2 Preliminaries

Intermediate propositional logics have been investigated extensively since the 1950s. Intermediate predicate logics are comparatively less well understood; however, they constitute an active area of research (see [Reference Gabbay, Shehtman and Skvortsov14]). We begin by collecting some preliminary definitions.

All the logics we consider are formulated in the standard language of intuitionistic and classical logic with propositional connectives $\land $ , $\lor $ , $\mathbin {\rightarrow }$ , the constant $\bot $ for absurdity, and the quantifiers $\forall $ and $\exists $ . $\lnot A$ is defined as $A \mathbin {\rightarrow } \bot $ and $\top $ as $\lnot \bot $ . Terms and atomic formulas are defined as usual. We allow $0$ -place predicate symbols, i.e., propositional variables. A formula containing no quantifiers is called quantifier-free and a quantifier-free formula with only $0$ -place predicate symbols is called propositional. For the most part we will consider pure logics, i.e., logics not involving the identity predicate $=$ . We assume that function symbols are available, however.

We use A, B, …, as metavariables for formulas, and t, s, …, as metavariables for terms. We write $A(x)$ to indicate that x occurs free in A. The result of substituting s for all (free) occurrences of x in a term t or a formula A is indicated by $t[s/x]$ or $A[s/x]$ . When it is clear which variable x is intended, we write $A(s)$ for $A[s/x]$ . The result of replacing every occurrence of a term t by a term s in a formula A is indicated by $A[s/t]$ . A substitution instance of a formula A is any formula resulting from A by uniformly replacing any number of atomic formulas $P(t_{1}, \dots , t_{n})$ by formulas $B[t_{1}/x_{1}, \dots , t_{n}/x_{n}]$ (in such a way that free variables of B are not captured by quantifiers in A of course). In particular, a substitution instance of a propositional formula A is any formula resulting from A by uniformly replacing any number of $0$ -place predicate symbols P by formulas B.

Definition 2.1. A propositional logic $\mathbf {L}$ is a set of propositional formulas closed under substitution and modus ponens, i.e., if $\mathbf {L}$ contains $A \mathbin {\rightarrow } B$ and B it also contains B. A predicate logic is a set of formulas closed under substitution, modus ponens, and the quantifier rules

which are subject to the eigenvariable condition: x must not be free in the conclusion.

Logics can be characterized by the formulas derivable from a set of axiom schemas by modus ponens. This is of course equivalent to closing the axioms under substitution and modus ponens. For instance, intuitionistic propositional logic $\mathbf {H}$ is obtained from the propositional axioms given in [Reference Troelstra and van Dalen28, 4.1]. Classical propositional logic $\mathbf {C}$ and predicate logic $\mathbf {Q}\mathbf {C}$ are obtained by adding $\lnot A \lor A$ (or alternatively $\lnot \lnot A \mathbin {\rightarrow } A$ ) as an axiom.

The following intermediate propositional logics will play important roles:

1. 1. $\mathbf {LC}$ , characterized alternatively as the formulas valid on linearly ordered Kripke frames or as infinite-valued Gödel logic,Footnote ² axiomatized over $\mathbf {H}{}$ using the schema

   (Lin) $$\begin{align} (A \mathbin{\rightarrow} B) \lor (B \mathbin{\rightarrow} A). \end{align}$$

2. 2. $\mathbf {LC}_{m} = \mathbf {LC} + {B_{m}}$ , characterized as formulas valid on linearly ordered Kripke frames of height $< m$ , or as the Gödel logic on m truth values, also known as $\textbf {S}_{m-1}$ [Reference Hosoi16]. Here, ${B_{m}}$ is:

   (Bm) $$\begin{align} (A_{1} \mathbin{\rightarrow} A_{2}) \lor (A_{2} \mathbin{\rightarrow} A_{3}) \lor \dots \lor (A_{m} \mathbin{\rightarrow} A_{m+1}). \end{align}$$

3. 3. $\mathbf {KC}$ , the logic of weak excluded middle [Reference Jankov19], axiomatized over $\mathbf {H}{}$ using the schema

   (J) $$\begin{align} \lnot A \lor \lnot\lnot A. \end{align}$$

If $\mathbf {L}$ is an intermediate propositional logic, we can consider the corresponding “elementary calculus,” i.e., the system obtained by replacing propositional variables with atomic formulas of a first-order language, with or without identity. We will be interested in formulas provable in such an elementary calculus from a set of assumptions $\Gamma $ .

Given a propositional logic $\mathbf {L}$ , and possibly a set of additional axiom schemas $\mathit {Ax}$ involving quantifiers, we can generate a predicate logic:

Definition 2.4. If $\mathbf {L}$ is a propositional logic, and $\mathit {Ax}$ is a set of formulas, then $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ is the smallest predicate logic containing $\mathbf {L}$ , the standard quantifier axioms

$$ \begin{align*} \forall x\, A(x) & \mathbin{\rightarrow} A(t) \quad \text{and}\\ A(t) & \mathbin{\rightarrow} \exists x\, A(x), \end{align*} $$

and the formulas in $\mathit {Ax}$ .

If $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ contains intuitionistic predicate logic $\mathbf {Q}\mathbf {H}$ and is contained in classical predicate logic $\mathbf {Q}\mathbf {C}$ , it is called an intermediate predicate logic.

A is a formula in $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ iff $\mathbf {Q}\mathbf {L} + \mathit {Ax} \vdash A$ . If $\mathbf {L}{}$ itself is characterized by a set of propositional axioms, it is enough to require substitution instances of axioms of $\mathbf {L}{}$ in the above definition. For instance, taking $\mathbf {H}{}$ as above, the proofs in intuitionistic predicate logic $\mathbf {Q}\mathbf {H}$ are just the proofs in the system $H_{2}$ -IQC of [Reference Troelstra and van Dalen28, 4.3].

If $\mathit {Ax}$ is empty, $\mathbf {Q}\mathbf {L} = \mathbf {Q}\mathbf {L} + \mathit {Ax}$ is the weakest pure intermediate predicate logic extending $\mathbf {L}$ . For instance, $\mathbf {Q}\mathbf {LC}$ is the weakest predicate logic obtained from $\mathbf {LC}$ , and is axiomatized by $\mathbf {Q}\mathbf {H} + \mathit {Lin}$ . It is complete for linearly ordered Kripke frames (see [Reference Corsi11, Reference Skvortsov26]).

It is possible to extend the weakest predicate logic of an intermediate propositional logic $\mathbf {L}$ by adding intuitionistically invalid schemas $\mathit {Ax}$ involving quantifiers. Some important examples are the constant domain principle,

(CD) $$\begin{align} \forall x(A(x) \lor B) & \mathbin{\rightarrow} (\forall x\,A(x) \lor B), \end{align}$$

the double negation shift (or Kuroda’s principle),

(K) $$\begin{align} \forall x\,\lnot\lnot A(x) & \mathbin{\rightarrow} \lnot\lnot \forall x\,A(x), \end{align}$$

and the quantifier shifts

(Q∃) $$\begin{align} (B \mathbin{\rightarrow} \exists x\,A(x)) & \mathbin{\rightarrow} \exists x(B \mathbin{\rightarrow} A(x)), \end{align}$$

(Q∀) $$\begin{align} (\forall x\,A(x) \mathbin{\rightarrow} B) & \mathbin{\rightarrow} \exists x(B \mathbin{\rightarrow} A(x)). \end{align}$$

$\mathbf {Q}\mathbf {LC} + \mathit {CD}$ axiomatizes the formulas valid on linearly ordered Kripke frames with constant domains, and also the first-order Gödel logic $\mathbf {G}_{\mathbb {R}}$ of formulas valid on the interval $[0, 1]$ . $\mathbf {Q}\mathbf {H} + K$ characterizes the formulas valid on Kripke frames with the McKinsey property and $\mathbf {Q}\mathbf {LC} + \mathit {CD} + K$ is the logic of linear Kripke frames with maximal element. It is also the first-order Gödel logic $\mathbf {G}_{0}$ of formulas valid on $\{0\} \cup [1/2,1]$ . $\mathbf {Q}\mathbf {LC}_{m}$ is not complete for linear Kripke frames of height $< m$ (contrary to what one might expect). However, $\mathbf {G}_{m} = \mathbf {Q}\mathbf {LC}_{m} + \mathit {CD}$ is complete for linearly ordered Kripke frames of height $<m$ with constant domains. It is also the m-valued first-order Gödel logic $\mathbf {G}_{m}$ .Footnote ³

3 ${\varepsilon \tau }$ -calculi for intermediate logics

Formulas and terms of the ${\varepsilon \tau }$ -calculus are defined by simultaneous induction, allowing that if $A(x)$ is a formula already defined, then $\varepsilon _{x}\,A(x)$ and $\tau _{x}\,A(x)$ are terms. In $\varepsilon _{x}\,A(x)$ and $\tau _{x}\,A(x)$ the variable x is bound. We call terms of the form $\varepsilon _{x}\,A(x)\ \varepsilon $ -terms and those of the form $\tau _{x}\,A(x)\ \tau $ -terms (collectively: ${\varepsilon \tau }$ -terms).

As usual, we consider $\varepsilon $ - and $\tau $ -terms to be identical up to renaming of bound variables, and define substitution of ${\varepsilon \tau }$ -terms into formulas, as in $A(\varepsilon _{x}\,A(x))$ , so that bound variables are tacitly renamed to avoid clashes.

Definition 3.2. Suppose $\mathbf {L}$ is an intermediate propositional logic. An ${\varepsilon \tau }$ -proof $\pi $ of B is a proof in the elementary calculus of $\mathbf {L}$ from critical formulas $\Gamma $ of the form

$$ \begin{align*} A(t) & \mathbin{\rightarrow} A(\varepsilon_{x}\,A(x)),\\ A(\tau_{x}\,A(x)) & \mathbin{\rightarrow} A(t). \end{align*} $$

We write $\mathbf {L}{\varepsilon \tau } \vdash B$ if such a $\pi $ exists, or $\Gamma \vdash _{\mathbf {L}{\varepsilon \tau }}^{\pi } B$ when we want to identify the critical formulas and the proof $\pi $ .

The pure ${\varepsilon \tau }$ -calculus $\mathbf {L}{\varepsilon \tau }$ of $\mathbf {L}$ is the set of quantifier-free formulas that have ${\varepsilon \tau }$ -proofs.

We are interested in the relationships between intermediate predicate logics $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ and the ${\varepsilon \tau }$ -calculus $\mathbf {L}{\varepsilon \tau }$ of their propositional fragment $\mathbf {L}$ . Since the language of $\mathbf {L}{\varepsilon \tau }$ does not contain quantifiers, we must define a translation of predicate formulas that do contain them into the language of the ${\varepsilon \tau }$ -calculus.

Definition 3.3. The ${\varepsilon \tau }$ -translation $A^{\varepsilon \tau }$ of a formula A is defined as follows:

$$ \begin{align*} A^{\varepsilon\tau} & = A \text{ if }A\text{ is atomic}.\\ (A \land B)^{\varepsilon\tau} & = A^{\varepsilon\tau} \land B^{\varepsilon\tau}, & (A \lor B)^{\varepsilon\tau} & = A^{\varepsilon\tau} \lor B^{\varepsilon\tau}, \\ (A \mathbin{\rightarrow} B)^{\varepsilon\tau} & = A^{\varepsilon\tau} \mathbin{\rightarrow} B^{\varepsilon\tau}, & (\lnot A)^{\varepsilon\tau} & = \lnot A^{\varepsilon\tau}, \\ (\exists x\, A(x))^{\varepsilon\tau} & = A^{\varepsilon\tau}(\varepsilon_{x}\,A(x)^{\varepsilon\tau}), & (\forall x\, A(x))^{\varepsilon\tau} & = A^{\varepsilon\tau}(\tau_{x}\,A(x)^{\varepsilon\tau}). \end{align*} $$

Again, substitution of ${\varepsilon \tau }$ -terms for variables must be understood modulo renaming of bound variables so as to avoid clashes. Clearly, if A contains no quantifiers, then $A^{\varepsilon \tau } = A$ .

The point of the classical $\varepsilon $ -calculus is that it can replace quantifiers and quantifier inferences. And indeed, in classical first-order logic, a first-order formula A is provable iff its translation $A^{\varepsilon }$ is provable in the pure $\varepsilon $ -calculus. The “if” direction is the content of the second $\varepsilon $ -theorem, while the “only if” direction follows more simply by translating derivations.

We defined the ${\varepsilon \tau }$ -calculus on the basis of a propositional logic $\mathbf {L}$ . It is also possible to define an “extended” ${\varepsilon \tau }$ -calculus by adding ${\varepsilon \tau }$ -terms and critical formulas to the full first-order language including quantifiers, and then considering proofs in $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ from critical formulas.

If A is quantifier-free and $\mathbf {L}{\varepsilon \tau } \vdash A$ then $(\mathbf {Q}\mathbf {L} + \mathit {Ax}){\varepsilon \tau } \vdash A$ . One may wonder, however, if $(\mathbf {Q}\mathbf {L} + \mathit {Ax}){\varepsilon \tau }$ is stronger than $\mathbf {L}{\varepsilon \tau }$ in the sense that for some formulas A, $(\mathbf {Q}\mathbf {L} + \mathit {Ax}){\varepsilon \tau } \vdash A$ but not $\mathbf {L}{\varepsilon \tau } \vdash A^{\varepsilon \tau }$ . This is not so as long as the ${\varepsilon \tau }$ -translations of the axioms $\mathit {Ax}$ of $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ are provable in $\mathbf {L}{\varepsilon \tau }$ ; then the extended $\varepsilon $ -calculus is conservative over the pure $\varepsilon $ -calculus.

Proof. By standard proof transformations we may assume that the proof $\pi $ in $(\mathbf {Q}\mathbf {L} + \mathit {Ax}){\varepsilon \tau }$ is such that every formula is used as a premise of at most one modus ponens or quantifier inference, and that the eigenvariables of all quantifier inferences are distinct (the proof is regular). The proof then proceeds by induction on the length of $\pi $ .

Any formula B in $\pi $ that is not the conclusion of an inference is either in $\mathbf {L}$ , a critical formula, in $\mathit {Ax}$ , or a standard quantifier axiom. Then $B^{\varepsilon \tau }$ is also either an axiom in $\mathbf {L}$ , a critical formula (as can easily be seen from the definition of the ${\varepsilon \tau }$ -translation), or, if $B \in \mathit {Ax}$ then $\mathbf {L}{\varepsilon \tau } \vdash B^{\varepsilon \tau }$ (by hypothesis). If B is a standard quantifier axiom, its ${\varepsilon \tau }$ -translation is a critical formula:

$$ \begin{align*} [A(t) \mathbin{\rightarrow} \exists x\,A(x)]^{\varepsilon\tau} &= A^{\varepsilon\tau}(t) \mathbin{\rightarrow} A^{\varepsilon\tau}(\varepsilon_{x}\,A^{\varepsilon\tau}(x)),\\ [\forall x\,A(x) \mathbin{\rightarrow} A(t)]^{\varepsilon\tau} &= A^{\varepsilon\tau}(\tau_{x}\,A^{\varepsilon\tau}(x)) \mathbin{\rightarrow} A^{\varepsilon\tau}(t). \end{align*} $$

If $B \mathbin {\rightarrow } A(x)$ is derivable (where the eigenvariable x is not free in B), then so is $B \mathbin {\rightarrow } A(\tau _{x}\,A(x))$ , by substituting $\tau _{x}\,A(x)$ everywhere x appears free in the part of $\pi $ leading to $B \mathbin {\rightarrow } A(x)$ , and renaming bound variables to avoid clashes. Thus, if B is the conclusion of a quantifier inference, $\mathbf {L}{\varepsilon \tau } \vdash B^{\varepsilon \tau }$ . Similarly, if $B(x) \mathbin {\rightarrow } A$ is derivable, so is $B(\varepsilon _{x}\,B(x)) \mathbin {\rightarrow } A$ . (Cf. [Reference Moser and Zach23, Lemma 7].)⊣

As we’ll see in Section 4, the quantifier axioms of the intermediate predicate logics considered in the preceding section satisfy the condition that the ${\varepsilon \tau }$ -translations of their additional quantifier axioms are derivable from critical formulas alone (i.e., already in $\mathbf {L}{\varepsilon \tau }$ ).

Proposition 3.7. For any intermediate predicate logic $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ ,

$$ \begin{align*} & (\mathbf{Q}\mathbf{L} + \mathit{Ax}){\varepsilon\tau} \vdash \forall x\, A(x) \mathbin{\leftrightarrow} A(\tau_{x}\,A(x)) \text{ and}\\ &(\mathbf{Q}\mathbf{L}+\mathit{Ax}){\varepsilon\tau} \vdash \exists x\, A(x) \mathbin{\leftrightarrow} A(\varepsilon_{x}\,A(x)). \end{align*} $$

4 Critical formulas and quantifier shifts

We will show later (Theorem 5.3) that any ${\varepsilon \tau }$ -calculus for an intermediate logic $\mathbf {L}$ is conservative over $\mathbf {L}$ . It is well-known that the ${\varepsilon \tau }$ -calculus over intuitionistic logic is not conservative over intuitionistic predicate logic. We’ll show now specifically that for any intermediate logic $\mathbf {L}$ , the ${\varepsilon \tau }$ -translations of all classically valid quantifier shift principles are provable from critical formulas.

Quantifier shift formulas divide into two kinds. On the one hand, we have conditionals the ${\varepsilon \tau }$ -translations of which are critical formulas, and are therefore provable in $\mathbf {L}{\varepsilon \tau }$ (see Table 3). On the other hand, we have formulas provable from critical formulas together with some propositional principles, all of which are intuitionistically valid and hence provable in all intermediate logics. For instance, to obtain the ${\varepsilon \tau }$ -translation of

$$ \begin{align*}(\forall x\, A(x) \lor B) \mathbin{\rightarrow} \forall x(A(x) \lor B), \end{align*} $$

take $A_{1} = A(\tau _{x}\,A(x))$ and $A_{2} = A(\tau _{x}\,(A(x) \lor B))$ . Then $A_{1} \mathbin {\rightarrow } A_{2}$ is a critical formula, viz.,

$$ \begin{align*} A(\tau_{x}\,A(x)) \mathbin{\rightarrow} A(\tau_{x}\,(A(x) \lor B)). \end{align*} $$

Apply modus ponens to it and the principle

$$ \begin{align*} (A_{1} \mathbin{\rightarrow} A_{2}) \mathbin{\rightarrow} ((A_{1} \lor B) \mathbin{\rightarrow} (A_{2} \lor B)). \end{align*} $$

This same pattern works in all cases, and the required critical formulas $A_{1} \mathbin {\rightarrow } A_{2}$ and propositional principles are given in Table 4.

Table 3 Quantifier shift formulas whose ${\varepsilon \tau }$ -translations are critical formulas. In each case, x is not free in B, and the ${\varepsilon \tau }$ -translation of the quantifier shift formula on the left is $C(t_{1}) \mathbin {\rightarrow } C(t_{2})$ .

Table 4 Proofs of ${\varepsilon \tau }$ -translations of quantifier shift formulas. In each case, x is not free in B, $A_{1} \mathbin {\rightarrow } A_{2}$ is a critical formula, and the ${\varepsilon \tau }$ -translation of the formula is given on the left. The propositional principle on the right is provable in intuitionistic logic, and the ${\varepsilon \tau }$ -translation of the quantifier shift formula follows by one application of modus ponens.

The most interesting quantifier shift formulas here are $\mathit {CD}$ , $Q_{\forall }$ , and $Q_{\exists }$ , since they are not intuitionistically valid. By contrast, we have:

The second $\varepsilon $ -theorem states that if $A^{\varepsilon }$ is provable in the pure $\varepsilon $ -calculus, then A is provable in classical predicate logic. The second $\varepsilon $ -theorem fails for any intermediate predicate logic $\mathbf {Q}\mathbf {L} +\mathit {Ax}$ , in which $A^{\varepsilon \tau }$ is provable in $\mathbf {L}{\varepsilon \tau }$ but A is not provable in $\mathbf {Q}\mathbf {L} +\mathit {Ax}$ , e.g., when $\mathbf {Q}\mathbf {L} +\mathit {Ax}$ does not prove one of K, $\mathit {CD}$ , $Q_{\exists }$ , or $Q_{\forall }$ .

Note that the only intuitionistically invalid De Morgan rule for quantifiers,

(Q) $$\begin{align} \lnot \forall x\,A(x) \mathbin{\rightarrow} \exists x\,\lnot A(x), \end{align}$$

is a special case of $Q_{\forall }$ , taking $\bot $ for B; $Q^{\varepsilon \tau }$ is a critical formula. The ${\varepsilon \tau }$ -translation of double negation shift $K^{\varepsilon \tau }$ is

$$ \begin{align*}\lnot\lnot A(\tau_{x}\,\lnot\lnot A(x)) \mathbin{\rightarrow} \lnot\lnot A(\tau_{x}\,A(x))\end{align*} $$

and is also a critical formula.

In classical first-order logic, both the addition of $\varepsilon $ -operators and critical formulas and the replacement of quantifiers by $\varepsilon $ -operators are conservative. The previous results show that for extensions of first-order intuitionistic logic, this is not the case: intuitionistically invalid quantified formulas (or their ${\varepsilon \tau }$ -translations) become provable. However, these quantifier shifts are provable in some intermediate logics, e.g., in some Gödel logics.

We might think of ${\varepsilon \tau }$ -terms semantically as terms for objects which serve the role of generics taking on the role of quantifiers, and indeed in classical logic this connection is very close. Because of the validity of

(Wel1) $$\begin{align} & \exists x(\exists y\,A(y) \mathbin{\rightarrow} A(x)) \text{ and} \end{align}$$

(Wel2) $$\begin{align} & \exists x(A(x) \mathbin{\rightarrow} \forall y\,A(y)) \end{align}$$

in classical logic, there always is an object x which behaves as an $\varepsilon $ -term ( $A(x)$ holds iff $\exists x\,A(x)$ holds), and an object x which behaves as a $\tau $ -term (i.e., $A(x)$ holds iff $\forall y\,A(y)$ holds). One might expect then that $\mathit {Wel}_{1}$ and $\mathit {Wel}_{2}$ , when added to $\mathbf {Q}\mathbf {H}$ , have the same effect as adding critical formulas, i.e., that all quantifier shifts become provable. Note that $\mathit {Wel}_{1}$ and $\mathit {Wel}_{2}$ are intuitionistically equivalent to

(Wel'1) $$\begin{align} & \exists x\forall y(A(y) \mathbin{\rightarrow} A(x)), \end{align}$$

(Wel'2) $$\begin{align} & \exists x\forall y(A(x) \mathbin{\rightarrow} A(y)). \end{align}$$

As is easily checked, $\mathbf {Q}\mathbf {H} \vdash \mathit {Wel}_{1} \mathbin {\leftrightarrow } Q_{\exists }$ and $\mathbf {Q}\mathbf {H} \vdash \mathit {Wel}_{2} \mathbin {\leftrightarrow } Q_{\forall }$ , and even $\mathbf {Q}\mathbf {H} \vdash \mathit {Wel}_{2} \mathbin {\rightarrow } \mathit {CD}$ . However, $\mathbf {Q}\mathbf {H} + \mathit {Wel}_{1} \nvdash \mathit {CD}$ .Footnote ⁴

5 $\mathbf {L}{\varepsilon \tau }$ is conservative over $\mathbf {L}$

The classical $\varepsilon $ -calculus is conservative over propositional logic. Work by Bell [Reference Bell9] and DeVidi [Reference DeVidi12] shows that, however, the addition of critical formulas to intuitionistic logic results in intuitionistically invalid propositional formulas becoming provable in certain simple theories. These results require the presence of identity axioms. One may wonder if these results can be strengthened to the pure logic and the ${\varepsilon \tau }$ -calculus alone. The following proposition shows that this is not the case. The addition of critical formulas to intermediate logics alone does not have any effects on the propositional level.

Definition 5.1. The shadow $A^{s}$ of a formula is defined as follows:

$$ \begin{align*} P(t_{1}, \ldots, t_{n})^{s} & = X_{P},\\ (t_{1} = t_{2})^{s} & = \top,\\ (A \land B)^{s} & = A^{s} \land B^{s}, & (A \lor B)^{s} & = A^{s} \lor B^{s}, \\ (A \mathbin{\rightarrow} B)^{s} & = A^{s} \mathbin{\rightarrow} B^{s}, & (\lnot A)^{s} & = \lnot A^{s}, \\ (\exists x\, A(x))^{s} & = A(x)^{s}, & (\forall x\, A(x))^{s} & = A(x)^{s}, \end{align*} $$

where $X_{P}$ is a propositional variable and $\top $ is any theorem of $\mathbf {L}$ .

The shadow of a proof $\pi = A_{1}$ , …, $A_{n}$ is $A_{1}^{s}$ , …, $A_{n}^{s}$ .

A first-order intermediate logic $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ (over a propositional base logic $\mathbf {L}$ ) is preserved under shadow if $\mathbf {L} \vdash B^{s}$ for all quantifier axioms $B \in \mathit {Ax}$ .

The shadow of a formula is a propositional formula obtained by disregarding all first-order structure. If an intermediate predicate logic $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ is preserved under shadow, the shadows of its theorems are already valid in $\mathbf {L}$ .

Proof. Consider a derivation $\pi $ in $(\mathbf {Q}\mathbf {L}+\mathit {Ax}){\varepsilon \tau }$ of B from $A_{1}$ , …, $A_{n}$ , and a formula C in $\pi $ not a conclusion of an inference, and not among $A_{1}$ , …, $A_{n}$ . If $C \in \mathbf {L}$ , then also $C^{s} \in \mathbf {L}$ . If C is a critical formula, then $C^{s}$ is of the form $A \mathbin {\rightarrow } A$ . If C is a standard quantifier axiom, we have $(\forall x\, A \mathbin {\rightarrow } A(t))^{s} \equiv (A(t) \mathbin {\rightarrow } \exists x\, A(x))^{s} \equiv A(x)^{s} \mathbin {\rightarrow } A(x)^{s}$ , which again is in $\mathbf {L}$ . (Clearly, $A(t)^{s} \equiv A(x)^{s}$ .)

If C is the conclusion of modus ponens from premises A and $A \mathbin {\rightarrow } C$ , then $C^{s}$ follows from $A^{s}$ and $(A \mathbin {\rightarrow } C)^{s}$ by modus ponens. If C is the conclusion of a quantifier rule, the shadows of premise and conclusion are identical, e.g.,

$$ \begin{align*} (B \mathbin{\rightarrow} A(x))^{s} \equiv B^{s} \mathbin{\rightarrow} A(x)^{s} \equiv (B \mathbin{\rightarrow} \forall x\, A(x))^{s}. \end{align*} $$

Thus we have shown that $\pi ^{s}$ is a derivation of $B^{s}$ from $A_{1}^{s}$ , …, $A_{n}^{s}$ in $\mathbf {L}$ .

This still holds if identity is present, as the shadows of identity axioms are: $(t = t)^{s} \equiv \top $ and

$$ \begin{align*} (t_{1} = t_{2} \mathbin{\rightarrow} (A(t_{1}) \mathbin{\rightarrow} A(t_{2})))^{s} \equiv \top \mathbin{\rightarrow} (A(t_{1})^{s} \mathbin{\rightarrow} A(t_{1})^{s}), \end{align*} $$

since $A(t_{1})^{s} \equiv A(t_{2})^{s}$ . Both are provable in $\mathbf {H}{}$ and thus in $\mathbf {L}$ .⊣

All intermediate predicate logics mentioned above are preserved under shadow. They are axiomatized by various quantifier shift principles. As we have seen in the preceding section, the ${\varepsilon \tau }$ -translations of all such quantifier shift principles become provable in the corresponding ${\varepsilon \tau }$ -calculus. However, the shadow of such a quantifier shift principle is a formula of the form $B \mathbin {\rightarrow } B$ . As a consequence, we have the following conservativity result for all intermediate ${\varepsilon \tau }$ -calculi:

Bell [Reference Bell9] claimed that in the extended intuitionistic $\varepsilon $ -calculus for $\mathbf {Q}\mathbf {H}$ with identity, we have $D \vdash _{\mathbf {Q}\mathbf {H}{\varepsilon \tau }} M$ , where M is

(M) $$\begin{align} \lnot(B \land C) \mathbin{\rightarrow} (\lnot B \lor \lnot C) \end{align}$$

and D is $\forall x(x = a \lor \lnot x =a)$ . M is an intuitionistically invalid direction of De Morgan’s laws. Since the shadow $D^{s}$ of D is the intuitionistically valid formula $\top \lor \lnot \top $ , this seems to contradict Lemma 5.2. The proof starts by asserting that

$$ \begin{align*} \forall x\,[(x = a \land B) \lor (x \neq a \land C)] \mathbin{\rightarrow} (B \land C) \end{align*} $$

is provable in $\mathbf {Q}\mathbf {H}$ with identity. This is false, however, as the formula is not true in any one-element model when B is true and C is false. Theorem 7 of [Reference DeVidi12] fails for the same reason. The results are correct with the additional assumption $a \neq b$ .Footnote ⁵

6 The extended first ${\varepsilon \tau }$ -theorem fails unless $\mathbf {L} \vdash B_{m}$

In classical first-order logic, the main result about the $\varepsilon $ -calculus is the extended first $\varepsilon $ -theorem. It states that if $A(e_{1}, \dots , e_{n})$ , where the $e_{i}$ are $\varepsilon $ -terms, is provable in the pure $\varepsilon $ -calculus, then there are $\varepsilon $ -free terms $t_{i}^{j}$ such that

$$ \begin{align*} A(t_{i}^{1}, \dots, t_{n}^{1}) \lor \dots \lor A(t_{i}^{k}, \dots, t_{n}^{k}) \end{align*} $$

is provable in classical propositional logic alone. Such $\varepsilon $ -terms $e_{i}$ appear as the result of translating $\exists x_{1}\dots \exists x_{n}\,A(x_{1}, \dots , x_{n})$ into the $\varepsilon $ -calculus.

In the context of intermediate logics, we may formulate the statement as follows:

Definition 6.1. An intermediate logic $\mathbf {L}$ has the extended first  ${\varepsilon \tau }$ -theorem, if, whenever $\mathbf {L}{\varepsilon \tau } \vdash A(e_{1}, \dots , e_{n})$ for some $\varepsilon $ - or $\tau $ -terms $e_{1}$ , …, $e_{n}$ , then there are ${\varepsilon \tau }$ -free terms $t_{i}^{j}$ such that

$$ \begin{align*} \mathbf{L} \vdash A(t_{i}^{1}, \dots, t_{n}^{1}) \lor \dots \lor A(t_{i}^{k}, \dots, t_{n}^{k}). \end{align*} $$

We obtain a first negative result: if $\mathbf {L}{\varepsilon \tau }$ has the extended first epsilon theorem, then an instance of ${B_{m}}$ , i.e.,

$$ \begin{align*} (A_{1} \mathbin{\rightarrow} A_{2}) \lor \cdots \lor (A_{m} \mathbin{\rightarrow} A_{m+1}) \end{align*} $$

for some $m \ge 2$ is provable already in the propositional fragment $\mathbf {L}$ .Footnote ⁶ This rules out an extended first ${\varepsilon \tau }$ -theorem for, e.g., ${\varepsilon \tau }$ -calculi for intuitionistic logic and infinite-valued Gödel–Dummett logic.

Proof. Consider $A(z) \equiv (P(f(z)) \mathbin {\rightarrow } P(z))$ and $\exists z\, A(z)$ , i.e., $\exists z(P(f(z)) \mathbin {\rightarrow } P(z))$ . Let $e \equiv \varepsilon _{z}\,(P(f(z)) \mathbin {\rightarrow } P(z))$ . The ${\varepsilon \tau }$ -translation of $\exists z\, A(z)$ is $P(f(e)) \mathbin {\rightarrow } P(e)$ , i.e.,

$$ \begin{align*} V \equiv P(f(\varepsilon_{z}\,(P(f(z)) \mathbin{\rightarrow} P(z)))) \mathbin{\rightarrow} P(\varepsilon_{z}\,(P(f(z)) \mathbin{\rightarrow} P(z))). \end{align*} $$

Let $U \equiv A(\varepsilon _{x}\,P(x)) \equiv P(f(\varepsilon _{x}\,P(x))) \mathbin {\rightarrow } P(\varepsilon _{x}\,P(x))$ . Note that U is of the form $P(t) \mathbin {\rightarrow } P(\varepsilon _{x}\,P(x))$ , so it is a critical formula. Also note that $U \mathbin {\rightarrow } V$ is of the form $A(t) \mathbin {\rightarrow } A(e)$ , and so $U \mathbin {\rightarrow } V$ is also a critical formula.

Since $\mathbf {L} \vdash (U \mathbin {\rightarrow } V) \mathbin {\rightarrow } (U \mathbin {\rightarrow } V)$ , and $U \mathbin {\rightarrow } V$ and U are critical formulas, $\mathbf {L}{\varepsilon \tau } \vdash V$ . By assumption, $\mathbf {L}{\varepsilon \tau }$ has the extended first ${\varepsilon \tau }$ -theorem, so $\mathbf {L}$ proves a disjunction of the form

$$ \begin{align*} (P(f(t_{1})) \mathbin{\rightarrow} P(t_{1})) \lor \dots \lor (P(f(t_{k})) \mathbin{\rightarrow} P(t_{k})) \end{align*} $$

for some terms $t_{1}$ , …, $t_{k}$ . (This is a Herbrand disjunction of $\exists z\, A(z)$ .) Each term $t_{i}$ is of the form $f^{i}(s)$ for some $i \ge 0$ and a term s which does not start with f. By rearranging the disjuncts to group disjuncts with the same innermost term s together (using commutativity of $\lor $ ) and by adding additional disjuncts as needed (using weakening), from this we obtain a formula

$$ \begin{align*} (P(f^{j_{1}+1}(s_{1})) \mathbin{\rightarrow} P(f^{j_{1}}(s_{1}))) \lor {} & \dots \lor (P(f(s_{1})) \mathbin{\rightarrow} P(s_{1})) \lor \\ & \vdots \\ (P(f^{j_{l}+1}(s_{l})) \mathbin{\rightarrow} P(f^{j_{l}}(s_{l}))) \lor {} & \dots \lor (P(f(s_{l})) \mathbin{\rightarrow} P(s_{l})). \end{align*} $$

Let j be the largest among $j_{1}$ , …, $j_{l}$ . By uniformly replacing $P(f^{i}(s_{j}))$ by $A_{j+2-i}$ in the proof of the last formula and contracting identical disjuncts, we obtain a proof in $\mathbf {L}$ of $(A_{1} \mathbin {\rightarrow } A_{2}) \lor \dots \lor (A_{j+1} \mathbin {\rightarrow } A_{j+2})$ . This is ${B_{m}}$ for $m=j+1$ , and since $j\ge 1$ , $m\ge 2$ .⊣

A formula of the form ${B_{m}}$ is provable in $\mathbf {L}$ iff $\mathbf {L}$ is a finite-valued Gödel logic $\mathbf {LC}_{n}$ (Proposition 6.5). By contrast, no ${B_{m}}$ is provable in intuitionistic logic $\mathbf {H}$ , Jankov logic $\mathbf {KC}$ , or infinite-valued Gödel logic $\mathbf {LC}$ (Proposition 6.3).

Proof. Simultaneously substitute A for $A_{i}$ if i is odd, and B for $A_{i}$ if i is even in ${B_{m}}$ . The result is one of

$$ \begin{align*} & (A \mathbin{\rightarrow} B) \lor (B \mathbin{\rightarrow} A) \lor \dots \lor (A \mathbin{\rightarrow} B) \text{ and}\\ & (A \mathbin{\rightarrow} B) \lor (B \mathbin{\rightarrow} A) \lor \dots \lor (B \mathbin{\rightarrow} A). \end{align*} $$

Both are equivalent in $\mathbf {H}{}$ to $(A \mathbin {\rightarrow } B) \lor (B \mathbin {\rightarrow } A)$ .⊣

Proof. Hosoi [Reference Hosoi16] showed that the n-valued Gödel logic is axiomatized by $\mathbf {H} + R_{n-1}$ , where $R_{n}$ is

$$ \begin{align*} A_{1} \lor (A_{1} \mathbin{\rightarrow} A_{2}) \lor \dots \lor (A_{n-1} \mathbin{\rightarrow} A_{n}) \lor \lnot A_{n}. \end{align*} $$

By simultaneously substituting $\top $ for $A_{1}$ , $\bot $ for $A_{n+1}$ , and $A_{i-1}$ for $A_{i}$ ( $i = 2$ , …, n) in ${B_{n}}$ , we obtain

$$ \begin{align*} (\top \mathbin{\rightarrow} A_{1}) \lor (A_{1} \mathbin{\rightarrow} A_{2}) \lor \dots \lor (A_{n-2} \mathbin{\rightarrow} A_{n-1}) \lor (A_{n-1} \mathbin{\rightarrow} \bot), \end{align*} $$

which is equivalent to $R_{n-1}$ in $\mathbf {H}$ . Hence, since $\mathbf {L} \vdash {B_{n}}$ , $\mathbf {LC}_{n} \subseteq \mathbf {L}$ .

Furthermore, Hosoi [Reference Hosoi17, Lemma 4.1] showed that if $\mathbf {L} \vdash \mathit {Lin}$ then $\mathbf {L} = \mathbf {LC}_{m}$ for some m or $L = \mathbf {LC}$ . Since $\mathbf {L} \vdash {B_{n}}$ , $\mathbf {L} \vdash \mathit {Lin}$ by Proposition 6.4. The result follows as $\mathbf {LC} \nvdash {B_{n}}$ and so $\mathbf {L} \neq \mathbf {LC}$ .Footnote ⁷ ⊣

We have restricted $\mathbf {L}$ here to be an intermediate propositional logic. However, it bears remarking that Theorem 6.2 does not require that $\mathbf {L}$ contains $\mathbf {H}$ . An inspection of the proof shows that all that is required is that $\mathbf {L} \vdash A \mathbin {\rightarrow } A$ , and in $\mathbf {L}$ , $\lor $ is provably commutative, associative, and idempotent, and has weakening ( $\mathbf {L} \vdash A \mathbin {\rightarrow } (A \lor B)$ ). Thus, Corollary 6.6 applies to any ${\varepsilon \tau }$ -calculus based on a logic which has these properties (such as, say, Łukasiewicz logic).

The extended first $\varepsilon $ -theorem in classical logic shows that if an existential formula $\exists x\,A(x)$ is provable, so is a disjunction of instances $\bigvee _{i} A(t_{i})$ . Clearly this is equivalent to: if $\forall x\,A(x) \mathbin {\rightarrow } B$ is provable so is $\bigwedge A(t_{i}) \mathbin {\rightarrow } B$ . Without the interdefinability of $\forall $ and $\exists $ , the question arises whether the alternative form of the $\varepsilon $ -theorem might hold in an intermediate ${\varepsilon \tau }$ -calculus even if the standard form does not. We’ll show that the versions are, in fact, equivalent even in intermediate logics.

Proof. (1) implies (3): Suppose

$$ \begin{align*} B(e^{\prime}) \vdash_{\mathbf{L}{\varepsilon\tau}} C(e). \end{align*} $$

By the deduction theorem,

$$ \begin{align*} \vdash_{\mathbf{L}{\varepsilon\tau}} B(e^{\prime}) \mathbin{\rightarrow} C(e). \end{align*} $$

By (1) we have terms ${s_{i}, t_{i}}$ so that

$$ \begin{align*} \vdash \bigvee_{i} (B(s_{i}) \mathbin{\rightarrow} C(t_{i})). \end{align*} $$

By intuitionistic logic,

$$ \begin{align*} & \vdash \bigwedge_{i} B(s_{i}) \mathbin{\rightarrow} \bigvee_{i} C(t_{i}) \text{ and so}\\ \bigwedge_{i} B(s_{i}) & \vdash \bigvee_{i} C(t_{i}), \end{align*} $$

by the deduction theorem.

(3) clearly implies (1) and (2).

(2) implies (1): Let X be a propositional variable. $A \vdash _{\mathbf {H}} (A \mathbin {\rightarrow } X) \mathbin {\rightarrow } X$ . So if $\mathbf {L}{\varepsilon \tau } \vdash A(e)$ then by the deduction theorem,

$$ \begin{align*} A(e) \mathbin{\rightarrow} X & \vdash_{\mathbf{L}{\varepsilon\tau}} X \text{ and by (2),}\\ \bigwedge_{i} (A(s_{i}) \mathbin{\rightarrow} X) & \vdash_{\mathbf{L}} X. \end{align*} $$

Now substitute $\bigvee _{i} A(s_{i})$ for X:

$$ \begin{align*} \bigwedge_{i} (A(s_{i}) \mathbin{\rightarrow} \bigvee_{i} A(s_{i})) \vdash_{\mathbf{L}} \bigvee_{i} A(s_{i}). \end{align*} $$

The formula on the left is provable intuitionistically.⊣

7 Elimination sets and excluded middle

The basic idea of Hilbert’s proof of the extended first $\varepsilon $ -theorem is this: Suppose we have a proof of $E \equiv D(e)$ from critical formulas $\Gamma , \Lambda (e)$ , where e is a critical $\varepsilon $ -term and $\Lambda (e)$ is a set of critical formulas belonging to e. Now we find terms $t_{1}$ , …, $t_{k}$ such that replacing e by $t_{i}$ allows us to remove the critical formulas $\Lambda (e)$ , while at the same time replacing the end-formula $D(e)$ by $\bigvee _{i=1}^{k} D(t_{i})$ and the remaining critical formulas by $\Gamma [t_{1}/e]$ , …, $\Gamma [t_{k}/e]$ . We repeat this procedure in such a way that eventually all critical formulas are removed and we are left with a disjunction of instances of E, as required by the first $\varepsilon $ -theorem. The difficulty of making this work lies in three challenges. The first is to find a suitable way of selecting $\varepsilon $ -terms e and corresponding critical formulas $\Lambda (e)$ so that the $\Lambda (e)$ can be removed. The second is to ensure that in passing from $\Gamma $ to $\Gamma [t_{i}/e]$ we again obtain critical formulas.Footnote ⁸ The third challenge is to guarantee that the process eventually terminates with no critical formulas remaining.

In this section we address the first challenge by considering the condition that suffices to overcome it: the existence of complete e-elimination sets (defined below) for every ${\varepsilon \tau }$ -term e. We then show why this condition is satisfied in classical logic, so we can clarify the role of excluded middle in the proof for the classical case, as well as how the proof for classical logic and those for intermediate logics given later correspond to one another. We will discuss the condition for intermediate logics in Section 8 and the remaining challenges in Section 9.

Definition 7.2. Suppose $\Gamma , \Lambda (e), \Lambda ^{\prime }(e) \vdash _{\mathbf {L}{\varepsilon \tau }}^{\pi } D(e)$ where $\Lambda (e) \cup \Lambda (e)^{\prime }$ are all critical formulas belonging to e. A set of terms $s_{1}$ , …, $s_{k}$ is an e-elimination set for $\pi $ and $\Lambda (e)$ if

$$ \begin{align*} \Gamma[s_{1}/e], \dots, \Gamma[s_{k}/e], \Lambda^{\prime}(e) & \vdash_{\mathbf{L}} D(s_{1}) \lor \dots \lor D(s_{k}). \end{align*} $$

If $\Lambda (e)$ is the set of all critical formulas belonging to e (i.e., $\Lambda ^{\prime }(e) = \emptyset $ ) then an e-elimination set for $\Lambda (e)$ is called a complete e-elimination set.

Here, $\Gamma [s_{i}/e]$ means the result of replacing, in each formula in $\Gamma $ , every occurrence of e by $s_{i}$ . If $T = \{s_{1}, \dots , s_{k}\}$ we write $\Gamma [T]$ for $\Gamma [s_{1}]$ , …, $\Gamma [s_{k}]$ . Note that we do not require in the definition of e-elimination sets that the formulas in $\Gamma [s_{i}/e]$ are actually critical formulas.

We are now in a position to apply the preceding lemmas and the concept of elimination sets to the case of classical logic. This elucidates how the first challenge is solved in the proof of the extended first ${\varepsilon \tau }$ -theorem for classical logic where $\tau $ -terms and corresponding critical formulas may also be present. (For Hilbert’s original proof for the $\varepsilon $ -calculus without $\tau $ -terms, see [Reference Hilbert and Bernays15] or [Reference Moser and Zach23].)

Proof. Suppose first that e is an $\varepsilon $ -term; then $C(e)$ is $A(s) \mathbin {\rightarrow } A(e)$ . Let $\Lambda ^{\prime }(e)$ be the critical formulas belonging to e other than $C(e)$ , and $\Gamma $ the remaining critical formulas for which e is not critical. So we have:

$$ \begin{align*} \Gamma, \Lambda^{\prime}(e), A(s) \mathbin{\rightarrow} A(e) & \vdash_{\mathbf{C}} D(e). \hspace{2pc} \end{align*} $$

On the one hand, by replacing e everywhere by s we get

$$ \begin{align*} \Gamma[s/e], \Lambda^{\prime}(s), A(s[s/e]) \mathbin{\rightarrow} A(s) & \vdash_{\mathbf{C}} D(s), \hspace{5.4pc} \end{align*} $$

and by Lemma 7.5(2), since A(s) $\vdash A(t[s/e]) \mathbin {\rightarrow } A(s) \in \Lambda ^{\prime }(s)$ and $A(s) \vdash C(s)$ ,

$$ \begin{align*} \hspace{1.6pc} \Gamma[s/e], A(s) & \vdash_{\mathbf{C}} D(s). \end{align*} $$

On the other hand, since $\lnot A(s) \vdash A(s) \mathbin {\rightarrow } A(e)$ ,

$$ \begin{align*} \Gamma, \lnot A(s) & \vdash_{\mathbf{C}} D(e) \text{ and so,}\\ \Gamma, \Gamma[s/e], A(s) \lor \lnot A(s)& \vdash_{\mathbf{C}} D(e) \lor D(s), \end{align*} $$

by Lemma 7.5(1). Since $\mathbf {C} \vdash A(s) \lor \lnot A(s)$ we have

$$ \begin{align*} \hspace{5.2pc} \Gamma, \Gamma[s/e] & \vdash_{\mathbf{C}} D(e) \lor D(s). \end{align*} $$

Thus, $\{e, s\}$ is an e-elimination set for the critical formula $C(e)$ .

Similarly, if e is a ${\tau }$ -term and $C(e)$ is $A(e) \mathbin {\rightarrow } A(s)$ we get

$$ \begin{align*} \Gamma[s/e], \Lambda^{\prime}(s), C(s) & \vdash_{\mathbf{C}} D(s), \hspace{1pc} \end{align*} $$

and by Lemma 7.5(2), since $\lnot A(s) \vdash C^{\prime } \in \Lambda ^{\prime }(e)$ and $\lnot A(s) \vdash C(s)$ ,

$$ \begin{align*} \hspace{1pc} \Gamma[s/e], \lnot A(s) & \vdash_{\mathbf{C}} D(s). \end{align*} $$

On the other hand, $A(s) \vdash C(e)$ , so

$$ \begin{align*} \Gamma, A(s) & \vdash_{\mathbf{C}} D(e) \text{ and}\\ \Gamma, \Gamma[s/e], \lnot A(s) \lor A(s)& \vdash_{\mathbf{C}} D(e) \lor D(s), \end{align*} $$

by Lemma 7.5(1).⊣

More generally, the set of all critical formulas belonging to e has a (complete) e-elimination set in $\mathbf {C}$ :

Proof. Let $C_{1} \equiv A(s_{1}) \mathbin {\rightarrow } A(e)$ , …, $C_{k} \equiv A(s_{k}) \mathbin {\rightarrow } A(e)$ be the critical formulas belonging to e if e is an $\varepsilon $ -term. Since

$$ \begin{align*} \Gamma, C_{1}(e), \dots, C_{k}(e) & \vdash_{\mathbf{C}} D(e) \text{, also}\\ \Gamma[s_{i}/e], C_{1}(s_{i}), \dots, C_{k}(s_{i}) & \vdash_{\mathbf{C}} D(s_{i}) \end{align*} $$

(writing ${C_{j}(s_{i})}$ for ${C_{j}[s_{i}/e]}$ ). Since ${A(s_{i}) \vdash A(s_{j}(s_{i})) \mathbin {\rightarrow } A(s_{i}) \equiv C_{j}(s_{i})}$ ,

$$ \begin{align*} \hspace{2.8pc} \Gamma[s_{i}/e], A(s_{i}) & \vdash_{\mathbf{C}} D(s_{i}), \end{align*} $$

by Lemma 7.5(2). By applying Lemma 7.5(1),

$$ \begin{align*} \Gamma[s_{1}/e], \dots, \Gamma[s_{k}/e], A(s_{1}) \lor \dots \lor A(s_{k}) & \vdash_{\mathbf{C}} D(s_{1}) \lor \dots \lor D(s_{k}). \hspace{1.4pc} \end{align*} $$

On the other hand, since ${\lnot A(s_{i}) \vdash A(s_{i}) \mathbin {\rightarrow } A(e)}$ , we get $\lnot A(s_{1}) \land \cdots \land \lnot A(s_{k}) \vdash C_{j}(e)$ for each ${j = 1, \dots , \textit {k}}$ , so we also have, from the first line by Lemma 7.5(2),

$$ \begin{align*} \Gamma, \lnot A(s_{1}) \land \cdots \land \lnot A(s_{k}) & \vdash_{\mathbf{C}} D(e). \hspace{1.2pc} \end{align*} $$

Since ${\bigvee _{i} A(s_{i}) \lor (\bigwedge _{i} \lnot A(s_{i}))}$ is an instance of excluded middle, we have

$$ \begin{align*} \hspace{7.4pc} \Gamma, \Gamma[s_{1}/e], \dots, \Gamma[s_{k}/e] & \vdash_{\mathbf{C}} D(e) \lor D(s_{1}) \lor \dots \lor D(s_{k}). \end{align*} $$

If e is a $\tau $ -term, then the critical formulas are of the form $C_{j}(e) \equiv A(e) \mathbin {\rightarrow } A(s_{j}(e))$ and consequently $C_{j}(s_{i})$ is $A(s_{i}) \mathbin {\rightarrow } A(s_{j}(s_{i}))$ . Each is implied by $\lnot A(s_{i})$ , so we have

$$ \begin{align*} \Gamma[s_{1}/e], \dots, \Gamma[s_{k}, e], \lnot A(s_{1}) \lor \dots \lor \lnot A(s_{k}) & \vdash_{\mathbf{C}} D(s_{1}) \lor \dots \lor D(s_{k}). \hspace{2.4pc} \end{align*} $$

On the other hand, $A(s_{1}) \land \dots \land A(s_{k}) \vdash A(e) \mathbin {\rightarrow } A(s_{j})$ , so

$$ \begin{align*} \Gamma, A(s_{1}) \land \dots \land A(s_{k}) & \vdash_{\mathbf{C}} D(e), \end{align*} $$

and consequently

$$ \begin{align*} \hspace{7.2pc} \Gamma, \Gamma[s_{1}/e], \dots, \Gamma[s_{k}/e] & \vdash_{\mathbf{C}} D(e) \lor D(s_{1}) \lor \dots \lor D(s_{k}), \end{align*} $$

since $\bigwedge _{i} A(s_{i}) \lor \bigvee _{i} \lnot A(s_{i})$ is a tautology.

In each case, $\{e, s_{1}, \dots , s_{k}\}$ is an e-elimination set.⊣

We know that intermediate logics other than $\mathbf {LC}_{m}$ do not have the extended first ${\varepsilon \tau }$ -theorem and so not every ${\varepsilon \tau }$ -term will have complete e-elimination sets. However, if the starting formula E is of a special form, they sometimes do. In the proof for the classical case above, this required excluded middle. But it need not. For instance, if E is negated, then weak excluded middle ( $\lnot A \lor \lnot \lnot A$ ) is enough.

Proof. Let $C_{1} \equiv A(s_{1}) \mathbin {\rightarrow } A(e)$ , …, $C_{k} \equiv A(s_{k}) \mathbin {\rightarrow } A(e)$ be the critical formulas belonging to e if e is an $\varepsilon $ -term. As before, we have

$$ \begin{align*} \Gamma[s_{i}/e], A(s_{i}) & \vdash_{\mathbf{L}} \bigvee_{j} \lnot D_{j}(s_{i}). \end{align*} $$

In ${\mathbf {KC}, B \mathbin {\rightarrow } (\lnot C_{1} \lor \lnot C_{2}) \vdash \lnot \lnot B \mathbin {\rightarrow } (\lnot C_{1} \lor \lnot C_{2})}$ , so

$$ \begin{align*} \Gamma[s_{i}/e], \lnot\lnot A(s_{i}) & \vdash_{\mathbf{L}} \bigvee_{j} \lnot D_{j}(s_{i}). \end{align*} $$

We obtain

$$ \begin{align*} \Gamma[s_{1}/e], \dots, \Gamma[s_{k}/e], &\\ \lnot \lnot A(s_{1}) \lor \dots \lor \lnot\lnot A(s_{k}) & \vdash_{\mathbf{L}} \bigvee_{j} \lnot D_{j}(s_{1}) \lor \dots \lor \bigvee_{j} \lnot D_{j}(s_{k}). \end{align*} $$

Again as before, we have

$$ \begin{align*} \Gamma, \lnot A(s_{1}) \land \cdots \land \lnot A(s_{k}) & \vdash_{\mathbf{L}} \bigvee_{j} \lnot D_{j}(e), \end{align*} $$

and together

$$ \begin{align*} \Gamma[s_{1}/e], \dots, \Gamma[s_{k}/e], \Gamma, &\\ (\lnot A(s_{1}) \land \cdots \land \lnot A(s_{k})) \lor {} &\\ \lnot \lnot A(s_{1}) \lor \dots \lor \lnot\lnot A(s_{k}) & \vdash_{\mathbf{L}} \bigvee_{j} \lnot D_{j}(e) \lor \bigvee_{j} \lnot D_{j}(s_{1}) \lor \dots \lor \bigvee_{j} \lnot D_{j}(s_{k}). \end{align*} $$

Since

$$ \begin{align*} (\lnot A(s_{1}) \land \cdots \land \lnot A(s_{k})) \lor {} \lnot \lnot A(s_{1}) \lor \dots \lor \lnot\lnot A(s_{k}) \end{align*} $$

is provable from weak excluded middle, the claim is proved.⊣

8 Elimination sets using ${B_{m}}$ and $\mathit {Lin}$

We’ve showed in Theorem 6.2 that the provability of ${B_{m}}$ for some $m \ge 2$ is a necessary condition for an intermediate logic to have the extended first ${\varepsilon \tau }$ -theorem. In this section, we show that it is also sufficient: if $\mathbf {L} \vdash B_{m}$ for some m, then every critical ${\varepsilon \tau }$ -term has a complete e-elimination set.

We use the notion of e-elimination sets developed in Section 7: the existence of an e-elimination set for a set of critical formulas $\Lambda $ guarantees that these critical formulas can be removed from a proof, while the end-formula is replaced by a disjunction of instances of the original end-formula. The proofs of the existence of e-elimination sets proceed by replacing an ${\varepsilon \tau }$ -term e in such a way that the disjunction of formulas $\Lambda ^{\prime }$ resulting from such replacements become (provable from) tautologies in the underlying propositional logic. In Proposition 7.6, we showed how to do this for a single critical formula $A(s) \mathbin {\rightarrow } A(e)$ in classical logic: We replace e first by s resulting in $A(s[s/e]) \mathbin {\rightarrow } A(s) \vdash D(s)$ , and then by itself (i.e., no replacement), resulting in $A(s) \mathbin {\rightarrow } A(e) \vdash D(e)$ . This gives

$$ \begin{align*} (A(s[s/e]) \mathbin{\rightarrow} A(s)) \lor (A(s) \mathbin{\rightarrow} A(e)) \vdash D(s) \lor D(e), \end{align*} $$

but the formula on the left is a classical tautology. When eliminating multiple critical formulas at the same time (e.g., all critical formulas belonging to a single ${\varepsilon \tau }$ -term), the resulting tautologies are more complicated. In the original proof, they are all equivalent to excluded middle, and so the proofs do not apply to intermediate propositional logics. Below, we show how this can nevertheless be done as long as the underlying propositional logic contains ${B_{m}}$ . We have to distinguish two kinds of critical formulas:

If all critical formulas are predicative, $\mathbf {L} \vdash \mathit {Lin}$ suffices (Lemma 8.5). If $\mathbf {L} \vdash {B_{m}}$ for some m, we can eliminate the impredicative critical formulas for some ${\varepsilon \tau }$ -term e in a similar way: by successively replacing e by suitable terms, we obtain a proof of a disjunction of instances of $D(e)$ from a formula provable from ${B_{m}}$ . The number m determines the number of necessary replacements. Once impredicative critical formulas are removed, we can use Lemma 8.5 to remove the remaining predicative critical formulas.

We prove the result for impredicative critical formulas first (Lemma 8.3). In preparation for the proof we first consider an example to illustrate the basic idea. Suppose $\mathbf {LC}_{3}{\varepsilon \tau } \vdash D(e)$ with the set of critical formulas

$$ \begin{align*} C_{s}(e) & \equiv A(s(e)) \mathbin{\rightarrow} A(e), \\[3pt] C_{t}(e) & \equiv A(t(e)) \mathbin{\rightarrow} A(e), \\ C_{u}(e) & \equiv A(u) \mathbin{\rightarrow} A(e), \\ C_{v}(e) & \equiv A(v) \mathbin{\rightarrow} A(e), \end{align*} $$

where terms u and v do not contain e. Let $C(e)$ be the conjunction of these critical formulas. We have $C_{s}(e), C_{t}(e), C_{u}(e), C_{v}(e) \vdash _{\mathbf {LC}_{3}} D(e)$ .

We’ll consider sets of terms $X_{i}$ where $X_{0} = \{e\}$ and $X_{i+1} = \{ s(x), t(x) : x \in X_{i}\}$ . For the sake of readability we will leave out parentheses when writing these terms, e.g., $s(t(e))$ is abbreviated as $ste$ .

For every $w \in X_{1}$ we have $C(w) \vdash D(w)$ . So, by applying Lemma 7.5(1) twice, we get:

$$ \begin{align*} C(se) \lor C_{s}(e),&\\ C(te) \lor C_{t}(e), &\\ C_{u}(e), C_{v}(e) & \vdash_{\mathbf{LC}_{3}} D(e) \lor D(se) \lor D(te). \end{align*} $$

By distributivity, C(se) ${\lor C_{s}(e)}$ is equivalent to

$$ \begin{align*} (C_{s}(se) \lor C_{s}(e)) & \land (C_{t}(se) \lor C_{s}(e)) \land {}\\ (C_{u}(se) \lor C_{s}(e)) & \land (C_{v}(se) \lor C_{s}(e)). \end{align*} $$

Thus we get

$$ \begin{align*} C_{s}(se) \lor C_{s}(e), C_{t}(se) \lor C_{s}(e),& \\ C_{u}(se) \lor C_{s}(e), C_{v}(se) \lor C_{s}(e),& \\ C(te) \lor C_{t}(e),&\\ C_{u}(e), C_{v}(e) &\vdash_{\mathbf{LC}_{3}} D(e) \lor D(se) \lor D(te). \end{align*} $$

We have

$$ \begin{align*} C_{u}(e) \equiv A(u) \mathbin{\rightarrow} A(e) \vdash_{\mathbf{LC}_{3}} (A(u) \mathbin{\rightarrow} A(se)) \lor (A(se) \mathbin{\rightarrow} A(e)) \equiv C_{u}(se) \lor C_{s}(e), \end{align*} $$

using $\mathit {Lin}$ . Similarly, $C_{v}(e) \vdash _{\mathbf {LC}_{3}} C_{v}(se) \lor C_{s}(e)$ , so we have by Lemma 7.5(2):

$$ \begin{align*} C_{s}(se) \lor C_{s}(e), C_{t}(se) \lor C_{s}(e),& \\ C(te) \lor C_{t}(e),&\\ C_{u}(e), C_{v}(e) & \vdash_{\mathbf{LC}_{3}} D(e) \lor D(se) \lor D(te). \end{align*} $$

Repeating this consideration with $C(te) \lor C_{t}(e)$ yields

$$ \begin{align*} C_{s}(se) \lor C_{s}(e), C_{t}(se) \lor C_{s}(e),& \\ C_{s}(te) \lor C_{t}(e), C_{t}(te) \lor C_{t}(e),& \\ C_{u}(e), C_{v}(e) &\vdash_{\mathbf{LC}_{3}} D(e) \lor D(se) \lor D(te). \end{align*} $$

Note that each of the four resulting disjunctions has as first disjunct a substitution instance of a critical formula of the form $C_{s}(w)$ or $C_{t}(w)$ where $w \in X_{1}$ . $X_{2}$ are the terms of the form $s(w)$ and $t(w)$ . So we can repeat the process, pairing $C(s(w))$ with $C_{s}(w)$ and $C(t(w))$ with $C_{t}(w)$ , i.e., obtaining $C(sse) \lor C_{s}(se) \lor C_{s}(e)$ , $C(tse) \lor C_{t}(se) \lor C_{s}(e)$ , etc. In each case, after distributing and removing conjuncts of the form $C_{u}(w) \lor \cdots $ we are left with now eight disjunctions:

$$ \begin{align*} & C_{s}(sse) \lor C_{s}(se) \lor C_{s}(e),\\ & C_{t}(sse) \lor C_{s}(se) \lor C_{s}(e),\\ & \vdots\\ & C_{t}(tte) \lor C_{t}(te) \lor C_{t}(e),\\ & C_{u}(e), C_{v}(e) \vdash_{\mathbf{LC}_{3}} D(e) \lor D(se) \lor D(te) \lor D(sse) \lor \dots \lor D(tte). \end{align*} $$

It remains to show that the formulas on the right of the turnstile are provable in $\mathbf {LC}_{3}$ . First, consider a formula of the form $C_{i}(w) \lor \dots \lor C_{j}(e)$ , e.g.,

$$ \begin{align*} & C_{s}(tse) \lor C_{t}(se) \lor C_{s}(e), \text{ i.e.,}\\ & (A(stse)) \mathbin{\rightarrow} A(tse)) \lor (A(tse) \mathbin{\rightarrow} A(se)) \lor (A(se) \mathbin{\rightarrow} A(e)). \end{align*} $$

In each disjunct, the consequent equals the antecedent of the disjunct immediately to the right, i.e., it is a substitution instance of

$$ \begin{align*} & (A_{1} \mathbin{\rightarrow} A_{2}) \lor (A_{2} \mathbin{\rightarrow} A_{3}) \lor (A_{3} \mathbin{\rightarrow} A_{4}), \end{align*} $$

i.e., of ${B_{3}}$ . Since $\mathbf {LC}_{3} \vdash {B_{3}}$ , these are all provable.

If we take $D^{\prime }(e)$ to be the disjunction obtained on the right, we have

$$ \begin{align*} C_{u}(e), C_{v}(e) & \vdash_{\mathbf{LC}_{3}} D^{\prime}(e) \text{ and thus also}\\ C_{u}(u), C_{v}(u) & \vdash_{\mathbf{LC}_{3}} D^{\prime}(u) \text{ and}\\ C_{u}(v), C_{v}(u) & \vdash_{\mathbf{LC}_{3}} D^{\prime}(v). \end{align*} $$

As ${C_{u}(u)}$ and ${C_{v}(v)}$ are of the form $A \mathbin {\rightarrow } A$ this reduces to

$$ \begin{align*} C_{u}(v) & \vdash_{\mathbf{LC}_{3}} D^{\prime}(u) \text{ and}\\ C_{v}(u) & \vdash_{\mathbf{LC}_{3}} D^{\prime}(v) \text{ and therefore:}\\ C_{u}(v) \lor C_{v}(u) & \vdash_{\mathbf{LC}_{3}} D^{\prime}(u) \lor D^{\prime}(v). \end{align*} $$

But $C_{u}(v) \lor C_{v}(u)$ is an instance of $\mathit {Lin}$ .

Lemma 8.2. If $\mathbf {L} \vdash \mathit {Lin}$ , then for all m,

$$ \begin{align*} A_{1} \mathbin{\rightarrow} A_{m+1} \vdash_{\mathbf{L}} (A_{1} \mathbin{\rightarrow} A_{2}) \lor \dots \lor (A_{m} \mathbin{\rightarrow} A_{m+1}).\end{align*} $$

Proof. By induction on m. If $m=1$ , this amounts to the claim: $A_{1} \mathbin {\rightarrow } A_{2} \vdash _{\mathbf {L}} A_{1} \mathbin {\rightarrow } A_{2}$ , which is trivial. Now suppose the claim holds for m. Then

$$ \begin{align*} A_{1} \mathbin{\rightarrow} A_{m+2} & \vdash_{\mathbf{L}} (A_{1} \mathbin{\rightarrow} A_{m+1}) \lor (A_{m+1} \mathbin{\rightarrow} A_{m+2}) \end{align*} $$

from the instance ${(A_{1} \mathbin {\rightarrow } A_{m+1}) \lor (A_{m+1} \mathbin {\rightarrow } A_{1})}$ of ${\mathit {Lin}}$ . By induction hypothesis,

$$ \begin{align*} A_{1} \mathbin{\rightarrow} A_{m+1} & \vdash_{\mathbf{L}} (A_{1} \mathbin{\rightarrow} A_{2}) \lor \dots \lor (A_{m} \mathbin{\rightarrow} A_{m+1}), \end{align*} $$

and the claim follows by $\mathbf {H}$ .⊣

Proof. Suppose $\Gamma , \Delta (e), \Pi (e) \vdash _{\mathbf {L}} D(e)$ , where $\Pi (e)$ is the set of predicative critical formulas belonging to e.

Suppose $\Pi (e)$ and $\Delta (e)$ consist of, respectively, the critical formulas

$$ \begin{align*} C_{u_{i}}(e) & \equiv A(u_{i}) \mathbin{\rightarrow} A(e),\\ C_{s_{i}}(e) & \equiv A(s_{i}(e)) \mathbin{\rightarrow} A(e). \end{align*} $$

The proof generalizes the preceding example: we successively substitute terms for e in such a way that a disjunction of instances of D is implied by substitution instances of the critical formulas in $\Gamma $ together with a disjunction of the form ${B_{m}}$ plus the predicative critical formulas $\Pi (e)$ . Once $k = m$ , the disjunction becomes provable from ${B_{m}}$ .

Let $T_{0} = \{e\}$ and $T_{i+1} = \{s_{j}(t) : t \in T_{i}, j \le r\}$ . Let $\Gamma (T) = \{C[t/e] : C \in \Gamma , t \in T\}$ . If w is a word over $\{s_{1}, \dots , s_{r}\}$ , i.e., $w = s_{i_{1}}\dots s_{i_{k}}$ then we write $w_{j}(t)$ for $s_{i_{j}}(s_{i_{j+1}}(\dots s_{i_{k}}(t)\dots ))$ . So, e.g., if $w=sst$ then $w_{1}(e)$ is $s(s(t(e)))$ , $w_{3}(e) = t(e)$ , and $w_{4}(e) = e$ . Let $W(m)$ be the set of all length m words over $s_{1}$ , …, $s_{r}$ .

We show by induction on m that

$$ \begin{align*} \Gamma, \Gamma(T_{1}), \dots, \Gamma(T_{m-1}), \Lambda_{m}, \Pi(e) & \vdash_{\mathbf{L}} \bigvee_{i=1}^{m} \bigvee_{t\in T_{i-1}} D(t), \end{align*} $$

where

$$ \begin{align*} \Lambda_{i} & = \left\{ \bigvee_{i=1}^{m} C(w,i) : w \in W(m)\right\} \text{ and}\\ C(w,i) & \equiv A(w_{i}(e)) \mathbin{\rightarrow} A(w_{i+1}(e)). \end{align*} $$

The induction basis is $m=1$ . Then

$$ \begin{align*} T_{1} & = \{s_{1}(e), \dots, s_{r}(e)\},\\ W(1) & = \{s_{1}, \dots, s_{r}\},\\ C(s_{j},1) & \equiv A(s_{j}(e)) \mathbin{\rightarrow} A(e), \end{align*} $$

and so each disjunction in $\Lambda _{i}$ is just one of the impredicative critical formulas $A(s_{j}(e)) \mathbin {\rightarrow } A(e))$ , i.e., $\Lambda _{1} = \Delta (e)$ . Likewise, the disjunction on the right is just $D(e)$ . So the claim holds by the assumption that $\Gamma , \Delta (e), \Pi (e) \vdash D(e)$ .

Now let $v = s_{i_{1}}\dots s_{i_{m}}$ be a length m word, and

$$ \begin{align*} \Lambda_{m}(v) & = \left\{ \bigvee_{i=1}^{m} (A(w_{i}(e)) \mathbin{\rightarrow} A(w_{i+1}(e)) : w \in W(m) \setminus \{v\}\right\},\\ C(v) & \equiv \bigvee_{i=1}^{m} (A(v_{i}(e)) \mathbin{\rightarrow} A(v_{i+1}(e)). \end{align*} $$

In other words, $\Lambda _{m} = \Lambda _{m}(v) \cup \{C(v)\}$ . We’ll abbreviate $\Gamma (T_{1})$ , …, $\Gamma (T_{m-1})$ as $\Gamma ^{\prime }$ , and $\bigvee _{i=1}^{m} \bigvee _{t\in T_{i-1}} D(t)$ as $D^{\prime }$ . The induction hypothesis can then be written as:

$$ \begin{align*} \Gamma, \Gamma^{\prime}, C(v), \Lambda_{m}(v), \Pi(e) & \vdash_{\mathbf{L}} D^{\prime}. \end{align*} $$

Take $t = v_{1}(e)$ , i.e., $s_{i_{1}}(\dots s_{i_{m}}(e))$ . By replacing e by t in $\pi $ , we have

$$ \begin{align*} \Gamma(t), \Lambda(t), \Pi(t) & \vdash_{\mathbf{L}} D(t) \text{ and so also}\\ \Gamma(t), \bigwedge \Lambda(t) \land \bigwedge \Pi(t) & \vdash_{\mathbf{L}} D(t). \end{align*} $$

Combining this with the induction hypothesis using Lemma 7.5(1) we have

$$ \begin{align*} \Gamma, \Gamma^{\prime}, \Gamma(t), &\\ \left(\bigwedge \Lambda(t) \land \bigwedge \Pi(t)\right) \lor C(v), &\\ \Lambda_{m}(v), \Pi(e) & \vdash_{\mathbf{L}} D^{\prime} \lor D(t). \end{align*} $$

If we write ${\Xi \lor G}$ for $\{F \lor G : F \in \Xi \}$ , by distributivity,

$$ \begin{align*} \Gamma, \Gamma^{\prime}, \Gamma(t),&\\ \Lambda(t) \lor C(v),&\\ \Pi(t) \lor C(v),&\\ \Lambda_{m}(v), \Pi(e) & \vdash_{\mathbf{L}} D^{\prime} \lor D(t). \end{align*} $$

Recall that $t \equiv v_{1}(e)$ where v is a word of length m. The formulas in $\Pi (t)$ are of the form $A(u_{i}) \mathbin {\rightarrow } A(v_{1}(e))$ , so a formula in $\Pi (t) \lor C(v)$ is of the form

$$ \begin{align*} (A(u_{i}) \mathbin{\rightarrow} A(v_{1}(e))) \lor (A(v_{1}(e)) \mathbin{\rightarrow} A(v_{2}(e))) \lor \dots \lor (A(v_{m}(e)) \mathbin{\rightarrow} A(e)). \end{align*} $$

Every such formula is implied by $A(u_{i}) \mathbin {\rightarrow } A(e)$ by Lemma 8.2, since $\mathbf {H} + {B_{m}} \vdash \mathit {Lin}$ by Proposition 6.4. Since $A(u_{i}) \mathbin {\rightarrow } A(e)$ is in $\Pi (e)$ , we get:

$$ \begin{align*} \Gamma, \Gamma^{\prime}, \Gamma(t), \Lambda(t) \lor C(v), \Lambda_{m}(v), \Pi(e) \vdash_{\mathbf{L}} D^{\prime} \lor D(t). \end{align*} $$

Every formula in $\Lambda (t) \lor C(v)$ is of the form ${B_{m}}$ , specifically,

$$ \begin{align*} (A(s_{i}(v_{1}(e))) \mathbin{\rightarrow} A(v_{1}(e))) \lor (A(v_{1}(e)) \mathbin{\rightarrow} A(v_{2}(e))) \lor \dots \lor (A(v_{m}(e)) \mathbin{\rightarrow} A(e)). \end{align*} $$

Since for each $i \le r$ , $A(s_{i}(v_{1}(e))) \mathbin {\rightarrow } A(v_{1}(e)) \in \Lambda (t)$ , $\Lambda (t) \lor C(v)$ is the set of all disjunctions

$$ \begin{align*} \bigvee_{i=1}^{m+1} (A(w_{i}(e)) \mathbin{\rightarrow} A(w_{i+1}(e))), \end{align*} $$

where $w = s_{i}v$ for some $i \le r$ . As every length $m+1$ word is of this form for some length m word v, repeating this process for all length m words v thus yields

$$ \begin{align*} \Gamma, \Gamma^{\prime}, \Gamma(T_{m}), \Lambda_{m+1}, \Pi(e) & \vdash_{\mathbf{L}} D^{\prime} \lor \bigvee_{t \in T_{m}} D(t). \end{align*} $$

As we’ve seen, a formula in $\Lambda _{m}$ is of the form ${B_{m}}$ , so in $\mathbf {L} + {B_{m}}$ , we have

$$ \begin{align*} \Gamma, \Gamma(T_{1}), \dots, \Gamma(T_{m-1}), \Pi(e) & \vdash_{\mathbf{L}} \bigvee_{i=1}^{m} \bigvee_{t\in T_{i-1}} D(t). \end{align*} $$

Thus, the claim follows by taking $T = \{e\} \cup T_{m-1}$ .

If e is a $\tau $ -term, the proof proceeds analogously. The resulting formulas in $\Lambda (e)$ are then of the form $(A_{m} \mathbin {\rightarrow } A_{m+1}) \lor \dots \lor (A_{1} \mathbin {\rightarrow } A_{2})$ which is equivalent to ${B_{m}}$ .⊣

Proof. First suppose e is an $\varepsilon $ -term. Replacing e by $u_{j}$ results in a proof showing

$$ \begin{align*} \Gamma[u_{j}/e], \bigwedge_{i=1}^{p} (A(u_{i}) \mathbin{\rightarrow} A(u_{j})) & \vdash_{\mathbf{L}} D(u_{j}). \end{align*} $$

By applying Lemma 7.5(1), we get

$$ \begin{align*} \bigcup_{j} \Gamma[u_{j}/e], \bigvee_{j=1}^{p} \bigwedge_{i=1}^{p} (A(u_{i}) \mathbin{\rightarrow} A(u_{j})) & \vdash_{\mathbf{L}} \bigvee_{j} D(u_{j}). \end{align*} $$

The disjunction of conjunctions on the left is provable in $\mathbf {L} + \mathit {Lin}$ by Lemma 8.4(1).

If e is a $\tau $ -term, we get

$$ \begin{align*} \bigcup_{j} \Gamma[u_{j}/e], \bigvee_{j=1}^{p} \bigwedge_{i=1}^{p} (A(u_{j}) \mathbin{\rightarrow} A(u_{i})) \vdash_{\mathbf{L}} \bigvee_{j} D(u_{j}), \end{align*} $$

and the claim follows by Lemma 8.4(2).⊣

9 The Hilbert–Bernays elimination procedure

Recall that the challenges in the proof of the extended first ${\varepsilon \tau }$ -theorem include, in addition to the existence of complete elimination sets, guarantees that the new sets $\Gamma [s_{i}/e]$ are in fact critical formulas (so eliminating a set $\Lambda (e)$ of critical formulas yields a correct $\mathbf {L}{\varepsilon \tau }$ -proof), and that the process eventually terminates. In Hilbert and Bernays’s original proof of the first $\varepsilon $ -theorem, this was ensured by processing sets of critical formulas in a specific order. We briefly review this proof, concentrating on its structure, since we’ll apply the same method to critical formulas in ${\varepsilon \tau }$ -proofs for intermediate logics.

An ${\varepsilon \tau }$ -elimination sequence is a sequence of ${\varepsilon \tau }$ -terms $e_{i}$ and sets of critical formulas $\Lambda (e_{i})$ belonging to it, such that eliminating $\Lambda _{i}(e_{i})$ from proof $\pi _{i}$ results in a new proof $\pi _{i+1}$ of a disjunction of instances of $D(e_{i})$ . This new proof proceeds from instances of the critical formulas for which $e_{i}$ is not critical and the remaining critical formulas belonging to $e_{i}$ . Since the definition requires $\pi _{i}$ to be an ${\varepsilon \tau }$ -proof, the formulas in $\Gamma [T_{i}]$ must actually be critical formulas.

If an ${\varepsilon \tau }$ -elimination sequence exists for a formula E and its ${\varepsilon \tau }$ -proof $\pi _{0}$ , then the extended first ${\varepsilon \tau }$ -theorem holds for E:

For the proof of the extended first ${\varepsilon \tau }$ -theorem, then, it is sufficient to show that suitable e-elimination sets always exist, and that ${\varepsilon \tau }$ -terms e and sets of associated critical formulas can be successively chosen in such a way as to yield an ${\varepsilon \tau }$ -elimination sequence for $\pi _{1}$ . Hilbert and Bernays did this by defining a well-ordering of ${\varepsilon \tau }$ -terms with the property that eliminating maximal ${\varepsilon \tau }$ -terms according to this ordering guarantees that the $\Gamma [T_{i}]$ are again critical formulas, and that in each step no critical ${\varepsilon \tau }$ -terms are newly introduced which are larger (in the ordering).

The ordering used is the lexicographic order on two complexity measures of ${\varepsilon \tau }$ -terms e. We say that e is nested in an ${\varepsilon \tau }$ -term $e^{\prime }$ if e is a proper subterm of e, i.e., if every occurrence of a variable which is free in e is also free in $e^{\prime }$ . An ${\varepsilon \tau }$ -term e is subordinate to $e^{\prime } \equiv \varepsilon _{x}\,A(x)$ or $\equiv \tau _{x}\,A(x)$ if e occurs in $e^{\prime }$ and x is free in e. The degree $\mathrm {deg}(e)$ of e is the maximal level of nesting of subterms of e; the rank $\mathrm {rk}(e)$ the maximal level of subordination. When C is a critical formula belonging to e, we let $\mathrm {deg}(C) = \mathrm {deg}(e)$ and $\mathrm {rk}(C) = \mathrm {rk}(e)$ .

The Hilbert–Bernays elimination order proceeds by always picking a critical formula of maximal degree among the critical formulas of maximal rank. Its success relies on the following two lemmas, which establish that (a) replacement of maximal ${\varepsilon \tau }$ -terms in a critical formula results in a critical formula, and (b) if new critical formulas are generated, they are of lower rank, or of the same rank but of lower degree.

The proofs of the preceding lemmas are as in [Reference Hilbert and Bernays15]; see also [Reference Moser and Zach23, Section 5], [Reference Zach, Ghosh and Prasad30, Section 4.1].

Proof. Take $\pi _{0} = \pi $ . Suppose $\pi _{i}$ has been defined. Let $e_{i}$ be a critical ${\varepsilon \tau }$ -term of $\pi _{i}$ of maximal degree among the critical terms of maximal rank. Let $\Lambda _{i}(e_{i})$ be all critical formulas belonging to $e_{i}$ , and $\Gamma _{i}$ the remaining critical formulas. We have

$$ \begin{align*} \Gamma_{i}, \Lambda_{i}(e_{i}) & \vdash_{\mathbf{L}{\varepsilon\tau}} D_{i}. \end{align*} $$

By assumption, there is a complete e-elimination set ${T_{i}}$ for ${\Lambda _{i}(e_{i})}$ and so we have ${\pi _{i+1}}$ showing that

$$ \begin{align*} \Gamma_{i}[T_{i}] & \vdash_{\mathbf{L}{\varepsilon\tau}} \bigvee_{t\in T_{i}} D_{i}[t/e_{i}]. \end{align*} $$

Each critical formula C in $\Gamma _{i}$ is not of higher rank than $e_{i}$ , and if it is of equal rank it is not of higher degree. So by Lemma 9.3, $C[t/e_{i}]$ is a critical formula. Hence $\pi _{i+1}$ is a correct $\mathbf {L}{\varepsilon \tau }$ -proof. Let $\Gamma _{i+1} = \Gamma _{i}[T_{i}]$ and $D_{i+1} \equiv \bigvee _{t\in T_{i}} D_{i}[t/e_{i}]$ .

Eventually, $\Gamma _{i} = \emptyset $ , since in each step, by Lemma 9.4, the maximal rank of critical ${\varepsilon \tau }$ -terms in $\Gamma _{i}$ does not increase, the maximal degree of critical ${\varepsilon \tau }$ -terms of maximal rank does not increase, and the number of critical ${\varepsilon \tau }$ -terms of maximal degree among those of maximal rank decreases.⊣

We are indebted to the referee for the Journal for the following observation:

10 Herbrand’s theorem and the second epsilon-theorem

The extended first ${\varepsilon \tau }$ -theorem implies Herbrand’s theorem for purely existential formulas. If $E \equiv \exists x_{1}\ldots \exists x_{n} E^{\prime }(x_{1}, \ldots , x_{n})$ is provable in predicate logic, then so is $E^{\varepsilon \tau } \equiv E^{\prime }(e_{1}, \ldots , e_{n})$ for some $\varepsilon $ -terms $e_{1}$ , …, $e_{n}$ . From the extended first $\varepsilon $ -theorem we then obtain a proof in propositional logic of a Herbrand disjunction

$$ \begin{align*} E^{\prime}(t_{1}^{1}, \ldots, t_{n}^{1}) \lor \cdots \lor A(t_{1}^{k}, \ldots, t_{n}^{k}) \end{align*} $$

for some terms $t_{i}^{j}$ . In predicate logic, we may now successively introduce existential quantifiers to obtain the original formula E. This holds in any intermediate predicate logic in which the first ${\varepsilon \tau }$ -theorem holds, since the only principles used in the last step (proving E from its Herbrand disjunction) are $A(t) \mathbin {\rightarrow } \exists x \,A(x)$ and $\exists x(A(x) \lor B) \mathbin {\rightarrow } (\exists x\,A(x) \lor B)$ , (x not free in B) which already hold in intuitionistic logic. (Despite the failure of the extended first ${\varepsilon \tau }$ -theorem in $\mathbf {LC}{\varepsilon \tau }$ , the Herbrand theorem for existential formulas does hold in $\mathbf {Q}\mathbf {LC}$ ; see [Reference Aschieri3]).

For classical predicate logic, the Herbrand theorem for existential formulas implies the Herbrand theorem for prenex formulas. If a prenex formula E has a proof in first-order logic, so does its (purely existential) Herbrand form $H(E) = \exists x_{1}\ldots \exists x_{n} E^{\prime }(x_{1}, \ldots , x_{n})$ . In classical predicate logic, we can obtain not just $H(E)$ from its Herbrand disjunction, but also the original prenex formula E. This requires that we introduce not just existential quantifiers, but also universal quantifiers. Consequently, we need not just the generalization rule $A(x) / \forall x\,A(x)$ but also a principle that allows us to shift universal quantifiers over disjunctions, viz.,

(CD) $$\begin{align} \forall x(A(x) \lor B) \mathbin{\rightarrow} (\forall x\,A(x) \lor B). \end{align}$$

This principle is not intuitionistically valid: it characterizes Kripke frames with constant domains. Since the implication $E \mathbin {\rightarrow } H(E)$ already holds in intuitionistic logic, any intermediate predicate logic in which $\mathit {CD}$ holds and which has the extended first $\varepsilon $ -theorem also has Herbrand’s theorem for prenex formulas:

Although the extended first $\varepsilon $ -theorem implies Herbrand’s theorem for prenex formulas if $\mathit {CD}$ is provable, the converse is not true. As we showed in Theorem 6.2, the extended first ${\varepsilon \tau }$ -theorem holds for $\mathbf {L}{\varepsilon \tau }$ only if $\mathbf {L} \vdash {B_{m}}$ for some m. In particular, it does not hold for infinite-valued first-order Gödel logics $\mathbf {G}_{\mathbb {R}} = \mathbf {Q}\mathbf {LC} + \mathit {CD}$ . However, Herbrand’s theorem for prenex formulas does hold for $\mathbf {G}_{\mathbb {R}}$ ; see [Reference Baaz, Preining and Zach6, Theorem 5.7], [Reference Baaz, Preining and Zach7, Theorem 7.8]. (Incidentally, the Herbrand theorem for prenex formulas also holds in intuitionistic logic despite the invalidity of $\mathit {CD}$ ; see [Reference Bowen10]).

In classical logic, the second $\varepsilon $ -theorem can be proved using the extended first $\varepsilon $ -theorem as follows: Suppose $\mathbf {C}{\varepsilon \tau } \vdash A^{\varepsilon \tau }$ . Since A is equivalent to a prenex formula $A^{p}$ in classical logic, we have $\mathbf {Q}\mathbf {C} \vdash A \mathbin {\rightarrow } A^{p}$ . Prenex formulas imply their Herbrand forms, i.e., $\mathbf {Q}\mathbf {C} \vdash A^{p} \mathbin {\rightarrow } H(A^{p})$ . Together we have $\mathbf {Q}\mathbf {C} \vdash A \mathbin {\rightarrow } H(A^{p})$ and by translating into the ${\varepsilon \tau }$ -calculus, $\mathbf {C}{\varepsilon \tau } \vdash A^{\varepsilon \tau } \mathbin {\rightarrow } H(A^{p})^{\varepsilon \tau }$ , so $\mathbf {C}{\varepsilon \tau } \vdash H(A^{p})^{\varepsilon \tau }$ . By the extended first ${\varepsilon \tau }$ -theorem, $H(A^{p})$ has a Herbrand disjunction, from which (in $\mathbf {Q}\mathbf {C}$ ) we can prove $A^{p}$ and hence A.

The steps that may fail in an intermediate predicate logic $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ , other than the extended first ${\varepsilon \tau }$ -theorem, are the provability of $A \mathbin {\leftrightarrow } A^{p}$ and proving $A^{p}$ from the Herbrand disjunction of $H(A^{p})$ . These steps do work provided all quantifier shifts can be carried out (i.e., in addition to $\mathit {CD}$ also the formulas $Q_{\forall }$ and $Q_{\exists }$ ). Thus, if $\mathbf {L}{\varepsilon \tau }$ has the extended first ${\varepsilon \tau }$ -theorem, any intermediate predicate logic $\mathbf {Q}\mathbf {L} + \mathit {Ax}$ in which all quantifier shifts are provable also has the second ${\varepsilon \tau }$ -theorem:

Infinite-valued first-order Gödel logic $\mathbf {G}_{\mathbb {R}} = \mathbf {Q}\mathbf {LC} + \mathit {CD}$ is an intermediate predicate logic which proves $\mathit {CD}$ but not $Q_{\forall }$ or $Q_{\exists }$ , and the extended first ${\varepsilon \tau }$ -theorem does not hold for $\mathbf {LC}{\varepsilon \tau }$ . The logics of linear Kripke frames with m worlds (and varying domains) are $\mathbf {Q}\mathbf {LC}_{m}$ . But $\mathbf {Q}\mathbf {LC}_{m}$ does not prove $\mathit {CD}$ , so it also does not have the second ${\varepsilon \tau }$ -theorem:

However, we have:

Herbrand’s theorem also yields other results, for instance, the following.