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PRI NC IP IA 23(1): 79–86 (2019) doi: 10.5007/1808-1711.2019v23n1p79
Published by NEL — Epistemology and Logic Research Group, Federal Universityof Santa Catarina (UFSC), Brazil.
ANA LYT IC AND SYNTHETIC BASED ON THE PARADOX OF
KNO WABIL ITY
NIC OLA D’ALFONSO
Independent Scholar, ITALY
nicola.dalfonso@hotmail.com
Abstract. The purpose of this paper is to show how the paradox of knowability loses its
paradoxical character when we correctly interpret one of its premises. It is then shown how
this new interpretation can be used to logically define analytical and synthetic truths. In this
way, the paradox of knowability is traced back to the harmless affirmation that, in order to
know every proposition with certainty, there must be no propositions whose truth is synthetic.
Keywords: Paradox of knowability •synthetic truths •analytical truths •principle of the
factivity •knowledge •certain knowledge.
RE CEI VE D: 17/09/2018 RE VIS ED: 15/01/2019 ACCEPTED: 12/03/2019
1. Introduction
In this paper I will first make a brief summary of the paradox of knowability, the
problematic aspects related to it and the way it has been dealt with in the literature.
Then I will show the kind of strategy I have followed to solve it, how I have overcome
its paradoxical aspects and how I have used it to define the synthetic and analytic
truths.
2. The paradox of knowability
The paradox of knowability (Hart 1979) is a theorem from which two important
conclusions can be drawn. The first conclusion states that if every truth is knowable
then every truth is known. The second is that if there are unknown truths, then it is
no longer true that every truth is knowable.
To understand the theorem we must know that K p is a knowledge operator and
stands for “someone knows that p”, with pthat is a proposition.
The theorem also makes use of the following premises.
The principle of the factivity of knowledge according to which if someone knows
that p, then p:
(K-FATT) K p →p
c2019 The author(s). Open access under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License.
80 Nicola D’Alfonso
The principle that knowledge distributes over conjunction according to which if
someone knows that p∧q, then knows that pand knows that q:
(K-DIST) K(p∧q)→K p ∧K q
The modal rule according to which if a proposition pis the result of a demonstra-
tion then it is necessary that p:
(NEC) if ⊢pthen □p
And finally the modal rule according to which if it is necessarily false that pthen
it is not possible that:
(MOE) ¬□p⊢◊¬p
That being said, we must first ask ourselves if it is possible to know that a given
proposition pis true, but no one knows it:
(KPE) ◊K(p∧ ¬K p)
Let us suppose by absurd that this is possible, and therefore that:
(1) K(p∧ ¬K p)assumption
If we do the following steps:
(2) K p ∧K¬K p (K-DIST) and assumption
(3) K p ∧ ¬K p (K-FATT) and (2)
we arrive at a contradiction. This means that our initial assumption was wrong and
therefore that:
(4) ¬K(p∧ ¬K p)discharging assumption
With this established we can do the following steps:
(5) □¬K(p∧ ¬K p)(4) and (NEC)
(6) ¬◊K(p∧ ¬K p)(5) and (MOE)
We have therefore seen that starting from:
◊K(p∧ ¬K p)(KPE)
we arrive at:
¬◊K(p∧ ¬K p)(KPE)
That is a contradiction.
The problem is to prevent such a contradiction from occurring.
PRI NC IP IA 23(1): 79–86 (2019)
Analytic and Synthetic 81
However, if we start from the assumption that we can know every true proposition
(principle of knowability):
(PK) ∀q(q→◊Kq)
the presence of a true proposition that no one knows:
(7) (p∧ ¬K p)
is sufficient to bring us back to that same contradiction:
(8) ◊K(p∧ ¬K p)(PK) and (7)
It follows that if there are true propositions that no one knows, the principle of
knowability can not be valid and therefore there will be true propositions that are
not knowable.
For this reason, if we want the principle of knowability to apply:
(PK) ∀q(q→◊Kq)
we must make sure that there are no true propositions that no one knows. Which
means that every time a proposition is true there must be someone who knows it:
(9) ∀q(q→Kq)
In conclusion, we can write:
∀q(q→◊Kq)⊢ ∀q(q→Kq)
3. Problematic aspects of the paradox of knowability
The first problematic aspect of the paradox of knowability is that if we want the
principle of knowability to apply:
∀q(q→◊Kq)
then every true proposition, besides being knowable, will also be known by someone
at some time:
∀q(q→Kq)
The problem is to understand how the possibility of knowing every true proposi-
tion implies that every true proposition is known at some time.
The second problematic aspect of the paradox of knowability is that if we suppose
there is a true proposition unknown at some time, that is, if we suppose that:
(p∧ ¬K p)
we must renounce the principle of knowability according to which every true propo-
sition is knowable:
PRINCIPIA 23(1): 79–86 (2019)
82 Nicola D’Alfonso
∀q(q→◊Kq)
The problem is to understand how an ignorance that is only contingent can lead
to an unknowability that is instead necessary.
4. Paradox of knowability in literature
There are two possible reactions to the conclusions of the knowability paradox. The
first is to accept them. One way to accept the conclusions of this paradox is to argue
that not everything is knowable because there are propositions that no one knows.
For example, Jenkins (2006) considers it quite obvious that there are unknowable
propositions if we take into consideration the particular propositions that are the
object of the paradox and have the form (p∧ ¬K p).
Among other things, it is possible to support this position without even having
to give up the principle of knowability. Provided we are willing to make some re-
strictions. According to Tennant (1997, pp.272–6), this can be done by limiting the
principle of knowability to only those propositions that, if known, do not generate
contradictions. Since in this way we can exclude precisely the propositions in the
form (p∧ ¬K p), whose knowledge as we have seen leads to a contradiction.
Another way to accept the conclusions of the paradox is to argue that everything
is knowable because all propositions are known by someone.
For example, Plantinga (1982) argues that what makes it possible for everything
to be known by someone is an omniscient being. In these terms, the knowability
paradox would provide evidence in favor of the thesis that such a being is necessary.
The second possible reaction to the conclusions of the knowability paradox is not
to accept them.
One way not to accept the conclusions of this paradox is to argue that a correct
interpretation of the principle of knowability would lead to other conclusions. For
example, Edgington (1985) argues that the possibility of knowing a truth should not
be limited to the present situation. This means that although propositions in the form
(p∧ ¬K p)cannot be known at present, as the paradox of knowability shows, they
could be known in a different situation from the present one.
Another way not to accept the conclusions of the paradox is to argue that the log-
ical steps used to reach them are not legitimate. For example, Wright (2001) argues
that the paradox of knowability provides an argument in favour of intuitionist logic.
For the precise reason that in intuitionist logic the absence of propositions in the form
(p∧ ¬K p)does not imply the presence of propositions in the form: (p→K p ). And
this allows us to consider all the propositions as knowable without all of them being
known.
PRINCIPIA 23(1): 79–86 (2019)
Analytic and Synthetic 83
5. Correct interpretation of the factuality of knowledge
In this paper, the conclusions of the knowability paradox are accepted without taking
any position on the existence of true propositions that no one knows.
All that needs to be done is to correctly interpret the premise of the factivity of
knowledge:
(K-FATT) K p →p
Which, being valid regardless of the content of the proposition p, is in fact placing
a constraint on the knowledge operator K. And when we attribute characteristics to
the knowledge operator K, we are also changing its meaning. Just like in formal
axiomatic systems, where the symbols used require one interpretation rather than
another depending on the axioms used to introduce them.
To affirm, as the principle we are considering here does, that the knowledge of
the proposition p also determines its truth, has a very precise consequence. Namely
that this knowledge must be assumed to be certain. Because only if a proposition is
known with certainty, that knowledge can be sufficient on its own to characterize the
proposition p as true.
This new interpretation of the knowledge operator Kobviously also changes the
meaning of the principle of knowability:
(PK) ∀q(q→◊Kq)
Which should no longer be interpreted as the possibility of knowing each proposi-
tion, but of knowing it with certainty. And it is precisely this new meaning that makes
the previous conclusions of the paradox of knowability harmless, as we are about to
see.
6. Solution of the second problem of the paradox of
knowability
We know that the premise (K-FATT) causes the following expression:
K p ∧K¬K p
to become:
K p ∧ ¬K p
producing a contradiction.
This means that if we consider the above premise valid, we must consider the
term K¬K p not compatible with K p.
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84 Nicola D’Alfonso
However, if we interpret the knowledge operator Kas certain knowledge, this
contradiction allows us to accept the conclusion of the paradox in a harmless way. In
fact, in that case, the expression K¬K p must be interpreted as knowing with certainty
that no one knows the proposition pwith certainty. But only if the proposition p
cannot be known with certainty (denial of the principle of knowability), can we be
certain that no one knows it with certainty. We can then write:
(10) K¬K p → ¬◊K p
7. Solution of the first problem of the paradox of knowability
According to the paradox of knowability, a world in which every proposition can be
known (PK):
∀q(q→◊Kq)
is also a world where everything is known (PK):
∀q(q→Kq)
However, if we interpret the operator of knowledge Kas certain knowledge, even
this conclusion becomes harmless.
For the simple reason that in a world where everything can be known with cer-
tainty there can be no room for something that no one knows with certainty. In fact,
the only way to know with certainty that there is something that no one knows with
certainty is that this something is part of what no one can know with certainty: as
shown to us by (10). But we are hypothesizing precisely a world in which this cannot
happen.
8. Introductions of synthetic and analytic truths
Once we understand that the principle of the factivity of knowledge (K-FATT) must
be referred to certain knowledge, the problem arises of how to introduce an operator
which refers to a different type of knowledge. For example, to a knowledge like the
scientific one, which we remember is never certain knowledge.
This problem is overcome by introducing the following operator of synthetic
knowledge Ks:
(11) Ksp↔K¬K p
according to which to know synthetically the proposition p implies to know with
certainty that no one knows it with certainty.
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Analytic and Synthetic 85
Thanks to (10) we will be able to write:
(12) Ksp→ ¬◊K p
that is, if we know something synthetically, we cannot have a certain knowledge of
it. The consequence of this is to have a synthetic knowledge that is completely equiv-
alent to ordinary beliefs, and also to make every possible distinction between them
depend on the justifications that have led us to consider them to be true. Justifica-
tions that in the case of scientific knowledge will be given by scientific practice, in
the case of religious knowledge by faith, and so on.
Similarly we can define the following operator of analytic knowledge Ka:
(13) Kap↔K K p
according to which to know analytically the proposition p implies to know with cer-
tainty that someone knows it with certainty.
Thanks to (K-FATT) we can write:
(14) K K p →K p
Moreover, if it is true that we are knowing a proposition p with certainty, then
we are also knowing with certainty that we are knowing it with certainty. And this
allows us to write:
(15) K p →K K p
and therefore:
(16) Kap↔K p
In this way we discover that knowing the proposition panalytically and knowing
it with certainty is the same thing.
We can finally note that if we introduce the following operator of inevitable
knowledge Ki:
(17) Kip↔Kap∨Ksp
the corresponding version of the principle of knowability:
(PKI) ∀p(p→◊Kip)
does not fall into the paradoxes of starting. In fact the following expression:
(18) (p∧ ¬Kip)
is always false because the following expression is always false:
(19) ¬Kip
To realize this, we just need to do the following steps from (19) and verify that
they lead to a contradiction:
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86 Nicola D’Alfonso
(20) ¬[Kap∨Ksp](17)
(21) ¬[(K K p)∨(K¬K p)] (20), (11) and (13)
(22) ¬[K p ∨ ¬K p](21) and (K-FATT)
(23) ¬K p ∧K p (22) and De Morgan ∨
9. Conclusion
We have seen that by interpreting the knowledge operator Kas certain knowledge, or
even as analytical knowledge, the paradox of knowability ceases to be a true paradox.
Allowing us to say that if we want to know everything with certainty there must not
be propositions whose truth is synthetic: as shown to us by (12).
References
Edgington, D. 1985. The Paradox of Knowability. Mind 94: 51–2.
Hart, W. D. 1979. The Epistemology of Abstract Objects: Access and Inference. Proceeding of
the Aristotelian Society 53: 153–65.
Jenkins, C. S. 2006. Review of Jonathan Kvanvig, The Knowability Paradox. Mind 11: 1141–7.
Plantinga, A. 1982. How to be an Antirealist. Proceedings of the American Philosophical Asso-
ciation 56: 47–70.
Tennant, N. 1997. The Taming of the True. Oxford: Oxford University Press.
Wright, C. 2001. On Being in a Quandary. Mind 110: 45–98
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