2.1 $\mathbf{G1h_{cd}}$

We present a multi-conclusion system based on a multi-conclusion calculus for intuitionistic logic:Footnote ² we call it $\mathbf {G1h}_{\mathbf {cd}}$ for Gentzen system for the logic $\mathbf {HYPE}$ with constant domains. Sequents are understood as pairs of multisets. We work with a language whose logical symbols are $\neg $ , $\vee $ , $\rightarrow $ , $\forall $ , $\bot $ . For $\Gamma =\gamma _1,\ldots ,\gamma _n$ a multiset $\neg \Gamma $ is the multiset $\neg \gamma _1,\ldots ,\neg \gamma _n$ . The logical constants $\wedge , \exists ,\leftrightarrow $ can be defined as usual and $\top $ is defined as $\neg \bot $ . Moreover, we can define ‘intuitionistic’ negation $\sim A$ as $A \rightarrow \bot $ , the material conditional $A\supset B$ as $\neg A\vee B$ , and material equivalence $A\equiv B$ as $(A\supset B)\land (B\supset A)$ . For A a formula, we write $\texttt {FV}(A)$ for the set of its free variables, and $\texttt {FV}(\Gamma )$ for the set of free variables in all formulas in $\Gamma $ .

The system $\mathbf {G1h}_{\mathbf {cd}}$ consists of the following initial sequents and rules:

We write $\texttt {rk}(A)$ for the logical complexity of A, defined as the number of nodes in the longest branch of its syntactic tree. For a derivation d we let

• • $\texttt {hgt}(d) := \texttt {sup}_{i<n} \{\texttt {hgt} (d_i) + 1\, |\, d_i \text { an immediate subderivation of } d\}$ (the height of the derivation), where $d_0, \ldots , d_{n}$ are the immediate subderivations of d.

We say that a formula A is derivable in a system, if the sequent $\Rightarrow A$ is derivable in it.

The next lemma collects some basic facts about $\mathbf {G1h}_{\mathbf {cd}}$ . They mostly concern the admissibility of some basic inferences in $\mathbf {G1h}_{\mathbf {cd}}$ .

We opted for this specific formulation of $\mathbf {G1h}_{\mathbf {cd}}$ mainly because it substantially simplifies the presentation of the results of the next sections 3–5, which are the main focus of the paper. From a proof-theoretic point of view, the calculus has some drawbacks even at the propositional level, as the rules ConCp and ClCp compromise the induction needed for cut-elimination. In the propositional case, even if one removes ConCp and ClCp and splits the contraposition rule of Lemma 1(ii) on a case by case manner, problems for cut-elimination remain [Reference Fischer9]. Moreover, when one moves to the quantificational system, there are deeper problems. The same counterexample that is employed to show that cut is not admissible in systems of intuitionistic logic with constant domains can be employed for the systems we are investigating.Footnote ³ Both problems can be addressed by employing techniques from Kashima and Shimura [Reference Kashima and Shimura17], which however rely on the extension of the systems with additional resources.

Since cut elimination is not the main focus of our paper, we opt for a more compact presentation of $\mathbf {G1h}_{\mathbf {cd}}$ that fits nicely our purpose of extending it with arithmetic and truth rules.

2.2 Semantics

In this section we present the semantics of $\mathbf {G1h}_{\mathbf {cd}}$ (and therefore of $\mathbf {HYPE}$ ) and sketch the completeness of $\mathbf {G1h}_{\mathbf {cd}}$ . We follow a simplification of the semantics in Leitgeb [Reference Leitgeb19] suggested by Speranski [Reference Speranski25]. Speranski connects $\mathbf {HYPE}$ -models with Routley semantics. A Routley frame $\mathfrak {F}$ is a triple $\langle W, \leq , * \rangle $ , where:

1. (i) W is a non-empty set of states;

2. (ii) $\leq $ is a preorder;

3. (iii) $*$ is a function from W to W, which is:

   • – antimonotone, i.e., for all $w,v \in W$ , if $w \leq v$ , then $v^* \leq w^*$ ;

   • – involutive, i.e., for all $w \in W$ , $w^{**} = w$ .

A constant domain model $\mathfrak {M}$ for $\mathbf {HYPE}$ is a triple $(\mathfrak {F}, {D},I)$ where $\mathfrak {F}$ is a Routley frame, ${D}$ is a non-empty set (the domain of the model), and I is an interpretation function. In particular, I assigns to every constant c an element of D and it associates with each state w and n-place predicate P a set $P^{w} \subseteq D^n$ . Constants are interpreted rigidly and, although domains do not grow, we impose the following hereditariness condition: for all $v,w \in W$ , if $v \leq w$ , then for all predicates P, $P^{v} \subseteq P^{w}$ .

Let $\mathfrak {M}$ be a constant domain model, $w \in W$ and $\sigma \colon \text {VAR}\to D$ a variable assignment on D, then the forcing relation $\mathfrak {M}, w, \sigma \Vdash A$ is defined inductively:

$$ \begin{align*} \begin{array}{lc@{\quad}l} \mathfrak{M}, w, \sigma \Vdash P (x_1,\ldots,x_n) & &\text{ iff } (\sigma (x_1),\ldots, \sigma(x_n)) \in P^{w}; \\ \mathfrak{M}, w, \sigma \Vdash \neg A & & \text{ iff } \mathfrak{M}, w^*, \sigma \nVdash A; \\ \mathfrak{M}, w, \sigma \Vdash A \vee B & & \text{ iff } \mathfrak{M}, w, \sigma \Vdash A \text{ or } \mathfrak{M}, w, \sigma{} \Vdash B; \\ \mathfrak{M}, w, \sigma \Vdash A \rightarrow B & & \text{ iff } \text{ for all } v, \text{ with } w \leq v \text{, if } \mathfrak{M}, v, \sigma \Vdash A \text{, then } \mathfrak{M}, v, \sigma \Vdash B; \\ \mathfrak{M}, w, \sigma \Vdash \forall x A & & \text{ iff } \text{ for all } x\text{-variants } \sigma' \text{ of } \sigma, \mathfrak{M}, w, \sigma' \Vdash A; \\ \mathfrak{M}, w, \sigma \nVdash \bot. \end{array} \end{align*} $$

Finally, we define logical consequence. We write, for sets of sentences:

• • iff: if $\mathfrak {M},w\Vdash \gamma $ for all $\gamma \in \Gamma $ , then for some ;

• • iff for all $\mathfrak {M},w$ : .

The system $\mathbf {G1h}_{\mathbf {cd}}$ is equivalent to the Hilbert-style system $\mathbf {QN}^{\circ }$ featuring the axiom schemata

$$ \begin{align*} &A\rightarrow (B\rightarrow A) &&A\rightarrow (B\rightarrow C)\rightarrow ((A\rightarrow B)\rightarrow (A\rightarrow C))\\ &A\land B\rightarrow A && A\land B\rightarrow B\\ &A\rightarrow A\vee B &&B\rightarrow A\vee B\\ &A\rightarrow (B\rightarrow A\land B) && (A\rightarrow C) \rightarrow ((B\rightarrow C)\rightarrow (A\vee B\rightarrow C))\\ & \neg\neg A\rightarrow A && A \rightarrow \neg \neg A\\ &\forall x A \rightarrow A(t)&& A(t)\rightarrow \exists x A \end{align*} $$

and the following rules of inference:

$\mathbf {QN}^{\circ }$ is a neater presentation of $\mathbf {HYPE}$ where a few redundant principles are dropped. The consequences of the two systems are identical.

That our system $\mathbf {G1h}_{\mathbf {cd}}$ is equivalent to $\mathbf {QN}^{\circ }$ can be seen as follows. $\mathbf {G1h}_{\mathbf {cd}}$ is an extension of intuitionistic logic (modulo the definition of $\sim A$ as $A\rightarrow \bot )$ . Therefore, since all axioms of $\mathbf {QN}^{\circ }$ except for the double negation axioms are intuitionistically valid, Lemma 1 enables us to show that all axioms of $\mathbf {QN}^{\circ }$ are consequences of $\mathbf {G1h}_{\mathbf {cd}}$ . Additionally, Lemma 1 shows that contraposition is admissible in $\mathbf {G1h}_{\mathbf {cd}}$ . Rules for quantifiers are easily established in $\mathbf {G1h}_{\mathbf {cd}}$ . For the other direction a proof on the length of the derivation is sufficient. The fact that the deduction theorem holds in $\mathbf {QN}^{\circ }$ renders the proof particularly simple. Therefore, we have:

Lemma 2 then entails that $\mathbf {G1h}_{\mathbf {cd}}$ is equivalent to Leitgeb’s $\mathbf {HYPE}$ .

Speranski [Reference Speranski25] establishes a strong completeness result (for countable signatures) for $\mathbf {QN}^{\circ }$ .Footnote ⁴ Speranski uses a Henkin-style proof similar to the strategy employed in Gabbay et al. [Reference Gabbay, Shehtman and Skvortsov11, sec. 7.2] for intuitionistic logic with constant domains. Leitgeb [Reference Leitgeb19] establishes a (weak) completeness proof for his Hilbert style system based on the work of Görnemann [Reference Görnemann12]. By Lemma 2 we can employ Speranski’s completeness result for our system $\mathbf {G1h}_{\mathbf {cd}}$ with respect to Routley semantics:

We now turn to investigating how much classical reasoning can be reproduced in our logic. Such questions will turn out to be essential components of the analysis of truth theories over $\mathbf {HYPE}$ .

2.3 HYPE and recapture

One of the desirable properties of the nonclassical logics employed in the debate on semantic paradoxes is the capability of recapturing classical reasoning in domains where there is no risk of paradoxicality, such as mathematics [Reference Field7, chap. 4].Footnote ⁵

The following lemma summarizes the recapture properties of $\mathbf {G1h}_{\mathbf {cd}}$ and extensions thereof. It essentially states that, in systems based on $\mathbf {G1h}_{\mathbf {cd}}$ , once we restrict our attention to a fragment of the language satisfying the excluded middle and/or explosion, the native HYPE-negation and conditional, as well as the defined intuitionistic negation, all behave classically.

Proof. We prove the claims for the crucial cases in which a conditional is involved:

For (i):

For (ii):

□

2.4 Equality

For our purposes it’s important to extend $\mathbf {G1h}_{\mathbf {cd}}$ with a theory of equality. $\mathbf {G1h_{cd}^=}$ is obtained by adding to $\mathbf {G1h}_{\mathbf {cd}}$ the following initial sequents for equality.

(Ref) $$\begin{align} & \Rightarrow t = t, \end{align} $$

(Rep) $$\begin{align} & s = t, A (s) \Rightarrow A (t). \end{align} $$

By an essential use of $\texttt {ConCp}$ , we can establish in $\mathbf {G1h_{cd}^=}$ that identity statements behave classically.

Proof. We use the identity sequents:

□

Lemma 4 reveals some subtle issues concerning the treatment of identity in subclassical logics generally employed to deal with semantical paradoxes. It tells us that identity is essentially treated as a classical notion in $\mathbf {G1h}_{\mathbf {cd}}^{=}$ . To obtain a similar phenomenon in absence of $\texttt {ConCp}$ and $\texttt {ClCp}$ , one would have to add the counterpositives of $\texttt {Rep}$ and $\texttt {Ref}$ to the system. A nonclassical treatment of identity would require some non-trivial changes to $\texttt {Rep}$ and $\texttt {Ref}$ . That identity is a classical notion is perfectly in line with our framework, in which identity is a non-semantic notion akin to mathematical notions.

3.1 Ordinals and transfinite induction

Our notational conventions for schemata generalize to schemata other than induction. A prominent role in the paper will be played by transfinite induction schemata. In order to introduce them, we need to assume a notation system $(\texttt {OT},\prec )$ for ordinals up to the Feferman–Schütte ordinal $\Gamma _0$ as it can be found, for instance, in [Reference Pohlers23, chap. 2]. $\texttt {OT}$ is a primitive recursive set of ordinal codes and $\prec $ a primitive recursive relation on $\texttt {OT}$ that is isomorphic to the usual ordering of ordinals up to $\Gamma _0$ . We distinguish between fixed ordinal codes, which we denote with $\alpha ,\beta ,\gamma ,\ldots $ , and $\zeta ,\eta ,\theta ,\xi ,\ldots $ as abbreviations for variables ranging over elements of OT. Our representation of ordinals satisfies all standard properties. In particular, we will make implicit use of the properties listed in [Reference Troelstra and Schwichtenberg28, p. 322].

We will make extensive use of the following abbreviations. We call a formula progressive if it is preserved upward by the ordinals:

$$ \begin{align*} \texttt{Prog}(A) := \forall \eta(\forall\zeta\prec\eta \, A(\zeta)\rightarrow A (\eta)), \end{align*} $$

where $\forall \zeta \prec \eta \, A(\zeta )$ is short for $\forall \zeta (\zeta \prec \eta \rightarrow A(\zeta ))$ . We will use this (standard) notational convention in several occasions in what follows. Similarly, we will write $\exists \zeta \prec \eta \, A(\zeta )$ for $\exists \zeta (\zeta \prec \eta \land A(\zeta ))$ .

This formulation of progressiveness is $\mathbf {HYA}$ -equivalent to a formulation as a sequent $\forall \zeta \prec \eta \, A(\zeta ) \Rightarrow A (\eta )$ . Moreover, if $A(x)\vee \neg A(x)$ is provable, then $\texttt {Prog}(A)$ is $\mathbf {HYA}$ -equivalent to:

(1) $$ \begin{align} \forall \eta(\forall\zeta\prec\eta \, A(\zeta)\supset A (\eta)). \end{align} $$

Transfinite induction up to the ordinal $\alpha \; (\prec \Gamma _0)$ will be formulated as the following sequent:

(TI _α(A)) $$\begin{align} \texttt{Prog} (A) \Rightarrow \forall \xi \prec \alpha \, A(\xi). \end{align} $$

An alternative would be to use a rule-formulation:

$\texttt {TI}_{\alpha }^{\texttt {r}}(A)$ differs from the standard rule formulation of transfinite induction (see, e.g., [Reference Halbach14]) in that its premise features only one formula in the succedent: this is because of the intuitionistic nature of the $\mathbf {HYPE}$ -conditional that is apparent in the rule $(\texttt {R}\!\rightarrow )$ .

The two formulations of induction just introduced are equivalent over $\mathbf {HYA}^-$ , i.e., given $\texttt {TI}_{\alpha }(A)$ , $\texttt {TI}_{\alpha }^{\texttt {r}}(A)$ is admissible, and given $\texttt {TI}_{\alpha }^{\texttt {r}}(A)$ , $\texttt {TI}_{\alpha }(A)$ is derivable.Footnote ⁶

$\texttt {TI}_{\alpha }(\mathcal {L})$ is short for $\texttt {TI}_{\alpha } (A)$ for every formula A of the language $\mathcal {L}$ . $\texttt {TI}_{< \alpha } (\mathcal {L})$ is short for $\texttt {TI}_{\beta } (\mathcal {L})$ for all $\beta \prec \alpha $ . The function $\omega _n$ is recursively defined in the standard way as: $\omega _0=1$ , $\omega _{n+1}=\omega ^{\omega _n}$ .

3.2 Transfinite induction and nonclassical predicates

Our main purpose in this paper is to study the proof-theoretic properties of extensions of $\mathbf {HYA}$ with additional predicates that may not behave classically—i.e., they may not satisfy Lemma 5. In fact, in the case of the pure arithmetical language, Lemma 5 gives us immediately that $\mathbf {HYA}$ derives $\texttt {TI}_{< \varepsilon _0} ( \mathcal {L}_{\mathbb {N}}^{\rightarrow } )$ . In this section we show directly that Gentzen’s original proof of $\texttt {TI}_{< \varepsilon _0} ( \mathcal {L}_{\mathbb {N}}^{\rightarrow } )$ can be carried out in $\mathbf {HYA}$ for suitable expansions of $\mathcal {L}_{\mathbb {N}}^{\rightarrow }$ .Footnote ⁷ To abuse slightly of notation, we will keep the label HYA for the versions of HYA formulated in languages with additional predicate symbols.

The rest of this subsection will be devoted to the proof of Theorem 1, which will involve several preliminary lemmata.

A key ingredient of Gentzen’s proof—which will also play an important role in sections 4 and 5—is Gentzen’s jump formula:

$$ \begin{align*} A^+(\theta) := \forall\xi(\forall\eta(\eta\prec\xi\rightarrow A(\eta))\rightarrow \forall\eta(\eta\prec\xi+\omega^{\theta}\rightarrow A(\eta))). \end{align*} $$

Proof. The argument is as follows: We assume $\texttt {Prog}(A)$ and we want to show $\texttt {Prog}(A^+)$ , i.e., $\forall \zeta \prec \theta \, A^+ (\zeta ) \rightarrow A^+ (\theta )$ . So we also assume $\forall \zeta \prec \theta \, A^+ (\zeta )$ and $\forall \zeta (\zeta \prec \xi \rightarrow A(\zeta ))$ and $\eta \prec \xi + \omega ^{\theta }$ to show $A(\eta )$ .

Informally, we make a case distinction: Either $\theta = 0$ or $\theta \succ 0$ .

Case 1: If $\theta = 0$ , then

(2) $$ \begin{align} \theta = 0, \eta\prec\xi+\omega^{\theta}\Rightarrow\eta\prec\xi\,\vee\,\eta=\xi. \end{align} $$

We have, by the reflexivity sequents and logical rules:

(3) $$ \begin{align} \forall\zeta \,(\zeta\prec\xi\rightarrow A (\zeta)), \eta\prec\xi \Rightarrow A (\eta). \end{align} $$

Again by reflexivity and the identity axioms:

(4) $$ \begin{align} \texttt{Prog}(A), \forall\zeta\,(\zeta\prec\xi\rightarrow A (\zeta)), \eta=\xi \Rightarrow A (\eta). \end{align} $$

By (2)–(4) and (Cut), we obtain

(5) $$ \begin{align} \theta = 0,\ \texttt{Prog}(A), \forall\zeta\,(\zeta\prec\xi\rightarrow A (\zeta)), \eta\prec\xi+\omega^{\theta} \Rightarrow A(\eta). \end{align} $$

Case 2: $\theta \succ 0$ . Then by a derivable version of Cantor’s Normal Form Theorem:

(†) $$ \begin{align} \theta \succ 0, \eta\prec\xi+\omega^{\theta}\Rightarrow \exists n \,\exists \theta_0\prec \theta (\eta\prec\xi+\omega^{\theta_0}\cdot n). \end{align} $$

Given that induction for ordinal notations up to $\omega $ is provable in $\mathbf {HYA}$ , we will show by induction on $n\prec \omega $ that

$$ \begin{align*} \forall \zeta \prec \, \theta \, A^+ (\zeta),\theta_0 \prec \theta \Rightarrow \forall \zeta (\zeta\prec \xi + \omega^{\theta_0} \cdot n \rightarrow A (\zeta)). \end{align*} $$

The base case is straightforward because the following is trivially derivable (by property (ord6) in [Reference Troelstra and Schwichtenberg28, p. 322]):

(6) $$ \begin{align} \forall \eta \prec \xi \, A (\eta) \Rightarrow (\forall \eta\prec\xi + \omega^{\theta_0}\cdot 0) \, A(\eta). \end{align} $$

For the induction step, we start by noticing that by instantiating $\xi $ in $A^+(\theta _0)$ with $\xi +\omega ^{\theta _0}\cdot n$ , we obtain

(7) $$ \begin{align} A^+(\theta_0) \Rightarrow \forall \zeta \prec \xi + \omega^{\theta_0} \cdot n \, A (\zeta) \rightarrow \forall \zeta \prec \xi + \omega^{\theta_0} \cdot (n +1) \, A (\zeta). \end{align} $$

As mentioned, by letting

$$ \begin{align*} B(x):= \forall \zeta\prec \xi +\omega^{\theta_0}\cdot x \,A(\zeta), \end{align*} $$

$\textbf {HYA}$ proves the $\omega $ -induction principle (with $n\prec \omega $ )

$$ \begin{align*} B(0), \forall n (B(n)\rightarrow B(n+1)) \Rightarrow \forall n\, B(n). \end{align*} $$

Therefore, by a series of cuts, we obtain

(8) $$ \begin{align} A^+(\theta_0), \forall \zeta \prec \xi \, A (\zeta) \Rightarrow \forall n \forall \zeta \prec \xi + \omega^{\theta_0} \cdot n \, A (\zeta). \end{align} $$

From (8) we obtain

(9) $$ \begin{align} \forall \zeta \prec \xi \, A (\zeta), \forall\zeta\prec\theta \, A^+(\zeta), \theta_0\prec\theta \Rightarrow \forall n \forall \zeta \prec \xi + \omega^{\theta_0} \cdot n \, A (\zeta). \end{align} $$

Therefore, we can instantiate n and $\zeta $ (with $\eta $ ), and move the antecedent of $\eta \prec \xi + \omega ^{\theta _0} \cdot n \rightarrow A (\eta )$ from the right-hand side to the left hand side of the sequent arrow. Since both n and $\eta $ are general, we can existentially generalize over them to get

(10) $$ \begin{align} \theta \succ 0, \forall \zeta \prec \xi \, A (\zeta), \forall\zeta\prec\theta \, A^+(\zeta), \exists n \,\exists \theta_0\prec\theta(\eta\prec\xi+\omega^{\theta_0}\cdot n) \Rightarrow A (\eta), \end{align} $$

which in turn by gives us

(11) $$ \begin{align} \theta \succ 0, \forall \zeta \prec \xi \, A (\zeta), \forall\zeta\prec\theta \, A^+(\zeta), \eta\prec\xi+\omega^{\theta} \Rightarrow A (\eta). \end{align} $$

Now we combine the two cases. Together with our (5) in Case 1, the last sequent enables us to derive

$$ \begin{align*} \theta = 0 \vee \theta \succ 0,\ \texttt{Prog}(A), \forall \zeta \prec \xi \, A (\zeta), \forall\zeta\prec\theta \, A^+(\zeta), \eta\prec\xi+\omega^{\theta} \Rightarrow A (\eta). \end{align*} $$

By the provability of $\theta = 0 \vee \theta \succ 0$ and applications of the rules $(\texttt {R} \rightarrow )$ and $(\texttt {R} \forall )$ we finally get

$$ \begin{align*} \texttt{Prog}(A) \Rightarrow \texttt{Prog}(A^+). \end{align*} $$

□

The progressiveness of Gentzen’s jump formula enables us then to establish:

Proof. We assume $\texttt {TI}_{\alpha } (\mathcal {L}^+)$ . Specifically we have

(12) $$ \begin{align} \texttt{Prog}(A^+) \Rightarrow \forall\xi\prec\alpha \,A^+(\xi). \end{align} $$

By the meaning of $\texttt {Prog}(A^+)$ , we obtain

(13) $$ \begin{align} \texttt{Prog}(A^+) \Rightarrow A^+(\alpha). \end{align} $$

By the previous Lemma 6 and cut we also have

(14) $$ \begin{align} \texttt{Prog}(A) \Rightarrow A^+(\alpha), \end{align} $$

which is

(15) $$ \begin{align} \texttt{Prog}(A) \Rightarrow \forall\xi(\forall\eta \prec\xi \, A(\eta) \rightarrow \forall\eta \prec\xi+\omega^{\alpha} \, A(\eta)). \end{align} $$

But also

(16) $$ \begin{align} \Rightarrow \forall \eta\prec 0 \,A(\eta), \end{align} $$

and therefore by (15) taking $\xi = 0$ , we obtain

$$ \begin{align*} \texttt{Prog}(A) \Rightarrow \forall\eta\prec\omega^{\alpha} A(\eta), \end{align*} $$

as desired.□

Corollary 2. If A is such that $\mathbf {HYA}$ proves $A(x)\vee \neg A(x)$ , we have that, if $\mathbf {HYA}$ proves the classical transfinite induction axiom schema for $\alpha $

$$ \begin{align*} (\forall \zeta\prec \eta A(\zeta)\supset A(\eta))\supset \forall \xi\prec \alpha \,A(\xi), \end{align*} $$

then $\mathbf {HYA}$ proves

$$ \begin{align*} (\forall \zeta\prec \eta A(\zeta)\supset A(\eta))\supset \forall \xi\prec \omega^{\alpha} \,A(\xi). \end{align*} $$

By Lemmas 3(ii) and 4, $\mathbf {HYA}$ proves $A(x)\vee \neg A(x)$ for any formula $A(x)$ of $\mathcal {L}_{\mathbb {N}}$ . Therefore, Corollary 2 entails the proof above can be carried out by replacing the $\mathbf {HYPE}$ -conditional with the material conditional when arithmetical properties are at stake.

All is set up to finally prove the main result of this section, the admissibility in $\mathbf {HYA}$ of the required schema of transfinite induction up to any ordinal $\alpha \prec \varepsilon _0$ .

Theorem 1 is key to our proof-theoretic analysis of a theory of truth over $\mathbf {HYPE}$ . We now turn to the definition of such a truth theory.

4.1 Semantics

The intended interpretation of our theory of truth is based on Kripke’s fixed point semantics [Reference Kripke18] and stems from the $\mathbf { HYPE}$ -models presented in Leitgeb [Reference Leitgeb19, sec. 7]. Our model will feature a state space, whose states are fixed-points of the usual monotone operator associated with the four-valued evaluation schema as stated in [Reference Visser29, Reference Woodruff30].

Let $\Phi \colon \mathcal {P}\omega \longrightarrow \mathcal {P}\omega $ be the monotone operator defined in [Reference Halbach14, lemma 15.6]: given some set X of codes of sentences of $\mathcal L_{\texttt {Tr}}$ , $\Phi (X)$ returns its closure under the $\mathbf {FDE}$ -evaluation. States will have the form $(\mathbb {N}, S)$ , where S is a fixed point of $\Phi $ . Since we are interested in constant domains and in keeping the interpretation of the arithmetical vocabulary fixed, we omit reference to $\mathbb {N}$ and identify states with the fixed points themselves. Therefore, we let

$$ \begin{align*} & \mathbb{W}:=\{ X \subseteq \texttt{Sent}_{\mathcal L_{\texttt{Tr}}}\;|\; \Phi(X) = X \}, \tag{17} \\ & S \leq_{\mathbb{W}} S' :\Leftrightarrow S \subseteq S', \tag{18} \\ & S^* := \omega \setminus \overline{S}, \text{ with } \overline{X}=\{\neg \varphi\;|\; \varphi\in X\}, \tag{19} \\ & \text{the interpretation of } \texttt{Tr} \text{ is denoted with } \texttt{Tr}^S := S. \tag{20} \end{align*} $$

The intended full model $\mathfrak {M}_{\Phi }$ is the $\mathbf {HYPE}$ model based on the frame $( \mathbb {W},\leq _{\mathbb {W}}, * )$ with the constant domain $\omega $ . The intended minimal model $\mathfrak {M}_{\Phi }^{\texttt {min}}$ is given by restricting the set of states to the minimal and maximal fixed points. By a straightforward induction on the height of the derivation in ${\textbf {KFL}}$ , we obtain:

4.2 Proof theory: lower bound

We show that ${\textbf {KFL}}$ can define (and therefore prove the well-foundedness of) Tarskian truth predicates for any $\alpha \prec \varepsilon _0$ . By the techniques employed in Feferman and Cantini’s analyses of the proof theory of $\textbf {KF}$ [Reference Cantini1, Reference Feferman4], this entails that ${\textbf {KFL}}$ can prove $\texttt {TI}_{<\varphi _{\varepsilon _0}0}(\mathcal {L}_{\mathbb {N}})$ .

We first define the Tarskian languages.

Definition 3. For $0\leq \alpha <\Gamma _0$ , we let

We then write

$$ \begin{align*} \texttt{Sent}_{\mathcal L_{\texttt{Tr}}}^{<\alpha}(x):\leftrightarrow\;& \exists \zeta \prec \alpha \,\texttt{Sent}_{\mathcal L_{\texttt{Tr}}}(\zeta,x), \\ \texttt{Tr}_{\alpha} (x) :\leftrightarrow \;& \texttt{Sent}^{< \alpha}_{\mathcal L_{\texttt{Tr}}} (x) \wedge \texttt{Tr} (x). \end{align*} $$

As we mentioned in Section 3, the arithmetical vocabulary behaves classically in ${\textbf {KFL}}$ .

The next two claims establish that the previous fact can be extended to all Tarskian languages whose indices can be proved to be well-founded. First, one shows that the claim ‘sentences in $\texttt {Sent}_{\mathcal L_{\texttt {Tr}}}^{<\eta }$ are either determinately true or determinately false’ is progressive.

Lemma 11. ${\textbf {KFL}}$ proves:

Proof. By the definition of $\texttt {OT}$ , ${\textbf {KFL}}$ proves that $\eta \in \texttt {OT}$ is either $0$ , or a successor ordinal, or a limit. By arguing informally in ${\textbf {KFL}}$ , we show that the statement of the lemma holds, thereby establishing the claim.

Lemma 10 gives us the base case. The limit case follows immediately by the definition of $\texttt {Sent}_{\mathcal L_{\texttt {Tr}}}(\lambda ,x)$ . For the successor step, one needs to establish (cf.[Reference Nicolai22, lemma 7]):

(21)

Claim (21) is obtained by a formal induction on the complexity of y. Crucially, the proof rests on the following ${\textbf {KFL}}$ -derivable claims, which provide the cases required by the induction:

□

By Theorem 1, we obtain:

Since, by Theorem 1, ${\textbf {KFL}}$ proves transfinite induction up to ordinals smaller than $\varepsilon _0$ , it follows that we are able to establish the fundamental properties of Tarskian truth predicates up to any ordinal smaller than $\varepsilon _0$ . For $\alpha <\Gamma _0$ , $\mathbf { RT}_{<\alpha }$ refers to the theory of ramified truth predicates up to $\alpha $ , as defined in [Reference Halbach14, sec. 9.1].

By the proof-theoretic equivalence of systems of ramified truth and ramified analysis established by Feferman [Reference Feferman2, Reference Feferman4], we obtain:

Feferman and Cantini established that $\textbf {KF}$ is proof-theoretically equivalent to $\mathbf {RT}_{<\varepsilon _0}$ . Our results so far then establish that ${\textbf {KFL}}$ is proof-theoretically at least as strong as $\textbf {KF}$ . In subsection 4.3, we will show that ${\textbf {KFL}}$ and $\textbf {KF}$ are in fact proof-theoretically equivalent.

4.3 Proof theory: upper bound

We interpret ${\textbf {KFL}}$ in the Kripke–Feferman system $\textbf {KF}$ . For definiteness, we consider the version of $\textbf {KF}$ formulated in a language $\mathcal {L}_{{\mathbb {T,F}}}$ featuring truth ( $\mathbb {T}$ ) and falsity ( $\mathbb {F}$ ) predicates. Such a version of $\textbf {KF}$ is basically the one presented in [Reference Cantini1, sec. 2], but without the consistency axiom that rules out truth-value gluts.

In order to interpret ${\textbf {KFL}}$ into $\textbf {KF}$ , we consider a two-layered translation that differentiates between the external and internal structures of $\mathcal L_{\texttt {Tr}}^{\rightarrow }$ -formulas. Essentially, the external translation fully commutes with negation, and translates the $\mathbf { HYPE}$ conditional as classical material implication. The internal translation treats negated truth ascriptions as falsity ascriptions, and is defined by an induction on the positive complexity of formulas that adheres to the semantic clauses of $\mathbf {FDE}$ -style fixed-point models. The internal translation therefore translates truth and non-truth of ${\textbf {KFL}}$ as truth and falsity of $\textbf {KF}$ , respectively. Since we want to uniformly translate formulas and their codes inside the truth predicate, we essentially employ the recursion theorem, as described for instance by [Reference Halbach14, sec. 5.3].

${\textbf {KFL}}$ -proofs can then be turned, by the translation $\sigma $ , into $\textbf {KF}$ -proofs, as the next proposition shows.

Proof. The proof is by induction on the height of the derivation in ${\textbf {KFL}}$ and follows almost directly from the definition of the translation $\sigma $ . Only the case of (KFL3) is slightly more involved: we require that (with $\equiv $ expressing material equivalence)

(26)

However, this can be proved by formal induction on the complexity of x.□

The combination of Proposition 2 and Corollary 4 yields that $\textbf {KF}$ and ${\textbf {KFL}}$ have the same arithmetical theorems, and in particular they have the same proof-theoretic ordinal—cf. [Reference Pohlers23, sec. 6.7].

In Section 5, we extend our results to schematic extensions of ${\textbf {KFL}}$ and $\textbf {KF}$ .

5.1 $\textbf {KFL}^{*}$ : rules and semantics

In this section we study the schematic extension of ${\textbf {KFL}}$ in the sense of [Reference Feferman4]. This is obtained by extending ${\textbf {KFL}}$ with a special substitution rule that enables us to uniformly replace the distinguished predicate P in arithmetical theorems $A(P)$ of our extended theory for arbitrary formulas of $\mathcal L_{\texttt {Tr}}^{\rightarrow }$ . More precisely, following Feferman, we will employ a schematic language $\mathcal L_{\texttt {Tr}}^{\rightarrow } (P)$ (and sub-languages thereof) featuring a fresh schematic predicate symbol P, which is assumed to behave classically.

The properties of ${\textbf {KFL}}$ expressed by Lemma 8 transfer directly to $\textbf {KFL}^{*}$ , and are proved in an analogous fashion.

The semantics given in Section 4.1 can be modified to provide a class of fixed-point models for $\textbf {KFL}^{*}$ . We call $\Phi _X$ the result of relativizing the operator from Section 4.1 to an arbitrary $X\subseteq \omega $ .Footnote ⁸ In particular, this means supplementing the positive inductive definition associated with $\Phi $ with the clause:

This modification clearly does not compromise the monotonicity of the operator. Therefore, let $\texttt {MIN}_{\Phi _X}$ be the minimal fixed point of $\Phi _X$ , and $\texttt {MAX}_{\Phi _X}$ its maximal one. For any X, we then obtain the minimal $\mathbf {HYPE}$ model

$$ \begin{align*} \mathfrak{M}^{\texttt{min}}_{\Phi_X}:=(\{\texttt{MIN}_{\Phi_X},\texttt{MAX}_{\Phi_X}\},\subseteq, *). \end{align*} $$

Again in $\mathfrak {M}^{\texttt {min}}_{\Phi _X}$ all arithmetical vocabulary is interpreted standardly at its two states (fixed-points). Only the interpretation of the truth predicate varies. Our notation reflects this.

Proof. By induction on the length of the derivation in $\textbf {KFL}^{*}$ .

We consider the case of the substitution rule applied to an arithmetical sequent

. That is, our proof ends with

with $B(x)$ an arbitrary formula of $\mathcal L_{\texttt {Tr}}^{\rightarrow }$ .

By induction hypothesis, for all X,

. Since

is arithmetical, for all interpretations Y of P,

. Then, following [Reference Feferman4], we can let Y be

to obtain that

□

5.2 Proof-theoretic analysis

We first consider the proof-theoretic lower-bound for $\textbf {KFL}^{*}$ . We adapt to the present setting the strategy outlined in [Reference Feferman and Strahm6, p. 84]. In particular, Feferman and Strahm formalize the notion of A-jump hierarchy, which is a hierarchy of sets of natural numbers obtained by iterating an arithmetical operator expressed by an arithmetical formula $A(X,\theta ,y)$ . An A-jump hierarchy is relativized when the starting point is a specific set of natural numbers expressed by some predicate P. The notion of A-jump hierarchy is quite general, and has as special cases familiar hierarchies such as the Turing-jump hierarchy.

For our purposes, it’s useful to consider A-jump hierarchies in which membership in second-order parameters is replaced by the notion of satisfaction. In order to achieve this, we employ Feferman’s strategy in [Reference Feferman4] in which the stages of the Turing jump-hierarchy are represented by means of suitable primitive recursive functions on codes of $\mathcal L_{\texttt {Tr}}$ -formulas. Specifically, we encode in suitable primitive recursive functions the stages of a hierarchy in which the formula A is the Veblen-jump formula that will be introduced shortly.

We denote with ${A}(\texttt {Tr} f^A,\eta ,y)$ the result of replacing every occurrence of $(u, v) \in X$ in $A(X,\eta ,y)$ with

$$ \begin{align*} \texttt{Tr} \;\text{ sub}(f^A_v,\ulcorner x\urcorner, \text{ num}(u)), \end{align*} $$

where the functions $f^A_x(y)$ are recursively defined as follows:

In the clause for $f^{A,\zeta }$ , the input x is intended to be an ordered pair $(x_0,x_1)$ . As in the definitions of translations $\sigma $ and $\tau $ above, the existence of the function $f^A$ can be obtained by employing the recursion theorem, as it needs to apply to its arithmetical code.

Recall the general pattern of the Gentzen jump formula—with $\rightarrow $ the $\mathbf {HYPE}$ conditional:

$$ \begin{align*} \mathcal{J} (B,\xi) := \forall \eta ( \forall \zeta \prec \eta \, B( \zeta ) \rightarrow \forall \zeta \prec \eta + \xi \, B (\zeta)). \end{align*} $$

We build our $\mathcal {A}$ -hierarchy on the more complex Veblen-jump formula $\mathcal {A}$ , as stated by Schütte in [Reference Schütte24, p. 185], which is crucial for the proof-theoretic analysis of predicative systems.

One first defines the functions:

$$ \begin{align*} \begin{array}{ll} \texttt{e}(0)=0 & \texttt{h}(0)=0\\ \texttt{e}(\omega^{\eta})=\eta & \texttt{h}(\omega^{\eta})=0\\ \texttt{e}(\omega^{\eta_1}+\cdots +\omega^{\eta_n})=\eta_n &\texttt{h}(\omega^{\eta_1}+\cdots +\omega^{\eta_n})=\omega^{\eta_1}+\cdots +\omega^{\eta_{n-1}} \end{array} \end{align*} $$

with $\eta _n\preceq \cdots \preceq \eta _1$ .

The Veblen-jump formula $\mathcal {A}$ is then the following:

$$ \begin{align*} \mathcal{A} (\texttt{Tr} f^{\mathcal{A}} ,\xi, y) := \forall \zeta (\texttt{h}(\xi) \preccurlyeq \zeta \prec \xi \, \mathcal{J} ( \texttt{Tr} f^{\mathcal{A}}_{\zeta} , \varphi_{\texttt{e}(\xi)}y)). \end{align*} $$

It expresses that, given some ordinal $\xi $ , the $\mathcal {A}$ -jump hierarchy in the interval between the ordinal $\texttt {h}(\xi )$ and $\xi $ itself is closed under the Gentzen jump relative to $\varphi _{\texttt {e}(\xi )}y$ (with y a parameter). In the following we will omit the superscripts specifying the formula, since we will keep $\mathcal A$ fixed.

An essential ingredient of the lower-bound proof for $\textbf {KFL}^{*}$ is the ‘disquotational’ behavior of our truth predicate for stages in the hierarchy that are provably well-founded.

Lemma 12. If we have $\texttt {TI}_{\alpha }(\mathcal L_{\texttt {Tr}}^{\rightarrow })$ , then for all $\eta $ , with $0 \prec \eta \prec \alpha $

Proof. For all $\eta $ and all n, we can show that $\texttt {Sent}_{\eta } (f^{\eta }((n_0,n_1)))$ and $\texttt {Sent}_{\eta } (f_{\eta } (n))$ by transfinite induction on $\eta $ making use of the properties of the ramified truth predicates such as, for $\lambda \prec \alpha $ limit:

$$ \begin{align*} \forall \zeta \prec \lambda \big( \texttt{Tr}_{\lambda} (\texttt{Tr}_{\zeta} t)\leftrightarrow \texttt{Tr}_{\zeta}\texttt{val}(t)\big). \end{align*} $$

Such properties just state that Tarskian truth predicates are fully compositional for ordinals for which we have transfinite induction ([Reference Feferman4], [Reference Halbach14, sec. 9.1]).

Since all truth predicates in are provably compositional by $\texttt {TI}_{\alpha }(\mathcal L_{\texttt {Tr}}^{\rightarrow })$ , the claim is obtained by the fact that full compositionality entails disquotation as shown by [Reference Tarski, Berka and Kreiser27].□

The disquotational properties allow us to establish some fundamental properties of the $\mathcal {A}$ -jump hierarchy. In particular, we show that the $\mathcal {A}$ -jump hierarchy can be elegantly expressed by truth ascriptions on the functions $f_{\alpha }$ . This is because the truth predicate enables us to quantify over the stages of the construction of the $\mathcal {A}$ -jump hierarchy, and also to retrieve specific information about all the previous stages of the hierarchy, thanks to the disquotational property.

Lemma 13. If $\texttt {TI}_{\alpha }(\mathcal L_{\texttt {Tr}}^{\rightarrow })$ is derivable in $\textbf {KFL}^{*}$ for some $\alpha> 0$ , then we can derive in $\textbf {KFL}^{*}$ :

$$ \begin{align*} \forall y ( P (y) \!\!\leftrightarrow\!\! \texttt{Tr} f_0 (y)) \wedge \forall\! \zeta [& 0 \prec \zeta \prec \alpha\!\!\rightarrow\!\! \forall\! y [ \texttt{Tr} f_{\zeta} (y) \!\!\leftrightarrow\!\! \forall z ( \texttt{h} (\zeta) \preccurlyeq z \prec \zeta \\&\quad\rightarrow\! \mathcal{J}(\texttt{Tr} f_z,\varphi_{\texttt{e}(\zeta)}y)) ]]. \end{align*} $$

Proof. By our disquotational axioms for P, it immediately follows that $\forall y ( P(y) \leftrightarrow \texttt {Tr} f_0 (y) )$ .

Let’s assume now that $0 \prec \zeta \prec \alpha $ . We have

the right hand side is equivalent by the disquotational property to

$$ \begin{align*} {\mathcal A} ( \texttt{Tr} f^{\zeta} , \zeta , y). \end{align*} $$

By the definition of $f^{\zeta }$ , the latter formula is in turn equivalent to

which is again equivalent to

(27)

By applying the disquotational property to (27), we obtain

$$ \begin{align*} \forall z (\texttt{h}(\zeta) \preccurlyeq z \prec \zeta \, \mathcal{J}( \texttt{Tr} f_{z} (x),\varphi_{\texttt{e}(\zeta)} y)). \end{align*} $$

□

Just like the Gentzen jump enabled us to establish the preservation of properties via ‘steps’ determined by fundamental sequence of ordinals $\omega _n$ up to $\varepsilon _0$ , the Veblen jump allows us to establish the preservation of properties along the ‘steps’ governed by a fundamental sequence of ordinals up to $\Gamma _0$ . The progressiveness of the Veblen jump is the main engine that enables us to climb the $\mathcal {A}$ -hierarchy: this is what the next lemma establishes, which is analogous to Schütte’s Lemma 9 in [Reference Schütte24, p. 186], a lemma that has become a standard tool in predicative proof-theory.

Lemma 14. If $\texttt {TI}_{\alpha }(\mathcal L_{\texttt {Tr}}^{\rightarrow })$ is provable in $\textbf {KFL}^{*}$ for $\alpha <\Gamma _0$ , then $\textbf {KFL}^{*}$ proves

$$ \begin{align*} \forall \zeta (0\prec \zeta\prec \alpha \land (\forall \theta\prec \zeta\, \texttt{Prog}(\texttt{Tr} f_{\theta})\rightarrow\texttt{Prog}({\texttt{Tr} f_{\zeta}}))). \end{align*} $$

Proof. Let $\texttt {l}(\cdot )$ be the function that keeps track of the syntactic complexity of an ordinal code. One first shows that the following claims

$$ \begin{align*} & \zeta\prec \alpha \tag{28} \\ & \forall x \prec \zeta \,\texttt{Prog} (\texttt{Tr} f_x) \tag{29} \\ & \forall y\prec \eta \,\texttt{Tr} f_{\zeta} (y) \tag{30} \\ & \forall y(\texttt{l}(y)<\texttt{l}(\theta)\rightarrow (y \prec \varphi_{\texttt{e}(\zeta)} \eta \rightarrow ( \forall z ( \texttt{h} (\zeta ) \preccurlyeq z \prec \zeta \rightarrow \mathcal{J} ( \texttt{Tr} f_z, y)))) \tag{31} \\ & \theta \prec \varphi_{\texttt{e} (\zeta)}\eta \tag{32} \\ & \texttt{h}(\zeta)\preceq \xi\prec \zeta \tag{33} \end{align*} $$

entail $\mathcal {J}(\texttt {Tr} f_{\xi },\theta )$ .

First we apply the principle of induction on the syntactic composition of ordinal codes [Reference Schütte24, theorem 20.10, p. 173] (provable in $\textbf {KFL}^{*}$ ):

(34) $$ \begin{align} \forall x(\forall y(\texttt{l}(y)<\texttt{l}(x)\rightarrow \phi(y))\rightarrow\phi(x))\rightarrow \phi(t) \end{align} $$

to the formula

$$ \begin{align*} \phi(u):\leftrightarrow u\prec \varphi_{\texttt{e}(\zeta)}\eta\rightarrow \forall z( \texttt{h}(\zeta)\preccurlyeq z \prec \zeta \;\mathcal{J}(\texttt{Tr} f_z,u)). \end{align*} $$

Since we can prove that

$$ \begin{align*} \forall x & \prec \zeta \,\texttt{Prog}(\texttt{Tr} f_x)\rightarrow \big( \forall y\prec \theta \, (\texttt{l}(y)<\texttt{l}(\theta) \rightarrow (y\prec \varphi_{\texttt{e}(\zeta)}\theta \rightarrow \forall z(\texttt{h}(\zeta)\preccurlyeq z \\ &\prec \zeta \;\mathcal{J}(\texttt{Tr} f_z,y))))\big), \end{align*} $$

we obtain by the above-mentioned induction principle

(35) $$ \begin{align} \forall x & \prec \zeta \,\texttt{Prog}(\texttt{Tr} f_x)\rightarrow ( \forall y\prec \eta \,\texttt{Tr} f_{\zeta}(y)\rightarrow ( \theta\prec \varphi_{\texttt{e}(\zeta)}\eta\rightarrow \forall z(\texttt{h}(\zeta)\preccurlyeq z\nonumber \\ &\prec \zeta \;\mathcal{J}(\texttt{Tr} f_z,\theta)) )). \end{align} $$

By the definition of progressiveness we obtain

(36) $$ \begin{align} \texttt{Prog}(\texttt{Tr} f_{\xi}) \rightarrow (\forall x \prec \varphi_{\texttt{e}(\zeta)}\eta \,\mathcal{J} (\texttt{Tr} f_{\xi}, x) \rightarrow \mathcal{J} ( \texttt{Tr} f_{\xi},\varphi_{\texttt{e}(\zeta)} \eta)). \end{align} $$

Therefore, by combining the previous two claims, we obtain

(37) $$ \begin{align} \forall x\prec \zeta \,\texttt{Prog}(\texttt{Tr} f_x)\rightarrow \big(\forall y\prec \eta \,\texttt{Tr} f_{\zeta}(y)\rightarrow\forall z(\texttt{h}(\zeta)\preccurlyeq z \prec \zeta \;\mathcal{J} (\texttt{Tr} f_z,\varphi_{\texttt{e}(\zeta)}\eta))\big). \end{align} $$

Finally, by Lemma 13 applied to (37), we can conclude that

(38) $$ \begin{align} \forall x\prec \zeta \,\texttt{Prog}(\texttt{Tr} f_x)\rightarrow \big(\forall y\prec \eta \,\texttt{Tr} f_{\zeta}(y)\rightarrow \texttt{Tr} f_{\zeta}(\eta))\big). \end{align} $$

□

Let’s characterize the fundamental series of ordinals $<\Gamma _0$ as functions of natural numbers in the standard way: $\gamma _0:=\omega $ , and $\gamma _{n+1}:= \varphi _{\gamma _n}0$ . Then, we have:

Proof. We assume $\texttt {Prog}(P)$ . If $\texttt {TI}_{\gamma _n} (P)$ is derivable in $\textbf {KFL}^{*}$ , then by Corollary 2 and the determinateness of P we can show $\texttt {TI}_{\omega ^{\gamma _n} +1} (P)$ . By the substitution rule we get that the hierarchy predicates are well-defined. Additionally we can prove that

(39) $$ \begin{align} \forall \zeta \prec\omega^{\gamma_n} +1 \;\forall x ( \texttt{Tr}_{\zeta}(x) \vee \neg \texttt{Tr}_{\zeta}(x)). \end{align} $$

Notice that by (39) we can reformulate this fragment of the hierarchy by replacing all the occurrences of the ${ \bf HYPE}$ -conditional by the material conditional. Therefore, we have

(40) $$ \begin{align} \forall \zeta \prec \omega^{\gamma_n} +1 ( \texttt{Tr} f_{\zeta} (x) \leftrightarrow {\mathcal A} (\texttt{Tr} f^{\zeta}, \zeta,x) ). \end{align} $$

By the previous lemma, for $a\prec \omega ^{\gamma _n}+1$ ,

(41) $$ \begin{align} \forall b\prec a \;\texttt{Prog}(\texttt{Tr} f_b)\rightarrow \texttt{Prog}(\texttt{Tr} f_a), \end{align} $$

and therefore, by the substitution rule applied to $\texttt {TI}_{\gamma _n} (P)$ and (41), we have that $\texttt {Prog}(\texttt {Tr} f_{\omega ^{\gamma _n}})$ , which entails $\texttt {Tr} f_{\omega ^{\gamma _n}}({0})$ .

Using (40), we have

(42) $$ \begin{align} \forall \zeta (\texttt{h}(\omega^{\gamma_n})\preccurlyeq \zeta \prec \omega^{\gamma_n} \,\mathcal{J}(\texttt{Tr} f_{\zeta},\varphi_{\texttt{e}(\omega^{\gamma_n})}0). \end{align} $$

However, since $\texttt {h}(\omega ^{\gamma _n}) = 0$ and $\texttt {e}(\omega ^{\gamma _n}) = \gamma _n$ we can then infer

(43) $$ \begin{align} \forall \zeta \prec \omega^{\gamma_n} ( \forall y ( \forall x \prec y \texttt{Tr} f_{\zeta} (x) \rightarrow \forall x \prec y + \varphi_{\gamma_n}0 \: \texttt{Tr} f_{\zeta} ( x ))). \end{align} $$

By letting $\zeta =y=0$ , we obtain $\forall x\prec \varphi _{\texttt {e}(\omega ^{\gamma _n})}0\;P(x)$ , as desired.□

Following the characterization of predicative analysis in terms of ramified systems given in [Reference Feferman2, Reference Feferman4], and the relationships between ramified truth and ramified analysis studied there, one can then conclude that the systems of ramified analysis below $\Gamma _0$ are proof-theoretically reducible to our system $\textbf {KFL}^{*}$ .

In subsection 4.3 we showed that ${\textbf {KFL}}$ can be proof-theoretically reduced—w.r.t. arithmetical sentences—to $\textbf {KF}$ . That argument can be lifted to $\textbf {KFL}^{*}$ . One can consider the system $\textbf {KF}^*$ — $\texttt {Ref}^{*}(\texttt {PA}(P))$ in [Reference Feferman4]—and slightly modify the translations $\sigma $ , $\tau $ from Definition 4: in particular, we let

$$ \begin{align*} & (P s)^{\sigma}= (Ps)^{\tau}= Ps && (\neg Ps)^{\tau}= \neg Ps. \end{align*} $$

Then, by induction on the length of proof in $\textbf {KFL}^{*}$ , we can prove:

Given the analysis of $\textbf {KF}^*$ given in [Reference Feferman4], the combination of Propositions 5 and 4 yields a sharp proof-theoretic analysis also for $\textbf {KFL}^{*}$ :