Surplus Value of Knowledge
The Surplus Value of Knowledge (4)
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The Issue
There is a well-known traditional account of the value of knowledge:
Knowledge has practical value; it enables us to better reach our aims.
Or in formal decision-theoretic terms: the expected value or utility of cost-free
relevant true information is always positive (Savage 1954).
Theorems of this kind are usually subsumed under the heading “the value of
knowledge” (Skyrms 1990). However, strictly speaking, they only explain
the value of true belief.
There is a more recent discussion in epistemology about the value of knowledge asking: what is the surplus value of knowledge over and above the
value of (justified) true belief? Clearly, this question presupposes an
account of what knowledge is.
So, it is also a Gettier-induced question. The threat behind it is: if we can’t find
a good answer, we need not care about knowledge and can in particular
forget about all this Gettier business.
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A Brief Word on the Gettier Business
As is well-known this business arose from Gettier’s criticism [which we also
find, e.g., in Russell and in 8th century Indian philosophy (Dharmottara)] of
the traditional JTB analysis that knowledge is justified true belief.
Knowledge is not just that, but something more, which we are well advised to
scrutinize. My only criticism of the obsessive scrutiny is that we have forgotten how unclear the necessary JTB conditions are:
!
What is the relevant notion of truth involved here? (Don’t say the
correspondistic one only because you don’t know what else to say.)
!
What at all is belief? (This is a very deep question, and it is surprising
how little clarity there is in the relevant literature.)
!
What is justification? (The relevant literature is shamefully missing
clear accounts of that.) (I tend to say that, from a rationalistic perspective, all beliefs are justified. If a belief were not at least subjectively justified, I would be irrational to hold it. However, this remark does
not absolve us from saying what justification is.)
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Modal Theories of Knowledge
The last remark only suggests that we won’t find a surplus value of knowledge over true belief within the JTB analysis. So we must attend to what
“something more” might mean.
Let us focus on the so-called modal theories of knowledge, for which a useful
and very general scheme may be provided (cf. Freitag 2013).
The scheme is this:
x knows that p iff:
(1)  In the actual world @ p holds as well as that x believes that p,
(2)  The material implication “if x believes that p, then p” – in short:
Bx(p) ⟶ p – is true in each world of the so-called warranty set K, to which
the actual world @ may or must belong.
They idea here is that, if x knows that p, her belief in p is guaranteed to be
true, or is necessarily true (where that guarantee or necessity is not absolute, but restricted to the warranty set K – which is a dummy so far).
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Nozick‘s Sensitivity Analysis I
The warranty set K may be interpreted in various ways. One possibility is the
sensitivity analysis of Nozick (1981) (which is close to Goldman’s (1967)
causal analysis).
According to it a condition beyond JTB for x's knowing that p is (where
stands for the conditional, however it is to be interpreted – see below):
(3)  if p had not been the case, x would not have believed that p – formally:
¬p
¬Bx(p) .
According to the received Stalnaker/Lewis truth conditions of counterfactuals,
(3) says that the warranty set K consists of the non-p-worlds closest or
most similar to the actual world @ and all worlds at least as close than
those the non-p-worlds (in which p is true). In all those worlds, (3) requires, ¬p ⟶ ¬Bx(p), i.e., Bx(p) ⟶ p must be true.
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Nozick‘s Sensitivity Analysis II
Nozick adds a second condition, often considered less important, which he
expresses as a factive subjunctive conditional and which I try to capture
thus (the intention, or hope, is here that “since” is less causally loaded
than “because”):
(4) since p has been the case, x believes that p – formally: p
Bx(p) .
According to the received semantics this means that in all closest p-worlds
Bxp is true as well.
Thus, according to scheme (2), clause (4) does not add any further truth
guarantee to (3), since p and hence the material implication Bx(p) ⟶ p is
true in all (closest) p-worlds, anyway. And if Centering is assumed in the
semantics of , (4) adds nothing to (1).
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Pritchard‘s Safety Analysis
Pritchard (2005) states his so-called safety analysis also with the help of a
factive subjunctive conditional, which I again try to capture with “since”:
(5)  since x believes that p, p is (has been) the case – formally:
Bx(p)
p.
According to the received semantics this means that in all closest Bx(p)worlds p is true as well.
Hence, according the received semantics and assuming Weak Centering, but
avoiding Centering (otherwise (5) would be trivial), (5) assumes the
warranty set K to be the set of all worlds closest to actuality (in which
Bx(p) ⟶ p is true, whether or not x believes that p in them).
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Freitag‘s Normality Analysis
According to Freitag (2013) the warranty set K consists in all normal worlds in
which the material implication Bx(p) ⟶ p is true. So knowledge is a belief
the truth of which is normally guaranteed. We cannot generally presuppose that the actual world is normal, i.e., that Weak Centering holds if the
similarity order is interpreted as a normality order. Hence, Freitag must
explicitly assume that the actual world is normal, i.e., that @ is in K (this
explains the ambiguity in (2)). So, according to his analysis, knowledge
requires:
(6)  if x believes that p, then p is (normally) the case – formally:
Bx(p)
p (though with a different interpretation of
than in (5)).
This is also a natural reading of indicative conditionals.
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A Discontent with Modal Theories I
Now my interest is not to discuss niceties of those three variants of modal
theories of knowledge. Nor is my interest to discuss whether other accounts also fall under the scheme (2). Let it suffice that we have a representative sample of quite plausible accounts of knowledge.
My concern is rather that all three variants heavily rely on the conditional
idiom (and partially even on an awkward factive version of the conditional
idiom). The conditional idiom is fundamental and ubiquitous; it is philosophically essential not only in accounts of knowledge, but almost everywhere. But it is still so ill understood that it is careless to simply assume it
as the basis of any philosophical analysis.
Thus, it is a problem for the knowledge theorist as well and not only for the
conditional theorist.
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A Discontent with Modal Theories II
One big problem is that the conditional idiom is so confusingly multifarious
that it does not seem to admit of a unified account – whence the multitude
of accounts.
Another big problem is that the very graphic geometry of similarity spheres –
on which the received semantics for subjunctive (not indicative) conditionals is based – hides how poorly we understand the similarity or
closeness involved here. Our relevant intuitions exclusively derive from
our intuitions which conditionals to accept.
The same concern applies to Freitag’s normality analysis. Normal conditions
are a special case of ceteris paribus conditions, and that’s a hornets’ nest
as well; philosophers of science are quarreling for decades about an appropriate analysis.
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The Epistemic Interpretation of Conditionals
I cannot discuss now conditionals (and normal conditions) in general. (I did so in
Spohn (2013).) Let me only say:
!  that in my view the best approach to conditionals is to see them as
expressing conditional belief and features thereof,
!
!
that we should hence turn to accounts of conditional belief like AGM
belief revision theory or, much better, ranking theory (see Spohn (2012),
where the ill-understood similarity orderings find an intelligible subjective
correlate in entrenchment orderings or cardinal ranking functions.
This seems to be a well explored strategy. But it is not. I seem to appeal to understand conditionals via the familiar Ramsey test (according to which conditionals indeed simply express conditional belief). However, conditi-nals go
far beyond the Ramsey test; they can be used to express other features of
conditional belief beyond conditional belief itself. And this is not well explored. The relevant feature for us is this:
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The „Circumstances are Such That“ Reading of Conditionals
According to the Ramsey test, what I do by uttering a conditional p
express my conditional belief in q given p, B(q | p).
q is to
Another and in my view more plausible reading is that I thereby express my
(unconditional) belief that the circumstances are such that I can maintain
my conditional belief, or, more precisely: that I believe s, B(s), where s is
the disjunction s1 … sn of all maximal circumstances si under which I
believe in q given p, i.e., B(q | p si) (where maximality is relative to some
contextually given more or less fine-grained partition of circumstances).
This entails that B(q | p s) and, only (!) given some further conditions, the
latter entails B(q | p), i.e., the Ramsey test.
I call this the “circumstances are such that” reading of conditionals. A special
case is the most wide-spread causal reading of conditionals (which is, in
fact, a “history is such that” reading).
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The Sensitivity Analysis Epistemically Interpreted I
Let’s apply this, for instance, to Nozick’s sensitivity analysis (3). According
to the epistemic interpretation we are not after the truth condition of a
knowledge ascription, but rather after which beliefs a ascriber z expresses by ascribing knowledge to the subject x.
According to Nozick, if z ascribes to x knowledge of p, z thereby claims: p,
Bx(p), and ¬p
¬Bx(p). And according to the “circumstances are such
that” reading of , z thereby expresses Bz(p), Bz(Bx(p)), Bz(b3), and
Bz(¬Bx(p) | ¬p b3) –
where b3 may be called the knowledge background, or rather the
knowledge ascription background, which is defined in the same way as
the disjunction s of maximal circumstances on the previous slide. (The
subscript “3” refers to Nozick’s condition (3).)
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The Sensitivity Analysis Epistemically Interpreted II
The knowledge background b3 is an objective proposition which is true or false,
but it is subjectively defined relative to the conditional beliefs of z. z takes it
to be true, but she may be wrong, of course.
The knowledge background b3 is characterized by a peculiar stability, which
consists in the fact that any additional information c, as long as it is logically compatible with b3, does not change the relevant conditional belief
of z, i.e., for any such c we have Bz(¬Bx(p) | ¬p b3 c).
This does not hold generally. In general, a conditional belief is not preserved
under additional logically compatible information. The reason why the
background b3 has this stability property lies in its definition as a disjunction of maximal circumstances. This stability will prove important for my
argument below.
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The Other Analyses Epistemically Interpreted
Similarly, Nozick’s condition (4) translates into the condition that the ascriber
z claims that p
Bx(p) and thus has some background b4, for which
Bz(b4) and Bz(Bx(p) | p b4).
Furthermore, Pritchard’s condition (5) translates into the condition that the
ascriber z claims that Bx(p)
p and thus has some background b5, for
which Bz(b5) and Bz(p | Bx(p) b5).
Finally, Freitag’s normality analysis yields something similar in an epistemic
interpretation of normal conditions. Condition (6) then translates into the
condition that the ascriber z claims that Bx(p) → p under normal conditions and thereby expresses Bz(b6) and Bz(p | Bx(p) b6) for some disjunction b6 of maximal normal (≈ not unexpected) conditions.
The backgrounds b3 – b6 may differ (because Nozick, Pritchard, and Freitag
subtly differ). But all of them have the stability mentioned. And it would be
an instructive exercise to find conditions under which they fall in one.
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Returning to the Surplus Value of Knowledge
Plato‘s Meno is often interpreted as offering a stability theory of knowledge:
“True opinions are … a fine thing and altogether good in their effects so long as
they stay with one. … they are not worth much until one ties them down by
reasoning out the explanation … then … they become items of knowledge,
and … permanent. And that’s what makes knowledge more valuable than
right opinion, and the way knowledge differs from right opinion is by being
tied down.”
This resembles the JTB analysis often associated with the Theaitetos, but goes
beyond it by adding the stability idea. However, reasons as such don’t help.
As I said, in a rationalistic perspectives beliefs always come with reasons.
And in general, reasons come and go and may be as unstable as beliefs.
So, how can the stability idea be understood within the framework I have developed? (Rott (2004) was the first to pursue this idea in the belief revision
framework.)
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The Surplus Value of Knowledge I
If an (empirical) proposition is true, we will eventually find reasons for taking it
to be true in Peirce’s ideal limit of inquiry (which is extremely counterfactual in various ways). (We may leave it open whether we can thereby
define a notion of truth, though I tend to think so.)
So, if I now believe p, I take p to be true – this is one and the same. Hence I
also believe that this belief will survive in the ideal limit of inquiry. It is my
everyday experience, though, that I have to give up and revise my beliefs,
although, when I believe p, I at the same time believe that this will not
happen to my belief in p. However, once I should have given up my belief
in p, I likewise do not expect it to return.
The same is true from the ascriber’s perspective. If she ascribes me a true belief, she simply shares my belief, we may both be wrong, and her belief is
subject to the same dynamic instabilities as mine.
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The Surplus Value of Knowledge II
How do things stand if the ascriber z ascribes me (= x) knowledge of p? Of
course, she may be wrong again. However, there is still a difference:
First, she again shares my belief; i.e.: Bz(p) and Bz(Bx(p)).
She does more, however. She also believes in the knowledge ascription background b (where I now neglect the differentiation by indices); i.e.: Bz(b).
And given this background, she believes Bx(p) given p and/or ¬Bx(p) given
¬p, or perhaps more neutrally, that p is a reason or positively relevant for
Bx(p) in the sense that her degree of belief in Bx(p) is higher given p than
given ¬p.
Moreover, this reason relation or positive relevance between p and Bx(p) is
stable in the sense that no information logically compatible with b can
change anything about it.
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The Surplus Value of Knowledge III
This puts the ascriber into a better position: If she simply takes p to be true (and believes that I do so, too), she does have some reason for that, but maybe a
wrong reason; and she also believes that eventually, in the ideal limit, there will
be true reasons for maintaining the belief in p.
However, if she ascribes knowledge of p to me, then she has a specific (conditional)
reason for p, namely Bx(p), that I believe in p, and moreover this reason will
persist, at least as long as her background b is not defeated.
This improves the warrant which the ascriber has for the truth of my belief. If the
ascriber merely believes p (perhaps because she trusts me), this belief may be
undermined by any reason against p. However, if she thinks that I know p, then
not any reason against p will undermine her belief in p. Rather, the entire background b would have to be defeated, before anything could count against p.
(It is a general advantage of explanations that they provide stable reasons in this
sense, and the ascriber has some particular explanation for my belief, if she
ascribes knowledge to me.)
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The Surplus Value of Knowledge IV
The final move is: The very same holds within the first-person perspective. I
myself can distinguish between believing p and believing to know p.
Well, almost the same: I cannot presently take the third-person perspective on
myself. I am conscious of the fact that I presently believe that p. And this
conscious fact bears no justificatory relations; all possible reasons pertain
only to the content p and not to that fact that I now believe that p.
However, concerning my past (and my future) beliefs I can very well take the
third-person perspective. And then I need not take each of my previously
acquired beliefs to be knowledge, precisely because I need not believe in
the knowledge background required for the belief to be knowledge. This is
how I can distinguish between believing p and believing to know p.
But if I believe in the relevant knowledge background for p, then my belief in p
is, as described, more stable than the mere belief in p. This, I take it, is the
surplus value of knowledge for me.
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Many thanks
For your attention!
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Bibliography
Freitag, Wolfgang (2013), I know. Modal Epistemology and Scepticism, Mentis, Münster.
Goldman, Alvin I. (1967), “A Causal Theory of Knowing”, Journal of Philosophy 64, 357-372.
Nozick, Robert (1981), Philosophical Explanations, Harvard University Press, Cambridge, Mass.
Pritchard, Duncan (2005), Epistemic Luck, Oxford, Oxford University Press.
Rott, H. (2004), „Stability, Strength and Sensitivity: Converting Belief into Knowledge“, Erkenntnis,
61, 469-493.
Savage, Leonard J. (1954), The Foundations of Statistics, New York: Wiley, 2nd ed.: Dover 1972.
Skyrms, Brian (1990), The Dynamics of Rational Deliberation, Cambridge, Mass.: MIT Press.
Spohn, Wolfgang (2012), The Laws of Belief. Ranking Theory and its Philosophical Applications,
Oxford University Press, Oxford.
Spohn, Wolfgang (2013), “A Ranking-Theoretic Approach to Conditionals”, Cognitive Science 37,
1074-1106.
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