Knowability Paradox

1. Introduction

The Knowability Paradox, often called Fitch’s Paradox of Knowability, is a result in epistemic and modal logic that appears to show that a seemingly modest thesis about truth and knowledge has surprisingly strong consequences. The modest thesis is the Knowability Principle: every truth is, at least in principle, knowable. The striking consequence, under standard logical assumptions, is the omniscience result: every truth is in fact known.

Philosophers have treated this as a paradox because each ingredient seems, taken separately, appealing:

• It appears plausible that there could be true propositions that no one ever actually comes to know (so there are unknown truths).
• It also appears attractive—especially to various forms of semantic anti‑realism and verificationism—to claim that truth is essentially tied to what could be known.
• Standard principles of epistemic logic, such as that knowledge is factive and distributes over conjunction, are widely used and independently motivated.

Fitch’s proof shows that these commitments, when combined in a precise way, generate a tension. If all truths are knowable, it seems to follow that there cannot be any unknown truths at all, contrary to common sense. This tension has made the Knowability Paradox a central reference point in debates about the nature of truth, the limits of knowledge, and the correct logical treatment of epistemic notions.

Subsequent sections present the origin of the paradox, its formal structure, its role in disputes between realism and anti‑realism, and the main strategies that have been proposed to resolve or dissolve it.

2. Origin and Attribution

The standard attribution of the paradox is to Frederic B. Fitch, who presented the key formal result in 1963 in an article on value concepts:

"

“From the principle that every truth is possibly known, it follows that every truth is, in fact, known.”

— Frederic B. Fitch, “A Logical Analysis of Some Value Concepts,” Journal of Symbolic Logic 28(2) (1963)

Although Fitch’s paper was primarily concerned with topics in value theory, one of its technical lemmas showed that a global knowability thesis entails a kind of omniscience. At the time, this was not developed as a standalone paradox; its broader philosophical significance emerged only later.

In the late 1970s and 1980s, philosophers such as Michael Dummett, Neil Tennant, and Crispin Wright recognized that Fitch’s result had important implications for semantic anti‑realism, which often characterizes truth in terms of potential knowability or verifiability. This retrospective interpretation led to the now standard label “Fitch’s Paradox of Knowability” or the Church–Fitch Knowability Paradox, sometimes acknowledging Alonzo Church for related unpublished or informal observations.

The following table summarizes main attributions:

While the exact historical priority of the core idea has been discussed, there is broad agreement that Fitch’s 1963 article introduced the formal argument that underpins what is now called the Knowability Paradox.

3. Historical and Philosophical Context

Fitch’s result emerged within mid‑20th‑century analytic philosophy, against a backdrop of rapid developments in modal and epistemic logic and ongoing debates about the legacy of logical positivism and verificationism.

Logical and Technical Context

By the early 1960s, systems of modal logic using operators for necessity and possibility had been largely standardized. At the same time, logicians were introducing explicit operators for knowledge and belief, giving rise to epistemic logic. Fitch’s argument uses both types of operators and relies on structural features common to these systems, such as normal modal rules and factive knowledge.

Semantic and Epistemological Context

Philosophically, the period was marked by:

• Continuing influence of verificationist ideas, according to which the meaning or truth of a statement is tied to its conditions of verification.
• Emerging forms of semantic anti‑realism, later articulated by figures like Dummett, which proposed that truth should be understood in terms of what is, in principle, knowable or justifiable.

These views often endorsed some version of the claim that every truth is knowable, at least in principle, and questioned the realist idea of truth transcending possible knowledge.

Later Reception

Fitch’s argument drew relatively little philosophical attention when first published. Its broader significance became prominent when:

• Dummett and others sought rigorous formulations of anti‑realist theses about truth.
• Logicians and philosophers of language examined the interaction between quantification, modality, and epistemic operators.

From this point on, the Knowability Paradox became a focal point where technical logic, philosophy of language, and metaphysics intersected, providing a test case for accounts of truth that link it closely to knowability.

4. The Knowability Principle Formulated

The central principle at issue in the Knowability Paradox is the Knowability Principle (KP). In its most familiar formal version, it is stated as:

"

KP: ∀p (p → ◊Kp)

Read informally, this says: for every proposition p, if p is true, then it is possible that p is known. Here, K is an epistemic operator (for “it is known that”), and ◊ is a modal operator (for “it is possible that”).

Motivations and Intended Readings

KP is typically motivated in two different ways:

• Epistemically modest reading: There are no truths that are absolutely beyond the reach of knowledge; any truth could be known by some suitably idealized knower under appropriate conditions.
• Anti‑realist/verificationist reading: Truth is constitutively tied to potential verification; to be true just is (or requires) being knowable in principle.

Different authors stress different glosses on “◊”:

These interpretations matter because the paradox’s force depends on how closely “knowable” in KP matches what various theories of truth or knowledge intend.

Global vs Restricted Formulations

The version used in the paradox is global: it quantifies over all propositions in the language. Some philosophers have later suggested restricted forms, for example:

• Only decidable propositions are claimed to be knowable.
• Only propositions from certain domains (e.g., mathematics, observation statements) are covered.

Those restrictions are discussed elsewhere; at this stage, the key point is that Fitch’s original argument takes KP in its simple, global form and combines it with standard assumptions to derive the paradoxical conclusion.

5. Formal Statement of Fitch’s Argument

Fitch’s argument is typically presented as a reductio ad absurdum that begins with the Knowability Principle and, under standard logical assumptions, derives the problematic conclusion that all truths are known.

Basic Notation

• Kp: it is actually known that p
• ◊p: it is possible that p
• p, q, r: arbitrary propositions
• KP: ∀p (p → ◊Kp)

Core Derivation

A widely used formal presentation proceeds along the following lines:

1. Assume KP: ∀p (p → ◊Kp).
2. Assume, for reductio, that there is some unknown truth: ∃p (p ∧ ¬Kp).
3. Let q be such that q ∧ ¬Kq is true. Define r := (q ∧ ¬Kq).
4. From (2–3), r is true: r.
5. By KP, from r we infer ◊K(r).
6. Assume for analysis that ◊K(r). So there is a possible situation in which K(r) holds.
7. In that situation, K(r) is K(q ∧ ¬Kq). Using distribution of K over conjunction, infer Kq ∧ K¬Kq.
8. By factivity of knowledge, from Kq infer q, and from K¬Kq infer ¬Kq.
9. Thus, in that possible situation, both Kq and ¬Kq hold, yielding a contradiction.
10. Therefore, K(r) is impossible, so ¬◊K(r).
11. But KP plus r gave ◊K(r), so we have both ◊K(r) and ¬◊K(r), a contradiction.

From here, the standard conclusion is that the assumption of an unknown truth (step 2) is incompatible with KP together with the background logic. Thus, given KP, one cannot coherently assert that there are unknown truths, and it follows that every truth must be known.

6. Logical Structure and Key Assumptions

Fitch’s result depends not only on the Knowability Principle but also on a specific logical structure and several background assumptions about knowledge and modality. These assumptions are widely used in epistemic logic, though some have been questioned in response to the paradox.

Overall Structure

The argument has the form of a reductio:

• Combine KP with the existence of at least one unknown truth.
• Show that this yields a contradiction given the logical rules governing K and ◊.
• Conclude that KP rules out the possibility of any unknown truths.

The central move is to construct a proposition that says, in effect, “there is a particular truth that is unknown,” and then ask what would happen if that proposition were knowable.

Key Logical Assumptions

Common presentations highlight the following assumptions:

The self‑referential nature of the constructed proposition r = (q ∧ ¬Kq) is crucial: it embeds an epistemic condition within a truth that is itself subject to KP.

Status of Assumptions

In subsequent literature, philosophers have examined which of these assumptions can be modified or rejected to avoid the paradoxical conclusion while preserving as much of KP as possible. Some focus on restricting the domain of quantification, others on altering the logic governing K, or the interpretation of the possibility operator ◊. These strategies are discussed in later sections; here, the point is that Fitch’s original derivation is generally regarded as formally valid relative to the background assumptions just listed.

7. Role in Realism vs Anti-Realism

The Knowability Paradox occupies a central place in debates between realists and anti‑realists about truth.

Anti-Realist Commitments

Many forms of semantic anti‑realism—inspired by verificationism or by Dummett’s work—seek to explain truth in terms of justification, verification, or possible knowledge. They often endorse a principle akin to KP, treating it not merely as an epistemic thesis but as a constitutive feature of truth: roughly, that truths cannot outrun what could, in principle, be established.

Realist Commitments

Realists about truth typically insist that truth is independent of our cognitive capacities: some truths might never be known, or even be knowable, by any human or idealized agent. They therefore reject any biconditional link between truth and knowability, though they may accept weaker claims (e.g., that many truths are knowable).

How the Paradox Intervenes

The paradox appears to place pressure on global anti‑realist theses:

• If truth is tied to knowability via KP, then Fitch’s argument seems to force the conclusion that, in effect, all truths are already known.
• This is often viewed as incompatible with the anti‑realist’s own practice, which typically allows for the existence of unknown but in‑principle knowable truths.

Realists use the paradox to argue that truth must be transcendent: there can be truths that no one ever could know. Anti‑realists, in turn, interpret the paradox as revealing the need to refine how knowability is understood, how it is formalized, or how widely it is applied.

Thus, the Knowability Paradox acts as a testing ground for competing conceptions of truth: views that equate truth with knowability must explain how they avoid collapse into omniscience, while realist views must account for the intuitive appeal of tying truth to practices of justification and verification.

8. Variations and Reformulations

While Fitch’s original presentation used specific logical resources, later authors have developed numerous variations and reformulations of the Knowability Paradox. These serve to clarify which aspects of the argument are essential and how robust the paradox is across different formalisms.

Syntactic and Semantic Reformulations

Some reconstructions work primarily at the syntactic level (proof systems), others at the semantic level (possible‑worlds or analogous models):

• In proof‑theoretic settings, the derivation is given as a sequence of inferences in a natural deduction or sequent calculus system for epistemic logic.
• In model‑theoretic settings, the paradox is framed in terms of properties of accessibility relations and truth conditions for K and ◊.

Alternative Operators and Notions

A range of operators have been used in place of, or alongside, K:

In some versions, the knowability thesis is formulated as “every truth is eventually known,” using temporal operators; others employ a fixed‑point construction to characterize knowable truths.

Quantificational and Domain Variants

Variants also differ on how quantifiers and domains are handled:

• Some formulations isolate finitary fragments of the language to assess whether paradoxical reasoning depends on higher‑order resources.
• Others use typed or stratified languages to manage self‑reference differently.

Conceptual Reformulations

Beyond technical changes, there are conceptual reformulations that replace “knowability” with:

• “Rational acceptability in the limit of inquiry”
• “Definiteness of truth‑conditions”
• “Availability of a verification procedure”

These are meant to capture more closely what specific philosophical positions (especially anti‑realist ones) mean by linking truth to epistemic capacities, and to examine whether the omniscience consequence persists under those interpretations.

Across these variations, a recurrent theme is that the core tension—between a strong global knowability claim and the existence of unknown truths—tends to reappear unless some substantive change is made to the logical, semantic, or conceptual background.

9. Standard Objections and Critiques

Philosophers have raised a range of objections to the Knowability Paradox as a challenge to global knowability theses. These do not all deny the formal validity of Fitch’s derivation; many instead question specific premises or background assumptions.

Objections to the Scope of KP

One influential line of criticism targets the unrestricted quantification in KP:

• Some argue that KP should hold only for decidable or effectively investigable propositions, not for all propositions expressible in the language.
• On this view, problematic self‑referential sentences like “this proposition is true and unknown” lie outside the intended domain of the knowability thesis.

Objections to the Interpretation of Possibility

Another cluster of objections questions the use of a simple ◊ operator:

• Critics contend that the modal operator in KP should encode epistemic or procedural possibility, rather than general metaphysical possibility.
• If “knowable” is tied to admissible verification procedures or to the limit of ideal inquiry, the standard modal treatment might misrepresent the intended notion, and Fitch’s derivation may no longer go through.

Objections to the Epistemic Logic

Some philosophers focus on the formal rules governing K:

• They question whether knowledge should distribute over conjunction in the way assumed, or whether it should be closed under all logical consequences.
• They also examine the plausibility of embedding claims about knowledge and ignorance within the same language in an unrestricted way.

Objections to the Unknown Truth Premise

A more radical critique questions the assumption that there are unknown truths:

• Certain anti‑realist perspectives maintain that truths that could never be known are unintelligible or incoherent.
• From that standpoint, the intuitive pull of unknown truths is said to derive from a realist conception that anti‑realists need not share.

These objections motivate alternative treatments of KP, of the underlying logic, or of the metaphysical and semantic assumptions built into the paradox, leading to a variety of responses and proposed resolutions.

10. Responses from Anti-Realists

Anti‑realist philosophers, especially those influenced by Dummett and verificationist traditions, have developed several strategies for reconciling their views with the Knowability Paradox, or for showing that the paradox does not genuinely threaten their positions.

Domain Restriction and Predicativity

Some anti‑realists accept a version of KP but restrict its domain:

• They may limit KP to propositions that are effectively decidable, or whose truth‑conditions are given by canonical verification procedures.
• Certain self‑referential constructions, such as “this proposition is true and unknown,” are then excluded as illegitimate or non‑statable within the anti‑realist’s preferred language.

On these approaches, the paradox is seen as exploiting propositions that fall outside the range of meaningful or assertible statements for anti‑realism.

Reinterpretation of Knowability

Other anti‑realists argue that the modal operator in KP should express idealized verifiability, not generic metaphysical possibility. They may tie knowability to:

• The eventual convergence of inquiry,
• The existence of a constructive proof,
• Or the availability of a publicly recognizable justification.

When knowability is understood this way, the step from “p is true” to “◊Kp” may be governed by constraints not captured in standard modal logic, and the derivation of omniscience can fail.

Revision of Epistemic Closure Principles

A further response is to revisit assumptions about knowledge itself:

• Some anti‑realists accept that knowledge is factive but deny certain closure or distribution principles used in the paradox.
• They may hold that knowledge of complex epistemic propositions (e.g., involving “unknown”) does not decompose in the same way as knowledge of simple propositions.

Attitudes to Unknown Truths

Finally, certain anti‑realists are prepared to reject the premise that there are unknown truths, maintaining that what cannot, in principle, be known does not count as true. For them, the tension produced by Fitch’s argument reveals the incompatibility of realist intuitions about unknown truths with an anti‑realist conception of truth.

These responses illustrate how anti‑realists can preserve a connection between truth and knowability, while modifying either the formal framework or the scope of their claims to avoid the paradox’s omniscience conclusion.

11. Non-Classical and Restricted Approaches

Beyond purely conceptual revisions, many responses to the Knowability Paradox employ non‑classical logics or restriction strategies within classical frameworks to block the derivation of omniscience.

Non-Classical Logics

Several approaches weaken or alter the logic underlying Fitch’s proof:

• Intuitionistic logic: By rejecting some classical principles (such as excluded middle), intuitionistic treatments may disrupt key inferential steps, especially those involving negation and double‑negation, reducing the force of the paradox for constructivist views of truth.
• Paraconsistent logic: Systems that tolerate some contradictions without triviality can respond differently to the inconsistent status of “this is known and not known,” potentially containing the problematic constructions.
• Substructural logics: By limiting structural rules (like contraction or distribution), some derivations involving knowledge of conjunctions or nested epistemic claims may no longer be valid.

In these settings, proponents explore whether a suitably formulated knowability principle avoids collapse into omniscience.

Restricted Epistemic Operators

Another family of strategies works within broadly classical settings but modifies the behaviour or scope of the epistemic operator K:

• Non‑normal epistemic logics restrict standard rules for K, such as closure under known logical equivalences or unrestricted distribution over conjunction.
• Hierarchical or typed knowledge operators distinguish between levels of knowledge (e.g., object‑level vs meta‑epistemic), preventing self‑referential embedding of “unknown” claims at the same level.

Domain and Language Restrictions

Restriction strategies also target the language and domain of quantification:

• Limiting KP to non‑epistemic propositions, or to propositions without certain forms of self‑reference, can prevent the construction of the paradox‑generating sentence.
• Using typed languages or predicative comprehension principles aims to avoid paradoxes akin to those arising in set theory or the theory of truth.

These non‑classical and restricted approaches treat Fitch’s argument as revealing the need for refined logical or linguistic tools, rather than as a decisive refutation of any link between truth and knowability.

12. Implications for Epistemic Logic

The Knowability Paradox has had a significant impact on the development and assessment of epistemic logic, prompting reconsideration of how knowledge and related notions should be formalized.

Scrutiny of Standard Principles

The paradox has led logicians to re‑examine familiar assumptions about K:

• Factivity is widely retained, but its interaction with other principles is analysed more carefully, especially in contexts involving nested epistemic operators and quantification over epistemic statements.
• The principle that knowledge distributes over conjunction, and broader closure under logical consequence, are evaluated in light of their role in enabling paradoxical constructions.

Richer Epistemic Languages

To handle self‑reference and meta‑epistemic claims, some approaches introduce:

• Multiple knowledge operators (e.g., for different agents, levels, or idealizations).
• Distinctions between actual knowledge, practicable knowability, and more abstract ideal knowability.

These enrichments aim to respect intuitive epistemic distinctions that the simple K operator of basic systems cannot capture.

Interaction with Modality and Quantification

The paradox underscores the importance of how modal and epistemic operators interact with quantifiers:

• Questions arise about whether knowledge claims involving quantification over all propositions (e.g., “all truths are knowable”) can be safely expressed within the same language.
• Model‑theoretic analyses explore constraints on accessibility relations and domain conditions that avoid paradoxical outcomes.

Influence on Epistemic Axiomatizations

Finally, Fitch‑style reasoning informs the choice and justification of axioms in epistemic logics used in computer science, game theory, and philosophical logic. Designers of such systems often consider whether to:

• Allow axioms that support knowability‑style principles,
• Or restrict them to avoid omniscience and related pathologies.

In these ways, the Knowability Paradox serves as both a cautionary example and a methodological tool for refining formal theories of knowledge.

13. Connections to Other Paradoxes

The Knowability Paradox is frequently discussed alongside other logical and semantic paradoxes, both for structural similarities and for shared themes about self‑reference, truth, and knowledge.

Liar and Semantic Paradoxes

There are notable parallels with the Liar Paradox (“This sentence is false”) and related semantic puzzles:

• Both involve sentences that make claims about their own status.
• In Fitch’s case, the crucial sentence effectively says “this truth is unknown,” intertwining truth and knowledge rather than truth and falsity.
• Responses that appeal to hierarchies of language or typed predicates echo strategies in the theory of truth.

Paradoxes of Omniscience and Foreknowledge

The omniscience result connects Fitch’s paradox to debates about:

• Logical omniscience in epistemic logic, where agents are unrealistically attributed knowledge of all logical consequences.
• Divine omniscience and foreknowledge in philosophy of religion, where tensions arise between what a deity knows and human freedom or ignorance.

While the settings differ, each involves questions about the limits of what can be known, in principle, by idealized knowers.

Modal and Set-Theoretic Paradoxes

Connections have also been drawn to:

• Modal paradoxes involving necessity and possibility, such as the Church–Fitch style arguments in other contexts, and puzzles about “non‑contingent existence” claims.
• Set‑theoretic paradoxes, in that unrestricted quantification over a single domain (e.g., all sets, all propositions) often enables paradoxical self‑reference; responses sometimes involve restricted comprehension or typed domains, similar to some reactions to Fitch.

By examining these connections, philosophers assess whether strategies successful for other paradoxes—such as hierarchy, restriction, or non‑classical logics—can be adapted to address the issues raised by knowability.

14. Contemporary Debates and Open Questions

Current discussions of the Knowability Paradox revolve around both technical and philosophical issues, many of which remain unsettled.

Status of the Knowability Principle

One central question is how, if at all, KP should be retained:

• Some authors argue for restricted or domain‑relative knowability theses; debates concern where the restrictions should fall (e.g., on self‑reference, undecidable statements, epistemic vocabulary).
• Others explore whether KP should be replaced by weaker principles, such as claims about the knowability of only some classes of truths or about the approximate rather than exact convergence of inquiry.

Appropriate Modal and Epistemic Framework

There is no consensus on the best formal framework for modelling knowability:

• Proposals range from refined Kripke‑style models with special epistemic accessibility relations, to topological models of knowledge, to proof‑theoretic accounts.
• An open issue is how to capture the intended anti‑realist connection between truth and warranted assertibility without reintroducing the omniscience result.

Role of Non-Classical Logics

The viability of non‑classical approaches remains debated:

• Supporters argue that intuitionistic, paraconsistent, or substructural logics align better with anti‑realist or verificationist intuitions.
• Critics question whether changing logic to avoid a paradox is methodologically legitimate, and whether these logics truly dissolve the problem rather than relocate it.

Metaphysical and Semantic Implications

At a more general level, the paradox continues to inform questions about:

• Whether truth transcends possible knowledge, and if so, in what sense.
• How to articulate a coherent notion of idealized knowers or limit‑of‑inquiry agents without generating paradoxical self‑reference.
• The relationship between meaning, assertibility, and truth, particularly in the wake of anti‑realist programs.

These ongoing debates indicate that the Knowability Paradox functions not only as a technical result in epistemic logic but also as a focal point for broader disputes about the nature of truth, knowledge, and logical methodology.

15. Legacy and Historical Significance

The Knowability Paradox has secured a lasting place in both logic and philosophy, influencing how theorists think about knowledge, truth, and formal methods.

Impact on Philosophy of Truth

In the philosophy of truth, the paradox is often cited as a major challenge to global semantic anti‑realism and verificationism. It has prompted:

• More nuanced formulations of anti‑realist theses, often domain‑restricted or tied to specific practices.
• Renewed arguments for realism about truth, which treat the paradox as evidence that truth can outstrip possible knowledge, even if many truths are in principle knowable.

Regardless of stance, most contemporary treatments of truth acknowledge Fitch‑style reasoning as a constraint any adequate theory must address.

Role in the Development of Epistemic and Modal Logic

In epistemic logic, the paradox has:

• Encouraged careful scrutiny of axioms about knowledge, especially closure principles and interactions with modality.
• Stimulated the development of logics that distinguish knowledge, belief, knowability, and related notions, sometimes using multiple operators or hierarchies.

In modal logic more broadly, it has highlighted the subtleties of mixing epistemic and alethic modalities with quantification over propositions.

Influence Across Subfields

The paradox’s legacy extends to:

• Philosophy of mathematics, where it interacts with debates about constructivism, intuitionism, and the knowability of mathematical truths.
• Philosophy of science, through discussions of whether all scientific truths are, in principle, discoverable.
• Formal epistemology and adjacent areas in computer science and game theory, where it informs the design of systems modelling idealized agents and information.

Historically, the Knowability Paradox illustrates how a relatively technical lemma in a 1960s logic paper could become, through later reinterpretation, a central node in wide‑ranging philosophical debates. Its enduring significance lies in forcing precise articulation of the relationship between what is true and what can, even in principle, be known.