Encoding many-valued logic in $\lambda$-calculus

We will extend the well-known Church encoding of Boolean logic into $\lambda$-calculus to an encoding of McCarthy's $3$-valued logic into a suitable infinitary extension of $\lambda$-calculus that identifies all unsolvables by $\bot$, where $\bot$ is a fresh constant. This encoding refines to $n$-valued logic for $n\in\{4,5\}$. Such encodings also exist for Church's original $\lambda\mathbf{I}$-calculus. By way of motivation we consider Russell's paradox, exploiting the fact that the same encoding allows us also to calculate truth values of infinite closed propositions in this infinitary setting.


In memory of Corrado Böhm
Böhm's theorem [Böhm68] was instrumental in proving the equivalence between an operational semantics and a denotational semantics of the λ-calculus and inspired Barendregt [Bar77a,Bar84] to define the concept of Böhm tree, a first version of which had been introduced by Böhm and Dezani [BDC74]. Böhm trees have later been redefined as the normal forms in a suitable infinitary extension of λ-calculus by Kennaway et al. [KKSdV97]. Böhm trees and their generalisations are now another established way to capture the semantic content of a λ-term [KdV03,SdV11a]. In this paper Böhm trees play a crucial role: we use Böhm trees to encode (even infinite) propositions in λ-calculus and to calculate their values.

Motivation and overview
In this paper we will extend the well-known Church encoding of Boolean logic into λ-calculus to an encoding of n-valued logic (for 3 ≤ n ≤ 5) into an appropriate infinitary extension of λ-calculus. The extension we use in case of n = 3 is the extension that identifies all unsolvables by ⊥ such that the normal forms of the lambda terms are their Böhm trees. By way of motivation we will now consider Russell's paradox. Any notation that is used in this section will be explained in Section 2 and 3.
1.2. Infinite λ-calculus and Böhm trees. In the past [KdV03,SdV11a] we have developed a family of infinitary λ-calculi, each depending on a set of meaningless terms U. The set of terms underlying these extensions is the set Λ ∞ ⊥ of lambda terms obtained by interpreting the usual λ-calculus syntax extended with one fresh symbol ⊥ coinductively. We use this set U ⊆ Λ ∞ ⊥ of meaningless terms to add a new rewrite rule to λ β that allows us to rewrite meaningless terms to ⊥. Cf. Section 2 for a precise description of this rule.
The set US of unsolvable λ-terms is the best known example of such a set of meaningless terms. The corresponding infinitary extension λ ∞ β⊥ U S of the finite λ-calculus λ β is confluent and normalising for a suitable notion of possibly infinite reduction. The Böhm tree of a finite λ-term is precisely its normal form in λ ∞ β⊥ U S . In particular the Böhm tree of an unsolvable is ⊥.
So with this encoding in the λ-calculus in mind we no longer need be afraid of infinite propositions. By inspecting the Böhm trees of the encoding of infinite closed propositions we will find that they are either lambda terms representing a Boolean or they are unsolvable.
1.3. Encoding three-valued logic in infinitary λ-calculus. Thus we are led to extend the Church encoding to an encoding of three-valued logic in infinitary λ-calculus λ ∞ β⊥ U S , by mapping the third value to ⊥. Inspection of the truth tables then reveals that the Church encoding of Boolean logic now has naturally been extended to a Church encoding of what is called McCarthy's three-valued logic [McC63]. In particular we find that the infinite term ¬(¬(¬(. . .))) that we encountered in our analysis of Russell's paradox is neither true nor false but ⊥.
1.4. Encoding four-and five-valued logic. We will further note that the set of unsolvable λ-terms that get identified by ⊥ can be split into three subsets closed under infinite reduction and substitution. Repeating the above construction now with three new truth values instead of ⊥ we find that the Church encoding also encodes a five-valued McCarthyan logic. That five-valued logic and its four-valued sub-logic have been studied earlier by Bergstra and Van de Pol [BvdP96,BvdP11]. 1.5. Church's λI-calculus. When Church started his work on λ-calculus around or before 1928, his motivation was to use the λ-calculus as the basis for a symbolic logic that could serve as the foundation of mathematics [Chu32]. Church's hope was that by using non-classical logic (in which he had shown an early interest [Chu28]) he could side step the Paradoxes without having to introduce Zermelo's set axioms or Russell's type theory, that he both judged as somewhat artificial.
This is not what happened. He discovered with his students Kleene and Rosser that the lambda definable functions corresponded exactly to the recursive functions [Kle36b,Kle36a,Chu36b]. In the build-up to that result Kleene and Rosser managed to prove the inconsistency of his logical system [KR35] while Church himself was still publicly hopeful that not only his system could be paradox-free but also escape Gödel's incompleteness theorem [Chu34]. A disaster. Fortunately, the λ-calculus itself was consistent by the Church-Rosser theorem [CR36]. Various papers under preparation had to be rewritten. Church rebounded almost immediately with his formulation of the Church-Turing thesis [Chu36b] and his negative solution of Hilbert's Entscheidungsproblem [Chu36a] (there is no algorithm that can decide whether a given formula of the first order arithmetic is provable or not).
Church's goal, a paradox free system of symbolic logic, led him to the choice of the λI-calculus in which an abstraction λx.M is only accepted as well-formed term if it contains x as a free variable. For him only terms with a finite normal form where significant and for this he rejected, what we now call, the classical lambda calculus which has terms that have a normal form although they also have subterms which do not [Chu41].
It is a natural question to ask whether an encoding of 3-valued logic is possible in the λI-calculus. We recall that there is a Church encoding for Boolean logic in the λI-calculus. Barendregt has shown that the unsolvable terms in λI-calculus are exactly the λI-terms without a finite normal form. This means that the Böhm tree of a λI-term is either its finite normal form or ⊥. No infinite terms or reductions are needed in case of λI-calculus to define Böhm trees. Thus the above encoding of McCarthy's three-valued logic can be quite simply repeated in Church's λI-calculus. For the details see Section 4.2.
Yet, while this encoding is undoubtedly well within Church's technical means, the Böhm tree concept seems in conflict with his intuition of meaning. The Böhm tree construction gives meaning to any term: the terms without a finite normal form which Church considers meaningless/insignificant are given the "meaning" ⊥ in this extension of λI-calculus with the ⊥-rule.as 1.6. Overview of this paper. In Section 2 we assume familiarity with the finite λ-calculus and briefly introduce relevant notation and facts from the infinitary λ-calculus. In Section 3 we recall the encoding of Boolean valued logic and explain how to extend this to an encoding of three-valued logic. Then we show how this encoding can be refined to four-and five valued logic. In Section 4 we discuss Böhm trees for Church's λI-calculus and show that three valued logic can also be encoded in λI-calculus. Finally Section 5 is a brief conclusion.

Infinite λ-calculus
We will recall notation, concepts and facts from infinitary λ-calculus, while assuming familiarity with λ β , by which we denote the finite λ-calculus with β-reduction and no η-reduction [Chu41,Bar84]. We will use → and → → for respectively one step β-reduction 25:4

F.J. de Vries
Vol. 17:2 and finite β-reduction. We will use ≡ to indicate syntactical identity modulo α. We will use the following special terms.
Θ ≡ (λxy.y(xxy)) λxy.y(xxy) We will now explain how to construct infinite extensions of the finite λ-calculus that are confluent and normalising. We begin with the observation that finite reduction is not finitely normalising: for instance, the finite term Θx has an infinite reduction This is a converging reduction (think of terms as trees and take the standard metric on trees) with an infinite term as limit: x(x(x(. . .))) We can add infinite λ-terms to the finite λ-terms by reading the usual syntax definition (where x ranges over some countable set of variables) of finite λ-terms coinductively: We will write Λ ∞ for this set of finite and infinite λ-terms. Using → → → for a possibly infinite converging reduction, we can now write Later in the paper we will encounter the infinite term λyλyλy . . . as the limit of the converging reduction ΘK → K(ΘK) → λy.ΘK → → λyλy.ΘK → → λyλyλy.ΘK → → → λyλyλy . . .
These two examples show that by adding infinite terms and infinite reductions to the finite lambda calculus, we obtain that some finite terms without a finite normal form now have converging reductions to an infinite normal form. But we have lost confluence of the finite λ-calculus. E.g. the finite term (λx.I(xx))(λx.I(xx)) has a finite reduction to Ω and an infinite converging reduction to I(I(I(. . .))). Both reducts have the property that they can only reduce to themselves. Hence they cannot be joined by either finite or converging reductions. This example also shows that this extension of the finite lambda calculus is not normalising.
Yet, it is possible to build (in fact many different) infinitary extensions of λ β which are confluent and normalising for finite and convergent reductions, and finite and infinite terms [KKSdV97,KvOdV99,KdV03,SdV11a]. We need to do three things. First, we add a new symbol ⊥ to the syntax of λ-terms and consider the set Λ ∞ ⊥ of finite and infinite terms over the extended coinductive syntax. Second, we choose a set U of λ-terms in Λ ∞ . Third, we add a new reduction ⊥ U -rule on Λ ∞ ⊥ that will allow us to identify the terms of U by the new symbol ⊥: For a given U we denote this infinite extension by λ ∞ β⊥ U . In a series of papers [KKSdV97, KvOdV99, KdV03, SdV11a] we have determined a collection of necessary and sufficient axioms that the set U must satisfy in order for λ ∞ β⊥ U to be a converging and normalising infinite λ-calculus. We call such sets sets of meaningless terms. The choice of a set U of meaningless terms is akin to the choice of a semantics for lambda calculus: together the normal forms in λ ∞ β⊥ U form a model of the λ-calculus. The intuition is that the elements of a meaningless set are undefined, that is, have no meaning or are insignificant. In order for such a model to be consistent the set U has to be a proper subset of Λ ∞ .
Definition 2.1 ( [SdV11a]). U ⊆ Λ ∞ is called a set of (finite or infinite) meaningless terms, if it satisfies the axioms of meaninglessness: (1) Axiom of Root-activeness: (3) Axiom of Closure under Substitution: If M ∈ U then any substitution instance of M is an element of U. (4) Axiom of (Weak) Overlap: Either for each λx.P ∈ U, there is some W ∈ U such that This construction is inspired by the definition of Böhm tree [Bar84]. If one takes for U the set US of unsolvables [Bar77b], then the resulting infinite λ-calculus λ ∞ β⊥ U S is confluent and normalising for β⊥ U S reduction. The Böhm tree of a finite λ-term M can equivalently be described as its unique normal form in λ ∞ β⊥ U S [KKSdV97]. Here a (possibly infinite) The set of unsolvables is the largest set for which this construction works. A λ-term is unsolvable if an only if it has no finite β-reduction to a head normal form [Bar84].
The smallest set of meaningless terms [KKSdV97,Ber96] is the set R of terms that are root-active (or mute). A λ-term M is root-active if any reduct of M can further reduce to a redex. The classical root-active term is Ω. The unsolvable ΩI is not root-active. Note that the definition of a root-active term allows for free variables. The normal forms in λ ∞ β⊥ R are exactly the Berarducci trees.
The Lévy-Longo trees can be obtained if one performs this construction over the set of terms without a weak head normal form. In general there are uncountably many sets of meaningless terms [SdV11a]. The collection of normal forms of each such λ ∞ β⊥ U is a model of the λ-calculus λ β . The axioms are chosen such that different sets of meaningless terms give rise to different consistent models.
Church considered the terms without finite normal form as insignificant [Chu41,Bar84]. We recognise that the set of terms in Λ ∞ without a finite normal form is not a set of meaningless terms [KvOdV99,KdV03] in the sense of Definition 2.1, because it is not closed under reduction. The term KIΩ has an infinite reduction, because its subterm Ω has. Yet KIΩ reduces to the finite normal form I. We will come back to this in Section 4.2.

Encoding many-valued logic in λ-calculus
In this section we will extend the familiar Church encoding of Boolean logic to many-valued logic using ideas from Böhm trees and infinitary λ-calculus. We don't know precise reference to the original Church encoding. As Landin remarks in [Lan64]: In particular Church and Curry, and McCarthy and the ALGOL 60 authors, are so large a part of the history of their respective disciplines as to make detailed attributions inevitably incomplete and probably impertinent. which evaluates only one of N 1 and N 2 according to whether M is true or false" and also his "desire for a programming language that would allow its use" in the period 1957-8. He also recalls "the conditional expression interpretation of Boolean connectives" as one of the characterising ideas of LISP. By this he means concretely the if-then-else construct (when applied to Boolean expressions only) which in combination with the truth values T and F can be used as a basis for propositional logic [McC60] with the following natural definitions: Barendregt's book [Bar84] records two elegant encodings of the Booleans and the ifthen-else construct. One encodes into the classical λ-calculus and the other into the more restricted λI-calculus preferred by Church [Chu32,Chu41]. The latter we will discuss in Section 4.2. The former is the simplest: It is easy to see that if-then-else behaves as intended in this encoding. When B reduces to T and F, we have respectively: With help of (3.2) it is straightforward to verify that the standard truth tables of Figure 1 for Boolean valued propositional logic hold in λ-calculus. Boolean logic commonly deals with It is not hard to prove by induction that all closed finite propositions have a unique finite normal form: Lemma 3.1. Let φ be a finite closed proposition. Then φ has a unique finite normal form, which is either T or F. Not all infinite propositions reduce to infinite left spines: for instance, the infinite proposition P 1 ≡ T ∧ (T ∧ . . .) ≡ T ∧ P 1 ≡ TP 1 T ≡ (λxy.x)P 1 T → P 1 is root-active. Also, some infinite propositions reduce just to T or F: for instance, the term These examples show that some infinite propositions reduce to a Boolean, but not all. The latter have in common that their Böhm tree is ⊥.
Theorem 3.2. Let φ be a finite or infinite closed proposition. Then the Böhm tree of φ is either T, F or ⊥.

Proof. By coinduction!
The missing detail in the above "proof" follows from the corollary of the next lemma:   McCarthy discovered left-sequential three-valued propositional logic in his search for a suitable formalism for a mathematical theory of computation [McC63]. In the context of a language for computational (partial) functions he introduced conditional expressions of the form (p 1 → e 1 , . . . , p n → e n ) where the p i are propositional expressions that evaluate to true or false. The idea is that the value of the whole conditional expression is the value of the expression e i for the first p i with value true. If all p i have value false then the conditional expression is undefined. To allow that the evaluation of an expression can be inconclusive, McCarthy stated the rule to evaluate conditional expressions more precisely: If an undefined p occurs before a true p or if all p's are false or if the e corresponding to the first true p is undefined, then the form is undefined. Otherwise, the value of the form is the value of the e corresponding to the first true p. Now the propositional connectives can be defined with help of conditional expressions.   Proof. After applying the definitions of the logical operators it remains to show that The argument now is by inspection. (1) • B 0 = F. Since Fxy = y, it is enough to show that B 2 = TB 2 F, which follows by Txy = x.
In the remainder of the paper we will ignore implication as it can be defined from ¬ and • ∨.

3.4.
Refining the encoding from three-valued to four-and five-valued logic. In the previous section we identified unsolvable λ-terms with ⊥, their (possibly infinite) normal form in the infinitary λ-calculus λ ∞ β⊥ U S . We used ⊥ as third truth value besides T and F. We can refine this idea using the observation of [SdV11b] that the set of unsolvables is the union of three pairwise disjoint sets, each closed under substitution and infinite reduction.
At the basis of this observation lies the simple and well known fact that any finite λ-term has one of two forms, where m, n range over natural numbers: The former expression is called a head normal form and the redex (λxP )Q in the latter is called the head redex. Wadsworth has shown that repeated head reduction of a term M terminates in a head normal form if and only if M has one, and also that having a head normal form is equivalent to being solvable. A reduction in which each step reduces a head redex is called a head reduction [Bar84, see Section 8.3] Hence any unsolvable term M has an infinite head reduction. One of the following scenarios must hold for M .
The three sets can be characterised alternatively using the notion of Berarducci tree which can reveal more detail of a term than Böhm trees do. The union of the HA, IL and O is the set of unsolvables. With help of these three sets we can refine the notion of Böhm reduction. We will represent each set by its own truth value. Instead of replacing unsolvable all λ-terms by ⊥ we will now replace the elements in HA, IL and O by, respectively, the constants ⊥ HA , ⊥ IL and ⊥ O , so that instead of one ⊥-reduction → ⊥ we have now three reduction rules, that we denote by → ⊥ HA , → ⊥ IL and → ⊥ O . We will use ⊥ HA , ⊥ IL and ⊥ O as truth values next to T and F to interpret five-valued propositional logic.
In the same fashion, if we split the unsolvables in only two sets HA and IL ∪ O and introduce besides ⊥ HA a single constant ⊥ IL∪O to replace the elements in IL ∪ O, we have the ingredients to interpret four-valued propositional logic.
These constructions work because of the following theorem.
(1) Let Λ ∞ ⊥ HA ⊥ IL ⊥ O be the set of finite and infinite λ-terms constructed with the symbols ⊥ HA , ⊥ IL and ⊥ O . Then the infinitary λ-calculus λ ∞ β⊥ HA ⊥ IL ⊥ O is confluent and normalising for (strongly) convergent reduction.
(2) Let Λ ∞ ⊥ HA ⊥ IL∪O be the set of finite and infinite λ-terms constructed with the symbols ⊥ HA and ⊥ IL∪O . Then the infinitary λ-calculus λ ∞ β⊥ HA ⊥ IL∪O is confluent and normalising for (strongly) convergent reduction.
Proof. Both follow from Lemma 3.7 and two facts from [KKSdV97], namely that λ ∞ β⊥ R is confluent and normalising, and that ⊥ R -reduction can be postponed over β-reduction.
We will now encode five-valued logic in λ-calculus using the same logical operators as before together with the five truth values from {T, F, ⊥ HA , ⊥ IL , ⊥ O }. Similarly using the four truth values from {T, F, ⊥ HA , ⊥ IL∪O } we will encode four-valued logic.
We need an analogue of Corollary 3.4. Proof. Immediate from the definitions. For instance, suppose U ∈ HA, that is suppose the Berarducci tree of U is of the form λx 1 . . . x n .⊥N m . . . N 1 . Then the Berarducci tree of ¬U is the Berarducci tree of (λx 1 . . . x n .⊥N m . . . N 1 )FT. One easily sees that ¬U is an element of HA.
and ⊥ X →N are all equal to ⊥ X for X ∈ {HA, IL, O}.
(2) The normal forms in λ ∞ β⊥ HA ⊥ IL∪O of ¬⊥ X , ⊥ X • ∧N , ⊥ X • ∨N and ⊥ X →N are all equal to ⊥ X for X ∈ {HA, IL ∪ O}. Then the normal form of φ in λ ∞ β⊥ HA ⊥ IL∪O is either T, F, ⊥ HA or ⊥ IL∪O .
Proof. By coinduction! Using Corollary 3.10, it is straightforward to calculate the truth tables for a four-valued logic encoded in λ-calculus: As it happens, this four-valued propositional logic has been studied by Bergstra and Van de Pol [BvdP96,BvdP11]. In the context of process algebra enriched with conditional statements the need for many-valued logic arises in case a condition evaluates to a truth value (e.g., error/exceptions and divergences) different from true or false. This led Bergstra and his colleagues to a study of a great many of versions of three-, four-and even five-valued logic [BBR95,BP98,BP00,BP99].
For the four-valued logic of Figure 4 Bergstra and Van de Pol gave a complete axiomatisation in [BvdP96,BvdP11]. See Figure 5. They use m (meaningless) for ⊥ HA and d (divergence) for ⊥ IL∪O . These names make some sense here as well. The terms in IL ∪ O can be called diverging as they have limits with infinite left spines. On the other hand, terms in HA reduce by definition to terms of the form M = λx 1 . . . x n .RP 1 . . . P k with R is root-active. This term R is meaningless, in the sense that it will not reveal any further information how long one may reduce it. (1) The axioms in Figure 5 have been selected carefully: each is independent of the others. They also note that Axiom (11) of Figure 3, does not hold in four-valued logic. We can recognise that in our context if we substitute ⊥ HA for x and ⊥ IL∪O for y. Then by Lemma 3.9 we see immediately that the left-hand side of the axiom reduces to ⊥ HA , while the right-hand side reduces to ⊥ IL∪O .
Similarly, the truth tables for the five-valued logic encoded in λ-calculus are as follows: Figure 6: Left-sequential five-valued propositional logic Finally using Corollary 3.10, it is also straightforward to calculate the truth tables of a five-valued logic encoded in λ-calculus. See Figure 6. This the five-valued logic that Bergstra and Van de Pol left implicit in their final remark in [BvdP11] that their complete axiomatisation generalises to five-and higher valued logics, as long as one adds axioms of the form ¬p = p for each new truth value p.

Encoding three-valued logic in the finite λI-calculus
The λ-calculus that Church used in his unfortunate attempt towards a foundation of mathematics was the λI-calculus. This calculus differs from the common λ-calculus λ β by a restriction on the set of λ-terms. Terms in the λI-calculus only contain abstractions of the form λx.M if x occurs free in M . For example the terms λxy.x and λxy.y that we used for the Booleans are now forbidden. So we cannot use the Church encoding of Boolean logic as before.
The consequence of this restriction is that terms in the λI-calculus have two properties Church deemed important: (i) if a term has a finite normal form, it cannot have an infinite reduction, and (ii) if a term has a finite normal form then all its subterms must also have a normal form [Chu41]. These properties don't hold in the classical λ-calculus.

4.1.
Another encoding of the Booleans. Barendregt gave in fact two encodings for the Booleans in his book [Bar84]. Besides the previous well-known encoding of the Booleans he also defined an encoding of the Booleans in the spirit of Church, because the new encodings of the Booleans are terms in the λI-calculus.
T In fact, Barendregt [Bar73] has shown that the unsolvable terms in the λI-calculus are precisely the terms without finite normal form. Klop [Klo75] gave a simpler proof. They did not consider the Böhm tree construction. But in setting of the λI-calculus the Böhm tree construction simplifies enormously. There is no need to consider infinite terms and infinite reductions. We just add the fresh symbol ⊥ to the syntax of the λI-calculus plus the rule M → ⊥ ⊥, whenever M [⊥ : = Ω] has no finite normal form.
Let us denote this extension of the λI-calculus by λI β⊥ . The extension λI β⊥ is confluent and normalising in the finitary sense, and the Böhm tree of any λI-term equals either ⊥ or is a finite ⊥-free normal form. In the past we have overlooked this construction, after we find that the finite λI-term λv.Θ(λxyz.xy)v has an infinite reduction to the infinite term λvλzλzλz . . . which is no longer a λI-term.
The above ⊥-rule resolves this problem, because now the term λv.Θ(λxyz.xy)v reduces in one step to ⊥. In λI β⊥ there is no need to consider infinite reduction as any finite λI-term has a finite reduction to finite Böhm tree in λI β⊥ . Hence we can encode three valued-logic in λI β⊥ if we take as truth values T I , F I and ⊥. Thus, despite a different encoding of the Booleans, we find the same truth tables of McCarthy's left-sequential three-valued propositional logic of Figure 2. Note that the earlier partition in Section 3.4 of the unsolvables based on the form of the left spine of their Berarducci tree applies verbatim to λI-terms. Hence, also this second encoding of the Booleans refines to an encoding of the same earlier four-and five-valued logics in the λI-calculus.
4.3. Why Curry's Paradox does not apply. We end with noting that Curry's Paradox does not apply to the finite λI-calculus because the above infinitary extension is consistent.
Contemporaneously with Church, Curry had been searching for a symbolic logic that could serve as foundation of mathematics. The technique by which Kleene and Rosser [KR35] found the inconsistency in the symbolic logic of Church also applied to some of the systems of illative combinatoric logic that Curry was exploring. In contrast to Church, Curry had not committed himself to an underlying philosophy. He considered the Kleene-Rosser paradox an helpful instrument in the search for "stronger and stronger systems which are consistent" as well as "weaker and weaker systems which are inconsistent" [CF58].
In 1942 Curry published a short and self-contained argument to show the inconsistency for the type of symbolic logics that he and Church were working on. Curry showed that that any combinatory complete system, like e.g. λ-calculus, with an implication operator satisfying: is inconsistent. The elegant short proof of the Curry's Paradox can be found in [CF58,Bar84].
As the infinitary extensions λ β and the λI-calculusλ-calculus are consistent (the normal forms of T and F are not equal in them) the Curry's Paradox does not apply to them. More direct: the implication X → Y does not satisfy the above two conditions for implication: if X is ⊥ then both expressions reduce to ⊥ for any value of Y .

Conclusion
The idea to solve Russell's paradox with three-valued logic is not at all new. Feferman gave various pointers in [Fef84]. But the conjunctions and disjunctions of the three-valued logics that are considered for that purpose all seem to be commutative in contrast to those in the left-sequential McCarthy logic that we use here. It is possible to further refine the encoding to an encoding of ∞-valued logic in λ-calculus. The new truth values then correspond to the different shapes of left spine that unsolvables can have. We see no further use for that.