A logical challenge to correlationism: the Church–Fitch paradox in Husserl’s account of fulfilment, truth, and meaning

Abstract

Husserl’s theory of fulfilment conceives of empty acts, such as symbolic thought, and fulfilling acts, such as sensory perceptions, in a strict parallel. This parallelism is the basis for Husserl’s semantics, epistemology, and conception of truth. It also entails that any true proposition can be known in principle, which Church and Fitch have shown to explode into the claim that every proposition is actually known. I assess this logical challenge and discuss a recent response by James Kinkaid. While Kinkaid’s proposal saves one direction of the parallel for semantics, it gives up the parallelism for truth. I spell out a different response which meshes naturally with Husserl’s account of meaning. If the parallelism is restricted to a class of basic propositions, the truth of non-basic propositions can be defined inductively, without leading to the paradox. I then discuss objections that have been raised against a similar proposal by Dummett. The result is that exegetically plausible and popular interpretations of Husserl’s correlationism are indeed challenged by Church and Fitch. But when taking into account the ‘logical adumbration’ of propositional blindspots, truth and possible fulfilment can be connected without paradox.

1 Introduction

As the stage empties, a name tag remains behind: ‘the president’. It refers to the same woman that everyone in the auditorium has just encountered as a charismatic leader. But while reading the name tag represents her emptily, the audience had an intuitively rich experience of the president ‘in the flesh’. Such pairings¹ between empty and fulfilling acts play a central role throughout Husserl’s phenomenology. In an 1893 manuscript, Husserl writes that symbolic representation exhibits a ‘deficit’, striving for fulfilment (Husserl, 1979, p. 292; p. 411; cf. Husserl, 2009, p. 39; Husserl, 1891, p. 363; Husserl, 1901, VI, §21, p. 607). Eight years later, in the breakthrough Logical Investigations, the pairing of empty and fulfilling acts continues to play a key role (Husserl, 1901, I, §§9f; also VI, §13; also Husserl, 2002b, p. 88) and forms the basis for Husserl’s account of perception (Madary, 2012, p. 148). As Rang puts it, conscious representation has to be based on and oriented towards a complementary, fulfilling intuition (Rang, 1975, p. 111). This pairing leads to different correlation theses for Husserl’s semantics, notion of truth, and epistemology and even serves as a guiding principle for phenomenology itself. A central commitment already before Husserl’s development of the phenomenological reduction and his endorsement of transcendental idealism, but also retained afterwards, the pairing and the correlation principles that follow from it are relevant for philosophers in Husserl’s tradition more broadly, including those who emphasize only the earlier or later period of his work.

To discuss this correlational metaphysics more precisely, I distinguish semantic and alethic correlations in terms of real and ideal possibilities of fulfilment. The limit of fulfilment is evidence (to be stated more precisely in the next section), and evidence is the strictest form of knowledge. That any true empty representation can be fulfilled therefore implies that every true proposition can be known. But from here it follows, as Church and Fitch have shown, that every true proposition is known: this is either called the paradox of knowability, or simply a philosophical “embarrassment” for those who take truth to be essentially knowable (Williamson, 2000, p. 271) . I spell out the Church–Fitch paradox and how it affects Husserl’s account before considering two ways to avoid such embarrassment. Kinkaid (2022) defends the semantic correlation by turning it into a thesis about imagination, which separates it from commitments about possible knowledge. While this avoids the Church–Fitch paradox, it also either drops or trivializes the correlation between truth and the possibility of evident fulfilment. To retain a substantive characterization of truth, I propose a restriction of Husserl’s correlation principles to a class of basic propositions. This approach fits naturally with Husserl’s inductive characterization of meaning and extends Husserl’s notion of ‘adumbrations’ to the treatment of propositional blindspots. The phenomenological context also allows to avoid objections to a structurally similar proposal that was formulated by Dummett (2001). The discussion shows not only how an embarrassment from paradox can be avoided, but also how the use of symbols extends our representational capacities.

2 Husserl’s pairing of empty and fulfilling acts

The pairing between empty and fulfilling acts leads to different correlation theses in Husserl’s conception of semantics, truth, and epistemology. And because the pairing ensures that the meanings of words can be clarified by intuition, it even underpins the phenomenological battle cry ‘to the things themselves’:

We can absolutely not rest content with ‘mere words’, i.e. with a merely symbolic understanding of words [...]. We must go back to the things ‘things themselves’. We desire to render self-evident in fully-fledged intuitions that what is here given in actually performed abstractions is what the word-meanings in our expressions of the law really and truly stand for. (Husserl, 1900, Introduction to vol. 2, §2, p. 168)

Wir wollen uns schlechterdings nicht mit “bloSSen Worten”, das ist mit einem bloSS symbolischen Wortverständnis zufrieden geben [$\dots$]. Wir wollen auf die “Sachen selbst” zurückgehen. An vollentwickelten Anschauungen wollen wir uns zur Evidenz bringen, [$\dots$] was die Wortbedeutungen im Gesetzesausdruck meinen [$\dots$]. (Husserl, 1901, Introduction, §2, p. 10)

We can absolutely not rest content with ‘mere words’, i.e. with a merely symbolic understanding of words [$\dots$]. We must go back to the things ‘things themselves’. We desire to render self-evident in fully-fledged intuitions that what is here given in actually performed abstractions is what the word-meanings in our expressions of the law really and truly stand for. (Husserl, 1900,Introduction to vol. 2, §2, p. 168)

This can be spelled out in terms of two different principles of correlation: one concerning the meanings, the other concerning truth. As long as our representations allow for intuitive fulfilment, we do not step beyond the boundaries of authentic thought; and within these bounds, we think about possibilities, not just words. If we want to find out whether our authentic thoughts are, in fact, true, we seek out what actually fulfils them. This metaphysical picture has come to be called ‘correlationism’ (originally in Beck, 1928, influentially in Meillassoux, 2006), but also ‘ideal verificationism’ (Smith, 2003, p. 186f.; Hopp, 2018, p. 632). These principles appear long before Husserl’s theory of the reduction and exert their influence also on philosophers who are cautious about Husserl’s later transcendental framework.

In semantics, the principle of correlation determines what constitutes a consistent representation. The basic unit in Husserl’s theory of meaning is not the written or spoken sentence, but act kinds that they motivate.² One can see the symbols of a foreign script without understanding, but when they are interpreted, we pass right through to what they represent. There is no intermediary step of visualizing (or even seeing) what one is reading about: written language offers a determinate content while remaining intuitively empty (‘sense-giving’ acts in Husserl, 1901, I, §14, p. 56). So we should distinguish the three levels of symbol, empty representation, and intuitive fulfilment. A perceptual act presents the symbols “a blackbird flies away”, which motivates an empty representation. That empty representation then can be fulfilled in the intuitive act of perceiving said blackbird.

Some perceptions of the blackbird are fuller than others, but there need not be a single act type that provides the most fulfilment to a representation. “A blackbird flies away” can be fulfilled by perceptual acts that present the blackbird from different angles. What determines what a particular empty act represents may therefore not be a single best fulfilling act, but a set of best fulfilling acts. What kind of acts are in this set depends on what is talked about; it may involve empathy, the understanding of a proof, or introspection, among others (Sokolowski, 1974, pp. 18f. lists many examples). The same holds for cases that are far removed from sensory perception. While sensory perception can present individual objects, such as a knife or a table, also the proposition ‘the knife is on the table’ can be intuitively fulfilled. This kind of fulfilment is not achieved by the senses alone, but requires categorial intuition. Categorial intuitions can be built atop other categorial intuitions, and ultimately account for fulfilment of propositions about abstract states of affairs, such as mathematical identities (Husserl, 1901, VI, §§40ff.), like $a+b = b+a$. Here again, we see the pairing between empty and full acts: either the equation can be understood emptily, as symbols that can be manipulated according to a set of rules that we remember; or the validity of that identity can be fulfilled in categorial intuition.

Using symbols for empty representations has the great benefit that we can recombine them into new representations. But how do we know that we created empty representations and not just a new graphical ornament? For written language, the rules of grammar determine which symbolic combinations lead to sentences. A sentence is more than an ornament, but this can fall short of representation in different ways. On the one hand, we can form grammatically correct sentences that are not meaningful, such as ‘he laughed asymptotically’ because it is not even clear how its parts combine. On the other hand, there are contradictions, like “A and not A” or “there is a triangular square”. While contradictions are not meaningless, they fail to express a “real” meaning (Husserl, 1901, I, §15, p. 61): that is, whatever meaning is attached to their symbols cannot correspond to a fulfilling act. With this further requirement, Husserl distinguishes authentic (eigentlich) from inauthentic (uneigentlich) thought (Husserl, 1901, VI, §63). Inauthentic thought moves in sentences that are grammatical, so not strictly meaningless. But inauthentic thought cannot be fulfilled. Authentic thought, on the other hand, recombines only those empty representations that also respect the possibility of their fulfilment. That all consistent propositions allow for their fulfilment is ensured by pure laws of meaning.

Die reinen Gesetze der Gültigkeit der Bedeutungen, der idealen Möglichkeit ihrer angemessenen Veranschaulichung, laufen offenbar den reinen Gesetzen parallel, welche die Verknüpfung und Umwandlung ihrer eigentlichen kategorialen Formen regeln. (Husserl (1901), VI, §63, p. 723)

The pure laws of the validity of meanings, the ideal possibility of their adequate intuitive illustration, obviously runs parallel to the pure laws governing the combination and transformation of authentic categorial forms. ( Husserl, 2001a, VI, §63, p. 313)

This is the semantic correlation. That a proposition is consistent means the same as it being metaphysically possible (Husserl, 1901, VI, §30, p. 633), only that ‘consistency’ emphasizes the relationship between partial meanings, and possibility concerns the compound meaning as a whole, together with its (intentional) object (Husserl, 1901, VI, §63, 117). The semantic correlation therefore expresses a metaphysical commitment: an empty representation without the possibility of a fulfilling act does not represent a metaphysical possibility.

Beyond semantics and modality, fulfilment also serves as the defining notion for Husserl’s theory of truth and evidence.

Die letzte Erfüllung repräsentiert ein Vollkommenheitsideal. Sie liegt allzeit in einer entsprechenden “Wahrnehmung” (wobei allerdings eine notwendige Erweiterung des Wahrnehmungsbegriffs über die Schranken der Sinnlichkeit hinaus vorausgesetzt ist). Die Erfüllungssynthesis dieses Falls ist die Evidenz oder Erkenntnis im prägnanten Wortsinn. Hier ist das Sein im Sinne der Wahrheit, der recht verstandenen “Übereinstimmung”, der “adaequatio rei ac intellectus” realisiert, hier ist sie selbst gegeben, direkt zu erschauen und zu ergreifen. (VI, introduction, p. 540)

Th[e] final fulfilment represents an ideal of perfection. It always consists in a corresponding percept (we of course take for granted a necessary widening of the notion of perception beyond the bounds of sense). The synthesis of fulfilment achieved in this limiting case is self-evidence or knowledge in the pregnant sense of the word.Here we have being in the sense of truth, ‘correspondence’ rightly understood, the ‘adaequatio rei ac intellectus’ is itself given, to be directly seized and gazed upon. (Husserl, 2001a, VI, introduction, p. 185)

We can “live through” the truth when an empty act represents an object in the identical fashion that a fulfilling act presents it. This limiting case of fulfilment is what Husserl calls ‘evidence’ or ‘knowledge in the pregnant sense of the word’ (cf. Husserl, 2003, p. 159). To call an empty representation true means that such fulfilling acts could be realized:

[D]ie Richtigkeit des Urteils im logischen Sinn des Satzes: der Satz “richtet” sich nach der Sache selbst; er sagt, so ist es, und es ist wirklich so. Darin ist aber die ideale, also generelle Möglichkeit ausgesprochen, daSS sich überhaupt ein Satz solcher Materie im Sinne strengster Adäquation erfüllen läSSt. (Husserl, 1901, VI, §39.4, Husserl, 1986cf. pp. 108, 120, Husserl, 1929b, §90),

[T]he proposition ‘directs’ itself to the thing itself, it says that it is so, and it really is so. In this we have the expression of the ideal, and therefore general, possibility that a proposition of such and such a ‘matter’ admits of fulfilment in the sense of the most rigorous adequation. (Husserl, 2001a, VI, §39.4)

Adequate fulfilment can only present the truth (and this itself is evident, Husserl, 1901, VI, §39, p. 656). The notion of evidence is hence not a probabilistic relation between propositions, as for the Bayesian, and it is not a feeling that accompanies judgements when they are justified. Evidence refers to a special class of acts in which the intended object is present and given how it actually is. What I am focusing on here is the principle that connects truth with possibilities of evident fulfilment. Such an alethic correlation has been ascribed to Husserl before (Tugendhat, 1970, p. 50f.; Soffer, 1991, pp. 75, 89; Smith, 2003, p. 186f.; Bernet, 2003, p. 165; Hardy, 2013, p. 100; Erhard 2014, p. 102; Zahavi, 2017, p. 71f.; Mulligan, 2017, p. 93; Hopp, 2020, p. 122) and it might be implicit in influential characterizations of a phenomenological conception of truth (Sokolowski, 2000, p. 159; Jacobs, 2016, p. 259). I offer an explicit statement of the semantic and alethic correlations below. But since Husserl’s conception of evidence has been much discussed—in particular, with respect to its changes—I first consider the alethic correlation after the Logical Investigations.

A main development is that the first notion of evidence is an act that identifies the content of empty and fulfilling acts (Husserl 1901, VI, §39). Husserl later calls the fulfilling act itself ‘evidence’ and gives it a form of priority.³ The later work also discusses that different subject matters might be given in acts of evidence with different structures (Husserl, 1939, p. 11f.). But these are considerations about distinctions between different types of evidence and their structure; they are orthogonal to the idea that every true empty representation has a (set of) complementary fulfilling presentations.

Husserl’s notions of adequate and apodictic evidence have also created much discussion (at book-length in Levin, 1970). I now have to briefly introduce some of these technical debates in Husserl scholarship to show that the alethic correlation, despite Husserl’s progression in these topics, continues to play an important role in his later work. First, this concerns the role of apodictic evidence; second, Husserl’s late transcendental foundation of logic, and third, the fallibility of evident judgement. To introduce the terminology, consider that fulfilment can be partial. A quick glimpse of the blackbird provides less complete fulfilment of ‘a blackbird flies away’ than undisturbed observation. Such increasing degrees of fullness specify a partial ordering of acts, at the limit of which Husserl locates adequate evidence (Husserl, 1901, VI, §16, p. 598). In the mathematical sense of ‘limit’, a series of numbers with the limit $\pi$ approximates $\pi$ to arbitrarily small differences, while $\pi$ never appears in that series. This is how adequate evidence can be the limit for evident experiences, even if there are always further perspectives on the same blackbird that would add to what we have seen so far (cf. Husserl, 1913, §143). Apodictic evidence on the other hand is such that we cannot conceive of a reason to doubt it (Husserl, 1929a, §6), similar to what other philosophers might call ‘certainty’. We may for example wonder whether far-fetched thought-experiments with brains in vats constitute such reason to doubt, and whether this undermines apodicticity. In the search for the ultimate foundation of knowledge, it becomes relevant whether apodictic evidence is possible without adequacy and whether it can be achieved at all (Heffernan, 1998, p. 62). But I am concerned here with the idea that the emptiness of a representation can always be fulfilled. This is compatible with the idea that the best possible fulfilment does not realize the ideals of adequacy or apodicticity, and it is compatible with not knowing whether any given fulfilment allows further improvement.

One may still wonder whether Husserl’s Formal and Transcendental Logic reveals a naïvety in formal logic that prohibits an explicit statement of the alethic correlation. Husserl considers formal logic a positive science, insofar as it presupposes the reality of the world and that there are absolute truths (Husserl, 1929b, §105). This creates a critical distance between Husserl and the notion of truth as it is used in a formal framework. My considerations, however, are focusing on Husserl’s understanding of propositional truth, not what he says about subpropositional relations of fulfilment. And even in the Formal and Transcendental Logic, truth remains centred around the ideal of adequation between empty intention and fulfilling intuition. In the case of formal logic, the principle of excluded middle is explicated as the claim that any meaningful statement can either be brought to adequate fulfilment or is evidently disappointing, a fulfilment of its negation (Husserl, 1929b, p. 90). Transcendental logic is ‘critical’ of such assumptions about truth, in the sense that it suspends the presumption of objective validity.

It is no small feat to draw out the consequences of suspending that presumption. At the idealistic end of possible interpretations, it might appear that to consider a judgement that p true, it has to be matched to the subjective activity which made it so, that p is true (Husserl, 1929b, §46). This would constitute a sea change about what underwrites the truth of our judgements—no mind-independent but knowable states of affairs, but sedimented activities of transcendental consciousness—but despite this change in truthmakers, truth would still be spelled out in terms of possible fulfilment. If anything, the connection would be even closer, since possible fulfilment were then not only correlated with truth, but constitute it like bricks constitute a wall. So even in this extreme case, any truth of an empty representation would remain coordinated with a fulfilling presentation. With a small exception, this will be all I wish to say here about transcendental logic and transcendental idealism.⁴ The rest of this essay operates at a level of naivety that is shared with formal logic and the empirical sciences, to discuss a consequence of Husserl’s thought at that level. This does not mean that there is an incompatibility with the Formal and Transcendental Logic: the critical project does not aim to ‘surrender’ (preisgeben) what formal logic and empirical science presuppose (cf. Husserl, 1929b, §80, p. 206).

Formal and Transcendental Logic also contains a famous remark that might appear to question the assumption that evidence implies truth:

Selbst eine sich als apodiktisch ausgebende Evidenz kann sich als Täuschung enthüllen und setzt doch dafür eine ähnliche Evidenz voraus, an der sie “zerschellt”. (Husserl, 1929b, §58)

Even an ostensibly apodictic evidence can become disclosed as deception and, in that event, presupposes a similar evidence by which it is “shattered”. (Husserl, 1969, §58)

But the second half is crucial: evidence disappoints only where we need to have found better, more complete evidence. This rather reaffirms the idea that every truth is connected with a possibility of evident fulfilment: If purported evidence disappoints, this means that we have not yet reached the last rung in the ladder of improving fulfilments. So even in Formal and Transcendental Logic, truth entails the possibility of an evident fulfilment, and if purported evidence disappoints, there must be more complete evidence that does not disappoint. This is in no tension with the idea that the best fulfilment presents the truth—what I call the alethic correlation. This best fulfilment need not amount to apodictic or adequate evidence: seeing the blackbird can present the truth intuitively, even if there are hidden sides of that bird, so that the limit of adequate fulfilment still points beyond it. For the lack of a better term, let me call the best achievable level of fulfilment ‘evident fulfilment’.

Now we can explicitly state the two correlation principles. The semantic correlation Corr-Sem is about any consistently combined meaning. The alethic correlation Corr-Al is about any true proposition.

Corr-Sem:

A proposition is consistent if and only if it is possible that an act intuitively fulfils that proposition. (ideal possibility of evident fulfilment)

Corr-Al:

A proposition is true if and only if it is possible that an act intuitively fulfils that proposition. (real possibility of evident fulfilment)

Husserl also correlates the existence of individual objects with the possibility of directly experiencing them (Husserl, 2001b, p. 157), and holds that any existing object is the subject of true propositions (Husserl, 2003b, p. 146). Moreover, he endorses a correspondence between true propositions and obtaining states of affairs⁵ (Husserl, 1906, §14, p. 53; Husserl, 1900, p. 230f; Husserl, 2013, p. 78f; Husserl, 2001b, p. 154). Together with Corr-Al, this leads to an ontological correlation that connects possible fulfilment with obtaining states of affairs, which is a good candidate for interpreting Husserl’s transcendental idealism without a subjective creation of truthmakers (Husserl, 2003b, p. 146). But I want to proceed in smaller increments, maintaining distinctions between what we can represent, what we mean by truth, and whether these boundaries of our representation coincide with those of reality. It is possible, for example, to endorse a correlation between truth and possible experience without endorsing a similar correlation between mind and world (a similar point is made for semantic anti-realism by Wright, 1992, p. 159, for Peircean verificationism by Misak, 1995, p. 118, for Husserl by Hopp, 2020, p. 291). In what follows, I put aside the ontological correlation and Husserl’s transcendental idealism. The focus is rather on the pairing of empty and fulfilling acts, which Husserl endorsed before and after his explicit turn to transcendental idealism.

The distinction between a semantic and an alethic correlation immediately shows that ‘possibility of evident fulfilment’ has to mean different things. I here use Husserl’s terminology as follows: Ideal possibilities ($\Diamond _{I}$) are only constrained by essential laws and concern anything that could be the case. Real possibilities ($\Diamond _{R}$) are a subset of the ideal possibilities. Soffer (1991, p. 75) uses ‘real possibility’ to refer to possibilities of knowledge within (practical) reach. This use of ‘real possibility’ might be common in the literature, but it does not allow to distinguish possible fulfilment in mere phantasy and the positing acts of evident fulfilment (Husserl, 2002a, §48, pp. 178ff.). And with that distinction, we can also separate the semantic and alethic correlations.

Husserl’s official account characterizes real possibilities as those ideal possibilities that have been motivated (Husserl, 2020, p. 203; Husserl, 1973, §84, p. 292f. Husserl, 1913, §47). I think there are serious problems with this proposal (see Bös, manuscript) although it can perhaps be radicalised into a “rejection of the hypostatization of possibilities” (Zhok, 2016, p. 232), according to which actual experience creates the room for possibilities. Fortunately, this need not be decided here, because after illustrating what these notions of possibility amount to, they can be treated as primitives.

Husserl describes real possibility as ideal possibilities with an additional constraint from reality: and this additional constraint just seems to be the way that the world actually is (Husserl, 2002a, §61, p. 269; see also Husserl, 1973, p. 292f.). If there is one apple left on the tree but nobody is looking, it is a real possibility of evidence to look and see that there is a single apple left. When somebody says ‘there are two apples on the tree’, there is no corresponding real possibility of fulfilment, hence this statement is false. But seeing two apples remains an ideal possibility of fulfilment, which distinguishes this statement from the lack of fulfilment of a contradiction.

After setting aside the ontological correlation and an issue about a distinction between modal terms, we can look at a new challenge to Corr-Al, which will occupy us for the rest of this essay. Evident fulfilment, even where it does not achieve adequacy, is a particular way of knowing, namely knowing with intuitive fullness (Husserl 1900, §6, p. 28). If every true empty representation can be brought to evident fulfilment, this implies a fortiori a thesis about the scope of what can be known: for every true proposition, it is possible to know that proposition, namely by realizing an act of evident fulfilment. The Church–Fitch paradox can be formulated as an argument to the effect that such possible knowledge entails that every truth is actually known. I first introduce the paradox as an argument in modal epistemic logic, commenting on the justification for each step. Then I illustrate the structure of the argument in natural language before discussing ways in which it can be addressed.

3 The Church–Fitch paradox

What is known as the Paradox of Knowability goes back to Fitch (1963), who, in beautiful irony, credits it to an anonymous 1945 referee report for a paper that was never published. In 2005 this referee was identified as Alonzo Church (Salerno, 2009, pp. 13, 36). The paradox challenges the conceptual connection between truth and possible knowledge. Let Kp stand for somebody, somewhere, at some time knows that p, and let $\Diamond$ stand for a suitable notion of possibility.⁶ The schema according to which any true proposition can be known,

• Universal Knowability $p \rightarrow \Diamond Kp$

then collapses into the omniscience schema,

• Omniscience $p\rightarrow Kp$

according to which every proposition is known. Since these are schemata, arbitrary propositions can take the place of p.

The Church–Fitch paradox does not depend on the scope of the modal operator, so we can leave aside the earlier complications about the relation between ideal and real possibilities. The only assumptions are that knowledge is factive: $Kp \rightarrow p$ and distributive: $\forall \,p,q:K(p \wedge q) \rightarrow Kp \wedge Kq$. Factivity simply means that one can only know propositions that are true. Distributivity means that knowing a conjunction entails knowing each conjunct. With these properties for the knowledge operator, it follows that there are ‘no known unknowns’, in the specific sense that it cannot be known that a proposition is true and never known.

1.:

$K (p \wedge \lnot Kp)$ (assumption)

2.:

$Kp \wedge K \lnot Kp$ (1., distributivity of K)

3.:

$Kp \wedge \lnot Kp$ (2., Factivity of K. Contradiction)

4.:

$\lnot K(p \wedge \lnot Kp)$ (1.–3., discharging 1.)

5.:

$\Box \lnot K(p \wedge \lnot Kp)$ (1.–4., theorems are necessary)

6.:

$\lnot \Diamond K (p \wedge \lnot Kp)$ (5., duality of $\Box$ and $\Diamond$)

The properties of K alone make it impossible to know a proposition $p \wedge \lnot Kp$, for arbitrary p. Like a proof of $720 = 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6$, this derivation can always be made. The ‘theorem’ it establishes therefore holds necessarily, which was used in step 5. The last line uses the equivalence of ‘necessarily not’ and ‘not possibly’. With this ‘no known unknowns’ lemma, Omniscience follows quickly.

• 1. $p \rightarrow \Diamond Kp$ (Universal Knowability)

The schema can be applied to arbitrary propositions, including for any proposition p, the more complex proposition $p \wedge \lnot Kp$.

• 2. $(p \wedge \lnot Kp) \rightarrow \Diamond K(p \wedge \lnot Kp)$ (from 1.)

The lemma we just derived did not depend on what p was, so we can use it here as well.

• 3. $\lnot \Diamond K(p \wedge \lnot Kp)$ (‘no known unknowns’ lemma)

• 4. $\lnot (p \wedge \lnot Kp)$ (from 2., 3.; modus tollens)

• 5. $p \rightarrow \lnot \lnot Kp$ ($\rightarrow$-introduction)⁷

• 6. $p \rightarrow Kp$(from 5.; double negation elimination. Omniscience)

We have shown how the thesis that every true proposition could be known leads to the claim that every truth is, was, or will be actually known. This is a much stronger coordination between mind and world than what Husserl could have wanted to endorse. Even in his attempt to prove transcendental idealism, certainly one of the places where Husserl’s idealism is the most unadulterated, he writes that objects are possible “without my [thinking] of it or anyone else’s thinking of it” (Husserl, 2003b, p. 146). Such objects would make propositions true that nobody knows in fact (although they all could be known). Speculating about an omniscient being does not help either, since the same problem would arise for a reinterpretation of K as ‘known by a finite being, at some point, at some time’.

As an informal illustration of the Church–Fitch paradox, consider Bolzano’s blossoming tree (Bolzano, 1837, §25, p. 84; Beyer, 2019, p. 69). Imagine that in a long-abandoned monastery, we find a dying almond tree, dropping its last petal as we enter the courtyard. “Nobody knows how many petals were on this tree last spring”, we think to ourselves. This thought itself has its determinate truth or falsity because it corresponds or fails to correspond to a state of affairs: We do not know the number of petals, nor does anyone else. Also in Husserl’s terms, there is no issue with thinking of a state of affairs that makes judging a particular n correct, and judging any other number incorrect. But it is enough for the Church–Fitch paradox to arise. If there is a truth about the number of the petals on the tree, and it is also true that nobody knows, has known, or will know this number, these form an unknowable conjunction: a proposition that represents an obtaining state of affairs but cannot be known (and therefore cannot be known with evident fulfilment). In phenomenological terms, we have constructed a symbolic expression which can be interpreted as an empty representation, but which does not allow for evident fulfilment. Yet this is not a case of inauthentic thought, because no contradiction is involved: for some p$_{n}$, the empty representation has a corresponding state of affairs. Unlike Meillassoux’ (2006) much discussed argument from ancestrality, the Church–Fitch paradox is not restricted to scientific statements about time before sentient life, but arises for all kinds of trivial propositions. For any true proposition that nobody actually knows in past, present and future, there is a logical conjunction that nobody could possibly know. We do not need to go as far as the Big Bang; challenges to correlationism already arise from the myriads of trivial truths that we never cared to ask about.

It is a classical topic in Husserl’s phenomenology to criticize formalized thought and its deadends (Husserl, 1936, §9). But Husserl’s countermeasure, echoed in the return to ‘the things themselves’, precisely presupposes the pairing of empty and fulfilling acts that leads to the Church–Fitch paradox. So I do not think that Husserl’s account of formalization in the Crisis will suffice, we need to substantially engage with it. Most responses fall into one of three camps, they either

• 1. Revise the logical properties of the epistemic operator,

• 2. Reject the inference principles of classical logic, or

• 3. Restrict the scope of knowability.

The epistemic operator in the literature on the Church–Fitch paradox is either the knowledge operator K or a generalized factive and distributive operator. In Sect. 2 I introduced evident fulfilment as a best possible case of fulfilment which does not need to amount to adequate or apodictic evidence. I have taken evident fulfilment to imply knowledge of what is evidently fulfilled, but the paradox could be formulated analogously as long as evident fulfilment is factive (ensuring the truth of what is evidently fulfilled) and distributive (evident fulfilment of (p and q) entails the evident fulfilment of p and the evident fulfilment of q). In the phenomenological context, the first strategy would be to revise the logical properties of fulfilment so that the best possible kind of fulfilment does not imply truth.⁸ This might be motivated by the late Husserl’s concerns about the possible disappointment of what appears to be apodictic evidence.

But I think it is important to distinguish here between first and second order knowledge. Undeniably, we sometimes believe to know something, and it turns out that we were wrong. But the correlation theses are about what we actually judge, not whether we always know if we have judged evidently or not. How well introspection can determine our own epistemic position is a different question. For fallibility to be relevant to the Church–Fitch paradox, we would have to take the best possible cases of evidence to be compatible with the falsity of what they support. This is only possible if the notion of evidence is completely separated from the notion of truth. It would then be simpler to give up Corr-Al entirely, since any connection between truth and evidence would need to be explained differently.⁹

Husserl writes that evidence is knowledge ‘in the narrowest sense’ (Husserl, 1900, p. 29; Husserl, 2001a, VI, introduction, p. 185), and I take evident fulfilment to be a form of intuitively full knowledge. But ‘knowledge’ is undeniably factive: “He knows that I am tall, but that is not true” is either contradictory or shows a confusion of knowledge and belief. That this is not as obvious in “She judges evidently that I am tall, but that is not true” is due to the ambiguity of the world “evidence”. In Sect. 4 I discuss a sense in which a logical revision of fulfilment can save a version of Corr-Sem, but Corr-Al requires a factive notion of evident fulfilment.

Instead of revising the logical properties of evident fulfilment, one could also seek the flaw in the rules of inference of classical logic. Such a response has been pursued for example by Dummett (2007, 2009) and expanded by Bermúdez (2009). In the formulation offered here, only the double negation elimination in the last step is not intuitionistically valid. Dummett, in any case, takes the penultimate line to express an intended intuitionist characterization of truth: That a proposition is true implies that there is no proof that it cannot be proved (Dummett, 2007, p. 348). In other words, the truth of a proposition implies that coming to know that truth always remains open.

Affinities between phenomenology and intuitionist logic appear in the work of Hermann Weyl and Oskar Becker. Given their connection to Husserl, one may therefore wonder if the intuitionist response is attractive as an interpretation of Husserl’s metaphysics. If Husserl can be considered a mathematical platonist, then surely not of a ‘naïve’ kind, as he attempts to spell out the metamathematical conditions under which mathematicians can symbolically construct new domains (see the double lecture, Husserl, 1891, pp. 430–451). But he understands mathematics and logic to study abstract objects and ideal states of affairs (Kinkaid, 2022, p. 1017). And the principles of formal logic that he endorses are classical, including the elimination of double negation (Husserl, 1901, VI, §34, p. 642; Husserl, 2001b, p. 243). Even Formal and Transcendental Logic reaffirms, at the formal level, the principle of excluded middle (Husserl, 1929b, §79; Mohanty, 1999, p. 192f.). Whereas Dummett’s intuitionist response fits with his wider revision of logic and his programme in the philosophy of mathematics, it would lack motivation in the context of Husserl’s philosophy.

A restriction, on the other hand, is less invasive. The general idea is that the paradox arises from the unrestricted quantification over all propositions, whether formulated as a principle for knowability or possible evidence. A main proponent of such a strategy is Tennant, who restricts the knowability thesis to propositions that can be known without contradiction (Tennant, 1997, p. 273). This kind of response is much more convincing if it does not just respond to a paradox, but follows from general principles. I spell out such a strategy in Sect. 5.

4 Fulfilment in Imaginative Illustration

Kinkaid observes that Husserl’s semantic correlation has a general problem with propositions which have come to be called ‘blindspots’ (after Sorensen, 1988). A blindspot proposition is consistent, but inaccessible to some propositional attitude. The most famous example is Moore’s “It is raining, but I do not believe it”, which is inaccessible to the propositional attitude of belief (Kinkaid, 2022, p. 1019). The Church–Fitch paradox exploits that $p \wedge \lnot Kp$ is a blindspot for knowledge. As a blindspot for knowledge, it is also a blindspot for evident fulfilment: so it is meaningful, possibly true, but cannot be judged evidently. In response, Kinkaid (2022, p. 1022) offers an elegant revised principle:

• K-Rev All consistent propositions can be intuitively illustrated.

This principle is weaker than Husserl’s original correlation between intention and fulfilment, which includes the ‘possible knowledge-use of meanings’ (‘mögliche Erkenntnisfunktion der Bedeutungen’, Husserl, 1900, I, §29, p. 227; Husserl, 1901, I, §29, p. 101). Kinkaid’s revision avoids the Church–Fitch paradox by substituting the factive possibility of evident fulfilment with the non-factive possibility of intuitive illustration.¹⁰ Intuitive illustration can happen through non-positing acts, such as imagining a state of affairs that would make the proposition true (Kinkaid, 2022, p. 1023). Whereas evident fulfilment vouches for what it presents, acts of imagination do not in general have such “positing” character. That imagination is not a positing act also means that imagination is not factive. In terms of the strategies outlined in Sect. 3, this is similar to a revision of the logical operator on the right-hand side of the correlation: the truth of a proposition entails that an act with that propositional content is possible, but this act has no bearing on what is the case.

The positing character of evidence concerns object and subject of experience alike. ‘I do not exist’ or ‘There is an unimagined tree’ can be intuitively illustrated by imagining a world in which there is nobody or where there is a tree but nobody who imagines it. One could describe this process as a separation of the phantasizing from the phantasized ego, which in phenomenology has been discussed as ‘ego-splitting’ Cavallaro (2017).¹¹

This is an interesting lesson about the scope of imagination, but it also reverses a priority between evidence and imagination. The official account has it that the non-positing acts of phantasy are understood as a modification of the positing acts of evidence (Husserl, 1996a, p. 115; Husserl, 1901, VI, §30, p. 633; Husserl, 1980, p. 590; Husserl, 1913, §104; Tugendhat, 1970, p. 40; Hopp, 2020, pp. 238, 291). Imagination allows us to learn about possibility because we can imagine what kind of evidence we could have. We are able to imagine the kind of evidence we would have if the door were green, even if the actual door is red. What we have really imagined is the possibility to have evidence that the door is green ($\Diamond Ep$, with $E$ as a propositional operator for ‘It is evident, that’). Since the evident judgement that p implies the truth of p, it follows that p is possible ($\Diamond Ep \vdash \Diamond p$). Propositional blindspots, however, show that this cannot be the full story. There are propositions we understand but that cannot be posited in an evident act: $\Diamond (p \wedge \lnot Ep)$ but not $\Diamond E(p \wedge \lnot Ep)$. Thus, blindspots illustrate that symbols can be combined into meaningful empty representations that we could not have arrived at by positing a state of affairs and then neutralizing that commitment.

Kinkaid interprets Corr-Sem along the lines of Dummett’s manifestation requirement. Its role is to ensure that we can demonstrate our understanding of the concepts we use by a manifestation of their meaning (Kinkaid, 2022, p. 1022). Intuitive illustration is a less demanding manifestation than evidential fulfilment. This avoids counterexamples from blindspot propositions for evidence, but it also drops the correlation between truth and possible fulfilment (together with any claim about the scope of knowledge). In short, Kinkaid’s revision of the semantic correlation does not save Corr-Al from paradox. But could the proposal offer a restriction that saves a version of Corr-Al? Kinkaid already states that he intends to “restrict ideal verificationism to acts to which the expression is accessible” (Kinkaid, 2022, p. 1019), where ideal verificationism is a label for Husserl’s semantic and alethic correlations. Although this phenomenological “restriction strategy” (Kinkaid, 2022, pp. 1012, 1018) does not restrict the domain of propositions over which Corr-Sem quantifies, Kinkaid has already identified blindspots as a class of problematic propositions. One could think that this entails a restriction strategy for Corr-Al: all true propositions have a possibility of evident fulfilment, except for blindspot propositions.

On considering the definition of a blindspot, however, this collapses into triviality. A proposition p is a blindspot with respect to an attitude A if p is consistent but inaccessible to A (Kinkaid, 2022, p. 1020). Sorensen writes a little more, stating that a proposition p is a knowledge blindspot for S at t if and only if p is consistent, true, the subject S is absolutely epistemically thorough at time t, and yet p cannot be known by subject S at time t (Sorensen, 1988, p. 52f.). Let us consider the blindspots that hold for all subjects at all times. If we exclude propositional blindspots from the correlation between truth and evident fulfilment ($Ep$), this would exclude all propositions for which $\lnot \Diamond Ep$. But as a restriction of the left-right direction of alethic correlationism, this yields:

$$\begin{aligned}\forall p: p \rightarrow \Diamond Ep \text { except when } \lnot \Diamond Ep\end{aligned}$$

which is not more informative than ‘I can lift every weight, except for the weights that I cannot lift’—nothing to boast about. Maybe this could be refined by distinguishing the modalities involved. In Sorensen’s account, the notion of possibility in question is determined by ‘background constraints’ that can include the laws of logic and physics or principles of psychology. But neither Kinkaid nor Sorensen get more specific than this, and a puzzle would remain in any case: if alethic correlationism characterizes an essential link between evidence and truth, why would this characterization not extend to true blindspot propositions? Fortunately, there is also a different restriction strategy for Corr-Al which grows naturally out of Husserl’s semantics.

5 A phenomenological restriction strategy and an inductive characterization of propositional truth

Husserl often focuses his investigation on the study of the ‘basic kinds’ of a domain. To elucidate the meaning of complex scientific statements, we need to follow a ‘guideline’ (Leitfaden) of basic objects (Husserl, 1996, p. 329; Husserl, 2003a, p. 222). The parallelism between empty acts and the possibility of fulfilment is a case in point. Husserl first affirms the parallel between symbolic representation and fulfilment for the symbolic connectives, which are ‘primitive types’ of categorial intuition (Husserl, 1901, VI, §63, p. 721). Once we have established such primitive types, they can be applied to arbitrary empty representations, such as ‘there is a dog’ and ‘there is no dog’. Husserl here separates grammatical correctness from possibility in terms of possible fulfilment. Grammatically correct but unfulfillable representations, like ‘A and not A’ are ‘inauthentic’ (uneigentlich) meanings. This should concede that we can distinguish between different kinds of contradictions (like Marty argued), so possible fulfilment is not really what determines meaningfulness (Husserl, 1901, I, §15, p. 61). But Husserl considers the fulfillable meanings to be those that are not contradictory; which for him means that they describe possible states of affairs. This requires that the possibility of fulfilment is not only important for justifying that a connective like ‘and’ can connect arbitrary propositions, but also that there will be corresponding complex fulfilments wherever that combination is not contradictory. And this is what we find in Husserl’s ‘ideal of a logically adequate [angemessenen] language’, for which the contradiction-free composition of empty meanings follows exactly the rules by which fulfilling presentations can be combined (Husserl, 2001a, VI, §63, p. 311, Husserl, 1901, VI, §63, p. 721).

Even before considering the Church–Fitch paradox, there are reasons to wonder if this more ambitious parallel should really obtain. Simple examples involve quantification over finite domains. Trapped in a hut during a hail storm, one may wonder whether all windows are still intact (Hopp, 2018, pp. 636f.). But no matter how fast one moves around, checking one window always precludes checking another. While empty representations about each individual window can be fulfilled, these fulfilments are not compossible. Nevertheless, there is no difficulty in symbolically representing all of this in the single sentence ‘all the windows are unbroken’, and of course this representation is not contradictory. While the hail storm case leaves open whether some kind of idealization or a fanciful mirror setup would allow for a single intuition to fulfil the empty intention, the Church–Fitch paradox takes this objection to an extreme where fulfilment is clearly impossible.

So there is reason to doubt the strict parallelism between the composition of empty meanings and intuitive fulfilment. But this parallelism is not always expressed so rigidly. In the 1905 lecture on meaning, Husserl allows en passant that a symbolic act may have its intuitive counterpart not in a united act, but a bundle of acts:

[Es] besteht die Eigentümlichkeit, dass zu jedem symbolischen Akt ein ihm entsprechender oder ein Bündel ihm entsprechender intuitiver Akte aufzuweisen ist derart, dass beiderseits dieselbe Gegenständlichkeit, und in völlig gleichem Sinn vorstellig ist. (Husserl, 2002b, p. 102)

[There is] the peculiarity, that for each symbolic act, we can present [aufweisen] a corresponding act or a bundle of corresponding acts, such that in both cases, the same object is represented, and it is represented in a completely equal manner. (translated by the author)

Such ‘bundle parallelism’ suggests that not every symbolic representation can be fulfilled in a single act; thus, there might be incompossible presentations of an emptily represented state of affairs. And in other contexts, phenomenologists take this for granted. In the phenomenology of perception, it is perhaps the central idea that the same intention (‘the house is red all over’) is only fulfilled in adumbrations, meaning that the fulfilment of one side of the house precludes the fulfilment of another. What the Church–Fitch paradox reveals is that there can be a structure of ‘logical adumbration’: rather than the perspectival nature of perception, it is the logical structure of a proposition that precludes its fulfilment in one blow. What I am proposing here is such a ‘bundle parallelism’ between propositions and their fulfilment that takes these logical adumbrations into account. This makes it easier to account for the possible fulfilment of logically complex statements (cf. Tugendhat, 1970, p. 135f.). For a house with six windows, the sentence ‘all windows are unbroken’ would not need to be fulfilled in a single act with content $p_{1}\wedge \dots \wedge p_{6}$, but in a bundle of acts that have the contents $\{p_{1}, \dots , p_{6}\}$. The parallelism between empty and fulfilling acts is then not understood as a claim about all symbolic representations, but as a thesis about the elements that can be combined. This leads to a different interpretation of the alethic correlation.

• (I) A basic proposition p is true if and only if it is a real possibility that an act evidently fulfils p.

Regarding the Church–Fitch paradox, this constitutes a phenomenological restriction strategy, in the sense that it restricts the scope of “all” in the correlation between truth and the real possibility of fulfilment. The truth-conditions for non-basic propositions need to be supplied by further principles; but these are anticipated in Husserl’s lectures on logic. Under the label propositional mathesis, Husserl develops a theory of all truths that are based in the concept of proposition.¹² Like algebra contains rules for the transformation of mathematical statements, the propositional mathesis spells out the rules by which propositions can be combined into new propositions. These compositional principles can also spell out how the truth of complex propositions depends on truths that are related to possible evidence. Nowadays one might say that they define truth-functionally complex propositions. Husserl speaks of a ‘functional form’ that does not change the meaning of the propositions on which it operates, but ‘functionally enriches them’ (Husserl, 2003a, p. 208). The liberal reformulations here follow the lectures on logic and the theory of judgement (Husserl, 2001b, p. 239f.; see also Husserl, 1896, §§36ff.; Husserl, 2002b, p. 152ff.).

• (II) A conjunction $p \wedge q$ is true if and only if p and q are true.¹³

• (III) A disjunction $p \vee q$ is true if and only if at least one of p and q is true.¹⁴

• (IV) A hypothetical $p \rightarrow q$ is true exactly if p, then q.¹⁵

• (V) A negation $\lnot p$ is true if and only if $p$ is not true.¹⁶

• (VI) An existential quantification $\exists ^{D} x: p(x)$ is true if and only if the domain D of the function $p(x)$ ranges over at least one value of x that yields a true proposition.¹⁷

• (VII) A universal quantification $\forall ^{D} x: p(x)$ is true if and only if the domain D of the function $p(x)$ ranges only over values of x that yield true propositions.¹⁸

By adding these additional clauses, Corr-Al can be restricted to a class of basic propositions, while leaving the truth of complex propositions well-defined. In particular, this means that $p \wedge \lnot Kp$ is not in the class of basic propositions, and therefore need not be fulfillable, and therefore step 4 in the Church–Fitch argument fails. The truth of ‘all windows are unbroken’, and Tugendhat’s worry about sentences like ‘there is no devil’ are also taken care of (1970, p. 135f.). But the new distinction between basic and non-basic propositions deserves clarification. Basicness is an ambiguous concept, and we can distinguish syntactic, semantic, and epistemic notions of basicness that need not coincide. Syntactic basicness is based in the structure of the symbols that make up our sentences. Semantic basicness would need to identify meanings by which more complex meanings can be expressed. Epistemically basic propositions could be understood as those that are immediately justified or those that serve the justification of epistemically non-basic propositions. Which kind of basicness is operative in the phenomenological restriction strategy?

Occasionally, Husserl marks out simple sentences as those that have only non-sentential parts (see Husserl, 2001b, p. 200; Husserl, 2003a, pp. 200 and 101) “If God exists, the sinners will be punished” is a composite sentence, because its part ‘the sinners will be punished’ by itself is a sentence. Despite Husserl’s general use of ‘Satz’ to refer to propositions (see footnote 12), this seems to express a form of syntactic basicness. So one may wonder whether non-basic propositions are exactly those that are expressed in syntactically non-basic sentences.

But the relation between the structure of the expressing symbols and the expressed propositions is more complex. Take the sentence ‘Carl is a grandparent’. This could be understood to express either a simple proposition, or a hidden complex proposition, like ‘Carl is a parent and one person, to whom Carl is a parent, is a parent’. Arguably, we could understand grandparenthood entirely in terms of parenthood. But the symbols that form the sentence ‘Carl is a grandparent’ reveal no such logical structure. And a simple syntactic form can be introduced at will. If we call the grandparent of a cousin who is not your own grandparent your ‘grousin’, then ‘Carl is a grousin’ is syntactically simple. Therefore, syntactic simplicity can serve at best as a necessary condition for propositional basicness. Nevertheless, similar restriction strategies have been classified as ‘syntactic’ responses to the paradox (Brogaard and Salerno, 2019). This description is natural within a propositional calculus, where the basic propositions are represented by a single propositional variable (p, q, ...), and non-basic propositions can contain multiple propositional variables ($p \wedge q, \ldots$). But such a formal calculus has already decided which propositions are basic and which are not.¹⁹

Since the syntactic form of the expressing sentence is insufficient to determine whether a proposition is basic, we have to look elsewhere. Epistemological basicness does not seem to fit either, as there are complex propositions that are easier to find out than their constituents. It is easier to know with evidence that ‘either I carry the virus, or the test has malfunctioned’ than to find out about either of these disjuncts.

What remains is a notion of propositional basicness that depends on the meanings involved. Language evolves, and new terms can be introduced on the basis of old ones, and the traces of such complication can disappear. An apparently basic proposition can reveal hidden complexity, once we inquire into the origin of its meaning. Such inquiries, which have been discussed under the labels of genetic and generative phenomenology (Steinbock, 1995), do not aim to establish psychological facts about a subject or a community. They rather constitute a re-living of the creation of meanings that have been bequeathed to a community, and on which formal logic and empirical science are built (Husserl, 1939, §11, p. 47f.). That re-living can either result in criticism or the taking of responsibility for the inherited meanings, which otherwise constitute an ‘anonymous’ presupposition (Landgrebe, 1982, p. 76). The restricted alethic correlation holds that all propositions can either be intuitively fulfilled or they are traceable to intuitively fulfillable meanings. But since the notion of propositional basicness cannot be applied mechanically, for example by specifying a function on the representing symbols, it should be expected that there will be disputable cases. These disputes will concern what we consider evident fulfilment and how new terms are introduced, in particular through scientific theories. I do not think that a correlation between truth and basic propositions thereby puts theoretical language into an empiricist straitjacket, for example by taking basic propositions to be assertions about Carnapian ‘Ur-experiences’ (Carnap, 1928), but this is to be discussed elsewhere. Restricting Corr-Al to basic propositions makes it less demanding about knowability; any issues with the semantics for scientific terms would have arisen equally for unrestricted correlationism.

Instead of turning to such general issues, I want to use the rest of this paper to discuss specific objections about this particular way of addressing the Church–Fitch paradox. In the context of his intuitionist project, Dummett (2001) originally defended a similar restriction strategy, before simply accepting the argument’s penultimate step ($\forall p: p \rightarrow \lnot \lnot Kp$) as the desired intuitionist characterization of truth (Dummett, 2007). In the period that Dummett endorsed this restriction strategy, objections were raised that need to be considered here as well.

Dummett’s proposal is originally presented on merely two pages, which invites the suspicion that it constitutes an ad hoc response to the paradox, rather than an account that is motivated in general principles. Husserl on the other hand outlined his propositional mathesis before the paradox was known to him or indeed anyone else. The phenomenological descriptions consistently prioritize basic cases of experience, and a restriction of Corr-Al to basic propositions also addresses issues with propositions that are not related to the paradox, like ‘all windows are unbroken’. Restricting Corr-Al to a class of basic propositions is therefore well motivated in Husserl’s semantics, and not just a targeted response to the Church–Fitch paradox.

More substantial objections begin by asking whether “it is a real possibility that an act evidently fulfils p” is itself a basic proposition. In formal shorthand, we can ask how the possibility operators $\Diamond _{R}, \Diamond _{I}$ and an operator $E$ for evident fulfilment interact with propositions. Unlike the other logical symbols we considered, the truth of $\Diamond _{I} p$, $\Diamond _{R} p$ and $E p$ cannot be expressed as a function of the truth of $p$. Adding a supplementary clause in the style of (II)–(VII) therefore would not seem to help. But Husserl has written about different methods for finding truths about possibility. The use of imaginative illustration, as in Kinkaid’s revised principle, is one of them. One can also begin with the essential laws that ground necessary truths, and then consider anything that is not in contradiction with these laws to be possible (Husserl, 1996, §43, p. 218f.). There are different ways to spell this out, but one may end up articulating a way in which it could become evident that $\Diamond _{I} p$. This could then be used to give an account of the truth of $\Diamond _{I} p$, for example:

(VIII$^{\Diamond {}?}$):

A statement about ideal possibility $\Diamond _{I} p$ for a basic proposition p is true if it is a real possibility ($\Diamond _{R}$) that p is completely intuitively illustrated.

Formally, (VIII$^{\Diamond {}?}$) looks like an instance of (I). But it might be preferable to treat it as a supplementary clause, since considering $\Diamond _{I} p$ as a basic proposition leads to recursion for propositions like $\Diamond _{I}\Diamond _{I}p$, $\Diamond _{I}\Diamond _{I}\Diamond _{I}p$, etc. I also becomes apparent here that the use of imagination and eidetic laws will only help us to learn about possibilities, not contingent facts. What more can be said about real possibilities remains a different, and in my view, still open question.

We can treat $Ep$ analogously to $\Diamond _{I}p$, by introducing a supplementary clause about acts that make the truth of $Ep$ evident.²⁰

(IX$^{E?}$):

A statement about evident fulfilment Ep is true if and only if it is a real possibility that an act of reflection or empathy evidently fulfils that Ep.

Formally, this would mean that $Ep \leftrightarrow \Diamond _{R}E(Ep)$. This might appear like a strong internalist commitment, but this appearance can be lessened, if one is so inclined, by distinguishing the truth-conditions for evidence from the truth-conditions for believing to have evidence. Take the evident judgement for the proposition ‘There is an apple in front of me’ to be based on an act of perception. Whenever this apple-perception occurs, we could reflect on what we are visually experiencing and judge with evident fulfilment that we are judging with evident fulfilment. But this is compatible with there being other scenarios, where we only believe that our judgement is evidently fulfilled. All that we need is that, in the good cases, a reflection on that act of judgement is possible. That reflection, however, would be a separate judgement about one’s own state of evidence. And there is still room to consider what counts as evident fulfilment beyond acts of reflection. Acts of empathy, for example, could serve as evident fulfilment of sentences about evident judgment, even for creatures who lack conceptual or reflexive capacities. There remain open questions, but these arise equally for the unrestricted version of Corr-Al.

More specifically relevant is the suggestion by Brogaard and Salerno (2002) that a restriction to basic propositions leads to new paradoxes. Their ‘revenge paradox’ depends on an inference $\Diamond \Diamond E(Ep) \vdash \Diamond E(Ep).$ In terms of contemporary modal logic, such an inference principle holds if the accessibility relation between possible worlds is transitive. It is not trivial to translate such a property from possible-world semantics into the phenomenological way of thinking about possibilities. But ‘intuitive illustration’ as discussed in Sect. 4 seems to make good sense of it. If I read about Eddie’s childhood and how he liked to read stories about being a knight in shining armour, it seems that I have inuitively illustrated a possibility in which somebody is intuitively illustrating a possibility. The intuitive illustration of this possibility seems to require filling in Eddie’s intuitive illustrations as well. Therefore, intuitive illustration is transitive: I could as well switch to imagine what Eddie is imagining; whereas I am at a loss when I try to illustrate ‘Eddie imagined something that nobody could imagine’. This suggests that $\Diamond _{I}\Diamond _{I}p \vdash \Diamond _{I}p$. What is possible in a possible world should also be possible in the actual world.

But only ideal possibilities are transitive in this sense—we have already seen that the real possibilities of evidence work differently. The possible worlds where we know more are subtly or not-so-subtly different from the actual world. Think of a pupil in a chemistry class. The instruction is to measure the concentration of ions in a liquid, and to determine if the saline solution has been saturated. The pupil knows that his solution is saturated when any added salt settles on the ground, rather than dissolving. As it happens, the solution is not saturated. It is a real possibility to trickle salt into the liquid and watch it disappear. At the same time, the student could also measure the saline concentration and read a value $\rho _{a}$ off the display: so it is a real possibility to determine that the density of solved particles in the solution is $\rho _{a}$. Now, given that it makes sense at all to talk about the real possibilities that obtain “within” or “from” other real possibilities, then the student who just watched salt dissolve can also measure the saline concentration, with result $\rho _{b}$. This does not mean that the actual student has a real possibility of measuring the concentration $\rho _{b}$—after all, the contingently unknown truth about the particle density was $\rho _{a}$.

‘Possible’ is transitive for ideal possibilities: what can be successfully imagined is not affected by where it is imagined from. Real possibilities are more fickle. They reflect how the contingent facts about the actual world select from the totality of ideal possibilities, and this selection can change with stepping from the actual world to a really possible world. So although the case of ideal possibilities might make it tempting to treat all possibility operators as transitive, this does not apply to the kind of possibility we need for correlating truth with evidence. Therefore, the revenge paradox does not arise.

Rosenkranz (2004) argues more generally that Brogaard and Salerno (2002) have demonstrated the incompatibility of two inference rules that they assumed in addition:

$\Diamond$-Factivity:

$\Diamond _{R} Ep \vdash p$

Closure:

$\Diamond _{R}Ep \wedge \Box (Ep \rightarrow Eq) \vdash \Diamond _{R} Eq$

We are looking for a correlation between truth and the possibility of evident judgement, so we also want the real possibility of evidence to imply the truth of its content. This is just $\Diamond$-Factivity, which should therefore be endorsed. The principle itself might require a form of restriction, since finding things out usually changes the world and it is not clear that the factivity of $\Diamond _{R}E$ is without problems (Schlöder, 2021; Heylen, 2022, p. 2251f.). But this is only to say that phenomenologists who want to endorse a correlation between truth and evidence have good reason to find a defensible version of $\Diamond$-Factivity, and this will greatly depend on the further theory of real possibilities of evidence.

Closure resembles a valid principle that relates possibilities and necessary conditionals: $\Diamond _{I} p \wedge \Box (p \rightarrow q) \vdash \Diamond _{I} q$, but like transitivity, Closure does not work for real possibility. Take a fresh lottery scratch card, which has not been scratched yet. You know that, once it has become evident that the ticket is a winner, it has also become evident that the ticket has been scratched, because that is the only way anyone could find out that a ticket is a winner. This conditional is a ‘real necessity’: there is no real possibility ($\lnot \Diamond _{R}\lnot$) for seeing that a ticket is a winning ticket without seeing that it has been scratched. Now looking at the unscratched ticket in front of you, a few rubs with a coin are all that stands in the way of seeing that the ticket is a winner. It is therefore a real possibility to know that the ticket is a winner. Given the real necessity of the mentioned conditional, Closure would allow to infer that it is a real possibility to see that the ticket has been scratched. But we have already endorsed factivity: real possibilities of evidence obtain only for finding out truths—and the reason we do not know that we have a winner is precisely because the ticket not been scratched yet.²¹ Insofar a correlation between truth and possible evidence is a reason to hold onto $\Diamond$-Factivity, Closure becomes suspicious. And without Closure, there remains little room where new paradoxes could arise.

6 Conclusion

Common and textually supported interpretations of Husserl’s correlationism indeed lead to a knowability principle which generates the Church–Fitch paradox. Kinkaid avoids this by considering only non-positing possibilities of intuitive illustration. Such a revised principle is intended to save a sense in which our grasp of consistent propositions can manifest experientially; but as such it only concerns fulfilment and possibility, not the relationship between truth and evidence. I proposed a restriction strategy which grows naturally out of Husserl’s inductive characterization of propositions. This response can be formulated in terms of possibilities of evidence, which allows to maintain an alethic correlation. Objections that were raised for a structurally similar proposal by Dummett have been addressed or do not apply. Alleged revenge paradoxes rely on transitivity and closure principles that phenomenologists have reason to do without. It remains ambitious to correlate truth and possible evidence, even after a restriction to basic propositions; but it is no longer paradoxical.