2.1 Two formulations of the Bullet connective

We adopt the following notational convention concerning derivational systems (in particular, natural deduction systems): when we write some derivations where the discharge of open assumptions and the composition of derivations are at play, as in this case

we denote with the index k the fact that the application of $\to _I$ discharges n occurrences of an open assumption A ( $n \geq 0$ ) annotated with k. In addition, $\mathcal {D}_1$ denotes a derivation from an open assumption A to the conclusion (or the end formula) B, possibly containing some other open assumptions. Similarly, $\mathcal {D}_2$ denotes a derivation of A, possibly containing some open assumptions. The right-hand derivation is then obtained by substituting n copies of $\mathcal {D}_2$ for n occurrences of an open assumption A in $\mathcal {D}_1$ .

Let us now recall two formulations of the introduction and elimination rules of $\bullet $ , which is a $0$ -ary connective, thus corresponding to an atomic formula. The first formulation is the pair $\bullet 1_I$ and $\bullet 1_E$ below (see [Reference Read55, sec. 7]); if we then define $\neg A$ as $A \to \bot $ , a second formulation of the rules of $\bullet $ can be given as the pair $\bullet 2_I$ and $\bullet 2_E$ (see [Reference Read54, sec. 2.8])Footnote ⁹ :

One can easily see that these two formulations are equivalent, namely, each pair of introduction and elimination rules is derivable from the other pair by means of the inference rules of $\to $ . The second formulation, however, gives a very intuitive idea of the behaviour of $\bullet $ : it corresponds to a formula which is logically equivalent to its negation. This is why Read interprets $\bullet $ as a ‘proof-conditional Liar sentence’ [Reference Read54, p. 142] or, alternatively, as corresponding to Russell’s paradox formula $r \in r,$ where $r = \{ x \mid x \not \in x \}$ ([Reference Read55, p. 574]; cf. also Tranchini [Reference Tranchini67, sec. 2] about this second interpretation). For this reason, one can take $\bullet $ to be ‘a paradoxical connective’ (we will better clarify this point later).

Each of these two formulations satisfies the inversion principle, as already shown by Read in [Reference Read54, Reference Read55] (in his terms, each satisfies harmony); also, each of them satisfies the recovery principle:

The fact that $\bullet $ satisfies both the inversion and the recovery principles makes it comparable to the standard connectives (like conjunction, disjunction, and implication), which also satisfy these two principles. Up to this point, $\bullet $ meets all the relevant conditions to be a ‘good’ candidate as a logical connective.Footnote ¹⁰

However, Read showed that each of the two formulations yields a contradiction:

Read also showed in [Reference Read55] that the reduction of the first contradictory derivation is non-terminating, and it is not hard to see the non-termination of the reduction of the second as well. The absence of contradiction has been traditionally considered (since Aristotle) one of the characteristic features of any logical and meaningful discourse. The fact that $\bullet $ brings about a contradiction seems then to make a point against its logicality and meaningfulness, although it satisfies the harmony condition.Footnote ¹¹

2.2 The circularity of the Bullet connective

By analysing the problems of contradiction and non-termination we just exposed, Dyckhoff identifies another issue concerning $\bullet $ from the point of view of proof-theoretic semantics [Reference Dyckhoff, Piecha and Schroeder-Heister20, p. 82]. He considers the first formulation of the inference rules for $\bullet $ , and following the meaning-explanation of propositions as given by [Reference Dummett19, Reference Martin-Löf34, Reference Martin-Löf36], he claims that ‘we understand a proposition when we understand what it means to have a canonical proof of it, i.e. what forms a canonical proof can take’ [Reference Dyckhoff, Piecha and Schroeder-Heister20, p. 82]. By a ‘canonical proof’, Dyckhoff means a derivation whose last rule is an introduction rule. According to him, the first formulation of the introduction rule for $\bullet $ attempts to explain what a canonical proof for $\bullet $ is: this rule says that a canonical proof for $\bullet $ is a proof obtained by applying $\bullet 1_I$ -rule to a derivation from $\bullet $ to $\bot $ . Dyckhoff treats the subderivation $\mathcal {D}$ as a method for transforming arbitrary proofs of $\bullet $ into proofs of $\bot $ , and he finds a circularity in the alleged explanation. That is, the above explanation of canonical proofs for $\bullet $ presupposes the notion of proof of $\bullet $ , in particular the notion of canonical proof for $\bullet $ .

What stands behind Dyckhoff’s remark is in fact a well-founded and hierarchical view of proofs. A typical instance of such a view is given by the Brouwer–Heyting–Kolmogorov interpretation (the BHK-interpretation, for short; see [Reference Troelstra and van Dalen70, pp. 9–10]). The BHK-interpretation is an informal explanation of the meaning of intuitionistic logic. In particular, proofs in this interpretation stand for any kind of objects provided by (idealised) mathematicians in order to give evidence to their claims. The domain of such proofs is constructed bottom-up, and this construction proceeds along a certain hierarchical structure. As explained by van Atten, this view essentially follows Brouwer’s intuitionism:

If we read Dyckhoff’s remark about the circularity of $\bullet $ in the light of this passage, we can interpret it as suggesting that a well-founded and hierarchical domain of proofs cannot be constructed—even by some idealised mathematician, like the Creative Subject—in the case of $\bullet $ .

More precisely, we spell out Dyckhoff’s idea by showing that if we try to define in a bottom-up way a domain of proofs by using the $\bullet $ -rules, we eventually end up with a circular definition. Since the BHK-interpretation is an informal one, we replace it by appealing to a formal notion, that of Prawitz’s validity. This notion indeed includes a well-founded and hierarchical perspective of proofs, as the one considered above, although proofs are taken here to be closed derivations in the natural deduction format (see the definition below). The formulation of validity which we adopt here is a simplified version of the one given in [Reference Prawitz and Fenstad50, Reference Schroeder-Heister56].Footnote ¹²

A derivation $\mathcal {D}$ is closed if it has no open assumption; otherwise, $\mathcal {D}$ is said to be open. An atomic system is a pair composed of an arbitrary set $\mathcal {P}$ of atomic formulas including $\bot $ and an arbitrary set $\mathcal {R}$ of inference rules of the form

such that

1. (i) $n \geq 0$ ,

2. (ii) $\{ A , A_1 ,\ldots , A_n \} \subseteq \mathcal {P}$ , and

3. (iii) no closed derivation of $\bot $ can be obtained by using the inference rules of $\mathcal {R}$ .

When $n = 0$ holds, the inference rule is an axiom which states that A holds. In the rest of this subsection, we consider formulas to be given relative to an atomic system $\mathcal {S}$ : formulas are constructed by a given set of logical connectives from atomic formulas in $\mathcal {S}$ . For an atomic system $\mathcal {S}$ , a derivation in $\mathcal {S}$ is a derivation which contains inference rules in $\mathcal {S}$ only. Closed derivations in an atomic system $\mathcal {S}$ are the basic proofs on which a domain of proofs is constructed in a bottom-up way. For a finite set $\{ A_1 , \ldots , A_n \}$ of formulas, we define its complexity as the number of all occurrences of connectives in this set. The complexity of a formula A is defined as the complexity of $\{ A \}$ .

In order to introduce Prawitz’s notion of validity for $\bullet $ , we first formulate the notion of validity for derivations which contain only the rules of a given atomic system $\mathcal {S}$ and the rules for $\to $ .

Using the notion of $\mathcal {S}$ -validity, we can now explain better the circularity of the $\bullet $ -rules. The inductive definition of validity breaks once we add the clause for defining the validity of a closed derivation of $\bullet $ : by adding this clause, we get a circular definition. Consider the first formulation of the $\bullet $ -rules, and suppose that we try to define the $\mathcal {S}$ -validity of a derivation by induction on the sum of the complexities of its open assumptions and the end formula. The similarity between $\bullet 1_I$ and $\to _I$ suggests considering the following clause:

1. (B) A closed derivation of $\bullet $ is $\mathcal {S}$ -valid if it reduces to a derivation of the form

   
   and for any closed $\mathcal {S}$ -valid derivation , the derivation
   
   is $\mathcal {S}$ -valid.

Here the $\mathcal {S}$ -validity of a closed derivation of $\bullet $ depends on the validity of arbitrary closed $\mathcal {S}$ -valid derivations having the same conclusion, i.e., . This implies that the inductive clause (B) does not decrease the joint complexity of the open assumptions and the end formula, since the derivations considered are all closed, valid, and have the same complexity as the conclusion. Therefore, the inductive definition breaks down, and the clause leads to a circularity, as point out by Dyckhoff. The second formulation of the $\bullet $ -rules breaks the inductive definition as well.

To sum up, the introduction and elimination rules for $\bullet $ satisfy both the inversion principle and the recovery principle (whenever we adopt the first formulation or the second formulation). However, these rules yield a contradiction, and derivations of such a contradiction do not admit a normal form. Moreover, as Dyckhoff remarks, when these rules are analysed from the point of proof-theoretic semantics, the $\bullet $ -rules present a form of circularity, which can be made explicit by appealing to Prawtiz’s notion of validity, as we have just shown.

Considering $\bullet $ through Prawitz’s validity, we can specify that deriving a contradiction is highly undesirable and makes it problematic to consider $\bullet $ to be a ( $0$ -ary) logical connective. Although $\bullet $ is harmonious, and thus satisfies a necessary condition to be a logical connective, the derivation of a contradiction makes it possible to produce non-valid derivations in the sense of Prawitz. The derivations of $\bot $ that we have seen on p. 7 are closed and non-canonical; hence, by applying the rule of ex falso quodlibet, it is possible to obtain closed and non-canonical derivations of any formula. In particular, we can obtain a closed derivation for any atomic formula. Among these formulas, there is also $\bullet $ (since a $0$ -ary connective corresponds to an atomic proposition), and such a derivation cannot be reduced to a derivation in any atomic system. We are considering logical validity and no atomic rules are given. Moreover, the rules of $\bullet $ (in both formulations) cannot themselves belong to any atomic system: in the first formulation, the $\bullet $ -introduction rule can discharge finitely many assumptions of the form $\bullet $ , while in the second formulation the connective $\neg $ occurs in the $\bullet $ -rules, so that in both cases we are not respecting the format of atomic rules given at p. 8.Footnote ¹³ Thus, this situation makes the first clause of Prawitz’s validity false. In fact, the situation would be problematic also if we consider arbitrary formulas instead of atomic ones. The application of the ex falso quodlibet to the two closed derivations of $\bot $ considered at p. 7 makes it possible to obtain a closed derivation of $A \to B$ , for any A and B. This is a derivation that cannot be reduced to a derivation ending with the $\to $ -introduction rule, so that in this case it is the second clause of Prawitz’s validity definition which would become false.

It could be objected that by dropping the ex falso quodlibet rule, the $\bullet $ -rules do not yield non-valid derivations. However, the circularity of the validity clauses for $\bullet $ would still be problematic, as it prevents an inductive construction of the domain of proofs. The issue is whether such a conception of the domain of proofs is essential. Considering that this conception is motivated by an intuitionistic point of view, we might want to abandon it. However, if we do so, we will not only abandon one of the sources of inspiration for proof-theoretic semantics, but also one of the characteristic features shaping it. The adoption of proof-theoretic semantics is usually linked with intuitionistic logic.Footnote ¹⁴ Hence, to avoid the hierarchical and well-founded view of proofs that we mentioned, one should be able to show that proof-theoretic semantics can be defined without necessarily having to adopt an intuitionistic conception. In other words, one should show that it is possible to have a proof-theoretic semantics that does not validate intuitionistic logic. This is not the approach we want to pursue. Note that, originally, the very idea of proof-theoretic semantics rests on the BHK-interpretation (cf. [Reference Schroeder-Heister, Zalta and Nodelman59]). We consider that such a hierarchical and well-founded notion of proofs is perfectly legitimate, and that $\bullet $ is not enough to question it.

The lesson we learn from the proof-theoretic analysis of $\bullet $ is actually another one, namely, that harmony, being only a necessary but not sufficient condition for logicality, is too weak as a requirement. Moreover, it is not even a sufficient condition for meaningfulness. It is only by satisfying Prawitz’s validity that we can say that a certain connective is really meaningful from a proof-theoretic semantics point of view. More precisely, if the rules of a certain connective allow one to define a notion of proof that satisfies validity, we can then say that this connective is meaningful. In particular, this connective allows one to define propositions, that is, meaningful linguistic entities that are assertible, and for which we can give evidence.Footnote ¹⁵ The fact that the $\bullet $ -rules do not allow one to define a notion of proof satisfying validity means that $\bullet $ does not allow one to form propositions.

The notion of validity is defined inductively; one of the induction parameters is the complexity of the conclusion of a proof. Now, according to what we just said, the conclusion of a proof is not simply a formula, but a proposition. This implies that the notion of proposition must be also inductively defined, in the sense that it must be well-founded. This is consistent with the notion of proof discussed above: proofs involve propositions, and both proofs and propositions are organised in a hierarchical and well-founded way.

We will revisit the notion of proposition in §5, comparing it with the notion of type, which we will discuss in the next section.

3.1 The Bullet connective in the light of a computational framework

We extend simply typed $\lambda $ -calculus $\lambda ^{\to }$ (see, e.g., [Reference Sørensen and Urzyczyn63]) by adding the Bullet connective to this calculus. Given infinitely many atomic types $P_0 , P_1 , \ldots $ and variables $x_1 , x_2 , \ldots $ , types and (untyped) terms of the extended calculus are defined as in Figure 1. We denote this calculus by $\lambda ^{\to \bullet}_1$ , since the constructor $\hat {x}. t$ and the destructor ${@} (t , s)$ correspond to the first version of the $\bullet $ -introduction and $\bullet $ -elimination rules, respectively.Footnote ¹⁶ In what follows, we abbreviate $\mathsf {app}(t , s)$ as $t \: s$ , and write ${@} (t , s)$ using the infix notation $t {@} s$ . Note that, in the term $\hat {x}. t$ , the variable x is bound as in the $\lambda $ -abstraction term $\lambda x. t$ . The expression $t[s / x]$ denotes the term obtained by substituting the term s for every free occurrence of the variable x in the term t. We often abbreviate $t[s / x]$ as $t[s]$ when it is obvious which variable is replaced. These notations can be extended to simultaneous substitution $t[s_1 / x_1 , \ldots , s_n / x_n]$ straightforwardly (which can thus be abbreviated as $t[s_1 , \ldots , s_n]$ ).

A pair $(x , A)$ composed of a variable and a type is called a declaration, and it is usually written as $x : A$ . If the variables present in a set of declarations are pairwise distinct, we call this set a context, and a context is denoted by a capital Greek letter, such as $\Gamma $ and $\Delta $ . Usual notations for contexts are adopted: for instance, a singleton $\{ x : A \}$ is abbreviated as $x : A$ , and we denote the union of two contexts $\Gamma $ and $\Delta $ by $\Gamma , \Delta $ .

Figure 2 $\lambda^{\to \bullet}_1$ -typing rules.

Figure 3 $\lambda^{\to \bullet}_2$ -typing rules.

A typing judgement is an expression of the form $\Gamma \vdash t : A$ , meaning that the term t is of type A with respect to the context $\Gamma $ .Footnote ¹⁷ The typing rules of $\lambda ^{\to \bullet}_1$ operate on this kind of judgement and are formulated in a natural deduction system, presented in a sequent-style format,Footnote ¹⁸ as in Figure 2. This allows for an alternative reading of the role of the terms, namely, that of keeping track of the natural deduction rules applied to form it, so that a judgement of the $\Gamma \vdash t : A$ can also be read as expressing the fact that t is the code of a derivation of A from the premises $\Gamma $ . If we adopt the usual natural deduction format (instead of the sequent-style format), as well as the same notational conventions adopted in §2.1, the typing rules for $\bullet $ can be presented in the following way, where the introduction and elimination rules for $\bullet $ are decorated with appropriate terms:

We can also present the second version of the $\bullet $ -rules as typing rules. The syntax of the system $\lambda ^{\to \bullet}_2$ , which is an extension of $\lambda ^{\to }$ with this second version of rules for $\bullet $ , is obtained by replacing the constructor $\hat {x}. t$ and the destructor $t {@} s$ with ${\mathsf {fold}} (t)$ and ${\mathsf {unfold}} (t)$ , respectively. The typing rules of $\lambda ^{\to \bullet}_2$ are those of $\lambda ^{\to }$ together with the ones in Figure 3. The reason for presenting the constructor of the $\bullet $ -introduction rule (resp. the destructor of the $\bullet $ -elimination rule) as $\mathsf {fold}$ (resp. as $\mathsf {unfold}$ ) in the second version will be explained in §3.2.

Figure 4 Reduction rules.

We define the reduction relation $\rhd $ on the terms of $\lambda ^{\to \bullet}_1$ and $\lambda ^{\to \bullet}_2$ as in Figure 4. In addition to the standard rules $(\beta )$ , $(\eta )$ , $(\xi )$ , $(\mathrm {ConL})$ , and $(\mathrm {ConR})$ coming from $\lambda $ -calculus, the set of $\lambda ^{\to \bullet}_1$ -reduction rules contains the rules $(\beta _{\bullet 1} )$ , $(\eta _{\bullet 1} )$ , $(\xi _{\bullet 1} )$ , $(\mathrm {ConL}_{\bullet 1})$ , and $(\mathrm {ConR}_{\bullet 1})$ , while the set of $\lambda ^{\to \bullet}_2$ -reduction rules contains $(\beta _{\bullet 2} )$ , $(\eta _{\bullet 2} )$ , $(\mathrm {ConF}_{\bullet 2})$ , and $(\mathrm {ConU}_{\bullet 2})$ . A $\beta $ -redex in $\lambda ^{\to \bullet}_1$ (resp. $\lambda ^{\to \bullet}_2$ ) is a term of the form $(\lambda x.t)s$ or $(\hat {x}.t) {@} s$ (resp. a term of the form $(\lambda x.t)s$ or ${\mathsf {unfold}} ({\mathsf {fold}} (t))$ ). Roughly speaking, we can thus identify the $\beta $ -redexes with those terms that can occur on the left-hand side of $\beta $ -reduction rules. The notion of $\eta $ -redex is defined in a similar way.

The $\beta $ -reduction rule $(\beta _{\bullet 1})$ corresponds to the inversion principle for $\bullet $ following the first formulation (cf. §2.1), and the $\eta $ -reduction rule $(\eta _{\bullet 1})$ is the symmetric counterpart of the recovery principle for $\bullet $ in the first formulation (since the recovery principle corresponds in general to the $\eta $ -expansion rule):

Similarly, in the second formulation of $\bullet $ in §2.1, the rule $(\beta _{\bullet 2})$ corresponds to the inversion principle for $\bullet $ , while the rule $(\eta _{\bullet 2})$ is the symmetric counterpart of the recovery principle for $\bullet $ :

As it has already been discussed in the literature (see [Reference Schroeder-Heister57, p. 80] and [Reference Tranchini67, p. 598]), the derivations of $\bot $ we presented in §2.1 can be decorated by means of a language expanding that of $\lambda $ -calculus, so that we can have a term corresponding to the proof of a contradiction. This term contains $\beta $ -redexes, but its reduction does not terminate. In particular, this term cannot be reduced to a $\beta $ -normal form; it cannot be reduced to another term that does not contain any $\beta $ -redex. For example, the contradictory derivation in the first formulation of $\bullet $ can be represented in $\lambda ^{\to \bullet}_1$ as follows:

One can define a non-terminating term in $\lambda ^{\to \bullet}_2$ in a similar way. Now, if terms are interpreted here as computable functions (or programs), the fact of reaching a normal form can be interpreted as the fact of obtaining a value. Terms like those we just presented, which keep on reducing without reaching a normal form, can thus be interpreted as partial (computable) functions (or partial programs, namely, programs that do not always give an output). Partial functions are not problematic from a purely computational point of view, which is the point of view that we want to adopt here: the standard models of computation—like recursive functions, $\lambda $ -calculus, Turing computability, etc.—take partiality into account. Partiality becomes problematic when we want to assign a semantic role to computational processes by interpreting them as ways of fixing the denotation of linguistic expressions.Footnote ¹⁹ According to such an interpretation, values are associated with denotations (see [Reference Martin-Löf, Jonathan Cohen, Łoś, Pfeiffer and Podewski33, p. 160]),Footnote ²⁰ and partiality is taken to be a symptom of paradoxical situations: when we have a paradox, we keep on computing without being able to reach a value, that is, the denotation of an expression (see [Reference Tranchini67, secs. 2 and 3] for more details, especially the idea that a canonical proof can be seen as the denotation of a proposition). Imposing totality of computable functions (or programs) would thus be a way of blocking some possible paradoxical situation; however, in this way, many interesting computational procedures could be lost.Footnote ²¹

Another approach would be to analyse computation independently of its possible semantics use. This is indeed something which naturally emerges from the analysis we are offering by considering typing systems. Consider the term $(\hat {x}. x {@} x) {@} (\hat {x}. x {@} x)$ , which we constructed above. This keeps track of all the rules applied in the first derivation of $\bot $ at p. 7. However, we cannot say that such a term codifies a proof since, as we have shown in the previous section, this derivation of $\bot $ is not valid in Prawitz’s sense (although it is closed). As a consequence, we cannot consider that the types involved in the derivation of the judgement $\vdash (\hat {x}. x {@} x) {@} (\hat {x}. x {@} x) : \bot $ correspond to propositions. At best, they can be interpreted as purely syntactic formulas, devoid of any meaning (if we consider that meaning is fixed by proof-conditions).Footnote ²²

In what follows, we will consider terms solely from a computational point of view, and we will show that there are interesting computational situations we can account for by means of the typing rules for the $\bullet $ connective, where partiality is involved. To do this, we consider a generalisation of the $\bullet $ connective given by Read in [Reference Read55, pp. 574–575].Footnote ²³ Note that he described this generalised $\bullet $ connective as a kind of proof-conditional Curry’s paradox. Although Read considers the generalisation of its first version only, one can generalise the second version in a similar way. His idea is to replace $\bot $ with an arbitrary formula (or type) A in the formulation of $\bullet $ . Hereafter the expression $\lambda ^{\to \bullet}_1$ (resp. $\lambda ^{\to \bullet}_2$ ) is used to denote the system obtained by extending simply typed $\lambda $ -calculus with the first version (resp. the second version) of the generalised Bullet connective. The syntax is defined as in Figure 5, while the inference and reduction rules for the first version of $\bullet ^A$ are formulated in Figure 6. The congruence rules $(\mathrm {ConL}_{\bullet 1})$ and $(\mathrm {ConR}_{\bullet 2})$ are generalised accordingly. For each type A, a non-terminating term of type A is obtained as in the case of $\bot $ . Note that this non-terminating term of type A is closed; hence, we have a closed term (i.e., a closed derivation) of A. A similar generalisation of $\bullet $ in the second formulation can be obtained as in Figure 7.

Figure 5 Types and terms of $\lambda ^{\to \bullet}_1$ and $\lambda ^{\to \bullet}_2$ .

Figure 6 Generalised $\lambda ^{\to \bullet}_1$ -typing and reduction rules.

Figure 7 Generalised $\lambda ^{\to \bullet}_2$ -typing and reduction rules.

It is straightforward to prove a weakening lemma for both $\lambda ^{\to \bullet}_1$ and $\lambda ^{\to \bullet}_2$ . This lemma allows us to simulate the weakening rule in the case of natural deduction rules presented in the sequent-style format.

We will now assign an explicit computational interpretation to the rules of this generalised version of $\bullet $ . We will show that this generalisation of $\bullet $ makes it possible to define a general version of a fixed-point operator.

3.2 Fixed-point operator and recursive types

The computational behaviour of the generalised Bullet connective $\bullet ^A$ reminds one of the general fixed-point operator ${\mathsf {fix}}$ discussed in the literature on Martin-Löf type theory (see, e.g., [Reference Martin-Löf, Martin-Löf and Mints35, Reference Palmgren43]).Footnote ²⁴ The typing and reduction rules for ${\mathsf {fix}}$ can be formulated as followsFootnote ²⁵ :

The variable x in the term f is bound in the term ${\mathsf {fix}} ((x)f)$ . As shown by these rules, the role of the operator ${\mathsf {fix}}$ is to provide such a term f of type A with its ‘fixed-point’ ${\mathsf {fix}} ((x)f)$ .

By using the typing rule for ${\mathsf {fix}}$ , we have the following closed term of type A:

for each type A. Hence, ${\mathsf {fix}}$ provides a closed derivation for each formula A, in the same way as the generalised Bullet connectives do. The resulting term ${\mathsf {fix}} ((x) x )$ is non-terminating as follows, where we denote syntactic identity by $\equiv $ .

$$ \begin{align*} {\mathsf{fix}} ((x) x ) \rhd x [{\mathsf{fix}} ((x) x ) / x] \equiv {\mathsf{fix}} ((x) x ) \rhd x [{\mathsf{fix}} ((x) x ) / x] \equiv {\mathsf{fix}} ((x) x ) \rhd \cdots. \end{align*} $$

Here, we show that $\lambda ^{\to \bullet}_1$ allows one to deduce the typing rule for ${\mathsf {fix}}$ so that its reduction rule is also derivable. In Appendix 1, we show that $\lambda ^{\to \bullet}_2$ allows one to do the same. Let a variable y be fresh for a context $\Gamma , x : A$ , then we have a $\lambda ^{\to \bullet}_1$ -derivation as the one shown in Figure 8 (for the sake of brevity, we omit the application of the $\mathrm {Var}$ -rule). We define

$$ \begin{align*} {\mathsf{fix}} ((x)f) :\equiv (\hat{y}^A. (\lambda x. f) (y {@}^A y)) {@}^A (\hat{y}^A. (\lambda x. f) (y {@}^A y)) , \end{align*} $$

where $:\equiv $ indicates that the term on the left is a syntactic abbreviation for the term on the right. The derivation in Figure 8 then shows that the typing rule for ${\mathsf {fix}}$ is derivable in $\lambda ^{\to \bullet}_1$ . Moreover, we have

$$ \begin{align*} {\mathsf{fix}} ((x)f) &\equiv (\hat{y}^A. (\lambda x. f) (y {@}^A y)) {@}^A (\hat{y}^A. (\lambda x. f) (y {@}^A y)) \\ &\rhd (\lambda x. f) ((\hat{y}^A. (\lambda x. f) (y {@}^A y)) {@}^A (\hat{y}^A. (\lambda x. f) (y {@}^A y))) \\ &\rhd f[((\hat{y}^A. (\lambda x. f) (y {@}^A y)) {@}^A (\hat{y}^A. (\lambda x. f) (y {@}^A y))) / x] \\ &\equiv f[{\mathsf{fix}} ((x)f) / x] , \end{align*} $$

so the reduction rule for ${\mathsf {fix}}$ is derivable.

Figure 8 Interpretation of ${\mathsf {fix}}$ in $\lambda ^{\to \bullet}_1$ .

The derivability of ${\mathsf {fix}}$ enables to solve recursive equations which have several applications. Let ${\stackrel {\ast }{\rhd }}$ be the reflexive and transitive closure of a given reduction relation $\rhd $ , while ${\stackrel {\ast }{\bowtie }}$ is defined as the reflexive, symmetric, and transitive closure of $\rhd $ . Then, the following proposition holds by a standard argument (cf., e.g., [Reference Barendregt, Abramsky, Gabbay and Maibaum5, theorem 2.1.9]).

Proof. We consider only the case of $\lambda ^{\to \bullet }_1$ here, since the case of $\lambda ^{\to \bullet }_2$ is similar. Let a term $\psi [f , x]$ be given with $\Gamma , f : B \to A , x : B \vdash \psi [f , x] : A$ , so that $\Gamma , f : B \to A \vdash \lambda x. \psi [f , x] : B \to A$ holds. By the typing rule of ${\mathsf {fix}}$ , we have $\Gamma \vdash {\mathsf {fix}} ((f)\lambda x. \psi [f , x]) : B \to A$ .

We define $\varphi : \equiv {\mathsf {fix}} ((f)\lambda x. \psi [f , x])$ . Then, for any $\Gamma \vdash u : B$ ,

$$ \begin{align*} \varphi \: u \equiv ({\mathsf{fix}} ((f)\lambda x. \psi [f , x])) u {\stackrel{\ast}{\rhd}} (\lambda x. \psi [\varphi , x])u \rhd \psi [\varphi , u] \end{align*} $$

holds.

To observe an example of the proposition above, we consider an extension of $\lambda ^{\to \bullet }_1$ or $\lambda ^{\to \bullet }_2$ with the integer type $\mathsf {Int}$ and the Boolean type $\mathsf {Bool}$ , and put $\psi $ in this proposition as follows:

$$ \begin{align*} \psi [f , x] : \equiv \mathtt{if} \;\; x \leq 0 \;\; \mathtt{then} \;\; 1 \;\; \mathtt{else} \;\; x \times f \: (x - 1) \end{align*} $$

with $f : \mathsf {Int} \to \mathsf {Int}$ and $x : \mathsf {Int}$ . We then have a term $\mathsf {fac} : \mathsf {Int} \to \mathsf {Int}$ satisfying the ‘equation’

$$ \begin{align*} \mathsf{fac} \: l {\stackrel{\ast}{\rhd}} \mathtt{if} \;\; l \leq 0 \;\; \mathtt{then} \;\; 1 \;\; \mathtt{else} \;\; l \times \mathsf{fac} \: (l - 1) , \end{align*} $$

which provides the factorial function whenever l is a positive integer.

Next, we show that the second version of the generalised Bullet connective is an instance of non-normalising recursive types. This implies that its first version is definable as a recursive type as well, because the inference rules of the first version are derivable from those of the second version. Below, we consider the non-normalising recursive types discussed, for instance, in [Reference Gunter24, chap. 7] and [Reference Pierce48, chap. IV]. These recursive types have several applications; they allow one to define not only the general fixed-point operator ${\mathsf {fix}}$ discussed above, but also many examples of datatypes, such as the type of natural numbers and the one of lists of natural numbers. Thus, one can provide the recursive definition of a function on these datatypes by solving the corresponding recursive equation with ${\mathsf {fix}}$ .

Figure 9 Syntax of ${\lambda ^{\to \mathsf {rec}}}$ .

Our base theory is simply typed $\lambda $ -calculus, and we denote by ${\lambda ^{\to \mathsf {rec}}}$ the system obtained by adding recursive types to simply typed $\lambda $ -calculus. Types and terms of ${\lambda ^{\to \mathsf {rec}}}$ are defined in Figure 9, where the vocabulary contains type variables $X_{0}, X_{1}, \dots $ instead of atomic types. In particular, a type variable X is bound in a type of the form $\mu X. A$ , which is a recursive type. Similarly to what we did for the substitution of a term variable, we denote by $A[B / X]$ the type resulting from the substitution of a type B for all occurrences of a type variable X in a type A.

The system ${\lambda ^{\to \mathsf {rec}}}$ has the typing and reduction rules for recursive types in Figure 10. These rules go together with the suitable congruence rules.

Figure 10 ${\lambda ^{\to \mathsf {rec}}}$ -typing and reduction rules.

Let $\bullet ^A :\equiv \mu X. X \to A$ , where the variable X is not free in A.Footnote ²⁶ Then, we have

$$ \begin{align*} \bullet^A \to A \equiv (\mu X. X \to A) \to A \equiv (X \to A)[\mu X. X \to A / X], \end{align*} $$

so that it is easy to see that the typing and reduction rules for the recursive type $(\mu X. X \to A) \to A$ are exactly those of the second version of the generalised Bullet connective. This means that the typing and reduction rules for the recursive type $(\mu X. X \to A) \to A$ also provide a non-terminating (or non-normalising) term.Footnote ²⁷ Furthermore, this explains why, in the second version of the Bullet connective, the introduction and elimination rules are called $\mathsf {fold}$ and $\mathsf {unfold}$ , respectively. If one considers the second version as a recursive type, its introduction rule ‘folds’ a type into the Bullet connective, and the elimination rule ‘unfolds’ the Bullet connective.

The comparison with recursive types sheds light on another feature of the Bullet connectives, corroborating the idea that they cannot be considered as logical connectives forming propositions. Given a recursive type $\mu X.A$ , its corresponding rules make it possible to pass from it to $A[\mu X.A/X]$ , and vice versa. One can thus analyse a recursive type $\mu X.A$ with respect to the position of the variable X in A, that is, the variable on which the substitution (with the recursive types themselves) is operated. More precisely, we first define a positive/negative occurrence of a type variable in a type of ${\lambda ^{\to \mathsf {rec}}}$ (cf. [Reference Altenkirch, Gottlob, Grandjean and Seyr1]).

Note that a type variable X may occur in A both positively and negatively: for instance, X occurs in $X \to X$ both positively and negatively. Next, we define a narrower notion of positive occurrence, namely, the notion of strictly positive occurrence.Footnote ²⁸

Note that a type variable X does not occur negatively in a type A whenever X occurs strictly positively in A.

The notion of strictly positive recursive type can then be defined by requiring that in a type $\mu X.A$ the variable X occurs strictly positively in A. Strict positiveness is usually imposed as a condition on a system with recursive types in order to guarantee the termination of the reduction relation in such a system (see, e.g., [Reference Coquand, Paulin, Martin-Löf and Mints15]). Note, however, that in the definition of $\bullet ^A$ as $\mu X. X \to A$ , the variable X does not occur in a strictly positive way in $X \to A$ (in fact, it occurs negatively). As Dyckhoff [Reference Dyckhoff, Piecha and Schroeder-Heister20, p. 83] remarks, this can be seen as yet another symptom of the circularity of the Bullet connective: by unfolding $\mu X.X \to A$ , one substitutes $\mu X.X \to A$ for X in $X \to A$ , and because of the position of X, this allows one to obtain the type $(\mu X.X \to A) \to A$ , which not only folds to $\mu X.X \to A$ , but can also be applied to $\mu X.X \to A$ itself (which is indeed the symptom of circularity). In this way, in particular, it is possible to provide a non-terminating closed term t for any arbitrary type A, which means that every type A is inhabited.Footnote ²⁹ As Pierce remarks:

Here, again, the problem is considering that a type can be interpreted as a proposition: $\vdash t: A$ (where t is closed) would mean that there is a closed derivation, and thus a proof, of A coded by t.Footnote ³⁰ The solution is to drop the identification between types and propositions, and to look at a term simply as a program. In this sense, as we already mentioned in §1.1, a type would correspond to the classification of a program. By assigning a type to a term (standing for a program), we describe its behaviour from a formal point of view. Accordingly, a typable term does not require being terminating, because non-terminating terms or programs are often useful as the general fixed-point operator ${\mathsf {fix}}$ is, and recursive types can be used to type such programs.

In fact, we can push the analysis of the Bullet connective further via recursive types. As we have seen, the rules for recursive types allow one to pass from a recursive type to its unfolded version, and vice versa. However, what is the exact relation between a recursive type and its unfolded version? One way of looking at it is to consider that they are isomorphic types (see [Reference Pierce48, pp. 276–277]). We will use this notion to compare $\bullet ^{A}$ with other recursive types, and show that they allow for different instances of type isomorphisms. We denote the identity $\lambda x. x$ of type $A \to A$ by $1_A$ , and define the composition $s \circ t$ of two terms $t : A \to B$ and $s : B \to C$ as $\lambda x. s \: (t \: x) : A \to C$ . In proof-theoretic terms, the notion of type isomorphism is defined as follows (cf. [Reference Bruce, Longo and Sedgewick10, Reference di Cosmo16]): two types A and B are provably isomorphic in a type system if, in this system, there are closed terms $t : A \to B$ and $s : B \to A$ such that

$$ \begin{align*} s \circ t {\stackrel{\ast}{\bowtie}} 1_A , \qquad t \circ s {\stackrel{\ast}{\bowtie}} 1_B \end{align*} $$

hold. Intuitively, the idea is that the term t is an ‘isomorphism’ from A to B.

While the system ${\lambda ^{\to \mathsf {rec}}}$ allows one to show that there is a type A provably isomorphic to $A \to A$ , the systems $\lambda ^{\to \bullet }_1$ and $\lambda ^{\to \bullet }_2$ allow one only to show that there is a provable isomorphism between $\bullet ^A$ and $\bullet ^A \to A$ , for an arbitrary type A. The reason is that the inference rules for $\bullet ^A$ relate it with the implication (or the arrow type) $\bullet ^A \to A$ , where the antecedent $\bullet ^A$ is always distinct from the succedent A. Let us first show that there is a type A provably isomorphic to $A \to A$ in ${\lambda ^{\to \mathsf {rec}}}$ . Let ${\mathsf {Ref}} :\equiv \mu X. X \to X$ and consider then the following proposition.Footnote ³¹ Note that ${\mathsf {Ref}}$ is defined by means of the formula $X \to X$ with two occurrences of X, one of which is negative and the other is positive.

Proof. Take the following ${\lambda ^{\to \mathsf {rec}}}$ -derivations, where the $\mathrm {Var}$ -rule is omitted:

Define then $\psi :\equiv \lambda x. {\mathsf {fold}} (x)$ and $\varphi :\equiv \lambda y. {\mathsf {unfold}} (y)$ . These closed terms $\psi , \varphi $ satisfy the following reduction relations:

$$ \begin{align*} \varphi \circ \psi &\equiv \lambda z. \varphi \: (\psi \: z) \equiv \lambda z. (\lambda y. {\mathsf{unfold}} (y))((\lambda x. {\mathsf{fold}} (x)) z) & \\ &\rhd \lambda z. (\lambda y. {\mathsf{unfold}} (y))({\mathsf{fold}} (z)) & [\text{by } \beta ] \\ &\rhd \lambda z. {\mathsf{unfold}} ({\mathsf{fold}} (z)) & [\text{by } \beta ] \\ &\rhd \lambda z. z & [\text{by } \beta_{\mu} ] \end{align*} $$

$$ \begin{align*} \psi \circ \varphi &\equiv \lambda w. \psi \: (\varphi \: w) \equiv \lambda w. (\lambda x. {\mathsf{fold}} (x))((\lambda y. {\mathsf{unfold}} (y)) w) & \\ &\rhd \lambda w. (\lambda x. {\mathsf{fold}} (x))({\mathsf{unfold}} (w)) & [\text{by } \beta ] \\ &\rhd \lambda w. {\mathsf{fold}} ({\mathsf{unfold}} (w)) & [\text{by } \beta ] \\ &\rhd \lambda w. w. & [\text{by } \eta_{\mu} ] \end{align*} $$

The type of the term $\lambda z.z$ is $({\mathsf {Ref}} \to {\mathsf {Ref}} ) \to ({\mathsf {Ref}} \to {\mathsf {Ref}})$ , thus this term corresponds to $1_{{\mathsf {Ref}} \to {\mathsf {Ref}}}$ , while the term $\lambda w. w$ is of type ${\mathsf {Ref}} \to {\mathsf {Ref}}$ and corresponds to $1_{{\mathsf {Ref}}}$ . Therefore, ${\mathsf {Ref}}$ and ${\mathsf {Ref}} \to {\mathsf {Ref}}$ are provably isomorphic.

Here in ${\lambda ^{\to \mathsf {rec}}}$ we use both $\beta $ -reduction and $\eta $ -reduction, which correspond to the inversion principle and the (converse of) recovery principle, respectively. An argument similar to the previous one allows us to prove the fact that in $\lambda ^{\to \bullet }_2$ , $\bullet ^A$ and $\bullet ^A \to A$ are provably isomorphic for any type A. This statement also holds in $\lambda ^{\to \bullet }_1$ , as shown below.

Proof. We construct an isomorphism as follows. Consider first the two derivations below:

Then, we have the following reductions, which preserve types:

$$ \begin{align*} (\lambda v. \lambda z. v {@}^{A} z) \circ (\lambda y. \hat{x}^{A}. y \: x) &\equiv \lambda w. (\lambda v. \lambda z. v {@}^{A} z)((\lambda y. \hat{x}^{A}. y \: x) w) & \\ &\rhd \lambda w. (\lambda v. \lambda z. v {@}^{A} z)(\hat{x}^{A}. w \: x) & [\text{by } \beta ] \\ &\rhd \lambda w. \lambda z. (\hat{x}^{A}. w \: x) {@}^{A} z & [\text{by } \beta ] \\ &\rhd \lambda w. \lambda z. w \: z & [\text{by } \beta_{\bullet 1} ] \\ &\rhd \lambda w. w & [\text{by } \eta ] \\ &\equiv 1_{\bullet^A \to A} & \end{align*} $$

$$ \begin{align*} (\lambda y. \hat{x}^{A}. y \: x) \circ (\lambda v. \lambda z. v {@}^{A} z) &\equiv \lambda w. (\lambda y. \hat{x}^{A}. y \: x) ((\lambda v. \lambda z. v {@}^{A} z) w) & \\ &\rhd \lambda w. (\lambda y. \hat{x}^{A}. y \: x) (\lambda z. w {@}^{A} z) & [\text{by } \beta ] \\ &\rhd \lambda w. \hat{x}^{A}. (\lambda z. w {@}^{A} z) x & [\text{by } \beta ] \\ &\rhd \lambda w. \hat{x}^{A}. w {@}^{A} x & [\text{by } \beta ] \\ &\rhd \lambda w. w & [\text{by } \eta_{\bullet 1} ] \\ &\equiv 1_{\bullet^A.} & \end{align*} $$

This is sufficient to obtain an isomorphism between $\bullet ^A \to A$ and $\bullet ^A$ .

As we said, $\bullet $ is just a special case of $\bullet ^{A}$ (i.e., when $A \equiv \bot $ ). This result provides another reason for considering $\bullet $ as a paradoxical connective. This connective is not only paradoxical in that it mimics, at the proposition level, the Liar paradox or Russell’s paradox. This connective meets a stronger criterion of paradoxicality, as exposed by Schroeder-Heister and Tranchini (see also Petrolo and Pistone’s exposition [Reference Petrolo and Pistone44, p. 607]):

Although the type isomorphism between $\bullet ^{A}$ and $\bullet ^{A} \to A$ allows us to have a deeper understanding of the Bullet connective (in particular by meeting the condition of paradoxality formulated by Schroeder-Heister and Tranchini), one can ask whether it is possible to make the behaviour of the Bullet connective more similar to that of recursive types in general. We show that it is possible to define a simple variant of the Bullet connective behaving like ${\mathsf {Ref}}$ with respect to type isomorphism. Moreover, thanks to this variant of the Bullet connective, it is possible to interpret type-free (i.e., pure) $\lambda $ -calculus, a framework which is powerful enough to characterise a Turing-complete model of computation.

3.3 Interpretation of type-free $\lambda $ -calculus

Recall that the original formulation of the inference rules for $\bullet $ —that is before considering the generalisation presented at §3.1—is the following:

In order to obtain a variant of $\bullet $ which makes it possible to interpret type-free $\lambda $ -calculus, it is sufficient to replace $\bot $ with $\bullet $ itself in these rules. We denote this variant by ${\bullet ^{\ast }}$ , and the system $\lambda ^{\to \bullet }_{1\ast }$ is formulated by extending simply typed $\lambda $ -calculus with the typing rules in Figure 11. Similarly, the system $\lambda ^{\to \bullet }_{2\ast }$ is obtained by extending simply typed $\lambda $ -calculus with the typing rules in Figure 12. The reduction rules of $\lambda ^{\to \bullet }_{1\ast }$ and $\lambda ^{\to \bullet }_{2\ast }$ are those of Figure 4.

Figure 11 $\lambda ^{\to \bullet }_{1\ast }$ -typing rules.

Figure 12 $\lambda ^{\to \bullet }_{2\ast }$ -typing rules.

One can prove the following proposition about type isomorphisms in these systems by adapting the proof of Proposition 3.5 to the case of $\lambda ^{\to \bullet }_{2\ast }$ , and that of Proposition 3.6 to the case of $\lambda ^{\to \bullet }_{1\ast }$ . Note that, in fact, these proofs allow one to show a stronger statement than the one corresponding to the definition of type isomorphism. They allow one to show the existence of two closed terms whose composition reduces to the identity.

Proposition 3.7. In both $\lambda ^{\to \bullet }_{1\ast }$ and $\lambda ^{\to \bullet }_{2\ast }$ , there are two closed terms $\psi : ({\bullet ^{\ast }} \to {\bullet ^{\ast }} ) \to {\bullet ^{\ast }}$ and $\varphi : {\bullet ^{\ast }} \to ({\bullet ^{\ast }} \to {\bullet ^{\ast }})$ with

$$ \begin{align*} \varphi \circ \psi {\stackrel{\ast}{\rhd}} 1_{{\bullet^{\ast}} \to {\bullet^{\ast}}} , \qquad \psi \circ \varphi {\stackrel{\ast}{\rhd}} 1_{{\bullet^{\ast}}}. \end{align*} $$

By applying a general result presented in [Reference Gunter24, chap. 8, proposition 8.1] to the case of ${\bullet ^{\ast }}$ , and by using Proposition 3.7, we can provide an interpretation of type-free $\lambda \beta \eta $ -calculus in $\lambda ^{\to \bullet }_{1\ast }$ , and in $\lambda ^{\to \bullet }_{2\ast }$ as well. First, we define by induction a translation $(\cdot )^{\ast }$ from terms of type-free $\lambda $ -calculus into terms of $\lambda ^{\to \bullet }_{1\ast }$ and $\lambda ^{\to \bullet }_{2\ast }$ :

• ▶ $x^{\ast } := x$ ;

• ▶ $(\lambda x. t)^{\ast } := \psi (\lambda x. t^{\ast })$ ;

• ▶ $(t \: u)^{\ast } := \varphi \: t^{\ast } \: u^{\ast }$ .

The lemma below guarantees that terms of type-free $\lambda $ -calculus are translated into terms which are typable with type ${\bullet ^{\ast }}$ . It is straightforward to prove the weakening lemma for $\lambda ^{\to \bullet }_{1\ast }$ and $\lambda ^{\to \bullet }_{2\ast }$ (cf. Lemma 3.1 above), which will be used implicitly below.

Lemma 3.9. Let t be an arbitrary term of type-free $\lambda $ -calculus such that the set of all free variables of t are $\{ x_1 , x_2 , \ldots , x_n \}$ . The translation $t^{\ast }$ of t satisfies the typing judgement

$$ \begin{align*} x_1 : {\bullet^{\ast}} , x_2 : {\bullet^{\ast}} , \ldots , x_n : {\bullet^{\ast}} \vdash t^{\ast} : {\bullet^{\ast}} \end{align*} $$

in both $\lambda ^{\to \bullet }_{1\ast }$ and $\lambda ^{\to \bullet }_{2\ast }$ .

Proof. By induction on t. The case of variables is trivial, so we consider the cases of abstraction and application only. By Proposition 3.7, in $\lambda ^{\to \bullet }_{1\ast }$ and $\lambda ^{\to \bullet }_{2\ast }$ , respectively, we have two closed terms $\psi : ({\bullet ^{\ast }} \to {\bullet ^{\ast }} ) \to {\bullet ^{\ast }}$ and $\varphi : {\bullet ^{\ast }} \to ({\bullet ^{\ast }} \to {\bullet ^{\ast }})$ satisfying

$$ \begin{align*} \varphi \circ \psi {\stackrel{\ast}{\rhd}} 1_{{\bullet^{\ast}} \to {\bullet^{\ast}}} , \qquad \psi \circ \varphi {\stackrel{\ast}{\rhd}} 1_{{\bullet^{\ast}}}. \end{align*} $$

Let t be of the form $\lambda x. u$ . Using the induction hypothesis with the weakening lemma, we can derive the desired typing judgement as follows:

Let t be an application term $u \: s$ . Using the induction hypothesis with the weakening lemma, we have

This completes the proof.

We can now prove that type-free $\lambda \beta \eta $ -calculus is interpretable in $\lambda ^{\to \bullet }_{1\ast }$ and $\lambda ^{\to \bullet }_{2\ast }$ , respectively.

Proof. We consider the $\beta $ -reduction rule and the $\eta $ -reduction rule only. The cases of the remaining reduction rules are straightforward.

For the $\beta $ -reduction rule,

$$ \begin{align*} (\lambda x. t) \: s \rhd t[s/x] , \end{align*} $$

we have the terms $((\lambda x. t) \: s)^{\ast }$ and $(t[s/x])^{\ast }$ of type ${\bullet ^{\ast }}$ , by Lemma 3.9. It is sufficient to verify that the former term reduces to the latter in $\lambda ^{\to \bullet }_{1\ast }$ and $\lambda ^{\to \bullet }_{2\ast }$ , respectively. We have

$$ \begin{align*} ((\lambda x. t) \: s)^{\ast} = (\varphi (\psi (\lambda x. t^{\ast}))) s^{\ast} {\stackrel{\ast}{\rhd}} (\lambda x. t^{\ast}) s^{\ast} \rhd t^{\ast}[s^{\ast} / x] = (t[s/x])^{\ast} , \end{align*} $$

where $t^{\ast }[s^{\ast } / x]$ is well-typed by Lemma 3.9. So the $\beta $ -reduction rule is derivable.

In the case of the $\eta $ -reduction rule, the reduction relation below

$$ \begin{align*} (\lambda x. t \: x)^{\ast} = \psi (\lambda x. (\varphi \: t^{\ast}) \: x) \rhd \psi (\varphi \: t^{\ast}) {\stackrel{\ast}{\rhd}} t^{\ast} \end{align*} $$

holds, so the $\eta $ -reduction rule is derivable as well.

Note that the interpretation of type-free $\lambda \beta \eta $ -calculus into $\lambda ^{\to \bullet }_{2\ast }$ we just presented can be reformulated so as to obtain an interpretation of type-free $\lambda \beta \eta $ -calculus into ${\lambda ^{\to \mathsf {rec}}}$ , simply by replacing ${\bullet ^{\ast }}$ with ${\mathsf {Ref}}$ .

3.4 Bullet connectives as a modality

As mentioned above, the generalised Bullet connective $\bullet ^A$ can be regarded as a proof-conditional Curry’s paradox (cf. Footnote 26). On the one hand, Curry’s paradox is known to have an argument structure similar to that of a proof of Löb’s theorem. Whereas the former provides any given formula A with a formula C such that $C \leftrightarrow (C \to A)$ holds, the latter shows that for any recursively enumerable extension T of first-order Peano arithmetic, and for any number-theoretic sentence A, there exists a sentence C such that $C \leftrightarrow (\mathrm {Prov}_{T} (\lceil C \rceil ) \to A)$ is derivable in T, where $\mathrm {Prov}_{T}$ is a provability predicate of T (see, e.g., [Reference van Benthem6]). On the other hand, Löb’s axiom, which is an axiom of Gödel–Löb provability logic corresponding to Löb’s theorem, is known to be satisfied by Nakano’s modality in typed $\lambda $ -calculus with recursion [Reference Nakano41]. We argue that these analogies can help shed light on the computational meaning of the Bullet connectives further. Specifically, we first consider the fragment of $\lambda ^{\to \bullet }_1$ and $\lambda ^{\to \bullet }_2$ obtained by omitting the $\bullet ^A$ -elimination rules, and show that one can derive the type corresponding to Nakano’s fixed-point operator in this fragment. We then note that this fragment can be interpreted in Nakano’s type system, which we denote by ${\lambda ^{\mu \bullet }}$ ; the interpretation of $\bullet ^{A}$ in Nakano’s system not only demonstrates the consistency of the $\bullet ^A$ -introduction rules, but also provides them with a computational meaning.

Nakano’s modality, which allows one to form the type $\bullet A$ for a given type A, approximates a specification for a recursive program. A successive sequence $S_0 , S_1 , \ldots $ of specifications approximating a given one is described by possible worlds and the $\bullet $ -accessibility relation in the sense of Kripke semantics.Footnote ³² It is this approximation in terms of the modality $\bullet A$ that enables Nakano to define recursive programs including the fixed-point operators in ${\lambda ^{\mu \bullet }}$ without rendering the system logically inconsistent.

The notion of ‘specification’ invoked by Nakano can be clarified by appealing to the characterisation found in [Reference Turner72, pp. 232–233]. According to this characterisation, a specification for a program is a formal condition about the way in which the inputs and outputs of a program are related. Thus, the specification of a program describes both its behaviour and its associated properties. In this sense, a specification should allow us to state what happens for each input, namely, whether the program terminates on that input and which output it produces if it does terminate. One can then consider a program’s type as expressing its specification. For instance, the type $\mathsf {Int} \to \mathsf {Int} \to \mathsf {Int}$ specifies that addition and multiplication on integers take two inputs of type $\mathsf {Int}$ and return an output of type $\mathsf {Int}$ . Moreover, in a type system satisfying the strong normalisation property, that is, the property that all typable terms terminate, assigning a type to a program guarantees its termination.

In [Reference Nakano41], Nakano takes types to express specifications and refines Scott’s construction of solutions to recursive equations [Reference Scott61, Reference Scott, Nielsen and Schmidt62]. In Scott’s construction, the solution or function f satisfying a recursive equation $f = F(f)$ is approximated by the sequence $\bot , F(\bot ) , F(F(\bot )) , \ldots $ , where F is a Scott-continuous function on a directed-complete partial order with the least element $\bot $ and so we have $\bot \sqsubseteq F(\bot ) \sqsubseteq F(F(\bot )) \sqsubseteq \dots $ . Here, functions can be viewed extensionally as sets of input-output pairs, so that each approximation $F^n (\bot )$ is a finite set of such pairs. Nakano similarly approximates the type (i.e., the specification) S of the solution f using $S_0 , S_1 , \ldots $ , and the modality is crucial for this latter approximation, as noted above. However, treating types as expressing program specifications involves more than simply claiming that types capture program behaviour. In this respect, Nakano’s notion of type is more demanding than the one adopted here.Footnote ³³ Our perspective in this paper (as explained further in the conclusion) is that a type is merely a way of classifying programs with respect to their behaviour; this does not entail that a type conveys additional properties enabling a more exact description of the program, such as termination on every input. Nakano’s conception of type as specification is closer to the notion of type employed in logic (see the Concluding Remarks for a more detailed discussion of the different meanings of the notion of type).

According to Nakano’s semantics, the modality $\bullet A$ informally means that A holds at the ‘previous time’. The system ${\lambda ^{\mu \bullet }}$ includes the following typing rules (see [Reference Nakano41] for details of the ${\lambda ^{\mu \bullet }}$ -typing rules):

where the notation $\bullet ^{n}$ denotes the n-fold consecutive applications of the modality $\bullet $ with $n \geq 0$ , and the context $\bullet \Gamma $ is of the form $x_1 : \bullet A_1 , \ldots , x_n : \bullet A_n$ with $\Gamma \equiv x_1 : A_1 , \ldots , x_n : A_n$ . In addition, ${\lambda ^{\mu \bullet }}$ contains recursive types $\mu X. A$ in its type expressions, but the occurrences of X in A must ensue within the scope of some occurrence of $\bullet $ (i.e., the occurrences of X must be guarded by $\bullet $ ).

In this system, Nakano defined the fixed-point operator

$$ \begin{align*} \xi :\equiv \lambda f. (\lambda x. f (x x)) (\lambda x. f (x x)) : (\bullet X \to X) \to X \end{align*} $$

for any type variable X, using the subtyping rule below:

Here the relation $\preceq $ is a subtype relation, and this rule states that if A is a subtype of B then a term of type A is also a term of type B. Subtyping judgements such as $\vdash A \preceq B$ are governed by the subtype relation rules of ${\lambda ^{\mu \bullet }}$ , which derive subtyping judgements without assigning terms in the judgements. Although the fixed-point operator is definable in ${\lambda ^{\mu \bullet }}$ , this system is logically consistent: there exists a type that cannot be inhabited in ${\lambda ^{\mu \bullet }}$ . This contrasts with our definition of the fixed-point operator by means of the introduction and elimination rules for the generalised Bullet connective $\bullet ^A$ ; the presence of both the $\bullet ^A$ -introduction and $\bullet ^A$ -elimination rules leads to an unrestricted recursive type that entails a logical contradiction. Nakano imposed a guardedness condition on recursive types so as to obtain a version of Bullet connectives (i.e., his modality $\bullet A$ ) that does not entail a logical contradiction.

As an example of a recursive program defined using this fixed-point operator, we consider streams of data of type X, which can be presented as

$$ \begin{align*} \lambda x. \xi (\lambda y. \langle x , y \rangle) : X \to \mu Y. X \times \bullet Y , \end{align*} $$

where $\times $ denotes the usual product type. This stream unfolds as

$$ \begin{align*} \lambda x. \xi (\lambda y. \langle x , y \rangle) \rhd \lambda x. \zeta \zeta \rhd \lambda x. (\lambda y. \langle x , y \rangle) (\zeta \zeta) \rhd \lambda x. \langle x , \zeta \zeta \rangle \rhd \cdots \end{align*} $$

with $\zeta :\equiv \lambda x_1. (\lambda y. \langle x , y \rangle ) (x_1 x_1 )$ .

Treating a modal type $\bullet A$ as $\bullet ^A$ , one can derive $(\bullet A \to A) \to A$ in the fragment of $\lambda ^{\to \bullet }_1$ and $\lambda ^{\to \bullet }_2$ obtained by dropping the $\bullet ^A$ -elimination rules (below we omit the application of the $\mathrm {Var}$ -rule):

In contrast to ${\lambda ^{\mu \bullet }}$ , this does not allow us to obtain a recursive program, because we lack the $\bullet ^A$ -elimination rules and therefore cannot compute the terms constructed by the $\bullet ^A$ -introduction rules.

Nevertheless, we can assign a computational meaning to the $\bullet ^A$ -introduction rules by interpreting the current fragment of $\lambda ^{\to \bullet }_1$ and $\lambda ^{\to \bullet }_2$ in ${\lambda ^{\mu \bullet }}$ . Since simply typed $\lambda $ -calculus is a subsystem of ${\lambda ^{\mu \bullet }}$ , it suffices to derive the $\bullet ^A$ -introduction rules in ${\lambda ^{\mu \bullet }}$ . Here, we consider only the $\bullet ^A$ -introduction rule of $\lambda ^{\to \bullet }_1$ , which is formulated using type variables (see below). First, let $B :\equiv \mu Y. \bullet Y \to X$ . We omit the derivations of subtyping judgements used below, as they are immediately derivable using the ${\lambda ^{\mu \bullet }}$ -subtype relation rules. In particular, the judgements $\vdash \bullet B \to X \preceq \bullet B$ and $\vdash \bullet B \preceq \bullet (\bullet B \to X)$ are derivable from the facts that $\bullet B \to X \preceq B \preceq \bullet B$ and $B \preceq \bullet B \to X$ hold, respectively. Defining $\chi :\equiv \lambda x. (\lambda y. t)(xx)$ , we have

In short, we have derived the $\bullet ^A$ -introduction rule of $\lambda ^{\to \bullet }_1$ as

Thus, we not only establish the consistency of the $\bullet ^A$ -introduction rules, but also provide them with a computational meaning in terms of Nakano’s fixed-point operator.

However, it is not clear what this argument establishes from the perspective of proof-theoretic semantics. Does the interpretation of the above fragment of $\lambda ^{\to \bullet }_1$ and $\lambda ^{\to \bullet }_2$ in ${\lambda ^{\mu \bullet }}$ provide the $\bullet ^A$ -introduction rules not only with a computational meaning but also with a proof-conditional meaning? The difficulty lies in the fact that ${\lambda ^{\mu \bullet }}$ includes the $\preceq $ -rule, which allows the use of subtyping judgements, and the rules governing subtype relations in [Reference Nakano41] are not formulated in terms of introduction and elimination rules. If we consider the inversion principle, it clearly holds for $\to $ in ${\lambda ^{\mu \bullet }}$ , and one can establish the corresponding recovery principle as follows:

However, the $\preceq $ -rule plays a crucial role in this derivation.

In fact, providing a precise definition of what constitutes an introduction or elimination rule and determining their admissible forms is a non-trivial task. Only a small number of works in the literature address it explicitly—for instance, Prawitz [Reference Prawitz, Suppes, Henkin, Joja and Moisil51], Dummett [Reference Dummett19, pp. 256–258], and Martin-Löf [Reference Martin-Löf34].Footnote ³⁴ The formulations of introduction and elimination rules proposed by Prawitz and Martin-Löf are such that they satisfy the inversion and recovery principles by definition. Consequently, as stated in the Introduction, they meet the necessary conditions for logicality. Because the rules governing the subtype relation do not follow this format, it is difficult to formulate inversion and recovery principles for them, and therefore difficult to assess whether they can be regarded as logical. We cannot pursue this issue further here, as doing so would require engaging in a separate and considerably more ambitious project, namely, an investigation into the general format of introduction and elimination rules and the manner in which this format shapes the formulation of the condition of logicality.