Algebra and Logic, Vol. 43, No. 3, 2004

NOVIKOV COMPLETE LOGICS:TRANSLATION TECHNIQUES

A. D. Yashin∗ UDC 510.64

Key words: Novikov complete logics.

We show how to construct new, explicit examples of Novikov complete logics from the knownones, using a certain version of the translation techniques.

A translation is the mapt : Fm(L)→ Fm(L′)

from a set, Fm(L), of propositions in a language (signature), L, into a set, Fm(L′), of propositions in alanguage, L′.

Translations should satisfy the following basic requirements:(1) a translation must be given effectively, i.e., given a proposition A ∈ Fm(L), its translation t(A) ∈

Fm(L′) ought to be constructed explicitly, and, moreover, given any formula in Fm(L′), it is required thatwe determine effectively whether that formula is a translation of some formula or not.

(2) a translation should, if possible, preserve the meaning of the propositions; specifically, if the languagesL and L′ share a part L ∩ L′ �= ∅, then propositions from it must not be transformed by a translation.

There are a lot of examples of the translation techniques applying in, for instance, axiomatic set theory,first-order theory, modal and non-classical logics.

As a rule, given a translation t : Fm(L) → Fm(L′), we construct either the image of a theory T ofsignature L or the preimage of a theory T ′ of signature L′. With a translation chosen properly, some usefulproperties of the initial theory will be inherited.

In the present paper, a version of the translation techniques is used to construct new examples of Novikovcomplete logics based on the known ones.

1. NOVIKOV COMPLETE LOGICS

1.1. Preliminary information. A language, L, of intuitionistic propositional logic contains propo-sitional variables, VAR = {pi, qj, . . .}, and standard connectives, ∧, ∨, →, and ¬. An equivalence ↔ isdefined thus: p ↔ r (p → q) ∧ (q → p). Sometimes it will also be convenient to introduce into alanguage standard constants 0 (falsity) and 1 (truth). Formulas are defined in the usual manner. The classof formulas in L is denoted by Fm.

The symbol Int denotes intuitionistic propositional logic. For this logic, different deductive systems, aswell as semantics, have been described in the literature (see, e.g., [1]). The laws of Int will be used belowwithout further comment.

∗Supported by FDD Foundation grant No. PM 884-02.

Translated from Algebra i Logika, Vol. 43, No. 3, pp. 364-378, May-June, 2004. Original article submitted January 17,

2003.

0002-5232/04/4303-0205 c© 2004 Plenum Publishing Corporation 205

1.2. ϕ-Logics. Suppose that the language L of intuitionistic propositional logic is extended by addingconnectives ϕ1, . . . , ϕn. Every connective ϕi has ki = ar(ϕi) arguments. Specifically, a connective withoutarguments is referred to as constant. An extended language is denoted by L(ϕ), and the class of formulasin the extended language — by Fm(ϕ). In this event Fm ⊆ Fm(ϕ). Formulas that do not contain extraconnectives are said to be pure. SubA stands for the set of subformulas of a formula A. In this instanceA ∈ SubA. By writing D(p1, . . . , pm) we mean that only variables in the list p1, . . . , pm are involved in aformula D.

A substitution in the language L(ϕ) is the map s : Fm(ϕ) −→ Fm(ϕ) preserving constants and com-muting with all logical connectives (cf. [2]), that is,

s(0) = 0, s(1) = 1, s(¬A) = ¬s(A),s(A ◦B) = s(A) ◦ s(B) for ◦ ∈ {→,∧,∨},s(ϕi(A1, . . . , Aki)) = ϕi(s(A1), . . . , s(Aki)).

In this case s(A) is called a substitutional instance of the formula A. In fact, to define a substitution, itsuffices to specify s(p) for every propositional variable p, and we write A(p|B, q|C, . . .) to show that alloccurrences of p are replaced by a formula B, and all occurrences of q — by a formula C, etc.

An equivalents replacement rule for a connective ϕi is expressed thus:

(A1 ↔ B1) ∧ . . . ∧ (Aki ↔ Bki)ϕi(A1, . . . , Aki)↔ ϕi(B1, . . . , Bki)

.

A ϕ-logic is an arbitrary set of formulas in L(ϕ) including Int and closed under modus ponens, substi-tutions, and replacement of equivalents, for every extra connective.

An equivalents replacement axiom for ϕi is a formula such as

ki∧j=1

(pj ↔ qj)→ (ϕi(p1, . . . , pki)↔ ϕi(q1, . . . , qki)).

An equivalents replacement scheme is a set of formulas of the form

m∧i=1

(pi ↔ qi)→ (D(p1, . . . , pm)↔ D(q1, . . . , qm)),

for every formula D(p1, . . . , pm).

Proposition 1.1. A ϕ-logic L contains an equivalents replacement scheme if and only if L contains anequivalents replacement axiom for every extra connective in ϕ.

Notice that if a ϕ-logic contains the equivalents replacement scheme then the equivalents replacementrule becomes unnecessary. Such ϕ-logics are said to be normal.

For brevity, we write A L↔ B instead of A ↔ B ∈ L. An outer connective in formulas is sometimesdenoted by an index shaped as a point.

1.3. Conservative ϕ-logics. A ϕ-logic L is said to be conservative (over Int) if D ∈ L⇒ D ∈ Int, forany pure formula D.

Let L be a normal ϕ-logic and Σ some set of formulas. An inference in L from the set Σ of premises isa finite sequence of formulas every element of which either is a formula in L, or a premise, or is obtainedfrom the two preceding formulas in this sequence via modus ponens. Inferability of a formula A in L fromΣ is denoted by Σ !L A.

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THEOREM 1.2 (deduction). For every normal ϕ-logic L and for all formulas A and B,

{A} !L B ⇐⇒ A→ B ∈ L.

Proof. A known proof of the deduction theorem for the classical (as well as intuitionistic [3]) proposi-tional calculus is based on the schemes

A→ A,

A→ (B → A),

(A→ (B → C))→ ((A→ B)→ (A→ C)),

and on the fact that modus ponens is the sole postulated inference rule. For the case under examination,such premises are all satisfied, and so the proof of the deduction theorem applies in it virtually withoutchanges.

Recall that L + Σ denotes a least ϕ-logic comprising L and Σ. If Σ = {A} then, for brevity, we writeL+A instead of L+ {A}.

THEOREM 1.3. L+ Σ coincides with a set of formulas inferable in L from the set SUBST(Σ) of allsubstitutional instances of the formulas in Σ.

Proof. Clearly, every formula inferable in L from SUBST(Σ) is contained in L+ Σ. In order to provethe inverse inclusion, it suffices to verify that the set

L′ = {A ∈ Fm | SUBST(Σ) !L A}

is itself a ϕ-logic, that is, it includes Int and is closed under modus ponens and substitutions. The firsttwo requirements are obviously satisfied; the property of being closed under substitutions is verified byinduction on the length of an inference using the fact that a composition of substitutions is a substitution.

THEOREM 1.4 (on being adjoinable [4]). Let L be a normal ϕ-logic conservative over Int. Then:(a) a set Σ is adjoinable to L iff every finite subset Σ′ ⊆ Σ is adjoinable to L;(b) a formula A is non-adjoinable to L iff, for some substitutions s1, . . . , sl on Fm(ϕ) of some pure

formula D /∈ Int,s1(A) ∧ . . . ∧ sl(A)→ D ∈ L;

(c) a constant formula E in the ϕ-language is non-adjoinable to L iff E → D ∈ L, for some pure formulaD /∈ Int.

2. NOVIKOV COMPLETE ϕ-LOGICS

2.1. New intuitionistic connectives. Let L be a ϕ-logic. Under which conditions can we say that Lspecifies new intuitionistic logical connectives? In other words, how can the intuitionistic sense and newnessbe expressed in terms of the properties of L?

Analyzing the various works in which this question is touched upon in one way or another, we may dealout the following (minimal) list of requirements:

— L is conservative over Int (this condition accentuates an intuitionistic character of the extra connec-tives);

— L is closed under replacement of equivalents;— L does not contain any formula of the form ϕi(p1, . . . , pki)↔ D for a pure D (newness).

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Remark. The lack of a particular explicit relation in L may turn out to be accidental. It is notimprobable that some extension of L, still conservative over Int, has an explicit relation (which is validatedby the so-called Kaminski logic describing properties of the strong future operator [5]).

Novikov’s approach, including this remark, may be re-worded into the following two definitions.

Definition. A normal ϕ-logic L is Novikov complete if it is conservative over Int, and for any formulaA /∈ L, the ϕ-logic L + A is non-conservative over Int. In other words, L does not admit of adjoining anynew formula.

An explicit relation for ϕi is a formula of the form

ϕi(p1, . . . , pki)↔ A,

where A does not contain ϕi.

Definition. The connectives ϕ1, . . . , ϕn are independent in the conservative ϕ-logic L if, for any ϕi ∈ ϕand for every formula A freed of occurrences of ϕi, the ϕ-logic L+ ϕi(p1, . . . , pki)↔ A is non-conservativeover Int. In other words, L does not admit of adjoining explicit relations.

Remark. It is worth noting an analogy with how the negation is treated in the forcing method: acondition a forces ¬A whenever no one of its correct extensions a′ ≥ a forces A [6].

If |ϕ| = 1, then the explicit relations for ϕ have the form ϕ(p̄)↔ B, for some pure formula B and somevariables p̄ = {p1, . . . , pk}. In this case we can say that L defines a new k-ary intuitionistic connective ϕ.

On this approach, of interest among all ϕ-logics are the normal ϕ-logics defying adjoining of explicitrelations, while complete ϕ-logics are most interesting among the latter.

2.2. Examples of Novikov complete logics. In connection with the definitions given above, it isworth mentioning a problem due to Novikov which calls for constructing explicit examples of the completeϕ-logics with independent connectives. Zorn’s lemma shows that, for any ϕ-logic conservative over Int,there exists at least one completion, but it says nothing about how to describe it explicitly. At the momentwe need to specify what is to be conceived of as an explicit example. According to A. V. Kuznetsov, anexplicit example is a ϕ-logic definable by an explicit finite list of extra axioms.∗ Moreover, a decidableϕ-logic may also be thought of as defined explicitly.

In what follows we make use of the following:

THEOREM 2.1 [4, 7]. For every tuple ϕ of extra constants, there exist explicitly defined instances ofϕ-logics, in each of which extra constants are independent.

As a particular example, we can point out the so-called Smetanich logic given thus:

Tϕ = Int +

ϕ(p)↔ ϕ(q),¬¬ϕ(p),ϕ(p)→ (q ∨ ¬q)

(cf. [8, 9]). The first axiom in this logic shows that ϕ(p) does not depend on the value of the argument,that is, ϕ may be treated as a logical constant. Therefore Tϕ can be naturally reformulated in the languagewith one extra constant ϕ. The resulting logic is defined thus:

Sm = Int +

{¬¬ϕ,ϕ→ (A ∨ ¬A)

(cf. [10]). There exists the unique Novikov complete extension Sm+ of the logic Sm (cf. [11]).

∗I came to know about this definition by courtesy of L. L. Maksimova (oral communication).

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3. NOVIKOV COMPLETE ϕ-LOGICS: TRANSLATION TECHNIQUES

3.1. Preimages of logics. Consider languages, L(ϕ) and L(ψ), containing tuples, ϕ = {ϕ1, . . . , ϕn}and ψ = {ψ1, . . . , ψm}, of extra connectives. Denote by Fm(ϕ) and Fm(ψ) the corresponding classes offormulas.

Let ki = ar(ϕi), i = 1, . . . , n. For every connective ϕi ∈ ϕ, fix a formula Di(p1, . . . , pki) ∈ Fm(ψ) withki variables, and define a translation such as

t : Fm(ϕ) −→ Fm(ψ)

using the following rule:

t(p) p for all p ∈ VAR;t(A ◦B) t(A) ◦ t(B) for ◦ ∈ {∧,∨,→};t(¬A) ¬t(A);

t(ϕi(A1, . . . , Aki)) Di(t(A1), . . . , t(Aki )).

The next lemma shows that t should be consistent with substitutions in the languages L(ϕ) and L(ψ).(The statement of the lemma can be called a direct condition for a translation to be consistent with substi-tutions.)

LEMMA 3.1. For any substitution s : Fm(ϕ) −→ Fm(ϕ), there is a substitution s′ : Fm(ψ) −→ Fm(ψ)such that for any formula A ∈ Fm(ϕ), it is true that s′(t(A)) = t(s(A)).

Proof. For every p ∈ VAR, put s′(p) t(s(p)). Further we use induction on the complexity ofA ∈ Fm(ϕ).

If A = p ∈ VAR then s′(t(p)) = s′(p) = t(s(p)) by definition.If A = B ◦ C then s′(t(B ◦ C)) = s′(t(B) ◦ t(C)) = s′(t(B)) ◦ s′(t(C)) = [inductive assumption]

t(s(B)) ◦ t(s(C)) = t(s(B) ◦ s(C)) = t(s(B ◦ C)).If A = ¬B then s′(t(¬B)) = s′(¬t(B)) = ¬s′(t(B)) = [inductive assumption] ¬t(s(B)) = t(¬s(B)) =

t(s(¬B)).If A = ϕi(B,C) (the case ar(ϕi) = 2 is treated for brevity) then s′(t(ϕi(B,C))) = s′(Di(t(B), t(C))) =

Di(s′(t(A)), s′(t(B))) = [inductive assumption] Di(t(s(B)), t(s(C))) = t(ϕi(s(B), s(C))) = t(s(ϕi(B,C))).Let L′ be a ψ-logic. The preimage of this logic relative to t is defined thus:

t−1(L′) {A ∈ Fm(ϕ) | t(A) ∈ L′}.

THEOREM 3.2. (1) L = t−1(L′) is a ϕ-logic.(2) If L′ is conservative then so is L.(3) If L′ contains an equivalents replacement scheme then so does L.Proof. By the definition of t, for every pure formula A, t(A) = A. Then Int ⊆ L and L is conservative.Let A, A → B ∈ L. Then t(A), t(A → B) ∈ L′. By the definition of t, t(A) → t(B) ∈ L′. By modus

ponens, t(B) ∈ L′, that is, B ∈ L. Thus L is closed under modus ponens.Let A ∈ L and s : Fm(ϕ) −→ Fm(ϕ) be a substitution. Then t(A) ∈ L′. There is a substitution

s′ : Fm(ψ) −→ Fm(ψ) such that s′(t(A)) = t(s(A)). In this event s′(t(A)) ∈ L′. Hence t(s(A)) ∈ L′, thatis, s(A) ∈ L. Thus L is closed under substitutions.

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Our goal is to verify that L is closed under replacement of equivalents. For brevity, we deal with abinary connective ϕi(·, ·). Let A↔ A1, B ↔ B1 ∈ L. Then

t(A↔ A1) = t(A)↔ t(A1) ∈ L′, t(B ↔ B1) = t(B)↔ t(B1) ∈ L′.

By the equivalents replacement rule, in L′ we have Di(t(A), t(B)) ↔ Di(t(A1), t(B1)) ∈ L′. By theproperties of t, t(ϕi(A,B)↔ ϕi(A1, B1)) ∈ L′, and so

ϕi(A,B)↔ ϕi(A1, B1) ∈ L.

Assume that L′ contains an equivalents replacement scheme. Consider a translation of the replacementaxiom for ϕi(·, ·) such as

t((p↔ p1) ∧ (q ↔ q1)→. (ϕi(p, q)↔ ϕi(p1, q1))) =(p↔ p1) ∧ (q ↔ q1)→. (Di(p, q)↔ Di(p1, q1))) ∈ L′

as a particular case of the equivalents replacement scheme. This implies that L contains the replacementaxiom for ϕi.

3.2. Surjective translations and Novikov completeness. The main objective of this subsectionis to clarify under which conditions the preimage of a complete ψ-logic is a complete ϕ-logic. Let L′ be aψ-logic.

An inverse condition for a translation t to be consistent with substitutions (relative to L′) is the following:for every substitution s′ : Fm(ψ) −→ Fm(ψ) and every formula A ∈ Fm(ϕ), there is a substitution

s : Fm(ϕ) −→ Fm(ϕ) such that s′(t(A))↔ t(s(A)) ∈ L′.

THEOREM 3.3. Let L′ be a complete normal ψ-logic and the translation t satisfy the inverse conditionof being consistent with substitutions. Then L = t−1(L′) is a complete normal ϕ-logic.

Proof. Let A ∈ Fm(ϕ) and A /∈ L. We claim that A is non-adjoinable to L. We have t(A) /∈ L′. SinceL′ is complete, the formula t(A) cannot be adjoined to L′. By Theorem 1.4(b), there are a pure formulaD /∈ Int and substitutions s′1, . . . , s′j on Fm(ψ) such that

j∧i=1

s′i(t(A))→. D ∈ L′.

By the inverse consistency condition, there are substitutions s1, . . . , sj on Fm(ϕ) for which

s′i(t(A))↔ t(si(A)) ∈ L′, i = 1, . . . , j.

By the equivalents replacement rule, in L′ we have

j∧i=1

t(si(A))→. D ∈ L′.

By the properties of t,

t

( j∧i=1

si(A)→ .D

)∈ L′, i.e.,

j∧i=1

si(A)→. D ∈ L.

Therefore A is non-adjoinable to L.

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Thus the availability of a complete ψ-logic L′ and a translation satisfying the inverse consistency con-dition relative to L′ makes it possible to furnish an example of a complete ϕ-logic.

We say that a translation t : Fm(ϕ) −→ Fm(ψ) is surjective relative to a ψ-logic L′ if, for every formulaA′ ∈ Fm(ψ), there is a formula A ∈ Fm(ϕ) such that t(A)↔ A′ ∈ L′.

THEOREM 3.4. Let t be surjective relative to L′. Then t satisfies the inverse condition of beingconsistent with substitutions relative to L′.

Proof. Consider a substitution s′ : Fm(ψ) −→ Fm(ψ). Put A′i s′(pi) for every variable pi.

By the hypothesis, there exist formulas Ai ∈ Fm(ϕ) such that t(Ai)↔ A′i ∈ L′. We define a substitution

s : Fm(ϕ) −→ Fm(ϕ) using the rule s(pi) Ai. Now, by induction on the complexity of A ∈ Fm(ϕ), weprove that t(s(A))↔ s′(t(A)) ∈ L′.

If A = pi then t(s(pi)) = t(Ai)L′↔ A′

i = s′(pi) = s′(t(pi)).

If A = B ◦ C then t(s(B ◦ C)) = t(s(B) ◦ s(C)) = t(s(B)) ◦ t(s(C)) L′↔ [inductive assumption andequivalents replacement rule] s′(t(B)) ◦ s′(t(C)) = s′(t(B) ◦ t(C)) = s′(t(B ◦ C)).

If A = ϕj(B,C) (the binary connective is treated for brevity) then t(s(ϕj(B,C)) = t(ϕi(s(B), s(C))) =

Dj(t(s(B)), t(s(C))) L′↔ [inductive assumption and equivalents replacement rule] Dj(s′(t(B)), s′(t(C))) =s′(Dj(t(B), t(C))) = s′(t(ϕj(B,C))).

The next proposition demonstrates that in order to verify that t is surjective it suffices to check on thevalidity of the so-called covering condition for extra connectives in the ψ-language.

Proposition 3.5. Suppose that for every connective ψi, there is a formula Ψi(p1, . . . , pki) ∈ Fm(ϕ)such that

ψi(p1, . . . , pki)↔ t(Ψi(p1, . . . , pki)) ∈ L′. (∗)Then t is surjective relative to L′.

Proof. For every formula A ∈ Fm(ψ), we construct the formula A∗ ∈ Fm(ϕ) so that t(A∗)↔ A ∈ L′.Further, we use induction on A.

If A is a variable or standard constant then we put A∗ A.If A = B ◦ C (◦ is a standard connective) then we put A∗ B∗ ◦ C∗, where B∗ and C∗ are as at the

previous step.For negation, the construction is similar.If A = ψi(B1, B2) (the binary connective ψi is treated for brevity) then we put A∗ Ψi(B∗

1 , B∗2), where

B∗1 and B∗

2 are as at the previous steps.Consider a substitution s on Fm(ψ) satisfying the conditions s(p1) B1 and s(p2) B2. We have

s(ψi(p1, p2)) ↔ s(t(Ψi(p1, p2))) ∈ L′ as a substitutional instance of (∗). The left part of the latter equiv-alence coincides with A. The right one is transformed into t(Ψi(B1, B2)) since the substitution is carriedover all the connectives, both standard and extra.

Remark. The question about explicit relations in t−1(L′) should be treated separately.

4. EXEMPLIFYING APPLICATIONS OF THE TRANSLATION TECHNIQUES

4.1. Complete extension of Bessonov’s logic. The Bessonov logic, Bes, is defined in a languagewith one extra unary connective, φ, as follows:

standard axiom schemes for Int,(A→ B)→ (φ(A)→ φ(B)),

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¬¬φ(A),φ(1),φ(0)→ (A ∨ ¬A),modus ponens as the sole inference rule

(cf. [12]). We define a translation t from the φ-language into a language with the sole extra constant ϕ, bysetting t(φ(p)) = ϕ ∨ p.

The constant ϕ is a translation of the formula φ(0), that is, the translation is surjective relative to anyϕ-logic. In order to find a Novikov complete extension of the Bessonov logic, it suffices to point out anexample of a complete ϕ-logic containing translations of the specific axioms in Bes.

Let Sm+ be a Novikov complete extension of the Smetanich ϕ-logic. Translations of the axioms in Besare of the following forms:

(t(A)→ t(B))→ ((t(A) ∨ ϕ)→ (t(B) ∨ ϕ)),¬¬(t(A) ∨ ϕ),1 ∨ ϕ,(0 ∨ ϕ)→ (t(A) ∨ ¬t(A)).

It is easy to see that the translations all belong to Sm and so to Sm+ of course. Thus t−1(Sm+) exemplifiesa Novikov complete extension of the Bessonov logic (cf. [13]).

4.2. Example of a complete logic with several independent unary connectives. Let ψ ={ψ1(·), . . . , ψn(·)} be a tuple of unary connectives and ϕ = {ϕ1, . . . , ϕn} be the tuple consisting of the samenumber of constants.

Define a translation t : Fm(ψ)→ Fm(ϕ) using the rule

t(ψi(A)) = t(A) ∨ ϕi, i = 1, . . . , n.

Here, we exercise the same technique as was used for the Bessonov logic. The translation above is surjectiverelative to any ϕ-logic.

Let L be an arbitrary Novikov complete ϕ-logic. By virtue of the main result, its preimage L′ = t−1(L)is a Novikov complete ψ-logic. We claim that if L lacks explicit relations then the same is true of L′.

Let ψ1(p) ↔ A(p, ψ2, . . . , ψn) ∈ L′. The translation of this explicit relation has the form (p ∨ ϕ1) ↔B(p, ϕ2, . . . , ϕn) (here, it is only important that B is freed of ϕ1).

Substituting 0 for p in the latter formula, we arrive at the following explicit relation for ϕ1 in L:

ϕ1 ↔ B(0, ϕ2, . . . , ϕn) ∈ L,

which is a contradiction.

4.3. Example of a complete logic with an n-ary connective. Let ψ be an n-ary connective andϕ a constant. We define a translation such as

t(ψ(A1, . . . , An)) = t(A1) ∨ . . . ∨ t(An) ∨ ϕ.

The translation is surjective since ϕ is a translation of the formula ψ(0, . . . , 0). By our main result, thepreimage L′ = t−1(L) of every complete ϕ-logic L is a complete ψ-logic.

We claim that if L does not contain an explicit relation then neither does L′. Assume to the contrarythat L′ contains ψ(p̄) ↔ A(p̄) as an explicit relation. Here, A is a pure formula. A translation for theexplicit relation has the form

(p1 ∨ . . . ∨ pn ∨ ϕ)↔ A(p1, . . . , pn).

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(Recall that translation does not affect pure formulas.) Substituting 0 for all pi in the latter formula yieldsan explicit relation for ϕ in L: ϕ↔ A(0, . . . , 0) ∈ L, which is a contradiction.

Moreover, all arguments of the connective ψ in L′ are essential, and namely, L′ does not contain formulasof the form ψ(p1, p2, . . . , pn)↔ ψ(q, p2, . . . , pn), where q is a new variable. Indeed, assume the contrary. Atranslation for the formula in question is the following:

(p1 ∨ p2 ∨ . . . ∨ pn ∨ ϕ)↔ (q ∨ p2 ∨ . . . ∨ pn ∨ ϕ).

Substituting 0 for all pi and 1 for q yields an explicit relation ϕ↔ 1 ∈ L, and we so arrive at a contradictionagain.

Finally, we show that the connective ψ in L′ is essentially n-ary, that is, L′ contains no formulas of theform ψ(p1, p2, . . . , pn) ↔ A(p2, . . . , pn). Assume the contrary, letting ψ(p1, p2, . . . , pn) ↔ A(p2, . . . , pn) ∈L′.

A translation for this formula shows up as follows:

(p1 ∨ p2 ∨ . . . ∨ pn ∨ ϕ)↔ t(A(p2, . . . , pn)) ∈ L.

Substitute 1 for p1. We see that the right part is not affected by this substitution, yielding 1 ↔t(A(p2, . . . , pn)) ∈ L, that is, t(A(p2, . . . , pn)) ∈ L. Hence A(p2, . . . , pn) ∈ L′, and so ψ(p1, p2, . . . , pn) ↔1 ∈ L′, which is false.

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