In my paper (Paseau 2008), I noted that Fitch's argument, which purports to show that if all truths are knowable then all truths are known, can be blocked by typing knowledge. If there is not one knowledge predicate, ‘K’, but infinitely many, ‘K1’, ‘K2’, … , then the type rules prevent application of the predicate ‘Ki’ to (names of) sentences containing ‘Ki’ such as ‘p ∧¬Kip⌝’. This provides a motivated response to Fitch's argument so long as knowledge typing is itself motivated. It was the burden of my paper to explore the case that knowledge typing is as motivated as truth typing by drawing on the parallels between epistemic paradoxes generated by sentences of the kind ‘this sentence is unknown’ and semantic paradoxes generated by sentences such as ‘this sentence is untrue’. Given that typing truth is one of the acknowledged options for solving semantic paradoxes, if the parity argument succeeds it follows that epistemic typing is as well-motivated as truth typing and that the typing response to Fitch's argument is correspondingly strong.

Halbach (2008) presents an apparent problem for this argument. Let ‘N’ and ‘P’, respectively, denote the necessity and possibility predicates, ‘K1’ the knowledge predicate of first type, and let ‘γ’ be a K-free sentence (i.e. a sentence of knowledge type 0) such that γ ↔¬PK1⌜γ⌝⌝; we know such a ‘γ’ exists by a standard diagonalization argument. If we assume that γ is knowable at the next knowledge type, that is, at type 1, and that the possibility typing does not interfere with the knowledge typing, a contradiction quickly ensues. Formally:

  1. γ ↔¬PK1⌜γ⌝⌝ (by diagonalization)

  2. γ →PK1⌜γ⌝⌝ (version of knowability assumed by Halbach)1

  3. ¬γ (by logic, from (1) and (2))

  4. ¬K1⌜γ⌝ (by logic, from (3) and K1⌜γ⌝ → γ)

  5. N⌜¬K1⌜γ⌝⌝ (by applying the necessitation rule to (4))

  6. ¬PK1⌜γ⌝⌝ (by the link between P and N, from (5))

  7. γ (by logic, from (1) and (6); contradicts (3))

I shall sketch two responses to Halbach's argument (which generalizes for any knowledge predicate ‘Kn’) below, each of which blocks it in a motivated way.

1. Cross-hierarchy restrictions

As both Halbach (2008: 117) and I (2008: §7) agree, anyone who is tempted by knowledge typing as a solution to epistemic paradoxes will go in for necessity typing in response to modal paradoxes of the form ‘this sentence is not possible’. Thus, a type theorist invokes not only a hierarchy of truth predicates ‘T1’, ‘T2’, … , and a hierarchy of knowledge predicates ‘K1’, ‘K2’, … , but also one of necessity predicates ‘N1’, ‘N2’, … . The motivations for each of these three hierarchies naturally give rise to typing rules. It is reasonable, for example, to assume that KmpKnp if n > m; but not that KmpKnp if m > n (see below). Likewise, it cannot in general be true that NmpNnp if m > n: for example, a proposition of N-type 1 may be necessary at type 2 but cannot be necessary at type 1 without violating type restrictions on necessity.

What of the predicates’ interactions? Do the truth, necessity and knowledge typings (and any others, but let us ignore them here) cut across one another? For example, can any truth predicate ‘Tn’ apply to the name of any sentence not containing a truth predicate of type ≥n, including ones of any epistemic type or necessity type – and similarly for knowledge and necessity predicates? Or are there restrictions akin to the single-hierarchy ones?2

It is natural to assume there are such restrictions. Indeed, the very idea of typing from Tarski onwards seems to require cross-hierarchy restrictions. Consider that the typed approach to truth maintains that, given some interpreted language L, to talk about truth in L one must add a new predicate letter ‘T1’ that applies precisely to the true sentences of L. The result is a new extended language L′. To talk about truth in L′ one adds a new predicate ‘T2’ forming a new language L′′; and so on. Once one has in play a number of different typed notions – truth, knowledge and necessity say – there are a number of ways of putting something like this idea in motion. For example, one could start with L, then add three new predicates ‘T1’, ‘K1’ and ‘N1’, forming a new language L′, then another three ‘T2’, ‘K2’ and ‘N2’, etc. Alternatively, one could add the predicates one at time. No matter: any way of putting the general idea into effect will lead to a hierarchy of languages with cross-typing restrictions. Indeed the very meaning of ‘typing’ seems to impose such restrictions: they flow from the well-ordering of the predicates with respect to which the hierarchy of languages is typed. This is why ‘typed’ and ‘hierarchical’ are used interchangeably in this context.

Now it is beyond the scope of this note to provide a theory of the relevant cross-typing rules. But even absent that, once the point is appreciated we may no longer assume the instance of knowability Halbach exploits. Being careful to type the necessity predicate turns the diagonal sentence into ‘γ ↔¬PnK1⌜γ⌝⌝’ for some n; say n = 2 for concreteness, so that the diagonal sentence becomes ‘γ ↔¬P2K1⌜γ⌝⌝’. But now why accept the instance ‘γ→P2K1⌜γ⌝⌝’ the reductio assumes? There is no good reason. Even without a full theory of the cross-type restrictions to hand and without assuming anything about knowability, we can show that ‘K1⌜γ⌝’ is not a type-2 possibility. If we suppose that K1⌜γ⌝ it follows from K1⌜γ⌝ → γ that γ, hence using the diagonal sentence it follows that it is a type-2 impossibility that K1⌜γ⌝.

Of course this is to run the argument in reverse – but in an apparently legitimate way. Once we notice, as we must, that the necessity predicate is typed, and therefore that something can be possible without being possible at type 2, there is no reason to assume a version of knowability that claims that for any γ, its being (n + 1)-known is an (n + 2)-possibility. This is not at all the ‘radical cure’ Halbach moots of not applying necessity predicates to sentences containing the knowledge predicate and vice-versa. The more modest restriction suggested here does the job of blocking the contradiction while still allowing knowledge and necessity predicates to apply to sentences containing one another. It blocks the alleged reductio by typing necessity as well as knowledge in a way that is entirely in keeping with the typed perspective.

In fact, Halbach's argument has nothing to do with typing or knowability. It uses N-necessitation (for N-free sentences), and ‘K⌜γ⌝ → γ’ (for K-free γ). These principles are no more plausible than K-necessitation (for K-free sentences), and ‘N⌜γ⌝ → γ’ (for N-free γ).3 Yet taken together the four principles are inconsistent:

  1. γ ↔¬NK⌜γ⌝⌝ (by diagonalization)

  2. NK⌜γ⌝⌝ (assumption)

  3. K⌜γ⌝ (from (2), using ‘N⌜γ⌝ → γ’)

  4. γ (from (3), using ‘K⌜γ⌝ → γ’)

  5. ¬NK⌜γ⌝⌝ (from (1) and (4))

  6. ¬NK⌜γ⌝⌝ (from no assumptions, using (2) and (5))

  7. γ from (1) to (6))

  8. K⌜γ⌝ (from (7), by K-necessitation)

  9. NK⌜γ⌝⌝ (from (8), by N-necessitation; contradicts (6))

A similar reductio can be run simply on the modal logic T (or the epistemic logic T):

(1) γ ↔¬N⌜γ⌝ (by diagonalization)

(2) N⌜γ⌝ → γ (the axiom ‘N⌜γ⌝ → γ’)

(3) γ (by logic, from (1) to (2))

(4) N⌜γ⌝ (by applying N-necessitation to (3))

(5) ¬γ (by logic, from (1) to (4); contradicts (3))

But now notice: the second and third reductios exploit neither typing nor knowability. Given the similarities between the three arguments, this strongly suggests that Halbach's argument does not depend on typing or knowability either.

Halbach agrees with me that a predicate treatment of necessity and knowledge is to be preferred to an operator one,4 and he elsewhere blocks the third reductio, of the predicate necessity logic T, by rejecting some of T's principles (Halbach et al. 2003). However, he avails himself of precisely parallel principles in the first reductio but then curiously goes on to pin the blame on the combination of typing and knowability.5 Yet the second and third reductios' reasoning are so similar to that of the first that we seem driven to the conclusion that the source of the contradiction lies not in typing nor knowability nor indeed their combination, but rather in the factivity axioms (‘K1⌜γ⌝ → γ’ in the first case, ‘N⌜γ⌝ → γ’ and ‘K1⌜γ⌝ → γ’ in the second, ‘N⌜γ⌝ → γ’ in the third) or the necessitation rules (N-necessitation in the first case, K- and N-necessitation in the second and N-necessitation in the third) or their conjunction. Halbach must agree, since he himself rejects these principles and the associated logics T (for necessity and knowledge) while holding on to a predicate treatment.

In sum, from within the typed perspective there are good grounds to reject one of the principles assumed in Halbach's reductio, but still hold on to knowability, with the necessity typing properly acknowledged. Indeed, there are good grounds from any perspective that takes necessity and knowledge as predicates, as Halbach elsewhere acknowledges. The problem has nothing to do with typing or knowability and everything to do with the predicate treatment of the notion of necessity (and similar ones such as knowledge) coupled with apparently cogent necessitation and factivity principles. Far from creating the problem, then, typing offers a solution to it, by striving to respect the predicate conception of necessity and knowledge as well as appropriately typed versions of the intuitively compelling principles. Proper restrictions, either cross-type ones (violated in the first and second argument) or single-type ones (violated in the third argument), block Halbach's argument in a motivated way.

2. Knowability at some knowledge type

The variant of knowability Halbach assumes is maximally strong: he assumes the schema ϕ → PKn⌜ϕ⌝ for the next n greater than ϕ's knowledge type.6 This variant goes far beyond knowability's tenet that any proposition is knowable, as it adds to it a specification of the types at which knowledge is achieved. Knowability itself requires only that any proposition is known at some knowledge type or other; we may cash it out as the claim that ϕ → PKn⌜ϕ⌝ for some n greater than ϕ's knowledge type. This observation blocks Halbach's argument, since the knowledge type in line (2) has to be the same as the knowledge type in line (1) to generate a contradiction. (Ditto for the necessity type in lines (1) and (2), as explained in the previous section.)

In general, the type at which a proposition is known depends on how the proposition came to be known, and there is no compelling antecedent reason to think that every proposition can be known at its very next type or at some specified higher type. The proponent of knowability is in fact familiar with the idea that sentences of type n are in certain circumstances only knownn+m (where m ≥ 2) but not knownn+1, say because they are derived by deduction or by inference to the best explanation using sentences of type ≥(n + 1). For example, I know2 that Giggs scored the winning penalty in the 2008 Champions League final if I infer it from the facts that you know1 that it was either Giggs or Ronaldo, and that Ronaldo missed his penalty (see my paper (Paseau 2008) for more on this point). The most that the alleged reductio shows (assuming an untyped notion of necessity) is that, for a handful of sentences, this is true in all circumstances. Needless to say, this does not affect the great majority of statements, which are knowable at the next higher type. In particular, the first conjunct in standard Fitch cases (e.g. empirical claims of the form ‘the number of hairs on my head is n’) is typically knowable at the next knowledge type.

An analogy helps to articulate the point. We have come to realize that some true arithmetical sentences can only be derived in theories stronger than standard arithmetic. It is a fact of mathematical life that a true arithmetical sentence such as Goodstein's theorem, although expressible in Peano Arithmetic, is only provable in a theory stronger than it, e.g. set theory. This is in contrast to the great majority of mathematically interesting arithmetical truths, which are provable in Peano Arithmetic. Similarly, the proponent of typed knowability takes it to be a fact of typed life that for any n, a handful of sentences of knowledge type n can only be known at types higher than (n + 1), and more generally that there is no simple rule along these lines for the (lowest) type at which any sentence of type n is knowable at. This respects the core commitment of her view: everything is knowable, she believes, though not always at the next higher type.

How to express the proponent of knowability's commitment from within the perspective of the typed languages is a moot point. If existential quantification over types is not permitted, perhaps the knowability theorist can block the Fitch reasoning and assert any given instance of knowability, but not the full thesis. Or perhaps she can assert the full thesis, but schematically. Consider the knowability function k, which applies to any sentence and outputs the type at which the sentence is (first) knowable. Perhaps the knowability theorist may convey her view by endorsing the schema ‘pPKk(p)p’. If the procedures for determining how one might come to know any given proposition are sufficiently determinate, the function ‘k’ can be understood in these terms and knowability is then conveyable by means of the schema. And if every sentence is indeed knowable, k is a total function and the schema is acceptable.7

An opponent might argue that, on the contrary, the procedures for determining how one might come to know an arbitrary proposition are not sufficiently determinate to allow us to grasp the purported knowability function k, so that the schema ‘pPKk(p)p’ is equivalent to an existential generalization over types. However, no dialectical ground has now been gained. The knowability proponent's claim has always been that we have good reason to think that any given truth is knowable. Whatever reasons ground her confidence in that claim may now be wheeled out to respond to the worry that our grip on the function k is not sufficiently determinate to allow her view to be captured by the suggested schema. This is simply the original debate reapplied to this context. After all, underlying her position is the thought that we have reason to believe that for any given proposition there are procedures such that if we were to follow them we could come to know it. Arguably, then, on her view we can grasp the function k without quantifying over types and indeed we have reason to think it is total.

In any case, however the knowability theorist's commitment in a typed logic is expressed or conveyed, the point is that this commitment falls short of the maximally strong thesis Halbach assumes on her behalf.

3. Conclusion

I have given two reasons for rejecting Halbach's alleged reductio, both of which emerge naturally from the perspective of typed knowability. The first is that cross-typing restrictions between necessity and knowledge block his reasoning without apparently damaging knowability. Halbach's reasoning anyway assumes principles unacceptable on any predicate conception, as he elsewhere acknowledges. The second is that he assumes too strong a version of knowability: one can maintain that any given proposition is knowable without assuming that it is knowable at the very next knowledge type.8

Halbach calls knowability ‘weak verificationism’; I shall stick to my original terminology.
Either in the sense that such sentences are not well-formed or are always false.
The subjects in question are logically omniscient, a standard assumption in epistemic logic.
Briefly: operator treatments lead to expressively weak logics, hence the hope that a viable predicate logic of knowledge and necessity can be found.
He remains neutral on which of the two is the culprit or whether it is only their combination that is responsible for the inconsistency.
To keep the discussion simple, we ignore the typing of the necessity predicate in this section, so we do not subscript ‘N’ or ‘P’.
Note that the argument cannot be resurrected using ‘γ↔¬PKk(γ)⌜γ⌝⌝’. To run the diagonal argument, we fix an n and then diagonalize on the predicate ‘¬PKn(x)’ to find a γ such that ‘γ↔¬PKn⌜γ⌝⌝’. This γ is such that k(γ) ≠ n.
I am grateful to Bruno Whittle and Tim Williamson for comments.


On a side effect of solving Fitch's paradox by typing knowledge
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Possible words semantics for modal notions conceived as predicates
Journal of Philosophical Logic
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Fitch's argument and typing knowledge
Notre Dame Journal of Formal Logic
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