## 1. Brief History



The literature on the knowability paradox emerges in response to a
proof first published by Frederic Fitch in his 1963 paper, "A
Logical Analysis of Some Value Concepts." Theorem 5, as it was
there called, threatens to collapse a number of modal and epistemic
differences. Let ignorance be the failure to know some truth. Then
Theorem 5 collapses a commitment to contingent ignorance into a
commitment to necessary ignorance. For it shows that the existence of
truths in fact unknown entails the existence of truths necessarily
unknown. Formally, 



\[\tag{Theorem 5}
\exists p(p \wedge \neg Kp) \vdash \exists p(p \wedge \neg \Diamond Kp).
\]


The converse of Theorem 5 is trivial (since truth entails
possibility), so Fitch goes most of the way toward erasing any logical
difference between the existence of contingent ignorance and the
existence of necessary unknowability. 



It is however the contrapositive of Theorem 5 that is usually referred
to as the paradox: 



\[\tag{K Paradox} 
\forall p(p \rightarrow \Diamond Kp) \vdash \forall p(p \rightarrow Kp).
\]


It tells us that if any truth can be known then it follows that every
truth is in fact known. 



The earliest version of the proof was conveyed to Fitch by an
anonymous referee in 1945. In 2005 we discovered that Alonzo Church
was that referee (Salerno 2009b). His reports are published in their
entirety in Church (2009). Fitch apparently did not take the result to
be paradoxical. He published the proof in 1963 to avert a kind of
"conditional fallacy" that threatened his informed-desire
analysis of value. The analysis roughly says: \(x\) is valuable
to \(s\) just in case there is a truth \(p\) such that were
\(s\) to known \(p\) then she would desire \(x\). The
existence of unknowable truths ultimately explains why he restricts
the propositional variables to knowable propositions. For an
unknowable truth provides for an impossible antecedent in Fitch's
counterfactual, and ultimately trivializes the analysis. Since Fitch's
theory of value is not the context in which the paradox is widely
discussed, we will say no more about it here. 



Rediscovered in Hart and McGinn (1976) and Hart (1979), the result was
taken to be a refutation of verificationism, the view that all
meaningful statements (and so all truths) are verifiable. After all,
if one accepts the knowability principle,
\(\forall p(p \rightarrow \Diamond Kp)\), she is
committed to the absurd claim that all truths are known. Mackie (1980)
and Routley (1981), among others at the time, point to difficulties
with this general position but ultimately agree that Fitch's result is
a refutation of the claim that all truths are knowable, and that
various forms of verificationism are imperiled for related reasons.
Since the early eighties, however, there has been considerable effort
to analyze the proof as paradoxical. Why after all should it be that
an epistemic theory of truth collapses possible knowledge into actual
knowledge? Intuitively, that truth is to be understood in terms of the
epistemic capacities of non-omniscient agents is at least a coherent
position--a position distinct from and more plausible than the
thesis that all truths are known. Moreover, it has been thought
strange that sophisticated versions of the epistemic theory of truth
should fall prey to such swift deduction. Hence, the Church-Fitch
proof has come to be known as the *paradox* of knowability.




There is no consensus about whether and where the proof goes wrong. We
use this entry to lay out the proof and explore a range of proposed
treatments.


## 2. The Paradox of Knowability



Fitch's reasoning involves quantifying into sentence position. Our
propositional variables \(p\) and \(q\) will take
declarative statements as substituends. Let \(K\) be the
epistemic operator 'it is known by someone at some time
that.' Let \(\Diamond\) be the modal operator 'it is possible
that'.



Suppose the knowability principle (KP)--that all truths are
knowable by somebody at some time:



\[\tag{KP}
\forall p(p \rightarrow \Diamond Kp).
\]


And suppose that collectively we are non-omniscient, that there is an
unknown truth:



\[\tag{NonO}
\exists p(p \wedge \neg Kp).
\]


If this existential claim is true, then so is an instance of it: 



\[\tag{1}
p \wedge \neg Kp.
\]


Now consider the instance of KP substituting line 1 for the variable
\(p\) in KP: 



\[\tag{2}
(p \wedge \neg Kp) \rightarrow \Diamond K(p \wedge \neg Kp)
\]


It follows trivially that it is possible to know the conjunction
expressed at line 1:



\[\tag{3}
\Diamond K(p \wedge \neg Kp)
\]


However, it can be shown independently that it is impossible to know
this conjunction. Line 3 is false. 



The independent result presupposes two very modest epistemic
principles: first, knowing a conjunction entails knowing each of the
conjuncts. Second, knowledge entails truth. Respectively,



\[\begin{align}
\tag{A} K(p \wedge q) &\vdash Kp \wedge Kq \\
\tag{B} Kp &\vdash p
\end{align}\]


Also presupposed are two modest modal principles: first, all theorems
are necessary. Second, necessarily \(\neg p\) entails that
\(p\) is impossible. Respectively,



\[\begin{align}
\tag{C} &\text{If } \vdash p, \text{then } \vdash \Box p. \\
\tag{D} &\Box \neg p \vdash \neg \Diamond p.
\end{align}\]



Consider the independent result:



\[\begin{align}
\tag{4} K(p \wedge \neg Kp) &\quad\text{Assumption [for reductio]} \\
\tag{5} Kp \wedge K\neg Kp &\quad\text{from 4, by (A)} \\
\tag{6} Kp \wedge \neg Kp &\quad\text{from 5, applying (B) to the right conjunct} \\
\tag{7} \neg K(p \wedge \neg Kp) &\quad\text{from 4-6, by reductio,} \\
 &\quad\quad\text{discharging assumption 4} \\
\tag{8}\Box \neg K(p \wedge \neg Kp) &\quad\text{from 7, by (C)} \\
\tag{9}\neg \Diamond K(p \wedge \neg Kp) &\quad\text{from 8, by (D)}
\end{align}\]


Line 9 contradicts line 3. So a contradiction follows from KP and
NonO. The advocate of the view that all truths are knowable must deny
that we are non-omniscient: 



\[\tag{10}
\neg \exists p(p \wedge \neg Kp).
\]


And it follows from that that all truths are actually known: 



\[\tag{11}
\forall p(p \rightarrow Kp).
\]


The ally of the view that all truths are knowable (by somebody at some
time) is forced absurdly to admit that every truth is known (by
somebody at some time). 


## 3. Logical Revisions



In this section we inspect the prospects for treating Fitch's
reasoning as invalid. Is Fitch's epistemic reasoning in order? Is the
logic of knowability classical logic? More to the point: does the
knowability principle carry with it special considerations that
warrant the revision of classical logic? If so, does this logical
revision invalidate Fitch's reasoning? And if the reasoning is invalid, are
there closely related paradoxes that threaten the knowability
principle without violating the relevant logical standards? 


### 3.1 Epistemic Revision



The problem with Fitch's reasoning is not with either of the epistemic
inferences A or B. Though some have argued that knowing a conjunction
does not entail knowing the conjuncts (Nozick 1981), Williamson (1993)
and Jago (2010) have shown that versions of the paradox do not require
this distributive assumption. And questions about the factivity of
\(K\) may be defused rather quickly, since related paradoxes
emerge replacing the factive operator "It is known that"
with a non-factive operators, such as 'It is rationally believed
that' (Mackie 1980: 92; Edgington 1985: 558-559; Tennant
1997: 252-259; Wright 2000: 357). 



Deep and interesting discussions of the epistemic operators and/or
temporal analogs in the context of Fitch's paradox appear in many
papers. Burgess (2009) considers the temporal analogs. van Benthem
(2004; 2009), van Ditmarsh, et al. (2012), Berto et al. (forthcoming)
and Holliday (2018) explore the problem in dynamic epistemic
frameworks. Palczewski (2007), Kelp and Pritchard (2009), Chase et al.
(2018), and Heylen (forthcoming) consider non-factive notions of
knowledge and knowability. Linsky (2009), Paseau (2008), Jago (2010),
Carrara et al. (2011), and Rosenblatt (2014) debate the prospects for
typing knowledge. 


### 3.2 Intuitionistic Revision



Williamson (1982) argues that Fitch's proof is not a refutation of
anti-realism, but rather a reason for the anti-realist to accept
intuitionistic logic. Owing to a verificationist (or constructivist)
reading of negation and existential quantification, intuitionistic
logic validates neither the elimination of double negation, 



\[
\neg \neg p \vdash p,
\]


nor the following quantifier exchange rule: 



\[
\neg \forall x\P[x] \vdash \exists x\neg\P[x].
\]


Without double negation elimination one cannot derive Fitch's
conclusion 'all truths are known' (at line 11) from
'there is not a truth that is unknown' (line 10). Consider
line 10, 



\[
\neg \exists p(p \wedge \neg Kp).
\]


From this we may intuitionistically derive 



\[
\forall p\neg(p \wedge \neg Kp).
\]


But notice without double negation elimination, 



\[
\neg(p \wedge \neg Kp)
\]


does not entail 



\[
p \rightarrow Kp.
\]


Suppose



\[
\neg(p \wedge \neg Kp)
\]


and suppose \(p\) for conditional introduction. And suppose
\(\neg Kp\) for *reductio*. We may conjoin \(p\) with
\(\neg Kp\) to get



\[
p \wedge \neg Kp
\]


This contradicts our primary assumption. So, by *reductio*,
\(\neg \neg Kp\). Without double negation elimination, we cannot
conclude \(Kp\), and so, may not introduce the conditional



\[
p \rightarrow Kp
\]


The intuitionist is however committed, by conditional introduction, to




\[
p \rightarrow \neg \neg Kp.
\]


There is some debate about whether this consequence is sufficiently
troublesome, but the intuitionistic anti-realist takes solace in the
fact that she is not committed to the blatantly absurd claim that all
truths are known. Very interesting discussion concerning the hopes and
dreams of intuitionistic antrealism in this context appears in Murzi
(2010; 2012), Murzi et al. (2009) and Zardini(2015).


### 3.3 Problems for Intuitionistic Revision



Since Fitch's reasoning is intuitionistically valid through line 10, the
intuitionist anti-realist must accept that no truths are unknown:
\(\neg \exists p(p \wedge \neg Kp)\). Arguably
this is harmful enough, for it appears that the anti-realist cannot
give credence to the truism that (individually and collectively) we
are non-omniscient. Williamson responds that the intuitionist
anti-realist may naturally express our non-omniscience as "not
all truths are known": 



\[\tag{12}
\neg \forall p(p \rightarrow Kp)
\]


This claim is classically, but not intuitionistically, equivalent to
the non-omniscience thesis, 



\[
\exists p(p \wedge \neg Kp).
\]


That is because in intuitionistic logic the quantifier exchange rule,
\(\neg \forall x\P[x] \vdash \exists x \neg\P[x],\) is not
unrestrictedly valid. Importantly the expression of non-omniscience at
line \(12, \neg \forall p(p \rightarrow Kp)\), is only classically,
and not intuitionistically, inconsistent with line \(10, \neg \exists
p(p \wedge \neg Kp)\). So the intuitionist anti-realist can
consistently give expression to the truism that we are non-omniscient
(with line 12) while accepting the intuitionistic consequence derived
at line 10. In effect, the anti-realist admits both that no truths are
unknown and that not all truths are known. The satisfiability of this
claim on intuitionistic grounds is demonstrated by Williamson (1988,
1992). 


### 3.4 The Undecidedness Paradox of Knowability



A deeper problem is said to remain for the intuitionist anti-realist.
Fitch's paradox rests on the assumption that there are unknown truths.
But consider the intuitionistically weaker assumption that there are
undecided statements, that is, some \(p\), such that \(p\)
is unknown and \(\neg p\) is unknown. Formally, 



\[\tag{Und}
\exists p(\neg Kp \wedge \neg K\neg p)
\]


If (Und) is true, then so is an instance of it: 



\[\tag{i}
\neg Kp \wedge \neg K\neg p.
\]


And notice that the intuitionistically acceptable conclusion at line
\(10, \neg \exists p(p \wedge \neg Kp)\), is
intuitionistically equivalent to the universal claim, 



\[\tag{ii}
\forall p(\neg Kp \rightarrow \neg p).
\]


Deriving \(\neg Kp \rightarrow \neg p\) and \(\neg K\neg p \rightarrow
\neg \neg p\) from (ii), and applying the conjuncts of (i),
respectively, gives us the contradiction \(\neg p \wedge \neg \neg
p\). The intuitionist anti-realist is forced to admit absurdly that
there are no undecided statements: 



\[\tag{iii}
\neg \exists p (\neg Kp \wedge \neg K \neg p)
\]


The above argument is given by Percival (1990: 185). Since it is
intutionistically acceptable, it is meant to show that the
intuitionist anti-realist is still in trouble. 



In reply, the anti-realist again may utilize Williamson's strategy
jointly to revise logic and reconstruct an expression of the epistemic
truism. Embrace only the intuitionistic consequences of KP (in this
case, that there are no undecided statements), and give expression to
the truism about undecidedness by claiming that not all statements are
decided:



\[\tag{iv}
\neg \forall p(Kp \vee K\neg p).
\]


The reinterpretation of the undecidedness intuition at line (iv) gives
us a claim that is classically, but not intuitionistically, equivalent
to Und. And so, it is only classically, and not intuitionistically,
inconsistent with the inference at line (iii).



Related undecidedness paradoxes of knowability are discussed in Wright
(1987: 311), Williamson (1988: 426) and Brogaard and Salerno (2002:
146-148). The undecidedness paradoxes give the anti-realist even
further reason to revise classical logic in favor of intuitionistic
logic. When accompanied by a reconstrual of our epistemically modest
intuitions, logical space for anti-realism is reclaimed.



What all this suggests is that intuitionist anti-realism is coherent.
But is the approach well motivated? Is either the revision of
classical logic or the clever reconstrual of our epistemic intuitions
*ad hoc*?



The anti-realist's purported right to give up classical logic in favor
of intuitionistic logic has been defended independently. The argument
finds its roots in Dummett (1976, and elsewhere). More recent
interpretations of the anti-realist's argument for logical revision
appear in Wright (1992: Chp. 2), Tennant (1997: Chp. 7), and Salerno
(2000). The details and success or failure of the arguments for
logical revision is a topic for another time. For now it is enough to
point out that the threat of Fitch's paradox is not the anti-realist's
sole motivation for favoring a non-classical logic.



What about the reconstrual of our epistemic intuitions? Is it well
motivated? According to Kvanvig (1995) it is not. Why should we grant
that the intuitionistic treatments of non-omniscience and
undecidedness are better than our initial common-sense treatments? And
how is the anti-realist to explain away the apparent triviality of
these common-sense treatments? These questions have not been
answered.



Moreover, some of the intuitionistic consequences of KP are thought to
be bad enough. Even if 'there are no unknown truths' or
'there are no undecided statements' are intuitionistically
tolerable, the following appears not to be: If \(p\) is unknown then
\(\neg p\). Formally, \(\neg Kp \rightarrow \neg p\). This claim
follows intuitionistically from \(p \rightarrow \neg \neg Kp\), which
we have already established as an intuitionistic consequence of
KP. But \(\neg Kp \rightarrow \neg p\) appears to be false for
empirical discourse. Why should the fact that nobody ever knows \(p\)
be sufficient for the falsity of \(p\)? See Percival (1990) and
Williamson (1988) for further discussions of this and related problems
surrounding the application of intuitionist anti-realism to empirical
discourse. DeVidi and Solomon (2001) disagree. They argue that the
intuitionistic consequences are not unacceptable to one interested in
an epistemic theory of truth--indeed they are central to an
epistemic theory of truth.



For these reasons an appeal to intuitionist logic, by itself, is
generally taken to be unsatisfying in dealing with the paradoxes of
knowability. Exceptions include Burmudez (2009), Dummett (2009),
Rasmussen (2009) and Maffezioli, Naibo & Negri (2013).


### 3.5 Paraconsistent Revision



Another challenge to the logic of Fitch's paradox is mentioned in
Routley (1981) and defended by Beall (2000). The thought is that the
correct logic of knowability is paraconsistent. In a paraconsistent
logic contradictions do not trivialize a theory, because they do not
'explode'. That is, in a paraconsistent logic the
inference from \(p \wedge \neg p\) to an arbitrary
conclusion \(r\) is not valid. Owing to this consideration some
contradictions are permitted and thought to be possible. 



Beall contends that (1) Fitch's proof turns on the assumption that,
for all statements \(p\), the contradiction \(Kp \wedge \neg Kp\) is impossible and (2) we have independent evidence
for thinking \(Kp \wedge \neg Kp\), for some \(p\).
The independent evidence lies in the
 paradox of the knower
 (not to be
mistaken with the paradox of knowability). The relevant version of the
knower paradox may be demonstrated by considering the following
self-referential sentence:



\[\tag{\(k\)}
k \text{ is unknown.}
\]


Assume for the sake of argument that \(k\) is known. Then,
presuming that knowledge entails truth, \(k\) is true. But
\(k\) says that \(k\) is unknown. So \(k\) is unknown.
Consequently, \(k\) is both known and unknown. But then our
assumption (i.e., that \(k\) is known) is false, and provably so.
And, granting that a proven falsehood is known to be false, it follows
that it is known that \(k\) is unknown. That is to say, it is
known that \(k\). But we have already shown that if it is known
that \(k\) then \(k\) is both known and unknown. So it is
proven that \(k\) is both known and unknown. It is provably the
case that the full description of our knowledge includes both
\(K(k)\) and \(\neg K(k)\). That is the
knower paradox.



Beall suggests that the knower gives us some independent evidence for
thinking \(Kp \wedge \neg Kp\), for some \(p\), that
the full description of human knowledge has the interesting feature of
being inconsistent. With a paraconsistent logic, one may accept this
without triviality. And so it is suggested that one go paraconsistent
and embrace \(Kp \wedge \neg Kp\) as a true consequence
of the knowability principle. Beall concludes that Fitch's reasoning,
without a proper reply to the knower, is ineffective against the
knowability principle. For Fitch's reasoning allegedly turns on the
assumption that, for all \(p\), it is impossible that \(Kp \wedge \neg Kp\).



Notice that our presentation of Fitch's reasoning makes no explicit
mention of the assumption that \(Kp \wedge \neg Kp\) is
impossible. So here we attempt to pinpoint exactly where Fitch's
reasoning goes wrong on the above account. It is claimed at line 9 (in
the first section of this entry) that \(K(p \wedge \neg Kp)\) is impossible. Of course \(K(p \wedge \neg Kp)\) entails the contradiction \(Kp \wedge \neg Kp\). And so, if the reasoning is that
\(K(p \wedge \neg Kp)\) is impossible because
contradictions are impossible, then Beall would be directly attacking
the argument presented herein. But notice the argument here is subtly
different. It goes like this. \(K(p \wedge \neg Kp)\) entails the contradiction \(Kp \wedge \neg Kp\). So, by *reductio*, \(K(p \wedge \neg Kp)\) is false. By necessitation, it follows that
\(K(p \wedge \neg Kp)\) is necessarily false.
Depending on the paraconsistent logic, the paraconsistist may object
to use of reductio, or may object to other inferences. The claim that
\(K(p \wedge \neg Kp)\) is impossible (at line
9) is inferred from this claim that \(K(p \wedge \neg Kp)\) is necessarily false. This may trouble the
paraconsistentist. By the lights of one living with contradiction, it
may not follow that an inconsistent statement is impossible even if it
is necessarily false. After all, on this account a necessarilly false
statement may be both false and true at some world, in which case the
statement is both necessarily false and possible. If this is right
then the inference from \(\Box \neg p\) to \(\neg \Diamond p\)
has counterexamples and may not be employed to infer
\(\neg \Diamond K(p \wedge \neg Kp)\) from
\(\Box \neg K(p \wedge \neg Kp)\).



Beall's insights turn on a number of things: (1) the strength of the
independent evidence for true epistemic contradictions, (2) the
adequacy of the proposed resolutions to the knower paradox, (3) the
question of whether Fitch's reasoning is ineffective without a
resolution to the knower and (4) an interpretation for \(\Box\) and
\(\Diamond\) that invalidates the relevant inference (from
\(\Box \neg p\) to \(\neg \Diamond p)\) while remaining true
to the role played by \(\Diamond\) in the knowability principle. We leave
these problems for further debate. But compare with Wansing (2002),
where a paraconsistent constructive relevant modal logic with strong
negation is proposed to block the paradox.



More recent developments of the paraconsistent approach appear in
Beall (2009) and Priest (2009).


## 4. Semantic Restrictions



The remainder of proposals are restriction strategies. They
reinterpret KP by restricting its universal quantifier. In effect, the
restriction strategies invalidate Fitch's reasoning by prohibiting the
substitution-instances of KP that lead to paradox. In this section we
examine semantic reasons for restricting the universal quantifer in
KP.


### 4.1 Situations and Rigid Operations



Edgington (1985) offers a situation-theoretic diagnosis of Fitch's
paradox. She claims that the problem lies with the failure to
distinguish between 'knowing in a situation that
\(p\)' and 'knowing that \(p\) is the case in a
situation'. In the latter case, the situation is (at least in
part) that which the knowledge is about. In the former case, the
situation is that in which the knowledge is had. For instance, I may
know in my actual situation that I would be in pain in a
counterfactual situation where my tooth is pulled. Importantly, the
situation in which the knowledge takes place may be different from the
situation that my knowledge is about. In a situation in which my tooth
is *not* pulled, I may know things that are about a situation
in which my tooth is pulled. 



What are situations? The above example seems to suggest that
situations are worlds. But situations may be less complete than
worlds. That is, they need not have truth-values fixed for statements
that are irrelevant to the context. Consider an example by
Linstom: I may know in a given perceptual situation \(s\)
that John (one of the participants of a card game) has the best hand
and that none of the participants knows this. In this case my
knowledge is of one situation \(s^\*\), the card game, but my
knowledge is acquired in a different situation \(s\), my
perceptual situation. Situation \(s^\*\) is not only determined,
but its relevant information is limited, by the context of the card
game. And \(s\) is fixed and limited by the context of the
perceptual situation. Edgington prefers talk of situations rather than
worlds, because knowledge of non-actual situations, unlike knowledge
of non-actual worlds, does not require knowledge of an infinite amount
of detail.



Making explicit the situation-theoretic distinction between
'knowing in' and 'knowing about', we may
reinterpret the knowability principle: for each statement \(p\)
and situation \(s\), if \(p\) is true in \(s\) then
there is a situation \(s^\*\) in which it is known that \(p\)
is true in \(s\). Edgington requires of knowability the less
general thesis: if \(p\) is true in an actual situation
\(s\) then there is a possible situation \(s^\*\) in which it
is known that \(p\) is true in \(s\). Call this
E-knowability or EKP:



\[\tag{EKP}
\A p \rightarrow \Diamond K\A p,
\]


where A is the actuality operator which may be read 'In some
actual situation', and \(\Diamond\) is the possibility operator to be
read 'In some possible situation'.



As we see, EKP restricts the knowability principle to actual truths,
by saying \(p\) is actually true only if there is a possible
situation in which it is known that \(p\) is actually true. 



The important suggestion is this. Just as there may be actual
knowledge of what is counterfactually the case, there may be
counterfactual knowledge of what is actually the case. In fact, in
light of Fitch's proof, E-knowability requires the existence of such
non-actual knowledge. Let us see why.



Actual truths of the form \(p \wedge \neg Kp\) will have
to be E-knowable. But \(p \wedge \neg Kp\) cannot be
actually known to be actually the case. The reasoning here is exactly
analogous to Fitch's reasoning. 



The lesson is this. Since, for some \(p, p \wedge \neg Kp\) is actually the case, E-knowability commits us to
possible knowledge that \(p \wedge \neg Kp\) is actually
the case. Since this knowledge cannot be actual, E-knowability
requires non-actual knowledge of what is actually the case.
E-knowability then denies the following assumption: given a statement
\(p\), if it is known that \(p\) in \(s\), then in
\(s\) it is known that \(p\). By Edgington's analysis it is
exactly this implicit assumption that leads Fitch's reasoning astray.
The paradox is blocked without it.


### 4.2 Problems for Situations



Since the actuality operator *rigidly* designates actual
situations, the truth-value of statements of the form \(\A p\) will
not vary across possible situations. '\(\A p\)' entails
'in every situation \(\A p\)'. Thus, as Edgington is
aware, if \(\A p\) then it is necessary that \(\A p\). This by
itself poses a problem for EKP. The criticism is that Edgington's
approach is not general enough. Anyone who is likely to endorse the
knowability principle is likely to think that it holds of all truths,
not just those necessary truths involving the actuality operator. EKP
appears to be a very limited thesis failing to specify an epistemic
constraint on contingent truth (Williamson 1987a).



Further criticisms emerge when we attempt to say something informative
about what constitutes non-actual knowledge of what is actually the
case. If there is such non-actual knowledge, there is non-actual
thought about an actual situation. So the non-actual thinker somehow
has a concept of an actual situation. But how is it possible for a
non-actual thinker to have a concept that is specifically about
situations in this the actual world. It will not do for the thinker to
express the thought 'actually \(p\)', since
'actually' will designate rigidly only situations in her
own world. Moreover, since there is no causal link between the actual
world \(w\_1\) and the relevant non-actual world
\(w\_2\), it is unclear how non-actual thought in
\(w\_2\) can be uniquely about \(w\_1\)
(Williamson, 1987a: 257-258). Therefore, it is unclear how there
might be non-actual knowledge about what is actually the case.



Of course actual knowledge about the non-actual is no better at
singling out worlds. The special problem for the non-actual knower is
that the content of her thought must be precisely the content that we
grasp when we consider the truth of \(\A p\). Being in the actual
world we are able to single out this world uniquely. When we consider
the truth of \(\A p\) our context fixes the content of A
specifically. So if it really is \(\A p\) that is knowable by a
non-actual knower, then it must be \(\A p\) that she
grasps--that is, it must be the very same concept that we grasp.
But how this is possible is precisely the problem.



Related and additional criticisms of Edgington's proposal appear in
Wright (1987), Williamson (1987b; 2000b) and Percival (1991). Formal
developments on the proposal, including points that address some of
these concerns appear in Rabinowicz and Segerberg (1994),
Lindstrom (1997), Ruckert (2003), Edgington (2010), Fara
(2010), Proietti and Sandu (2010), and Schloder (forthcoming).


### 4.3 Modal Fallacies and Non-Rigid Statements



Kvanvig (1995) accuses Fitch of a modal fallacy. The fallacy is an
illicit substitution into a modal context. Consider a familiar modal
fallacy. For all persons \(x\), there is a possible world in
which \(x\) is not the inventor of bifocals. (Even Ben Franklin,
the actual inventor of bifocals, might not have invented them.)
Therefore, there is a possible world in which the inventor of bifocals
is not the inventor of bifocals. We can represent the argument
formally. Let our quantifiers range over persons, and let
'\(i\)' be the non-rigid designator 'the
inventor of bifocals'. Consider the argument: 



\[\begin{align\*}
&\forall x \Diamond \neg(x = i) \\
&\quad\text{Therefore,} \\
&\Diamond \neg(i = i)
\end{align\*}\]


Although anyone might not have been the inventor of bifocals, it does
not follow (in fact it is false) that it is possible that the inventor
of bifocals is not identical to the inventor of bifocals. After all,
it is necessary that the inventor of bifocals is the inventor of
bifocals. 



The lesson is that we may not substitute unrestrictedly into modal
contexts. Substitution into modal contexts, we might say, is permitted
only if the substituting terms are rigid designators. In the case of
Fitch's result, our terms are sentences. The knowability principle,
\(\forall p(p \rightarrow \Diamond Kp)\), apparently
allows us to substitute any sentence whatsoever for \(p\). But
notice that our quantifier has wide scope relative to \(\Diamond\). We would
expect that the lessons of quantified modal logic carry over to
quantified propositional modal logic. If so, then we may not
substitute for \(p\) any statement that does not designate
rigidly.



On Kvanvig's diagnosis, the problem with Fitch's reasoning is that
when he substituted the conjunction \(p \wedge \neg Kp\)
for \(p\) in KP (at line 2 of the result), he did not stop to
determine whether \(p \wedge \neg Kp\) is rigid. Kvanvig
maintains that \(p \wedge \neg Kp\) is not rigid. So
Fitch's result is fallacious owing to an illicit substitution into a
modal context. But we may reconstrue \(p \wedge \neg Kp\)
as rigid. And when we do, the paradox evaporates.



Kvanvig proposes that quantified expressions are non-rigid. The reason
he gives is that quantifiers designate different objects in different
possible worlds. "Everyone in Jon's Logic class must take the
final" is about different students in different possible worlds.
Were Sussie to have taken the class, the expression would have been
about her. But she decided not to take the class, so actually it is
not about her. \(Kp\) is an abbreviation for 'it is known
by somebody at some time that \(p\).' So, \(Kp\) is
implicitly quantified. Explicitly it reads
\(\exists x\exists t(Kxpt),\) which says there is
a being \(x\) and a time \(t\) such that \(x\) knows
that \(p\) at \(t\). Accordingly, on this account, the
quantified expression that \(Kp\) abbreviates is non-rigid.
\(\exists x\exists t(Kxpt)\) is about different
beings and times in different modal contexts. For instance, the
expression \(\exists x\exists t(Kxpt)\) is about
actual beings and times. But embedded in a modal context, e.g.,
\(\Diamond \exists x\exists t(Kxpt),\) the expression
is about possible beings and times. It says, 'there is a
possible world in which there is a being \(x\) and time
\(t\) such that \(x\) knows that \(p\) at
\(t\).'



Now consider the relevant instance of Fitch's non-omniscience thesis:
\(p \wedge \neg Kp\). Unabbreviated it reads, \(p \wedge \neg \exists x\exists t(Kxpt),\) which
says \(p\) is true but nobody ever knows that \(p\). The
quantified expression is, on this view, a non-rigid designator.
Uttered in the actual world, it is about actual beings and times. But,
it is argued, embedded within the scope of a possibility operator the
designation varies to be about possible beings and times. When Fitch
substituted the true conjunction, \(p \wedge \neg \exists x\exists t(Kxpt),\) for \(p\)
in the knowability principle, he substituted for \(p\) a
non-rigid designator, thereby altering the reference of the
conjunction and perpetrating a modal fallacy.



Alternatively, Kvanvig suggests, we may characterize \(Kp\)
rigidly to say, 'there is an actual being \(x\) and actual
time \(t\) such that it is known by \(x\) at \(t\) that
\(p\).' Since this expression designates rigidly (i.e., it
makes reference to the actual world regardless of the modal context in
which it appears), it may be substituted for \(p\) in the
knowability principle. The reinterpreted conjunction does not change
its designation when embedded within the scope of \(\Diamond\). Moreover, on
this reading of the conjunction, the paradox dissolves. It is possible
to know that the reinterpreted conjunction is true. There is no
contradiction in supposing that some possible being at some possible
time knows that \(p\) is true but never known by an actual being
at an actual time. The paradox dissolves.



Further discussion of modal fallacies and non-rigid statements appears
in Brogaard and Salerno (2008) and Kennedy (2014).


### 4.4 Problems for Non-Rigidity



Williamson (2000b) defends Fitch's reasoning against Kvanvig's charge.
He suggests that Kvanvig has no grounds for thinking that Fitch's
conjunction \(p \wedge \neg \exists x\exists t(Kxpt)\) does not
designate rigidly. The reason Williamson gives is this. An expression
is non-rigid if, when uttered in a fixed context, it varies its
reference with the circumstances under which it is evaluated. But
Kvanvig gives no convincing reason for thinking that Fitch's
conjunction, as uttered in a fixed context, varies its reference in
this way. At best Kvanvig has shown that the conjunction varies its
reference when uttered in *varying* contexts, since his
argument is that a quantified sentence, when uttered at different
worlds, will be about different objects. To think that this is
sufficient for non-rigidity, Williamson complains, is to confuse
non-rigidity for indexicality. Importantly, indexicality does not
imply non-rigidity. For instance, 'I am tired' is about
me, and continues to be about me when I evaluate its truth-value in
counterfactual circumstances. The sentence might have been false. Had
I gotten enough sleep, I would not be tired. Uttered in a fixed
context, 'I' rigidly designates, even though it is an
indexical. That is, even though, had it been uttered in a different
context by somebody else, it would have been about somebody other than
me. Analogously, even if quantified expressions are indexicals, it
does not follow that they are non-rigid. And so, even if Fitch's
conjunction is an indexical expression, we have not been given a
reason to think that it is non-rigid. If this is correct, then we have
no grounds for thinking that Fitch has committed the modal fallacy in
question.



Kvanvig (2006) replies and develops other interesting themes in the
*Knowability Paradox*, which is the only monograph to date
dedicated to the topic.


## 5. Syntactic Restrictions



The foregoing restriction strategies involved semantic reasons for
limiting universal quantification. In those cases, KP was restricted
in light of considerations about situations, possible worlds, or rigid
designation. Another kind of restriction strategy is syntactic. It
limits the scope of universal quantification to those formulas that
have a certain logical form or stand in a certain provability
relation. Most generally, 



\[
p \rightarrow \Diamond Kp, \text{ where } p \text{ has
logical property } F.
\]


\(F\) should then be a logical property that restricts the
quantifier in some principled way. 


### 5.1 Cartesian Statements



Tennant (1997) focuses on the property of being Cartesian: A statement
\(p\) is Cartesian if and only if \(Kp\) is not provably
inconsistent. Accordingly, he restricts the principle of knowability
to Cartesian statements. Call this restricted knowability principle
T-knowability or TKP: 



\[\tag{TKP}
p \rightarrow \Diamond Kp, \text{ where } p \text{ is Cartesian.}
\]


Notice that T-knowability is free of the paradoxes that we have
discussed. It is free of Fitch's paradox and the related
 undecidedness paradox.
 For both results substitute the problematic Fitch conjunction,
\(p \wedge \neg Kp\), for the variable in \(p \rightarrow \Diamond Kp\), giving us \((p \wedge \neg Kp)
\rightarrow \Diamond K(p \wedge \neg Kp)\). That is,
they require that \(p \wedge \neg Kp\) be knowable if
true (line 2 of Fitch's result). But \(p \wedge \neg Kp\)
is not Cartesian, since \(K(p \wedge \neg Kp)\)
is provably inconsistent (entailing the contradiction at line 6 of
Fitch's result). In effect, TKP offers the most tolerant restriction
needed to prohibit the bothersome substitution. For it only prohibits
substituting those statements for which it is logically impossible to
know. 


### 5.2 Basic Statements



Dummett (2001) agrees that the knowability theorist's error lies in
providing a blanket, rather than a restricted, knowability principle.
And he agrees that the restriction should be syntactic. Dummett
restricts the principle of knowability to "basic"
statements and characterizes truth inductively from there. For
Dummett,



\[\begin{array}{ll}
 p &
 \text{iff } \Diamond Kp, \text{ where } p \text{ is basic.} \\
 p \text{ and } q
 &\text{iff } p \wedge q; \\
 p \text{ or } q
 &\text{iff } p \vee q; \\
 \text{if } p \text{ then } q
 &\text{iff } p \rightarrow q; \\
 \text{it is not the case that } p
 &\text{iff } \neg p; \\
 F[\text{Something}]
 &\text{iff } \exists x F[x]; \\
 F[\text{Everything}]
 &\text{iff } \forall x F[x],
\end{array}\]


where the logical constant on the right-hand side of each
biconditional clause is understood as subject to the laws of
intuitionistic logic.



Dummett's knowability principle or DKP, like Tennant's, is not
threatened by the knowability paradoxes, and for the same reason. It
restricts the class of statements that are subject to knowability. For
Dummett's case, the problematic Fitch conjunction, \(p \wedge \neg Kp\), being compound, and so not basic, cannot replace the
variable in \(p \rightarrow \Diamond Kp\). The paradox is
consequently averted.


### 5.3 Problems for the Syntactic Restrictions



The relative merits of these two restrictions are weighed by Tennant
(2002). Tennant's restriction is the less demanding of the two, since
it bars only the substitution of those statements that are logically
unknowable, and so, only those statements that are responsible for the
paradoxes. Dummett's restriction, by comparison, bars not only the
substitution of these propositions, but also the substitution of
logically complex proposition that are clearly knowable. Tennant also
points out, if the knowability principle is the primary anti-realist
motivation for revising classical logic, restricting that principle to basic
statements can undermine arguments against a classical treatment of
complex statements. 



The main objections to the restriction strategies fall into two camps.
In the first camp we find the charge that a given syntactic
restriction on the knowability principle is not principled. From the
second camp arise formulations of Fitch-like paradoxes that are not
averted by the syntactic restrictions on knowable truth.



From the first camp Hand and Kvanvig (1999) protest that TKP has not
been restricted in a principled manner--in effect, that we have
been given no good reason, other than the threat of paradox, to
restrict the principle to Cartesian statements. (An analogous claim
may be made of Dummett's DKP.) Tennant (2001b) replies to Hand and
Kvanvig with general discussion about the admissibility of
restrictions in the practice of conceptual analysis and philosophical
clarification. By drawing analogies between his own restriction and
others that are clearly admissible, he maintains that the Cartesian
restriction is not *ad-hoc*. He also points out that TKP,
rather than the unrestricted KP, serves as the more interesting point
of contention between the semantic realist and anti-realist. The
realist believes that it is possible for truth to be unknowable in
principle. Fitch's reasoning, at best, shows us that there is
structural unknowability, that is, unknowability that is a function of
logical considerations alone. But is there a more substantial kind of
unknowability, for instance, unknowability that is a function of the
recognition-transcendence of the non-logical subject-matter? A realist
decrying the *ad hoc* nature of TKP (or DKP) fails to engage
the knowability theorist at the heart of the realism debate.



Other complaints that Tennant's restriction strategy is not principled
appear in DeVidi and Kenyon (2003) and Hand (2003). Hand offers a way
of restricting knowability in a principled manner.



These concerns may be waived upon noticing versions of the paradox
that do not violate the proposed restrictions on the knowability
principle. Williamson (2000a) asks us to consider the following
paradox. Let \(p\) be the decidable sentence 'There is a
fragment of Roman pottery at that spot.' Let \(n\) rigidly
designate the number of books actually now on my desk. Let \(E\)
be the predicate 'is even.' Williamson constructs the
conjunction,



\[
p \wedge(Kp \rightarrow En),
\]


and contends that it is Cartesian. Knowing it, apparently, does not
entail a contradiction. If he is right, we can apply to it TKP, giving



1. \((p \wedge(Kp \rightarrow En)) \rightarrow \Diamond K(p \wedge(Kp \rightarrow En))\).



Additionally, if \(p\) is true and \(Kp\) is false, then
Williamson's conjunction is true. So, 


2. \((p \wedge \neg Kp) \rightarrow(p \wedge (Kp \rightarrow En))\).



Lines (1) and (2) yield 


3. \((p \wedge \neg Kp) \rightarrow \Diamond K(p \wedge(Kp \rightarrow En))\).



Accepting the modest epistemic resources found in Fitch's reasoning,
one can prove the following theorem: 


4. \(K(p \wedge(Kp \rightarrow En)) \rightarrow En\).



Here is why. A conjunction is known only if its conjuncts are known.
So, if \(K(p \wedge(Kp \rightarrow En))\), then \(Kp\). And only
truths can be known. So, if \(K(p \wedge(Kp \rightarrow En))\), then
\(Kp \rightarrow En\). Of course, \(Kp\) and \(Kp \rightarrow En\)
jointly entail \(En\). So theorem 4 is valid, if Fitch's
epistemic resources are. Now, 4 is a theorem, and so, holds in all
possible worlds. So its consequent is possible if its antecedent is
possible:


5. \(\Diamond K(p \wedge(Kp \rightarrow En)) \rightarrow \Diamond En\).



From lines 3 and 5 we derive 


6. \((p \wedge \neg Kp) \rightarrow \Diamond En\).



Since \(n\) designates rigidly, it is not contingent whether
\(n\) is even. It follows then that line 6 yields 


7. \((p \wedge \neg Kp) \rightarrow En\).



An analogous argument replacing 'odd' for
'even' gives us 


8. \((p \wedge \neg Kp) \rightarrow \neg En\).



But then we have a contradiction resting on TKP and Fitch's
conjunction, \(p \wedge \neg Kp\). The result involves substitutions
of \(p \wedge(Kp \rightarrow En)\) and \(p \wedge(Kp \rightarrow \neg
En)\) for \(p\) in TKP, but Williamson maintains that neither violate
the Cartesian restriction. Paradox regained.



Tennant (2001a) takes issue with Williamson's claim that \(p
\wedge(Kp \rightarrow En)\) is Cartesian. In the case where \(n\) is
odd, \(En\) expresses a necessary falsehood (for instance, '13
is even'). But then, line 4 tells us that \(K(p \wedge(Kp
\rightarrow En))\) implies something that is necessarily false. And if
the falsity of '13 is even' is a matter
of *logical* necessity, then \(p \wedge(Kp \rightarrow En)\)
cannot be consistently known and therefore is not Cartesian. Hence,
when \(n\) is odd, the first part of Williamson's proof
(involving the predicate 'is even') does in fact violate
the Cartesian restriction. By contrast, Williamson's conjunction
is Cartesian when \(En\) is true. But, analogously, if the truth of
\(En\) is a matter of logical necessity, then \(p \wedge(Kp
\rightarrow \neg En)\) cannot be consistently known and is therefore
not Cartesian. Hence, when \(n\) is even, the second part of
Williamson's proof (involving the predicate 'is
odd') violates the Cartesian restriction. Either way, Tennant
argues, Williamson has not shown that TKP is an inadequate treatment
of Fitch's paradox.



The debate continues in Williamson (2009) and Tennant (2010).



Brogaard and Salerno (2002) develop other Fitch-like paradoxes against
the restriction strategies. Note that Dummett's knowability principle
is a biconditional: \(p \leftrightarrow \Diamond Kp\), where
\(p\) is basic. Tennant (2002) agrees that the knowability
principle should preserve the factive nature of \(\Diamond\)K. So Brogaard
and Salerno begin with the following strengthened knowability
principle:



\[\begin{align}\tag{SKP}
p \leftrightarrow \Diamond Kp,\,&\text{where } p \text{ satisfies the} \\
&\text{relevant syntactic condition}
\end{align}\]


Moreover, pending further discussion of the logic of \(K\), it is
not implausible that the intuitionist knowability theorist wishes to
validate the KK-principle: 



\[\tag{KK}
\Box(Kp \rightarrow KKp).
\]


The principle says, necessarily, if \(p\) is known then it is
known that \(p\) is known. One other resource is used, namely,
the closure principle which says that the antecedent of a necessary
conditional is possible only if the consequent is possible.



If these commitments are granted, one can derive Fitch's result
without violating Tennant's Cartesian restriction:



\[\begin{array}{lll}
1. &
 p \wedge \neg Kp &
 \text{Assumption (Fitch conjunction)} \\
2. &
 Kp \rightarrow KKp &
 \text{from KK} \\
3. &
 p \rightarrow \Diamond Kp &
 \text{from SKP (left-to-right)} \\
4. &
 \Diamond Kp &
 \text{from 1 and 3} \\
5. &
 \Diamond KKp &
 \text{from 4 and 2, by closure} \\
6. &
 \Diamond KKp \rightarrow Kp &
 \text{from SKP (right-to-left)} \\
7. &
 Kp &
 \text{from 5 and 6} \\
8. &
 Kp \wedge \neg Kp &
 \text{from 1 and 7}
\end{array}\]


SKP is applied at lines 3 and 6 to \(p\) and \(Kp\),
respectively. And these substituends do not violate the Cartesian
restriction. Neither \(Kp\), nor \(KKp\), is
self-contradictory. Nonetheless, the anti-realist is forced absurdly
to admit that no truth is unknown. 



Arguably, this result threatens Dummett's restricted knowability
principle as well. But that depends on whether we have applied the
principle to basic statements only. \(p\) is basic, but Dummett's
characterization of truth underdetermines the status of \(Kp\).
Perhaps it is basic, since \(Kp\) is not truth-functionally
complex. Nonetheless, the issue cannot be resolved without a
discussion of \(K\).



Brogaard and Salerno demonstrate other paradoxes against the
restriction strategies. Those further results do not presuppose a
commitment to the KK-principle. They hinge ultimately on the
knowability theorist's interpretation of \(\Diamond\). When \(\Diamond\) is
metaphysical possibility or governed by a logic at least as strong as
S4, the strong knowability principle (appropriately restricted), and
taken as a necessary thesis, entails that there are no unknown truths.
When \(\Diamond\) is epistemic possibility, and the knowability principle is
treated as a necessary thesis that is known, the knowability principle
entails that, necessarily, there are no undecided statements. Unlike
the undecidedness paradoxes of Wright (1987), Williamson (1988), and
Percival (1990), the reasoning provided by Brogaard and Salerno does
not violate Tennant's Cartesian restriction. A response to Brogaard
and Salerno appears in Rosenkranz (2004). Further discussion of the
Cartesian restriction appears in Brogaard and Salerno (2006, 2008).
Tennant (2009) is a further development and defense of the Cartesian
strategy. A defense of the restriction strategy appears in Fischer
(2013).



Much of what has been written on the knowability paradox comes in the
form of attempts to express the relevant form of anti-realism without
paradox. Proposals include Chalmers (2012), Dummett (2009), Edgington
(2010), Fara (2010), Hand (2009, 2010), Jenkins (2005), Kelp and
Pritchard (2009), Linsky (2009), Hudson (2009), Restall (2009),
Tennant (2009), Alexander (2013), Dean & Kurokawa (2010), Proietti
(2016). 



Chalmers (2002, 2012: chap. 2), for instance, defends the idea that
given enough qualitative information about the world we could in
principle know the truth value of any claim. More specifically, his
*scrutability thesis* says, if \(D\) is a complete qualitative
description of the world, then for all \(T\), it is knowable a priori that
\(D\) (materially) implies \(T\). Importantly, the knowability paradox does
not threaten the claim that true Fitch-conjunctions are derivable a
priori from a complete description of the world. 



Dummett takes \(\forall p(p \rightarrow \neg \neg Kp)\) to be the best expression of his brand of
anti-realism and embraces its intuitionistic consequences with open
arms. Edgington defends her knowability principle (viz., if actually \(p\)
then it is possible to know that actually \(p)\) by making the case for
some transworld knowability--specifically, in those cases where
the merely possible knower shares the relevant causal history with the
actual world. Hand defends anti-realism by pointing to the distinction
between a verification-type and its token performances, and argues
that the existence of a verification-type doesn't entail its
performability. The lesson there is that the anti-realist should think
about truth less in terms of the performability of verification
procedures and more in terms of the existence of verification-types.
And Linsky regiments the epistemic principles and reasoning with a
theory of types. The debates surrounding the proper characterization
of semantic anti-realism go far beyond the scope of this entry. As for
the knowability proof itself, there continues to be no consensus on
whether and where it goes wrong. 



Niche discussions of the paradox that don't fit well into any of the
above sections include Salerno's (2018) new paradox of happiness;
Kvanvig's (2010) argument that the paradox threatens Christianity
itself owing to its doctrine of the incarnation of Christ; and
Cresto's (2017) argument that the knowability paradox raises doubts
about Reflection Principle as a requirement on rationality.