Discovering knowability: a semantic analysis

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SyntheseAn International Journal forEpistemology, Methodology andPhilosophy of Science ISSN 0039-7857Volume 190Number 16 Synthese (2013) 190:3349-3376DOI 10.1007/s11229-012-0168-x

Discovering knowability: a semanticanalysis

Sergei Artemov & Tudor Protopopescu

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Synthese (2013) 190:3349–3376DOI 10.1007/s11229-012-0168-x

Discovering knowability: a semantic analysis

Sergei Artemov · Tudor Protopopescu

Received: 20 October 2010 / Accepted: 31 July 2012 / Published online: 28 August 2012© Springer Science+Business Media B.V. 2012

Abstract In this paper, we provide a semantic analysis of the well-known know-ability paradox stemming from the Church–Fitch observation that the meaningfulknowability principle all truths are knowable, when expressed as a bi-modal princi-ple F → ♦KF , yields an unacceptable omniscience property all truths are known.We offer an alternative semantic proof of this fact independent of the Church–Fitchargument. This shows that the knowability paradox is not intrinsically related to theChurch–Fitch proof, nor to the Moore sentence upon which it relies, but rather to theknowability principle itself. Further, we show that, from a verifiability perspective,the knowability principle fails in the classical logic setting because it is missing theexplicit incorporation of a hidden assumption of stability: ‘the proposition in questiondoes not change from true to false in the process of discovery.’ Once stability is takeninto account, the resulting stable knowability principle and its nuanced versions moreaccurately represent verification-based knowability and do not yield omniscience.

Keywords Knowability paradox · Church–Fitch · Modal logic · Intuitionism

1 Introduction

The Church–Fitch ‘knowability paradox’ reveals the problem with the meaningful(verificationist) principle all truths are knowable, in its commonly accepted bi-modalformulation

F → ♦KF, (VK)

S. Artemov (B) · T. ProtopopescuThe CUNY Graduate Center, 365 Fifth Avenue, rm. 4329, New York City, NY 10016, USAe-mail: [email protected]

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which we call the (verificationist) knowability principle (VK) (cf. Hart and McGinn1976). The well known Church–Fitch proof shows that it yields the omniscience prin-ciple

F → KF, (OMN)

which commits one to the obviously false position that all truths are known.In discussions of the knowability paradox there is little formal analysis of its seman-

tics1. Typically ♦ is read as a grade of alethic or metaphysical possibility, K as knowl-edge or ‘someone at sometime knows’ and ♦KF as ‘there is a state/possible worldsuch that someone at some time knows F .’ In this paper, we attempt a systematic studyof knowability and related notions via the formal possible worlds semantics which isnative for the bi-modal language used in VK.

We present a possible worlds analysis of the knowability principle and offer a newsemantic proof that knowability yields omniscience. Specifically, we show that in anymodel2 of knowability, each state has to be omniscient, i.e., at each state all its truthsare known. Our proof does not use the Church–Fitch argument; this demonstratesthat neither the Church–Fitch proof, nor the Moore sentence upon which it relies, areresponsible for the knowability paradox, but rather the structure of the knowabilityprinciple itself.

We then analyze the notion of knowability from the point of view of an informalsemantics of verifiability. We focus on a reading of the modalities which represents, inan abstract way, a discovery/verification process: � can be read as ‘at all states of thediscovery process’ and ♦ as ‘there is a state of the discovery process,’ ♦KF reads as‘in some state of discovery F becomes known.’ The knowability principle VK, hence,states that

if F is true then in some state of discovery, F becomes known.

Such a reading is, arguably, more in keeping with the spirit of verificationist approachesto truth since it references explicitly the relation between verification, knowledge, andtruth.

How restrictive is this ‘discovery’ reading of knowability? At first sight, it mightlook less general than the generic ‘there is a state/possible world such that someone atsome time knows F .’ However, from the modal logic perspective, such a ‘discovery’reading of knowability is as general as its generic reading. Since any Kripke structurewith two accessibility relations, R� for the alethic modality3 �, and RK for the epi-stemic modality K, can be regarded as an abstract representation of the correspondingdiscovery process, our approach is as general as it can be within the classical possibleworlds semantics.

Such a broad approach is very flexible in its reading of discovery processes as pre-scribed by the Kripke frames. In particular, the process of discovery is not

1 Some exceptions: Costa-Leite (2004, 2006), Negri et al. (2012), Fischer (2010), more frequently there issemantic analysis of proposals to escape the paradox, see Priest (2009), Beall (2009), Restall (2009).2 For the sake of simplicity, we state this fact for finite possible worlds models.3 Subject to generic reflexivity and, sometimes, transitivity conditions.

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deterministic, it can go back and forth indefinitely when the states are mutually acces-sible via R�, or stay at a given state indefinitely4, and information can be revised orlost in the process of discovery, unless specifically postulated otherwise, etc.

A natural question to ask, hence, is whether the knowability principle holds upunder such a ‘discovery’ reading of the bi-modal language and related Kripke struc-tures. We observe that in general the knowability principle F → ♦KF is not validif the proposition F in question can change from true to false during the process ofdiscovery. Indeed, even under a valid discovery process, because a proposition is truein a given actual circumstance it does not follow that its truth will be discovered insome (possibly other) circumstance, unless it is guaranteed that the proposition willnot change from true to false in moving from the first to the second.

These arguments, elaborated in Sect. 3, together with the semantic proof directlylinking VK with omniscience, suggest that despite its superficial plausibility, the stan-dard representation of verificationist knowability as VK is defective in the classicalbi-modal setting. This conclusion does not extend to other settings, e.g., intuitionisticlogic, in which VK could well be an adequate form for the knowability principle.5

However, there is an overwhelming body of evidence, to which we are contributing,that in the classical bi-modal setting the standard formulation of the knowability prin-ciple as VK is not adequate. It appears the proponents of the knowability principle tryto put more semantics into VK than is supported by the classical bi-modal framework.Our further goal will be to make these hidden assumptions, mostly stemming from theinformal constructive reading of the entities involved, explicit, to the extent allowedby the bi-modal language, and to propose classical logic knowability principles incor-porating them.

In the big picture, we do not analyze VK in the setting of constructive logic, whichis arguably the most natural environment for the knowability principle. What we do isto analyze and resolve the tension between VK and classical modal logic, which actu-ally constitutes the Church–Fitch ‘knowability paradox.’ A straightforward attemptto formalize the knowability principle as VK in the classical bi-modal logic fails andneeds to be altered.

One solution (Sect. 4) presents itself as a natural product of our semantical analysis:it is only stable truths, those that remain true in the process of discovery, which can besafely assumed to be verifiable. This yields one proposal for an alternative, non-defec-tive, formalization of the link between truth and verification: the stable knowabilityprinciple (SK),

�F → ♦KF,

if F holds at all states of discovery, then at some state F becomes known. We showthat this principle does not entail omniscience.

4 In particular, we do not limit � and R� to specific temporal modalities, see Sect. 3.5 But see Rasmussen (2009) for a caveat.

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The idea of stability as a condition for knowability is supported by considering theintuitionistic approach to knowability. In an intuitionistic setting, the Church–Fitchproof proves only the intuitionistic knowability principle (IK)

F → ¬¬KF.

Some, most notably Dummett, consider this to be the correct, intuitionistic, formal-ization of the connection between the possibility of knowing and truth. Since this isan intuitionistic principle it is not directly comparable to either stable or verificationistknowability. To make such a comparison, we invoke Gödel’s translation of intuition-istic logic into classical modal logic to find the classical counterpart to intuitionisticknowability.

Gödel’s translation of an intuitionistic formula F consists of prefixing each sub-formula of F with � which represents the constructive element under analysis, e.g.,provability, realizability, existence of a constructive witness, etc. Via this translation,Gödel suggests incorporating a constructive analysis into the classical setting andthereby defining well justified classical counterparts of constructive formulas. Weshow (Sect. 5) that the classical counterpart of the intuitionistic VK is equivalent tothe monotonic knowability principle (MK):

�F → ♦�KF,

if F holds at all states of discovery, then at some state F becomes and remains known.Monotonic knowability implies stable knowability, since both concern stable truths,

but differs from it by further assuming that once a verification is achieved it is alwaysavailable subsequently. This assumption is implicit in the intuitionistic understandingof truth and reflects the intuitionistic view that proofs once produced are never lost.What the principle of monotonic knowability shows is that the stability of truth isimplicit in the intuitionistic, and hence the verificationist, conception of the connec-tion between truth and knowability. Stability turns out to be the manner in which toexpress, in a classical language, the fact that a truth is constructively true. Monotonicknowability, then, is another option for the verificationist, and it too is shown not toentail omniscience.

However, not all truths are stable, and stable knowability and monotonic knowabilitymight be considered too narrow to adequately express the idea that all truths are verifi-able/knowable. To include non-stable truths we consider (Sect. 6) the total knowabilityprinciple (TK):

♦KF ∨ ♦K¬F,

either at some state of discovery F becomes known or at some sate of discovery¬F becomes known. We show that total knowability is strictly stronger than stableknowability, though it too does not entail omniscience.

It is not our purpose to restrict or otherwise modify the knowability principle inorder to avoid the “knowability paradox”. Rather we seek to analyze knowability in

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the classical modal setting in a manner which corresponds to an intuitive verification-based semantics, and is, hence, presumably implicit in what F → ♦KF is supposedto express. In doing so we find meaningful knowability principles which do not entailomniscience. What we offer is the beginning of a bi-modal framework for studiesof knowability, with an accompanying model theory, which opens the door to thesystematic study of the interaction between verification and knowledge.6

Our analysis of these logical principles is mostly semantic. By ‘principle A yieldsprinciple B’ we understand ‘for any model M, if all instances of A hold in M, thenall instances of B hold in M as well.’ This corresponds to reasoning within modalsystems with the usual necessitation rules ‘if F is derivable, then �F and KF are alsoderivable.’ In order to avoid excessive formalism, we take the freedom not to specifymodal assumptions unless it is necessary for the argument.

The main results of this paper were presented as a technical report (Artemov andProtopopescu 2010).

2 Semantical analysis of the knowability paradox

By the well-known Church–Fitch construction (Church 2009; Fitch 1963), assumingonly that knowledge is factive and distributes across conjunction, along with a mini-mal amount of modal and classical logic7, the standard bi-modal formulation VK ofthe knowability principle yields the omniscience principle OMN.

Theorem 1 (Church–Fitch) VK yields OMN.

Proof The idea is to consider the instance of VK with F being the well-known Mooresentence p ∧ ¬K p (which we will call Moore):

Moore → ♦K(Moore). (VK(Moore))

We first establish (1–5) that Moore cannot be known.

(1) K(Moore) → K p: since K(X ∧ Y ) → KX ;(2) K(Moore) → (p ∧ ¬K p) → ¬K p: factivity of knowledge.

Therefore, K(Moore) yields a contradiction K p and ¬K p, hence(3) K(Moore) → ⊥ where ⊥ is the the propositional constant false.Moreover, the Moore sentence cannot possibly be known:(4) ♦K(Moore) → ♦⊥: from 3, by modal reasoning;(5) ¬♦K(Moore): from 4, since ♦⊥ → ⊥.Now we apply these findings to VK(Moore):(6) ¬Moore: from 5 and VK(Moore);(7) p → K p: from 6, since ¬(X ∧ ¬Y ) yields X → Y in classical logic.

The last line is nothing but the omniscience principle OMN. ��

6 See Beall (2009), Priest (2009), Restall (2009), Costa-Leite (2006) for other bi-modal, Rabinowicz andSegerberg (1994), Costa-Leite (2004) for two-dimensional, approaches to the knowability paradox.7 The logic we use is no stronger than that used in the Church–Fitch proof; in most cases, T (sometimesalso called KT) suffices.

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Given this, a stronger result is provable.

Corollary 1 Schemas VK and OMN are equivalent.

Proof It remains to show that OMN yields VK. By OMN, F → KF . Since KF →♦KF for reflexive modality �, F → ♦KF . ��

2.1 The knowability paradox, semantically

We offer an alternative, semantic proof that knowability, VK, is equivalent to theomniscience principle, OMN.

To keep our analysis concise, we consider finite models, which is normally sufficientin the modal setting, since standard modal logics enjoy the finite model property. LetM be a model and u a state. As usual, u � F denotes the fact that F holds (is true)at u. By

u � F

we mean that each instance F ′ of F holds at u, M � F means that for each stateu, u � F .

The following theorem states that at each node that respects the knowabilityprinciple VK the omniscience principle OMN also holds.

Theorem 2 For a finite model M and each state u,

if u � VK, then u � OMN .

Proof Let M = (W, R�, RK, �) be a finite model. Define an absolute indistin-guishability relation ‘∼’ on states from W : u ∼ v iff the same formulas hold at u andv:

for all F, u � F if and only if v � F.

Let [u] denote the set of states absolutely indistinguishable from u. By definitions, ifv ∈ [u] then there is a sentence Fu,v such that u � Fu,v and v � ¬Fu,v . Define Fu as

∧

v ∈[u]Fu,v.

From the construction of Fu , it is clear that

u � Fu and for any v ∈ [u], v � ¬Fu .

Intuitively, formula Fu serves as an indicator of [u], namely, Fu holds at each statefrom [u] and does not hold at any state not from [u].

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Fix u such that u � V K . We claim that for each v,

u RKv yields u ∼ v, (1)

i.e., that states that are epistemically similar to u are absolutely indistinguishable fromu. Indeed, suppose the opposite, i.e., that for some v ∈ W, u RKv but u ∼ v. We claimthat then some instance of VK fails at u, namely,

u � Fu → ♦KFu

which contradicts the assumptions. Indeed, u � Fu , by the definition of Fu . To dem-onstrate that u � ♦KFu , it suffices to show that for any w, w � KFu . Consider threecases:

(1) w = u. Then w � KFu , since u RKv and v � Fu , hence u � KFu ;(2) w ∈ [u]. Then w � KFu since u � KFu and w ∼ u;(3) w ∈ [u]. Then w � KFu since w � Fu .

To complete the proof of the theorem, consider an arbitrary instance of OMN, F →KF , and check that u � F → KF . Indeed, suppose u � F . To establish that u � KF ,consider any v such that u RKv. By (1), u ∼ v, hence v � F as well. ��

Call a state u in a model omniscient if all the truths at u are known at u; a modelis omniscient if all its states are omniscient. Technically, omniscient states are theones for which all epistemically similar states (u RKv) are absolutely indistinguish-able (u ∼ v). Theorem 2 may be regarded as a semantical version of the Church–Fitchtheorem stating that any state that validates knowability, VK, is omniscient.

A natural class of omniscient models is the one in which epistemic similarity isdegenerated into identity, i.e., u RKv iff u = v. In such models, at any state all thetruths are immediately known. Naturally, both VK8 and OMN hold in such models.9

Note that instances of the Moore sentence play no role in Theorem 2 which showsthat the knowability paradox is not intrinsically related to the Church–Fitch proof,nor to the Moore sentence, but rather to the structure of the verificationist knowabilityprinciple VK itself. Our analysis confirms that the Church–Fitch construction is indeedvalid: VK really is equivalent to OMN, and that, strictly speaking, there is no ‘para-dox’ just an unexpected result which shows that VK is not an adequate formalizationof knowability in the classical bi-modal framework.

2.2 Diagnosis

So, what has gone wrong? From the Church–Fitch proof, it appears that Moore is, ifnot the main culprit of the “mystery of the disappearing diamond” (Jenkins 2009),

8 Assuming that R� is reflexive.9 One can easily produce an example of an omniscient model with a non-degenerated epistemic relation:W = {a, b}, R� = RK = W × W , and a, b � p for all atoms p. In this model, states a and b are bothepistemically similar and absolutely indistinguishable.

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then at least a willing accomplice; that it is integral to the “modal collapse” which(Kvanvig 2006, 2009) argues is the heart of the paradox. We showed in Sect. 2.1 thatthis is not so, and that the problem stems from VK itself.

Though not guilty of generating the paradox, the presence of Moore is instructive,indeed an unknowable sentence, like Moore, for knowability principles is like a crashtest for vehicles: it is not a test under normal conditions, but a test of its behaviorduring a disaster. It takes quite a different set of specifications and tests to checkwhether a vehicle does its normal job. The semantical analysis of Sect. 2.1 was the testunder normal conditions, and we saw that the knowability paradox bears no intrinsicconnection to Moore.

Let us, then, examine the results of the crash test to see how exactly Moore revealsthe structural weakness in VK. In the Church–Fitch proof, VK has been applied to theMoore sentence which cannot be known. Indeed, K(Moore) is inconsistent in anylogic of knowledge10: K(Moore) yields both K p and ¬K p,11 so ¬K(Moore). Thisobservation allows us to see where the collapse occurs. We have seen (Theorem 1, line4) that K(Moore) → ⊥. Since ⊥ implies anything,

K(Moore) ↔ ⊥.

This is the moment when “the diamond disappears.” In any normal modal logic, “thediamond disappears” from the constant false:

⊥ ↔ ♦⊥,

and this is exactly what happens here:

K(Moore) ↔ ♦K(Moore)

as well. Therefore, VK(Moore) of the form

Moore → ♦K(Moore)

is logically equivalent to

Moore → K(Moore),

which is equivalent to

Moore → ⊥,

i.e.,¬Moore.

10 In this respect our analysis agrees with Tennant (1997, 2009), see Sect. 7.2 below. See also Williamson(1994) for a similar argument.11 See Theorem 1.

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It is mentioned frequently that Church–Fitch result holds not just for K but forany factive and conjunction-distributive operator (as Fitch himself makes clear), e.g.,Fara (2009), Kvanvig (2006, 2009), Tennant (1997), Williamson (2000a). The “modalcollapse” on ⊥ is a well-known phenomenon and hence this explanation holds for anyother factive and distributive operator, O, applied to its un-O-able and un-O-ed Mooresentence p ∧ ¬Op.

3 Verificationist knowability is not intuitively valid

We turn now to consider the validity of VK , and begin to uncover what is wrong withit. We argue that the basic defect in VK is that it is missing the specification that thetruths of which it is supposed to hold are stable, i.e. that they stay true during thediscovery process.

Consider a scenario in which the classical version of VK appears to fail.

Consider a correct, i.e., knowledge- and knowability-producing discovery pro-cedure, Ver, which begins to be applied in a state where a proposition F istrue. However, during (or, perhaps, due to) the process of discovery, F changesits truth value and Ver eventually certifies that F is false. In such a scenarioVK will not hold despite the fact that a correct verification procedure has beenapplied to a true proposition and terminates with a definitive answer.

Note, that in the constructive/intuitionistic framework, such a scenario does notseem to be acceptable since intuitionistic truths are generally assumed to remain truethrough the process of discovery. However, in the classical framework, the aforemen-tioned scenario is not ruled out by the current formulation of VK.

More formally, call a proposition F stable in a given model, if at each state u,

u � F → �F.

Note that for a reflexive modality �, F is stable if and only if F ↔ �F holds every-where in the model. A stable sentence can be false at some (or even all) states of agiven model, but once it is true at a state, it remains true at all �-accessible states, andhence is available for verification. There are sentences, e.g., propositional constants (true) and ⊥ (false) which are stable in any model. Sentences �F are stable in anytransitive model.

To make this precise, consider the following bi-modal model, M1 = (W, R�, RK,

�). M1 has three states W = {u, v, w} and two accessibility relations, a verificationaccessibility relation R� (represented by arrows) which is reflexive and according towhich w is accessible from u and v, and an epistemic accessibility relation RK (repre-sented by ovals) which is an equivalence relation on W such that u and v are equivalent(epistemically indistinguishable) and w is equivalent only to itself. We assume thatu � F , and v, w � ¬F (Fig. 1).

A requirement for the truth of KF is just that F be true at all epistemically, RK,indistinguishable states. As R� represents the process of discovery, carrying out averification procedure may take us from one state to another, e.g. from u to w in M1,

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Fig. 1 Model M1 where VK fails

but this does not necessarily represent a forward movement in time. Accordingly R�should not be read as a temporal relation. For instance in the models M2 and M3(see Theorems 7 and 11 respectively, below), the discovery process moves back andforth between mutually �-accessible states. Or consider models where each state is areflexive point with respect to R�, so verification only takes that state into account.These do not represent movement in time but stages of the verification process. Accord-ingly each state should be thought of as a state of information during a process ofdiscovery or investigation. At some states propositions are true, at others false, atsome they are known, at others not, depending on which other states are epistemicallyaccessible. Unless we stipulate otherwise there is no reason why a proposition which istrue at a state cannot become false at a �-accessible state. Unlike intuitionistic Kripkemodels (a well known class of models where states are seen as states of information,and R as a process of verification) in a bi-modal classical setting there is no restrictionon information being revised/lost, or from the discovery process ‘going backward’.

We can think of M1 as modeling the process of verification, moving from igno-rance to knowledge by shrinking one’s epistemic possibilities.12 As model M1 shows,F is not stable: though it is true at u, it is not true at w which is �-accessible fromu. At u and v, Ver cannot come to a definitive conclusion concerning F : u and v areepistemically indistinguishable so F is not known at either of them. At w, Ver canonly conclude that F is false, since w � ¬F . In M1, ¬KF holds at all words andhence u � ¬♦KF, accordingly

u � F → ♦KF.

Despite the assumption that the agent possesses a correct verification procedure Verin all of the states, and so has the means to verify, come to know, that a truth holds, theverification procedure cannot verify F . At the initial state, Ver does not have enoughinformation to rule out as impossible some states in which ¬F . By proceeding with thediscovery process Ver gathers new information which helps Ver to shrink epistemicpossibilities, but all this new information yields ¬F .

Is M1 a legitimate scenario in which to consider the validity of VK? From theperspective of our discovery reading of � and ♦ there is nothing that forbids us toconsider a proposition F that holds at the initial state but is not known there yet, andis false at all other states. There is nothing a priori illegitimate in the idea that some

12 We can also consider M′1, which is S5 with respect to �, by setting R� = W 2 in M1.

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propositions can change their truth value, and that this change might occur during theverification process.

Speaking generally a discovery scenario modelled by the frame underlying M1may be regarded as a schematic description of any situation where the truth-value ofa proposition is not known, one then engages in investigation (a process of discov-ery/verification) and finally comes to a state where the process comes to a definitiveanswer. What is defective with VK is that it is not valid in any such scenario where theproposition we are interested in changes its truth value, that is unless the propositionin question is stable.13

As an illustration consider the following scenario. Say the actual state is u and Freads as ‘it rains’. At u Holmes asks Watson whether it is raining. As a matter of factat u it is raining but Watson does not know it. It would seem obvious to say that at u ‘itrains’ is something Watson can know. Suppose that Watson’s process of discovery islooking outside; at u Watson’s procedure is not able to yield a definitive answer. Nowsuppose also that by the time he looks outside the rain has stopped; that is by the timethe discovery process reaches the state w where it is able to yield a definitive answer itcertifies that F is false. Even though looking outside is a perfectly legitimate (thoughobviously not the only) discovery process, one in virtue of which it should be possiblefor Watson to know that it is raining, if it is, it is not in fact possible for Watson toknow that it is raining. When he begins his investigation he does not know it, thoughit is true, and by the time he has enough evidence to constitute knowledge it is false,and so he does not know it. Since there is no state at which he knows it is raining itis not possible for him to know it. Though true, that it is raining is not verifiable bythe procedure at hand. If we had supposed instead that F does not change its truthvalue from u to w there would be a state where Watson knows it is raining, and henceit is possible for him to know it, if it is true, i.e. at u. But this is only because of thesupposition that F is stable at u; VK holds in such scenarios only if F is stable.

It might be objected that the reason Watson cannot know it is raining, if it is, is dueto the weakness of his method of discovery. It does not yield an answer immediately,and so of course things can change. If he had a more powerful, more immediate, pro-cedure that it is raining is knowable. Let us suppose, then, that Watson has access to anoracle which can tell him everything which is true at his current state; it follows thatfor anyone with access to such an oracle every truth is knowable. Instead of lookingoutside Watson consults the oracle which tells him F: it was raining when Holmesasked. As a result of this process Watson has learned something new, F—but F didnot change its truth value during the process of discovery. In the natural Kripke modelfor this scenario, the discovery process starts in the real state u which has an epistemi-cally accessible v in which F is false. Hence F is true but unknown at u. In consultingthe oracle Watson’s process of discovery immediately takes him to the final state w,

13 It has been noted that non-stability, in particular empirical contingency, is a thorn in the side of ver-ificationists, see Percival (1990), Rasmussen (2009), Williamson (1994), Wright (1983). Verificationisttheories of meaning, replacing truth with warranted assertibility, seem not to be able to handle satisfactorilythe semantics of sentences whose warranted assertibilty changes between states. The lack of an explicitassumption of stability suggests an explanation of these problems.

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in which F is already known.14 Indeed supposing Watson to have access to such apowerful discovery method does make it knowable that it is raining, if it is, but whathe learns by such a method is stable! Note that what he learns is not ‘it rains’ but, ineffect, ‘it rains in state u (when Holmes asked)’. But this cannot change in movingfrom u to w.

Theorem 2 and model M1 suggest that VK is a defective formalization of the infor-mal idea that all truths are knowable. It does not seem open to the verificationist to saythat scenarios like the one modeled by the frame of M1 are not legitimate. Scenarioswith this structure are schematic descriptions of any kind of inquiry. To give them upwould seem to be to give up on the idea that we can gain knowledge through research,and to endorse a picture of knowledge that really has no room for ignorance, where ateach stage of investigation one knows everything which is true there.15

4 Stable knowability

Our semantic analysis showed that the non-stability of F invalidates the principle ofverificationist knowability. Without the stability assumption stated explicitly, the ver-ificationist knowability principle does not represent verificationist approaches fairly.In this section, we show that with the stability assumption made explicit, no paradoxresults.

Clearly, not all truths are stable, so to distinguish between the knowability of stableand non-stable truths, we need to explicitly incorporate stability into a knowabil-ity principle. Since the possibility of knowing stable versus non-stable truths capturessome of the distinction between the possibility of knowing non-contingent versus con-tingent truths, this would go some way to distinguishing between, e.g., mathematicaland empirical verifiability.

How might one represent the knowability of stable truths? Consider the principleall stable truths are knowable, e.g., if F is a stable truth (F → �F and F), itholds at all states of verification, then F is knowable, at some state verification yieldsknowledge (♦KF):

[(F → �F) ∧ F] → ♦KF. (2)

Since for reflexive modality �, the formula (F → �F) ∧ F is logically equiva-lent to �F , the principle (2) can be equivalently presented as �F → ♦KF . Theseconsiderations prompt the following definition of the principle of stable knowability

�F → ♦KF . (SK)

Let SK(F), VK(F), etc., denote principles SK, VK, etc., for a given proposition F .Note that one could envision a seemingly stronger version of SK, e.g., the principle SK ′:

14 That is model M1 with the arrow from v to w deleted and F true at w.15 The intuitionist’s ideal mathematician constructing the mathematical universe, would be a paradigmexample of such a knower; indeed such a picture expresses a very hardcore constructivism—the constructedtruths are all the truths there are. On such a view it is a truism that one knows all truths, and supposingthere is an unknown truth contradicts this notion of truth, of which Theorem 1 is just a straightforwardjustification.

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�F → ♦K�F.

However, SK ′ as a schema has the same strength as SK. Indeed, SK(�F) = ��F →♦K�F which is logically equivalent to SK ′(F).16

We can also consider the principle of eventual stable knowability (ESK),

♦�F → ♦KF.

Informally ESK expresses that the knowability of F can be concluded from the assump-tion that there is a state after which F holds everywhere. It can be shown that �(SK)

implies ESK which implies SK. Hence principle ESK is equivalent to SK as well.17

A stronger principle of monotonic knowability (MK), appears as the result of theGödel translation of intuitionistic knowability into the classical bi-modal language,which we discuss in Sect. 5:

�F → ♦�KF.

The justification of MK requires some additional assumptions concerning the relationbetween the verification modality � and the epistemic modality K and this restrictsthe domain of situations in which such knowability holds.

The semantical analysis in Sect. 2.1 sheds some light on how stable knowability SKescapes the knowability paradox: it allows non-omniscient (and meaningful) modelsand hence does not yield the omniscience principle OMN.

Theorem 3 For any model M with the frame in Fig. 218, M � SK.

Fig. 2 Non-omniscient frame for SK

Proof Indeed, if �X holds at some state in M, then w � X as well, hence w � KX .Since w is �-accessible from each state, ♦KX holds at each state.19 ��

16 For transitive �’s.17 ESK might be considered a rendering of specifically constructive knowability. In a constructive setting,a state after which a proposition holds everywhere can be regarded as a state where the proposition has aproof, in which case ESK can be read as saying that if it is possible that F has a proof, then F is knowable.18 e.g., model M1.19 This argument prompts a general characterization of SK-models: every state has a �-accessible omni-scient state.

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We can now show that stable knowability does not suffer from the knowabilityparadox, in particular, SK does not yield omniscience.

Corollary 2 Schema SK does not yield OMN.

Proof By Theorem 3, all instances of SK hold in M1, but OMN(F) does not. HenceSK does not yield omniscience. ��

5 Intuitionistic versus stable knowability

Given its intuitionistic inspiration, the natural logical perspective to have on VK isintuitionistic. Indeed one of the first reactions to the knowability paradox, Williamson(1982), showed that in intuitionistic logic the Church–Fitch construction yielded onlyp → ¬¬K p, which, when read from the intuitionistic view point, is not absurd.Indeed some, (De Vidi and Solomon 2001; Dummett 2009; Rasmussen 2009),20 mostnotably Dummett, argue that for the verificationist it is superior to VK as a repre-sentation of their sense of knowability. Accordingly, let us try to read verificationistknowability (VK) as an intuitionistic principle.

Even a brief perusal of the literature on intuitionistic modal logic shows there is nowidely accepted interpretation of intuitionistic modality, and many logics and fami-lies of logics have been constructed (see for instance Gabbay et al. 1998; Wolter andZakharyaschev 1997, 1999a,b for some overviews). Some systems consider indepen-dent ♦ or �, and some do not. Since Kripke semantics shows that intuitionistic logicitself can be regarded as a fragment of the classical modal logic S4 with the modal-ity �, introducing a new intuitionistic modality ♦ adds more modality than seemsto be needed to express what the verificationist wants to say (Rasmussen in 2009makes this a central point of his Mapping Objection). For a discussion of intuitionis-tic modality bearing on the knowability paradox, and verificationism more generally,see Williamson (1992); see also De Vidi and Solomon (1997), Ewald (1986), Prior(1957). To be precise, by intuitionistic modal logic in this paper we understand stan-dard modal systems containing only a reflexive � and intuitionistic logic. The ♦ isviewed merely as an abbreviation for ¬�¬X ; we do not, in general, endorse, theduality of intuitionistic ♦ and �.

Theorem 4 With intuitionistic logic, principles VK and p → ¬¬K p are equivalent.

Proof We follow the first six steps of the proof of Theorem 1 and verify that each ofthem is made according to intuitionistic logical rules (we repeat these steps here forthe reader’s convenience):

20 Both Dummett and Rasmussen (via Dummett), attribute this view to Bernhard Weiss, who appearsnot to have published this. Incidentally, Dummett’s informal reading of IK is “the possibility that F willcome to be known always remains open,” which fits nicely with the frame conditions characterizing SK, seefootnote 19. See Percival (1990) for objections to IK on the grounds that it yields that no truths are unde-cided (see also Brogaard and Salerno forthcoming; Wright 1983 for related discussions), and see Brogaardand Salerno (forthcoming), De Vidi and Solomon (2001), Williamson (1988) for responses on behalf ofthe intuitionistically inclined. It is not our purpose to defend IK, but to find its proper relation to otherknowability principles.

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(1) K(Moore) → K p;(2) K(Moore) → (p ∧ ¬K p) → ¬K p;(3) K(Moore) → ⊥;(4) ♦K(Moore) → ♦⊥ → ⊥;(5) ¬♦K(Moore);(6) ¬Moore.

Since intuitionistically ¬(X ∧ Y ) yields X → ¬Y , we conclude

(7) p → ¬¬K p.

Here is the proof of the other direction.

(1) p → ¬¬K p;(2) �¬K p → ¬K p: by reflexivity of �;(3) ¬¬K p → ¬�¬K p: since X → Y yields ¬Y → ¬X ;(4) p → ¬�¬K p: from 1 and 3, by syllogism;(5) p → ♦K p: where ♦ is an abbreviation for ¬�¬. ��

The aforementioned arguments and endorsements justify the following definition.By intuitionistic knowability we mean the following principle within intuitionisticlogic with epistemic modality K:

F → ¬¬KF. (IK)

In the 1930s Gödel (1986) found a faithful embedding of intuitionistic logic into modallogic with the S4-style modality: one translates intuitionistic formulas by means of therule box every sub-formula.21 By g(F) we denote the Gödel translation of formula F .The Gödel translation provides a complete characterization of intuitionistic validity:a formula F is intuitionistically valid if and only if its translation g(F) is valid in S4.

Gödel’s motivation resulted from the provability reading of the � modality, henceboxing a formula G forces a constructive reading of it as G is provable rather than clas-sically as G is true. The Kripke semantics for intuitionistic and modal logics revealedthat on the semantic level, the Gödel translation specifies intuitionistic logic as a frag-ment of the classical modal logic S4 satisfying the stability condition: what is trueremains true (see Sect. 5.1 for more discussion on stability and constructive seman-tics). Via the Gödel embedding we see that stability is a faithful modal reincarnationof provability. Indeed, once a proof of F becomes available at a state u, in all furtherstates F holds true. Conversely, if there is no proof of F at u, then a state at which Ffails is consistent with u and hence, in principle, possible at u. Stability is provabilityexpressed in a modal language.22

Note that the Gödel translation of IK(p) is

g(IK(p)) = �(�p → �¬�¬�K�p) = �(�p → �♦�K�p).

21 Gödel in 1986 offered two translations, each of which is essentially equivalent in S4 to the rule boxevery sub-formula.22 In the Logic of Proofs which combines the relational and provability readings of intuitionistic logic,this argument is captured by the Realization Theorem and the notion of fully explanatory models (Artemov2001; Fitting 2005).

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As a schema, g(IK) is equivalent to the following principle of monotonic knowability:

�F → ♦�KF. (MK)

Theorem 5 Principles g(IK) and MK are equivalent.

Proof It is an easy exercise in modal logic to show that for a reflexive �,

g(IK)(F) → MK(F)

which yields that as a schema, g(IK) implies MK.To show the converse, assume MK and consider MK(�F) = ��F → ♦�K�F .Then

(1) �F → ♦�K�F : by transitivity;(2) �(�F → ♦�K�F): by necessitation;(3) ��F → �♦�K�F : by distribution;(4) �F → �♦�K�F : by transitivity;(5) �(�F → �♦�K�F) which is nothing but g(IK)(F): by necessitation. ��By the following theorem it follows that MK does not yield omniscience.

Theorem 6 Principle MK is strictly weaker than VK.

Proof VK implies MK . By Theorem 1, VK yields F → KF . By necessitation anddistributivity, �F → �KF . By reflexivity, �KF → ♦�KF , hence �F → ♦�KF .

To see the converse does not hold consider that each instance of MK holds inmodel M1. Indeed, if �X holds anywhere in M1 then w � X , hence w � KX , andw � �KX . Therefore, ♦�KX holds at each node of M1. As before, VK(F) fails inM1, in particular, u � F and u � ♦KF . ��

We will now show that MK is stronger than SK and reveal the additional assumptionswhich, given SK, should be made to justify MK.

Theorem 7 Principle MK is strictly stronger than SK.

Proof It is easy to see that MK logically implies SK for a reflexive �. Let us show ina model that SK as a schema does not yield MK. Consider model M2 in Fig. 3.

Fig. 3 Model M2 where MK fails

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It is immediate that any instance of SK holds in M2. Indeed, if �X holds at somenode, then w � X , hence w � KX and ♦KX holds at each node.

Let us show that MK fails in M2. Indeed, u � �F . On the other hand, u � KF ,hence �KF fails at both u and w. Therefore, ♦�KF fails at both u and w since thesenodes are only �-accessible from each other. Hence �F → ♦�KF fails at u. ��

Apparently, schema g(IK) (i.e., MK) incorporates, along with stability, some otherspecifically intuitionistic assumptions concerning � and K. Stable knowability (SK)states that given a stable truth F , there is a possible state (moment of the discoveryprocess) in which the verification procedure confirms F . Intuitionistic knowabilityg(IK)/MK seemingly states the same: there is a possible state in which the verifica-tion procedure confirms F . However, in an intuitionistic universe, once KF becomestrue, it stays true, i.e., �KF holds.23 This additional assumption appears because ofthe limited expressive power of the intuitionistic language and semantics, where thisassumption is implicit in the semantics of �. The classical modal language is moreflexible and is able to express both SK and g(IK)/MK.

In this respect, model M2 is not an intuitionistic universe: KF holds at w, but doesnot stay true there, i.e., w � �KF .24

Analogous to the Gödel embedding, the explicit assumption of stability via SK hasthe effect of making explicit some of the constructive meaning of VK in the classi-cal language. VK is supposed to express a constructive notion of truth in a classicallanguage, but it is the constructive nature of the truth of F which makes it knowable.Stability expresses in a classical language that we are talking about constructive truths;without this VK says, in effect,

if F is classically true, then it is (constructively) knowable.

In 2009 Rasmussen calls VK “an amphibious hybrid between the points of view ofrealism and of anti-realism”; we see just how apt a description this is.

What we observe here is the remarkably robust character of stable knowabilityand its ability to represent the constructive intent of verificationist knowability and itscompatibility with intuitionistic knowability.25

5.1 Stability versus constructive semantics

Now we want to address the issue of whether our analysis of the role of the stabil-ity requirement gives us a deeper understanding of the verificationist conception ofknowability.

23 See Proietti (2001) for a reading of K in an intuitionistic setting.24 As an illustration of M2 consider the following scenario. Holmes is on a case; his discovery procedureis interviewing suspects. At the real world u, F holds but is not known, since Holmes regards v a possibil-ity. Interviewing a suspect takes Holmes to w where he already knows F as the testimony eliminates thepossibility that not F . However, if Holmes continues investigation beyond this point, it is conceivable thathe will shuttle back and forth between w and u, further interviews may re-introduce ¬F as an epistemicpossibility. If the real world is v, then F does not hold. However, the discovery process takes Holmes to w

in which F becomes both true and known.25 See Fischer (2010) for an approach to the knowability paradox taking stability into account.

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The history of studies of constructive, e.g., intuitionistic, logic has two distinguishedtraditions. First, there is the constructive ‘witness’ semantics, which originated in Brou-wer’s works and manifested itself in the Brouwer-Heyting-Kolmogorov semantics inwhich witnesses are viewed as proofs. Within this tradition, F is true is understood as

there is a proof of F,

i.e., via the informal existential quantifier over proofs.The second tradition can be traced back to intuitionistic Kripke semantics, accord-

ing to which F is true means

F holds in all possible situations,

i.e., via the informal universal quantifier over possible states. Intuitionistic truth inthis second ‘universal’ setting is stable: if F is true at u, F stays true at all other statesaccessible from u.

Reconciling these two traditions in a comprehensive formal model has been a long-time challenge in the area of constructive semantics. The first steps were made byGödel’s embedding discussed above. Later Gödel (1995) sketched a way to assignproof-like objects to each occurrence of modality in S4; this project of Gödel’s wascompleted in the Logic of Proofs (Artemov 2001) which connected the ‘existential’and ‘universal’ intuitionistic semantics: a formula F is true in the monotone ‘universal’semantics if and only if F is true in the ‘existential’ semantics of proofs.

Our analysis suggests that similar developments can occur in the study of verifi-cationist knowability. The standard verificationist justification of VK is based on the‘existential’ witness semantics of constructive truth. What we offer in our analysis isa Kripke-style ‘universal’ semantics of knowability with the core stability conditionleading to SK and rely on the intrinsic connection of stability and provability to connectit informally to the ‘existential’ semantics.

We wish to think that principles incorporating stability provide alternative formal-izations to VK because they incorporate a key component of constructive truth whichis missing in VK. In this respect the problem of finding principles which adequatelyexpress the verificationist view of knowability have plausible candidates in SK and MK.The natural next step would be to capture the ‘existential’ witness side of knowabilityand we hope that SK could play a role there too.

Despite the seemingly different character of our approach to the traditional ver-ificationist justification of the knowability principle, the aforementioned history ofreconciling the two semantic traditions for intuitionistic logic gives us a certain hopethat a similar reconciliation over stable knowability principles is possible as well. Wehope that this will become the subject of future studies.26

26 One may try some other format of representing this notion of knowability. Such a possibility is investi-gated by Dean and Kurokawa (2010), who consider the principle ‘if F is true, then there exists a proof ofF’ instead of VK and use the framework of the Quantified Logic of Proofs (see Artemov 2001, 2008 andFitting 2008).

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6 A bigger picture of knowability

From a general point of view, knowability might be thought of as a generalization ofdecidability.27 To say ‘F is knowable’ is to say that one knows of a procedure which,if carried out in an appropriate situation, would yield a decision on the truth of F. Sothe knowability of F would amount to the possibility of knowing F or the possibilityof knowing ¬F. Accordingly, the formalization of the claim that F is knowable in thissense is the principle of total knowability:

♦KF ∨ ♦K¬F. (TK)

The principle of total knowability asserts that, for any proposition, there is some statein which verification yields a definitive answer as to whether F or ¬F . From thispoint of view TK is a meaningful principle of possible knowledge. We show that TKis strictly stronger than stable knowability, but TK also does not entail omniscience.

How should we understand TK? TK represents a distinct conception of knowability,not compatible with the usual verificationist position. Hence, unlike Melia (1991), wedo not argue that TK should be viewed as a re-interpretation of the informal contentof VK, which preserves the verificationist’s constructive motivations but does not suc-cumb to the knowability paradox. We will show that the latter point is indeed correct,but the former should be resisted.

The lack of compatibility between TK and verificationist motivations has beenargued in Rückert (2004), Salerno (1999), Williamson (2000a), Wright (2001). Wil-liamson and Rückert point out that TK is consistent with there being truths whichare unknowable (we prove this in Theorem 8), which means it is inconsistent withthe verificationist’s manifestation requirement; the requirement that the understand-ing of a proposition be manifested in an (in principle) ability to verify it if true.Indeed, according to Dummett, denying TK is a necessary condition for even statingthe verificationist’s position28 vis a vis the realist. The tension between verificationism(as expressed by VK) and TK is further brought out by Wright and Salerno; accordingto them it is evident that we do not know ourselves to possess the means for decidingevery proposition. Hence the verificationist is committed to the possibility that TKis false alongside also holding that VK is true. Accordingly the verificationist cannotaccept TK as a better expression of the intent behind VK.29

We have no reason to disagree with any of this: TK represents a broadly constructiveposition distinct from verificationism, and which applies to all types of propositions,not just stable ones. We prove formally that TK does not succumb to the knowabilityparadox.

27 A classic statement “a closed formula A is (formally) decidable, if A is either provable or refutable, i.e.if either � A or � ¬A” (Kleene 1952).28 Salerno (1999) makes this point.29 In fact VK implies TK, given excluded middle, (see Theorem 9), hence the possibility that TK is falseyields the possibility that verificationism, VK, is false. The resolution of this problem, according to theverificationist, is to reject excluded middle, see Salerno (1999) for a more careful statement of this line ofthought.

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What is the relation between TK and VK?

Theorem 8 Principle TK does not yield VK.

Proof Consider again model M1. First, we note that since w forms a singleton withrespect to RK, for each formula X , either w � KX or w � K¬X . Since w is accessiblefrom each node, ♦KX ∨ ♦K¬X holds at each node. Therefore all instances of TKhold in M1. However, M1 � VK as it was noticed in Sect. 3. ��

On the other hand the converse does hold.

Theorem 9 Principle VK yields TK.30

Proof (1) F → ♦KF : an instance of VK;(2) ¬F → ♦K¬F : an instance of VK;(3) F ∨ ¬F : excluded middle;(4) ♦KF ∨ ♦K¬F : from 1 to 3, by propositional reasoning. ��

What is the relation between TK and SK?

Theorem 10 Principle TK yields SK.

Proof More specifically, we establish that TK(F) yields SK(F).

(1) K¬F → ¬F : factivity of knowledge;(2) F → ¬K¬F : contrapositive of 1 and double negation principle;(3) �F → �¬K¬F : from 2, by �-necessitation, distribution;(4) �F → ¬♦K¬F : from 3, converting �¬ into ¬♦;(5) ¬♦K¬F → ♦KF : conversion of X ∨ Y into ¬Y → X in TK(F);(6) �F → ♦KF : from 4, 5, by propositional reasoning. The latter is SK(F). ��The converse however does not hold.

Theorem 11 Principle SK does not yield TK.

Proof Consider model M3 in Fig. 4.

Fig. 4 Model M3 where SK holds but TK fails

It is easy to see that in M3 all instances of SK hold. Indeed, for any propositionX , if �X holds at some node, then X holds in the whole of M3, hence both KX and♦KX hold in M3. Therefore �X → ♦KX holds in M3. It now suffices to showthat TK(F) fails in M3. Indeed, u � KF and u � K¬F and hence u � ♦KF andu � ♦K¬F , so u � TK(F) ��

30 Substantially the same proof can be found in Wright (2001).

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There is an informal explanation of the scenario represented by model M3. Wehave a thorough discovery/verification procedure, i.e., it takes into account all episte-mically possible states, and so does not confirm F because it takes into account that Fdoes not hold in all epistemic states, hence TK fails. But because it is thorough it seesthat if F held in all states it would be known, hence SK is vacuously true at each node.Unlike TK, SK does not have any knowability obligations with respect to non-stabletruths.31

Theorem 12 Principle MK does not yield TK.

Proof All instances of MK hold in model M3. Indeed, for any proposition X , if �Xholds at some node, then X holds in the whole of M3, hence, KX , �KX , and ♦�KXhold in M3. As shown in Theorem 11, TK fails in M3. ��Theorem 13 Principle TK does not yield MK.

Proof Consider model M2 in Fig. 3. According to Theorem 7, MK fails in M2. Onthe other hand, each instance of TK holds in M2. Indeed, w is �-accessible from eachnode and w is omniscient hence, for each X , either KX or K¬X hold at w. ��

6.1 Knowability diamond

Figure 5 provides the diagram of relationships between the knowability principles.

Fig. 5 “Knowability Diamond”

Arrows represent generality/strength of the corresponding principles as schemes:X −→ Y reads as each model of X is a model for Y, or, equivalently, principle X yieldsprinciple Y . A more general principle is less strong logically. None of the conversearrows holds. So, VK is the most restrictive of the four—it holds only in omniscientmodels—and SK is the most general, has the most models of the four.

We would like to think that SK is the most plausible candidate for a formalizationof verificationism which does not yield omniscience.

One could argue that both TK and IK/MK change the problem rather than solveit. The incompatibility between verificationism and TK has already been noted (see

31 Holmes and Watson have no clues yet. Their discovery procedure is Holmes’ super-human powers ofobservation and ‘deduction’. Not knowing which state of information is the real one Holmes cannot drawany conclusions, it is not possible for him to know one way or the other, but if some information were truein all accessible epistemic states he would know it and be able to draw some conclusions.

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Sect. 6). The same holds for IK which is expressed in a different logic, with a quitedifferent semantics,32 and its translation, MK, into the classical modal language doesnot match the original format of VK. It is not clear, in what sense a paradox associatedwith VK can be solved without providing a satisfactory explanation of the problemswith VK and by considering other principles instead. SK seems to be a better candidatesince (a) SK provides a definitive answer to the core question of what went wrong withVK—the stability assumption was missing; (b) SK is well-principled since it elimi-nates from consideration only non-stable truths which has been definitively diagnosedas the reason for the failure of VK; (c) SK has the same format as VK hence does notinvoke any new scenarios, does not change the problem.

6.2 Safe knowability principles

It is immediate from Fig. 5 that neither TK, nor SK, nor MK fall into the scope of theChurch–Fitch paradox, i.e., none of them yield OMN.

Let us take a closer look at how with stable knowability (SK) instead of verification-ist knowability (VK) the knowability paradox disappears. A simple repetition of theChurch–Fitch argument with SK(Moore) only proves that ¬�(p∧¬K p), i.e., ♦OMN,which is strictly weaker than OMN.33 One can equivalently rewrite ¬�(p ∧¬K p) as

�p → ♦K p

which incidentally is SK(p). This conclusion does not appear paradoxical, indeed it isentirely what one would expect to follow from the idea that all truths areknowable.34

7 Comparisons

In analyzing why VK yields the omniscience principle we found the problem to be thatthe assumption that truth is stable was missing from the formalization of verificationistknowability. That this is a legitimate assumption for the verificationist to make is sug-gested by our semantic analysis and its connection with constructivity. Incorporatingstability should not be seen as simply a restriction imposed on VK in order to getaround the paradox, but rather as an independently motivated formalization of verific-ationist ideas. Unlike those who put forward a version of what is called ‘the restrictionstrategy’ we are not concerned to save VK in some fashion or other, but to offer a log-ical framework for studying different kinds of verifiability. Nevertheless, SK and MK

32 For criticisms of IK see the references in note 20 above.33 This result is adumbrated in Rückert (2004).34 Moreover, consider what happens if F is not stable. Let us assume that all non-stable truths are knowable,i.e., (F ∧ ¬�F) → ♦KF . A repetition of the Church–Fitch proof yields (p ∧ ¬K p) → �(p ∧ ¬K p). Ifall non-stable truths are knowable, then ignorance cannot disappear in the process of verification. This israther counter-intuitive and provides further confirmation that the knowability paradox is due to the missingstability assumption.

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invite comparison with restriction approaches, and an examination of some prominentexamples is instructive.

7.1 Edgington

In Edgington (1985, 2010), argues that only actual truths are knowable. She proposesthat the knowability principle should be read as

AF → ♦KAF,

where ‘A’ is the modal operator ‘actually.’ Methodologically, this approach lookscompatible with incorporating stability. However, within the natural formalization of‘actuality’ in Edgington (1985) as

u � Aw F if and only if w � F,

Edgington’s proposal turns out to be trivial. In any model, AF is equivalent to a prop-ositional constant (true) or ⊥ (false). Therefore, assuming the ‘actual knowability’schema above is equivalent to assuming VK for the propositional constants and ⊥,which trivially holds in each model regardless of any knowability assumptions.35

7.2 Tennant and Dummett

Tennant in (1997, 2001, 2009) argues that only Cartesian propositions are knowablewhere “a Cartesian proposition is a proposition F such that KF is consistent” (Tennant2001). The Cartesian restriction provides a correct negative test for the knowabilityof a proposition—if a proposition is not Cartesian, then it is not knowable. Indeed, ifF is non-Cartesian, then KF is formally inconsistent, hence in an appropriate logicL, L � ¬KF . Then by necessitation, L � �¬KF and hence L � ¬♦KF .

However, being Cartesian does not necessarily imply being knowable. In particular,a true Cartesian proposition may still not be knowable, and hence Cartesian-ness doesnot give the positive conditions under which a proposition is knowable, if true.36 Con-sider VK(p) with a propositional letter p for F ; this instance of F is clearly Cartesian.Consider model M-p in Fig. 6.

The same argument as accompanied M1 shows that M-p is a scenario in which pshould be knowable but in which VK(p) fails. Model M-p shows that being Carte-sian and true, without also being stable, is insufficient to guarantee the truth of VK.

35 An interesting development of Edgington’s proposal can be found in Rabinowicz and Segerberg (1994),which considers a non-standard semantics for actuality and knowledge.36 Hand (2009), Hand and Kvanvig (1999), Kvanvig (2009), Williamson (2000b) argue that the Cartesianrestriction is ad hoc, our analysis (in 2.2) shows that while the restriction does not provide a positive testfor knowability, it may not be as ad hoc as they argue.

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Fig. 6 Model M-p where VK(p) fails

The validity of VK(F) depends on the stability of F , as we have seen, and Cartesianpropositions are not necessarily stable.37

Note that if p is ‘basic’, then the above argument shows that the restriction on VKproposed in Dummett (2001) is also too weak.

7.3 Fischer

More recently Fischer (2010) has proposed restricting VK to uniform instances, wherea formula, F , is uniform only if for each of the the sentence letters, p, occurring in Fevery occurrence of p is only within the scope of an even number of negation signs,or only within the scope of an uneven number of negation signs. With this restrictionMoore sentences are ruled out, and hence the Church–Fitch result excluded. Fischerargues that his proposal comports with verificationist ideas because uniform formulasare monotonic, and hence stable in our sense.

While similar to stability, and hence closer to the constructive spirit of VK, unifor-mity is a less general explanation than stability. First, its motivation is to avoid theMoore sentence, which we have shown not to be the cause of the knowability para-dox; and second, uniformity is weaker than stability, since not all stable formulas areuniform, e.g. p ∨ ¬p.

8 Conclusion

In van Benthem (2009) van Benthem argues that “what one really wants is a newsystematic viewpoint” from which to approach the knowability paradox rather thanjust attempting to avoid it by weakening either the logic in the Church–Fitch proof orVK itself.38 We argue that our semantic analysis achieves this.

The contribution of this work can be summarized as follows.

• An alternative, semantic proof that VK=OMN. This is an argument that the paradoxis due to the principle VK itself rather than the Church–Fitch proof.

• A case has been made that the key requirement of the verificationist argument infavor of VK, stability of the truth in question, is missing in VK and its lack aloneis a sufficient explanation for the paradox.

37 In the same vein, Williamson (2000b, 2009) argues that a Cartesian proposition can still yield OMN viaa more complex proof than the one by Church and Fitch (Theorem 1). But see Tennant (2010) for a reply.38 One might argue that we do the latter, but our analysis justifies why this is legitimate.

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• An alternative correct and justified formalization of verificationist knowability, thestable knowability principle SK, was offered and shown to be paradox-free. Thisis our suggested alternative to VK.

• The monotonic knowability principle MK was introduced and shown to be a faith-ful classical counterpart of intuitionistic knowability (IK). This is another plausiblealternative to VK.

• A formal logical framework for studying knowability in the bi-modal setting hasbeen developed. The resulting “Knowability Diamond” answers the question ofthe relative strength/generality of all four knowability principles considered.

We see that there is no need to adopt a non-classical logic39 or to reject any ofthe epistemic principles used in the Church–Fitch proof. Indeed, the simplicity andplausibility of the principles appealed to in the proof is what makes the derivation ofOMN so ‘paradoxical.’ Our approach preserves them all.

Our framework helps to clarify debates about the nature of verificationist knowa-bility and its tenability. Due to the knowability paradox and the fact that VK yields theunacceptable OMN, there is a need for new bi-modal principles reflecting the construc-tive, verifiability, content of verificationism. Our framework offers three possibilities,none of which leads to the knowability paradox.

• Stable knowability (SK). This approach mirrors the format of VK but uncovers thehidden assumption of stability.

• Monotonic knowability (MK), which reflects the specifically intuitionistic readingof knowability. MK is stronger than SK, but more restrictive; it stipulates that thereis a state after which F stays known.

• Knowability in a general setting reflected by the principle of total knowability(TK). This approach attempts to preserve the idea that all truths are knowable.

The value of the framework comes from the bigger explanatory picture it provides.It becomes possible not only to establish logical dependencies between principles, butalso to definitively certify the absence of such dependencies which was not possibleoutside a rigorous logical framework. Such a framework allows us to begin system-atically studying a concept which pervades epistemology. The question of whether ornot a proposition is knowable is central to debates about skepticism. The existence ofa priori knowledge turns on the possibility or impossibility of knowing independentlyof experience.40 To make more palatable the closure properties of the epistemic oper-ator K, formal epistemological approaches sometimes gloss it as knowable (De Vidiand Kenyon 2006; Shapiro 2006), or as is entitled to know (Fitting and Mendelsohn1998), or as potential knowledge (Fitting 2005). Verificationism, of course, puts thepossibilities of knowledge, via verifiability, at its center. Given how central know-ability is to so many core epistemological topics, a direct investigation of it and itsproperties is warranted. We think the above analyzes and principles constitute a stepin this direction.

39 See for instance Beall (2009), Priest (2009), Wansing (2002), Williamson (1982, 1988).40 See Kripke (1980), Peacocke (2000).

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Acknowledgments The authors are grateful to Melvin Fitting, Franz-Peter Griesmaier, Hidenori Kurok-awa, Elena Nogina, Gillman Payette, and Graham Priest, and anonymous referees for inspiring discussionsand useful suggestions. The research is supported in part by NSF Grant 0830450. Supported in part by aDoctoral Student Research Grant from the CUNY Graduate Center.

References

Artemov, S. N. (2001). Explicit provability and constructive semantics. Bulletin of Symbolic Logic, 7(1),1–36.

Artemov, S. N. (2008). The logic of justification. Review of Symbolic Logic, 1(4), 477–513.Artemov, S. N., & Protopopescu, T. (2010). Knowability from a logical point of view. Technical Report

TR-2010008. CUNY PhD Program in Computer Science.Beall, J. C. (2009). Knowability and possible epistemic oddities. In J. Salerno (Ed.), New essays on the

knowability paradox (pp. 105–127). Oxford: Oxford University Press.Brogaard, B., & Salerno, J. (forthcoming). Fitch’s paradox of knowability. In E. N. Zalta (Ed.) The

Stanford Encyclopedia of Philosophy (Fall 2012 Edn.). http://plato.stanford.edu/archives/fall/2012/entries/fitchparadox/.

Brogaard, B., & Salerno, J. (2002). Fitch’s paradox of knowability. Analysis, 62(2), 143–151.Church, A. (2009). Referee reports on Fitch’s “a definition of value.” In J. Salerno (Ed.), New essays

on the knowability paradox (pp. 13–20). Oxford: Oxford University Press.Costa-Leite, A. (2004). Combining possiblity and knowledge. In F. Carnielli, F. Dionisio, & P. Mateus

(Eds.), Proceedings of the workshop on the combination of logics: Theory and applications. Lis-bon: IST-Department of Mathematics, Center for Logic and Computation.

Costa-Leite, A. (2006). Fusions of Modal Logic in Fitch’s Paradox. Croatian Journal of Philosophy,17, 281–290.

De Vidi, T., & Kenyon, D. (2006). Introduction. In T. De Vidi & D. Kenyon (Eds.), A Logical approachto philosophy: Essays in honour of Graham Solomon (pp. 1–21). Berlin: Springer.

De Vidi, T., & Solomon, G. (1997). � in intuitionistic modal logic. Australasian Journal of Philoso-phy, 75(2), 201–213.

De Vidi, T., & Solomon, G. (2001). Knowability and intuitionistic logic. Philosophia, 28, 319–334.Dean, W., & Kurokawa, H. (2010). From the knowability paradox to the existence of proofs. Syn-

these, 175(2), 177–225.Dummett, M. (2001). Victor’s error. Analysis, 61, 1–2.Dummett, M. (2009). Fitch’s paradox of knowability. In J. Salerno (Ed.), New essays on the knowability

paradox, Chapter 4 (pp. 51–52). Oxford: Oxford University Press.Edgington, D. (1985). The paradox of knowability. Mind, 94(376), 557–568.Edgington, D. (2010). Possible knowledge of unknown truth. Synthese, 173(1), 41–52.Ewald, W. B. (1963). Intuitionistic tense and modal logic. Journal of Symbolic Logic, 51(1), 166–179.Fara, M. (2009). Knowability and the capacity to knowability. Synthese, 173(1), 53–73.Fischer, M. (2010). Some remarks on restricting the knowability principle. Synthese, 1–26. doi:10.1007/

s11229-010-9833-0.Fitch, F. (1963). A logical analysis of some value concepts. Journal of Symbolic Logic, 28(2),

135–142.Fitting, M. (2005). The logic of proofs, semantically. Annals of Pure and Applied Logic, 132, 1–25.Fitting, M. (2008). A quantified logic of evidence. Annals of Pure and Applied Logic, 152, 67–83.Fitting, M., & Mendelsohn, R. (1998). First-order modal logic. Dordrecht: Kluwer Academic Publishers.Gabbay, D. M., Kurucz, A., Wolter, F., & Zakharyaschev, M. (1998). Multi-dimensional modal logic:

Theory and application. Studies in logic and the foundations of mathematics (Vol. 148). Amsterdam:Elsevier.

Gödel, K. (1986). Eine Interpretation des intuitionistischen Aussagenkalkuls. Ergebnisse Math. Kolloq.,4, 39–40, 1933. English translation in: (S. Feferman et al., Eds.), Kurt Gödel collected works(Vol. 1, pp. 301–303). Oxford: Oxford University Press, 1986.

Gödel, K. (1995). Vortrag bei Zilsel/Lecture at Zilsel’s (1938a). In S. Feferman, J. W. Dawson, Jr,W. Goldfarb, C. Parsons, & R. M. Solovay (Eds.), Unpublished essays and lectures, volume IIIof Kurt Gödel collected works (pp. 86–113). Oxford: Oxford University Press.

123


http://plato.stanford.edu/archives/fall/2012/entries/fitchparadox/

http://plato.stanford.edu/archives/fall/2012/entries/fitchparadox/

http://dx.doi.org/10.1007/s11229-010-9833-0

http://dx.doi.org/10.1007/s11229-010-9833-0

Synthese (2013) 190:3349–3376 3375

Hand, M. (2009). Performance and paradox. In J. Salerno (Ed.), New essays on the knowability paradox,Chapter 17 (pp. 283–301). Oxford: Oxford University Press.

Hand, M., & Kvanvig, J. L. (1999). Tennant on knowability. Australasian Journal of Philosophy, 77(4),422–428.

Hart, W. D., & McGinn, C. (1976). Knowledge and necessity. Journal of Philosophical Logic, 5(2),205–208.

Jenkins, C. (2009). The mystery of the disappearing diamond. In J. Salerno (Ed.), New essays on theknowability paradox (pp. 205–222). New York: Oxford University Press.

Kleene, S. (1952). Introduction to metamathematics. New York: Ishi Press.Kripke, S. (1980). Naming and necessity. Oxford: Basil Blackwell.Kvanvig, J. (2006). The knowability paradox. Oxford: Oxford University Press.Kvanvig, J. (2009). Restriction strategies for knowability. In J. Salerno (Ed.), New essays on the

knowability paradox (pp. 205–222). New York: Oxford University Press.Melia, J. (1991). Anti-realism untouched. Mind, 100(3), 341–342.Negri, S., Maffezioli, P., & Naibo, A. (2012). The Church–Fitch knowability paradox in the light of

structural proof theory. Synthese, 1–40. doi:10.1007/s11229-012-0061-7.Peacocke, P., & Boghossian, C. (2000). Introduction. In New essays on the a priori (pp. 1–10). Oxford:

Oxford University Press.Percival, P. (1990). Fitch and intuitionistic knowability. Analysis, 50(3), 182–187.Priest, G. (2009). Beyond the limits of knowledge. In J. Salerno (Ed.), New essays on the knowability

paradox (pp. 93–104). Oxford: Oxford University Press.Prior, A. N. (1957). Time and Modality. Oxford: Oxford University Press.Proietti, C. (2001). Intuitionistic epistemic logic, Kripke models and Fitch’s paradox. Journal of

Philosophical Logic, 1–24. doi:10.1007/s10992-011-9207-1.Rabinowicz, W., & Segerberg, K. (1994). Actual truth, possible knowledge. Topoi, 13(2),

101–116.Rasmussen, S. A. (2009). The paradox of knowability and the mapping objection. In J. Salerno (Ed.), New

essays on the knowability paradox (pp. 53–75). Oxford: Oxford University Press.Restall, G. (2009). Not every truth is knowable: At least not all at once. In J. Salerno (Ed.), New essays

on the knowability paradox (pp. 339–354). Oxford: Oxford University Press.Rückert, H. (2004). A solution to Fitch’s paradox of knowability. In S. Rahman & J. Symons (Eds.), Logic,

epistemology and the unity of science (pp. 351–380). Dordrecht: Kluwer Academic Publishers.Salerno, J. (1999). Revising the logic of logical revision. Philosophical Studies, 99, 2211–2227.Shapiro, S. (2006). Externalism, anti-realism, and the kk-thesis. In T. De Vidi & D. Kenyon (Eds.), A

logical approach to philosophy (pp. 22–35). Berlin: Springer.Tennant, N. (1997). The taming of the true. Oxford: Oxford University Press.Tennant, N. (2001). Is every truth knowable: Reply to Williamson. Ratio, 14, 263–280.Tennant, N. (2009). Revamping the restriction strategy. In J Salerno (Ed.), New essays on the knowability

paradox (pp. 223–238). New York: Oxford University Press.Tennant, N. (2010). Williamson’s woes. Synthese, 173(1), 9–23.van Benthem, J. (2009). Actions that make us know. In J. Salerno (Ed.), New essays on the knowability

paradox (pp. 129–146). Oxford: Oxford University Press.Wansing, H. (2002). Diamonds are a philosopher’s best friend: The knowability paradox and modal

epistemic relevance logic. Journal of Philosophical Logic, 36(6), 591–612.Williamson, T. (1982). Intuitionism disproved?. Analysis, 42(4), 203–207.Williamson, T. (1988). Knowability and constructivism. Philosophical Quarterly, 38(153), 422–432.Williamson, T. (1992). On intuitionistic epistemic modal logic. Journal of Philosophical Logic, 21, 63–89.Williamson, T. (1994). Never say never. Topoi, 13, 135–145.Williamson, T. (2000a). Knowledge and its limits. Oxford: Oxford University Press.Williamson, T. (2000b). Tennant on knowable truth. Ratio, XIII, 99–114.Williamson, T. (2009). Tennant’s troubles. In J. Salerno (Ed.), New essays on the knowability para-

dox (pp. 183–205). New York: Oxford University Press.Wolter, F., & Zakharyaschev, M. (1997). On the relation between intuitionistic and classical modal

logics. Algebra and Logic, 36, 121–155.Wolter, F., & Zakharyaschev, M. (1999a). Intuitionistic modal logics as fragments of classical bimodal

logics. In E. Orlowska (Ed.), Logic at work (pp. 168–186). Berlin: Springer.

123


http://dx.doi.org/10.1007/s11229-012-0061-7

http://dx.doi.org/10.1007/s10992-011-9207-1

3376 Synthese (2013) 190:3349–3376

Wolter, F., & Zakharyaschev, M. (1999b). Intuitionistic modal logic. In A. Cantini, E. Casari, &P. Minari (Eds.), Logic and foundations of mathematics (pp. 227–238). Dordrecht: Kluwer Aca-demic Publishers.

Wright, C. (1983). Can a Davidsonian meaning-theory be construed in terms of warranted assertibility?In C. Wright (Ed.), Realism, meaning and truth (pp. 286–315). Oxford: Blackwell.

Wright, C. (2001). On being in a quandry: Relativism, vagueness and logical revisionism. Mind,110(437), 45–98.

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Discovering knowability: a semantic analysis