A Translation of Intuitionistic Predicate Logic into ...

A Translation of Intuitionistic Predicate Logic into Basic Predicate Logic Mohammad Ardeshir Abstract. Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula ϕ in the language {∨, ∧, →, >, ⊥, ∀, ∃}, we associate two sequences of formulas < ϕ0 , ϕ1 , · · · > and < ϕ0 , ϕ1 , · · · > in the same languge. We prove that for every sequent ϕ⇒ψ, there are natural numbers m, n, such that IQC ` ϕ⇒ψ iff BQC ` ϕn ⇒ψ m . Some applications of this translation are mentioned.

Basic Propositional Logic, BPL, was invented by A. Visser in 1981 [Vi81], in an attempt to interpret implication as formal provability. To protect the system from the Liar Paradox, he weakened Modus Ponens. W. Ruitenburg [Ru91], [Ru93] reintroduced BPL with a philosophical motivation and extended BPL to the first order logic, BQC. In [Ru96], Ruitenburg corrected errors in [Ru91]. The axiomatization of BQC is presented in section 1(see [Ru96]). In section 2, we introduce the semantics of BQC. In section 3, we prove the main theorem in which Intuitionistic Predicate Calculus, IQC, can be interpreted in BQC. This theorem is very useful. It can be used to reprove some properties of intuitionistic logic. 1. The Language and Syntax of BQC. The language of BQC is L = {∨, ∧, →, >, ⊥, ∀, ∃}. The small Greek letters ϕ, ψ, · · · are used to denote the formulas in L. The notion of “formula” is defined as usual, except for the universally quantified formula which is in the form ∀x: ϕ(x).ψ(x), where x =< x1 , · · · , xn > is a finite sequence of variables(see [Ru91]). The intended meaning of ∀x: ϕ(x).ψ(x) is Ex ∧ ϕ(x)→ψ(x), where E is the extent operator in the sense of Scott and Heyting [Sc79]. If x does not contain a free variable in ϕ and ψ, the intended meaning of ∀x: ϕ(x).ψ(x) is exactly ϕ→ψ. So, in this case, we assume ∀x: ϕ(x).ψ(x) = ϕ→ψ. We use the sequent notation for the axiomatization of BQC, i.e., we consider a sequent to be in the form of ϕ⇒ψ, in which ϕ and ψ are formulas in L . By this convention, in the following list of the axioms and rules of BQC, the rule (13) and axioms (14), (15) and (16) have the following propositional counterparts: 1

∧ ψ⇒η (13p) ϕ ϕ⇒ψ→η (14p) (ϕ→ψ) ∧ (ψ→η)⇒ϕ→η (15p) (ϕ→ψ) ∧ (ϕ→η)⇒ϕ→ψ ∧ η (16p) (ϕ→ψ) ∧ (η→ψ)⇒ϕ ∨ η→ψ. Basic Propositional Calculus, BPC, consists of axioms and rules (1)-(7) and (13p)-(16p). In the axioms and rules of BQC, a double horizontal line means that we have two-direction rules.

Axioms and Rules of BQC (1) ϕ⇒ϕ

(2) ϕ⇒>

ψ⇒η (5) ϕ⇒ψ; ϕ⇒η

(6)

(3) ⊥⇒ϕ ϕ⇒ψ; ϕ⇒η ϕ⇒ψ ∧ η

(4) ϕ ∧ (ψ ∨ η)⇒(ϕ ∧ ψ) ∨ (ϕ ∧ η) (7)

ϕ⇒ψ; η⇒ψ ϕ ∨ η⇒ψ

(8)

ϕ(x)⇒ψ(x) , no variable in t is bound. ϕ(t)⇒ψ(t)

(9)

ϕ⇒ψ , the variables x are not free in ψ. ∃xϕ⇒ψ

(10) ϕ ∧ ∃xψ⇒∃x(ϕ ∧ ψ), the variables x are not free in ϕ. (11) >⇒x = x (12) x = y ∧ ϕ(x)⇒ϕ(y), the variable y is not bound. ϕ ∧ ψ⇒η , the variables x are not free in ϕ. (13) ϕ⇒∀x: ψ.η (14) (15) (16) (17) side and

∀x: ϕ.ψ ∧ ∀x: ψ.η⇒∀x: ϕ.η ∀x: ϕ.ψ ∧ ∀x: ϕ.η⇒∀x: ϕ.(ψ ∧ η) ∀x: ϕ.ψ ∧ ∀x: η.ψ⇒∀x: (ϕ ∨ η).ψ ∀x: ϕ(x).ψ(x)⇒∀y: ϕ(t).ψ(t), the variables y are not free on the left no variable in t is bound. 2

(18) ∀yx: ϕ(y, x).ψ(y)⇒∀y: (∃xϕ(y, x)).ψ(y), the variables x are not free in ψ. This ends the axiomatization of BQC. Let Γ be a set of sequents. We say a sequent ϕ⇒ψ is deducible from Γ, written Γ ` ϕ⇒ψ, when ϕ⇒ψ can be obtained, after finitely many applications of the BQC rules, from the BQC axiom schemas plus the sequents of Γ. 1.1 Remark. Let IQCS be BQC plus the axiom schema >→ϕ⇒ϕ. There is a bi-translation between the language of BQC and of the standard language of Intuitionistic Predicate logic, IQC(see, e.g., [TD88], Vol. I), such that under this bi-translation IQC is equivalent to IQCS. To pass from the languge of BQC to the language of IQC, for every fomula ϕ in the language of BQC, let ϕi denotes the formula obtained from ϕ by replacing every occurrence of a subformula ∀x: η(x).δ(x) by ∀x(η(x)→δ(x)). It is easy to verify that for every sequent ϕ⇒ψ, if IQCS ` ϕ⇒ψ then IQC ` ϕi →ψ i . For the other way around, for every formula ϕ in the standard language of IQC, let ϕb denotes the formula obtained from ϕ by replacing every occurrence of a subformula ∀xη(x) by ∀x : >.η(x). Then for every formula ϕ, if IQC ` ϕ then IQCS ` >⇒ϕb . We use the notation ϕ for >⇒ϕ when confusion is unlikely. The following two propositions are proved by induction on the length of the derivation(see [Ru96]): 1.2 Proposition. Let {ϕ1 ⇒ψ1 , · · · , ϕn ⇒ψn } ` ϕ⇒ψ in BQC. Then BQC ` ∀x1: ϕ1 .ψ1 ∧ · · · ∧ ∀xn: ϕn .ψn ⇒∀x: ϕ.ψ. 1.3 Proposition. Let Γ be a set of sequents, ϕ a formula and ψ⇒η a sequent. Then Γ ∪ {ϕ} ` ψ⇒η iff Γ ` ϕ ∧ ψ⇒η. 2. The semantics of BQC. The semantics of BQC is essentially based on Kripke models. A Kripke model for BQC is like a Kripke model for IQC (see [TD88], pages 75-92), except for BQC, the accessibility relation “≺ ” on the set K of nodes is not necessairily reflexive. The forcing relation, k−, is defined as for IQC, except for the implication and the universal quantifier which are defined as follows:

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αk−ϕ→ψ iff for all β  α, if βk−ϕ then βk−ψ, and αk−∀x : ϕ(x).ψ(x) iff for all β  α and all b ∈ D(β), if βk−ϕ(b) then βk−ψ(b). Also we define αk−ϕ(x) iff for all β  α and all d ∈ D(β), βk−ϕ(d), and αk−ϕ⇒ψ iff for all β  α, if βk−ϕ then βk−ψ. BQC is sound and strongly complete with respect to the class of Kripke models. The proof of soundness is routine and as for the completeness, the usual Henkin method for IQC works for BQC as well(for details, see [Ru96]). 2.1 Theorem. BQC is strongly complete with respect to the class of Kripke models. The Smorynski operation on Kripke models(see [Sm72], pages 127-130) gives the following result which we need in this paper: 2.2 Corollary. For every formula ϕ, if BQC ` >→ϕ then BQC ` ϕ. 3. A Translation of IQC to BQC. In this section we will prove a useful theorem about a “translation” of intuitionistic predicate calculus into basic predicate calculus. To do this, we need a suitable cut-free axiomatization of IQC with unisuccedent. We use a variant of Kleene’s G3 [Kl52] axiomatization of Intuitionistic Logic (see [TD88], Vol. 2, page 551). In [TD88] this system is called K. To distinguish it from the modal system K (it is worth to know that there is a translation from the propositional part of Basic Logic, BPC, into modal system K4. see [AR95], [Vi81]), we refer this system KG3. In the following system, Γ is a finite set of formulas. (If we interpret Γ as a multiset, we obtain an equivalent system.)

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KG3 Axioms : Γ⇒ϕ, for ϕ ∈ Γ. Rules : Γ, ϕ→ψ, ψ⇒η (→⇒) Γ, ϕ→ψ⇒ϕ Γ, ϕ→ψ⇒η

ϕ⇒ψ (⇒→) ΓΓ, ⇒ϕ→ψ

∧ ψ, ϕ, ψ⇒η (∧ ⇒) Γ, ϕ Γ, ϕ ∧ ψ⇒η

⇒ϕ Γ⇒ψ (⇒ ∧) Γ Γ ⇒ϕ∧ψ

ϕ⇒η Γ, ϕ ∨ ψ, ψ⇒η (∨⇒) Γ, ϕ ∨ ψ,Γ, ϕ∨ψ ⇒η

Γ⇒ϕ Γ⇒ψ (⇒ ∨) Γ⇒ϕ ∨ ψ Γ⇒ϕ ∨ ψ Γ⇒ϕ(t) (⇒∃) Γ⇒∃xϕ(x) Γ ⇒ ϕ(y) (⇒∀) Γ⇒∀xϕ(x)

Γ, ∃xϕ(x), ϕ(y)⇒η Γ, ∃xϕ(x)⇒η Γ, ∀xϕ(x), ϕ(t)⇒η (∀⇒) Γ, ∀xϕ(x)⇒η

(∃⇒)

⇒⊥ (⊥) ΓΓ⇒η

In rule (∃⇒), the variable y is not free in Γ ∪ {η} and in rule (⇒∀), the variable y is not free in Γ. 3.1 Remark. For every sequent Γ⇒ϕ in the language of KG3, we have V if KG3 ` Γ⇒ϕ then IQCS ` Γb ⇒ϕb , where Γb = {ϕb : ϕ ∈ Γ}, and ϕb is defined in Remark 1.1. On the other hand, for every sequent ϕ⇒ψ in the language of BQC, if IQCS ` ϕ⇒ψ then KG3 ` {ϕi }⇒ψ i . To aviod the complicated notations, in the rest of the paper, we tacitly use the immediate bi-translation between the languages mentioned in Remarks 1.1 and 3.1. 3.2 Definition. We define operations ( )1 and ( )1 on the set of all formulas of BQC simultaneously by the following clauses : (p)1 = >→p, for atomic p, (ϕ ∧ ψ)1 = ϕ1 ∧ ψ 1 , (ϕ ∨ ψ)1 = >→ϕ1 ∨ ψ 1 ,

(p)1 = p, for atomic p, (ϕ ∧ ψ)1 = ϕ1 ∧ ψ1 , (ϕ ∨ ψ)1 = ϕ1 ∨ ψ1 , 5

(ϕ→ψ)1 = ϕ1 →ψ 1 , (∃xϕ)1 = >→∃xϕ1 , (∀x: ϕ.ψ)1 = ∀x: ϕ1 .ψ 1 , (>)1 = >, (⊥)1 = >→⊥,

(ϕ→ψ)1 = ϕ1 →ψ1 , (∃xϕ)1 = ∃xϕ1 , (∀x: ϕ.ψ)1 = ∀x: ϕ1 .ψ1 (>)1 = >, (⊥)1 = ⊥.

For every formula ϕ, we define ϕ0 = ϕ and ϕ0 = ϕ and for all n ∈ ω, (ϕ)n+1 = (ϕn )1 and (ϕ)n+1 = (ϕn )1 . Define recursively, for every formula ϕ, >0 ϕ = ϕ, >n+1 ϕ = >→>n ϕ. 3.3 Lemma. For every formula ϕ and every natural number n, (>n ϕ)1 = > ϕ1 . n

Proof. By induction on n. For n = 0, (>0 ϕ)1 = ϕ1 = >0 ϕ1 . Induction step: (>n+1 ϕ)1 = (>→>n ϕ)1 = (>)1 →(>n ϕ)1 = >→(>n ϕ)1 = >→>n ϕ1 = >n+1 ϕ1 . 2 Similarly we can show (>n ϕ)1 = >n ϕ1 . 3.4 Lemma. For every natural number n, we have : (p)n = >n p, for atomic p, (ϕ ∧ ψ)n = ϕn ∧ ψ n , (ϕ ∨ ψ)n = >n (ϕn ∨ ψ n ), (ϕ→ψ)n = ϕn →ψ n , (∃xϕ)n = >n ∃xϕn , (∀x: ϕ.ψ)n = ∀x: ϕn .ψ n , (>)n = >, (⊥)n = >n ⊥, Proof.

(p)n = p, for atomic p, (ϕ ∧ ψ)n = ϕn ∧ ψn , (ϕ ∨ ψ)n = ϕn ∨ ψn , (ϕ→ψ)n = ϕn →ψn , (∃xϕ)n = ∃xϕn , (∀x: ϕ.ψ)n = ∀x: ϕn .ψn , (>)n = >, (⊥)n = ⊥.

The proof is by induction on n for every case. 2

3.5 Lemma. For every formula ϕ and for every natural number n, (i) BQC ` ϕn+1 ⇒ϕn , (ii) BQC ` ϕn ⇒ϕn+1 . Proof. It suffices to prove these for n = 0. The proof is by induction on the complexity of ϕ. We use simultaneous induction in (i) and (ii). Consider (ii). 6

(a) When ϕ is atomic it is trivial by definition. (b) Suppose ϕ = ψ∧η. Then (ψ∧η)1 = ψ 1 ∧η 1 . By induction hypothesis, BQC` ψ⇒ψ 1 and BQC` η⇒η 1 . Then BQC` ψ ∧ η⇒ψ 1 ∧ η 1 . (c) Suppose ϕ = ψ ∨ η. Then (ψ ∨ η)1 = >→ψ 1 ∨ η 1 . By induction hypothesis, BQC` ψ⇒ψ 1 and BQC` η⇒η 1 . Then BQC` ψ ∨ η⇒ψ 1 ∨ η 1 . So BQC` ψ ∨ η⇒>→ψ 1 ∨ η 1 . (d) Suppose ϕ = ψ→η. Then (ψ→η)1 = ψ1 →η 1 . By induction hypothesis, BQC` ψ1 ⇒ψ, η⇒η 1 . Then BQC` ψ→η⇒(ψ1 →ψ) ∧ (ψ→η) ∧ (η→η 1 ). So BQC` ψ→η⇒ψ1 →η 1 . (e) Suppose ϕ = ∃xψ. Then (∃xψ)1 = >1 ∃xψ 1 . By induction hypothesis, BQC` ψ⇒ψ 1 . Then BQC` ∃xψ⇒∃xψ 1 . So BQC` ∃xψ⇒>1 ∃xψ 1 . (f) Suppose ϕ = ∀x: ψ.η. Then (∀x: ψ.η)1 = ∀x: ψ1 .η 1 . By induction hypothesis, BQC` ψ1 ⇒ψ and BQC` η⇒η 1 . Then BQC` ∀x : ψ1 .ψ and BQC` ∀x: η.η 1 . So BQC` ∀x: ψ.η⇒∀x: ψ1 .ψ ∧ ∀x: ψ.η ∧ ∀x: η.η 1 . Then BQC` ∀x: ψ.η⇒∀x: ψ1 .η 1 . Similar arguments work for (i). 2 3.6 Corollary. For every formula ϕ and all natural numbers n, m, (i) BQC ` ϕm ⇒ϕn , when m ≤ n. (ii) BQC ` ϕn ⇒ϕm , when m ≤ n. (iii) BQC ` ϕn ⇒ϕm . 2 3.7 Corollary. Let ϕ⇒ψ be a sequent. Then for all natural numbers n1 , n2 , m1 , m2 such that n1 ≤ n2 , m1 ≤ m2 ; in BQC, ϕn1 ⇒ψ m1 ` ϕn2 ⇒ψ m2 . 2 3.8 Lemma. For every formula ϕ, BQC ` >→ϕ⇒ϕ1 . Proof. By induction on the complexity of ϕ. (a) ϕ = p is an atomic formula. Then (p)1 = >→p and BQC ` >→p⇒>→p. (b) ϕ = ψ ∧ η. By induction hypothesis, BQC` >→ψ⇒ψ 1 and BQC` >→η⇒η 1 . Then BQC` >→ψ ∧ η⇒ψ 1 ∧ η 1 , where (ψ ∧ η)1 = ψ 1 ∧ η 1 . 7

(c) ϕ = ψ ∨ η. By Corollary 3.6, BQC` ψ⇒ψ 1 and BQC` η⇒η 1 . Then BQC` >→ψ ∨ η⇒>1 (ψ 1 ∨ η 1 ). So BQC` >→ψ ∨ η⇒(ψ ∨ η)1 . (d) ϕ = ψ→η. Since (ψ→η)1 = ψ1 →η 1 , we must show BQC` >→(ψ→η) ⇒ψ1 →η 1 . By induction hypothesis BQC` >→η⇒η 1 . By Corollary 3.6, BQC` ψ1 ⇒ψ, so BQC` ψ1 ⇒>→ψ. Thus BQC` ⇒ψ1 →>1 ψ. Moreover, by Proposition 1.1, BQC` >→(ψ→η)⇒(>→ψ)→(>→η). Then BQC` >→(ψ→η)⇒(ψ1 →(>→ψ)) ∧ ((>→ψ)→(>→η)) ∧ ((>→η)→η 1 ). So BQC` >→(ψ→η)⇒ψ1 →η 1 . (e) ϕ = ∃xψ. By definition (∃xψ)1 = >1 ∃xψ 1 . By Lemma 3.5, BQC` ψ⇒ψ 1 . Then BQC` ∃xψ⇒∃xψ 1 . Thus BQC` >→∃xψ⇒>1 ∃xψ 1 . So BQC ` >→∃xψ⇒(∃xψ)1 . (f) ϕ = ∀x: ψ.η. By definition, (∀x: ψ.η)1 = ∀x: ψ1 .η 1 . We must show that BQC` >→∀x : ψ.η⇒∀x : ψ1 .η 1 . By Propositions 1.2 and 1.3, we get BQC` >→∀x : ψ.η⇒∀x : (>→ψ).(>→η). By Corollary 3.6 and using rule (13), BQC` ∀x: ψ1 .(>→ψ) and BQC` ∀x: (>→η).η 1 . Then BQC` >→∀x: ψ.η⇒∀x: ψ1 .η 1 . 2 3.9 Corollary. For every formula ϕ and every natural number n, BQC ` >→ϕn ⇒ϕn+1 . Proof. By Lemma 3.8, BQC` >→ϕn ⇒(ϕn )1 . So BQC` >→ϕn ⇒ϕn+1 . 2 3.10 Lemma. For every formula ϕ and every natural number n, IQC ` ϕ⇒ϕn and IQC ` ϕn ⇒ϕ. Proof. This trivially follows from the axiom schema >→ϕ⇒ϕ of IQC. 2 Before we can state and prove the embedding Theorem, we need some V notations and conventions. For Γ, a finite set of formulas, Γ = γ1 ∧ · · · ∧ γn , V where Γ = {γ1 , · · · , γn }. When Γ = ∅, take Γ = >. Every derivation D of a sequent Γ⇒ϕ in KG3 can be arranged in the form of a tree T . To every branch B of this tree we associate a natural number mB that is the number of applications of the rules (→⇒) and (∀⇒) 8

on the branch B. We say m is the index of the tree T iff m is the maximum of {mB : B ∈ T }. 3.11 Theorem (Translation of Intuitionistic Predicate Logic into Basic Predicate Logic). Let Γ⇒ϕ be a sequent. Then KG3 ` Γ⇒ϕ iff V there are natural numbers m, n such that BQC ` ( Γ)n ⇒ϕm . Moreover, in the “only if ” direction, m is at most the index of the tree of the derivation KG3 ` Γ⇒ϕ, and n ≤ m. Proof. The “if” direction is immediate by Remarks 1.1, 3.1 and Lemma 3.10. For the converse, suppose KG3 ` Γ⇒ϕ. We complete the proof by induction on the height of the derivation of Γ⇒ϕ in KG3. Case (1) : Γ⇒ϕ is an axiom, i.e., ϕ ∈ Γ. Then take n = m = 0. Case (2) : Γ⇒ϕ is derived by (→⇒), i.e., Γ = Γ0 ∪ {ψ→η} and 0

0

Γ , ψ→η⇒ψ Γ , ψ→η, η⇒ϕ . 0 Γ , ψ→η⇒ϕ 0

0

By induction BQC` ( Γ )n1 ∧ (ψ n1 →ηn1 )⇒ψ m1 and BQC` ( Γ )n2 ∧ V 0 (ψ n2 →ηn2 ) ∧ ηn2 ⇒ϕm2 . Since n1 ≤ m1 , BQC` ( Γ )m1 ∧ (ψ m1 →ηm1 )⇒ψ m1 . V 0 V 0 Let n = max(m1 , n2 ). BQC` ( Γ )n ∧ (ψ n →ηn )⇒ψ n and BQC` ( Γ )n ∧ V 0 (ψ n →ηn ) ∧ ηn ⇒ϕm2 . Then BQC` ( Γ )n ∧ (ψ n →ηn )⇒>→ηn and BQC` V 0 V 0 ( Γ )n ∧ (ψ n →ηn ) ∧ (>→ηn )⇒>→ϕm2 . Thus BQC` ( Γ )n ∧ (ψ n →ηn )⇒> V →ϕm2 . By Corollary 3.9, BQC` >→ϕm2 ⇒ϕm2 +1 . So BQC` ( Γ)n ⇒ϕm , where m = m2 + 1. To show the conditions on n, m, consider two cases : (2-i) m1 ≤ m2 . Then clearly m = m2 + 1 is at most the least number of applications of rules (→⇒) and (∀⇒) in the derivation of Γ⇒ϕ in KG3. Now, since n = max(m1 , n2 ) and n2 ≤ m2 , we get n ≤ m. (2-ii) m2 < m1 . Then we can replace m by m1 . V

V

Γ, ψ⇒η . Case (3) : Γ⇒ϕ is derived by rule (⇒→), i.e., ϕ = ψ→η and Γ⇒ψ→η By induction, BQC` ( Γ)n ∧ ψn ⇒η m . By Corollary 3.7, BQC` ( Γ)m ∧ V V ψm ⇒η m . Then BQC` ( Γ)m ⇒ψm →η m . So BQC` ( Γ)m ⇒(ψ →η)m . V

V

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Case (4) : Γ⇒ϕ is derived by the rule (∧⇒), i.e., Γ = Γ0 ∪ {ψ ∧ η} and Γ , ψ ∧ η, ψ, η⇒ϕ . 0 Γ , ψ ∧ η⇒ϕ 0

0

By induction hypothesis, there are n, m such that BQC` ( Γ )n ∧ (ψ ∧ V η)n ∧ ψn ∧ ηn ⇒ϕm . Then BQC` ( Γ)n ⇒ϕm . V

Case (5) : Γ⇒ϕ is derived by the rule (⇒∧), i.e., ϕ = ψ ∧ η and Γ⇒ψ Γ⇒η . Γ⇒ψ ∧ η By induction BQC` ( Γ)n1 ⇒ψ m1 and BQC` ( Γ)n2 ⇒η m2 . Let n = V max(n1 , n2 ) and m = max(m1 , m2 ). Then by Corollary 3.7, BQC` ( Γ)n ⇒ψ m V V V and BQC` ( Γ)n ⇒η m . So BQC` ( Γ)n ⇒ψ m ∧η m . Thus BQC` ( Γ)n ⇒(ψ∧ η)m . V

V

0

Case (6) : Γ⇒ϕ is derived by rule (∨⇒), i.e., Γ = Γ ∪ {ψ ∨ η} and 0 0 Γ , ψ ∨ η, ψ⇒ϕ Γ , ψ ∨ η, η⇒ϕ . 0 Γ , ψ ∨ η⇒ϕ 0

0

By induction BQC` ( Γ )n1 ∧ (ψ ∨ η)n1 ∧ ψn1 ⇒ϕm1 and BQC` ( Γ )n2 ∧ (ψ ∨ η)n2 ∧ ηn2 ⇒ϕm2 . Let n = max(n1 , n2 ) and m = max(m1 , m2 ). Then V 0 V 0 by Corollary 3.7, BQC` ( Γ )n ∧ (ψ ∨ η)n ∧ ψn ⇒ϕm and BQC` ( Γ )n ∧ V 0 (ψ ∨ η)n ∧ ηn ⇒ϕm . Then BQC` ( Γ )n ∧ (ψ ∨ η)n ∧ (ψn ∨ ηn )⇒ϕm . So V BQC` ( Γ)n ⇒ϕm . V

V

Γ⇒ψ or Case (7) : Γ⇒ϕ is derived by rule (⇒∨), i.e., ϕ = ψ ∨ η and Γ⇒ψ ∨η Γ⇒η . We consider only the first case. Γ⇒ψ ∨ η By induction BQC` ( Γ)n ⇒ψ m . Then BQC` ( Γ)n ⇒ψ m ∨ η m and so V V BQC` ( Γ)n ⇒>m (ψ m ∨ η m ). Then BQC` ( Γ)n ⇒(ψ ∨ η)m . V

V

0

Case (8) : Γ⇒ϕ is derived by (∃⇒), i.e., Γ = Γ ∪ {∃xψ} and Γ , ∃xψ, ψ(y)⇒ϕ , where y is not free in Γ ∪ {ϕ}. 0 Γ , ∃xψ⇒ϕ V 0 By induction hypothesis, BQC` ( Γ )n ∧ (∃xψ)n ∧ (ψ(y))n ⇒ϕm . Then V 0 V BQC` ( Γ )n ∧ (∃xψ)n ⇒ϕm . So BQC ` ( Γ)n ⇒ϕm . 0

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Γ⇒ψ(t) Case (9) : Γ⇒ϕ is derived by (⇒∃), i.e., ϕ = ∃xψ and Γ⇒∃xψ . V V By induction hypothesis, BQC` ( Γ)n ⇒(ψ(t))m . Then BQC` ( Γ)n ⇒ V V ∃xψ m and thus BQC` ( Γ)n ⇒>m ∃xψ m . This means that BQC` ( Γ)n ⇒(∃xψ)m . 0

Case (10) : Γ⇒ϕ is derived by (∀⇒), i.e., Γ = Γ ∪ {∀xψ} and Γ , ∀xψ, ψ(t)⇒ϕ . 0 Γ , ∀xψ⇒ϕ V 0 By induction hypothesis, BQC` ( Γ )n ∧(∀x : >.ψ)n ∧(ψ(t))n ⇒ϕl . Then V 0 V 0 BQC` ( Γ )n ∧ (∀x : >.ψ)n ∧ (>→ψ(t))n ⇒>→ϕl . So BQC` ( Γ )n ∧ (∀x: V 0 >.ψ)n ∧ (>→ψ(t))n ⇒>→ϕl . Then By Corollary 3.9, BQC` ( Γ )n ∧ (∀x: V 0 >.ψ)n ∧ (>→ψ(t))n ⇒ϕl+1 . So BQC` ( Γ )n ∧ (∀x : >.ψ)n ⇒ϕl+1 . Then V BQC` ( Γ)n ⇒ϕm , where m = l + 1. 0

Γ⇒ψ(y) Case (11) : Γ⇒ϕ is derived by (⇒∀), i.e., ϕ = ∀xψ and Γ⇒∀xψ , where y is not free in Γ. V V By induction hypothesis, BQC` ( Γ)n ⇒(ψ(y))m . Then BQC` ( Γ)n ⇒∀x: V >.ψ m . So BQC` ( Γ)n ⇒ϕm . Case (12) : Γ⇒ϕ is derived by rule (⊥). Then by induction hypothesis, V V BQC ` ( Γ)n ⇒⊥m . By Lemma 4.4, BQC` ( Γ)n ⇒>m ⊥. Since BQC` V >m ⊥⇒>m ϕ, we have BQC` ( Γ)n ⇒ϕm . 2 We can state the Translation Theorem in the following form. 3.12 Theorem (Translation of IQC into BQC). Let ϕ⇒ψ be a sequent. Then IQC ` ϕ⇒ψ iff there are natural numbers m, n such that BQC ` ϕn ⇒ψ m . Proof. Use Remarks 1.1, 3.1 and Theorem 3.11. 2 We can state Theorem 3.12 for the propositional parts of intuitionistic logic and basic logic. 3.13 Theorem (Translation of IPC into BPC). Let ϕ⇒ψ be a sequent. Then IPC ` ϕ⇒ψ iff there are natural numbers m, n such that BPC ` ϕn ⇒ψ m . 2

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By using Corollary 3.6 and pulling the existential quantifier outside the biconditional, we have: 3.14 Theorem. Let ϕ⇒ψ be a sequent. Then there is a natural number n such that IQC ` ϕ⇒ψ iff BQC ` ϕn ⇒ψ n . 2 The Translation Theorem is very useful. We can reprove many properties of IQC via the same properties of BQC. It is proven that BPC has interpolation property [Ar95] and disjunction property [Ru91], [AR95], and BQC has disjunction and explicit definability properties [Ru96] and the Craig interpolation property [Ar97]. By Translation Theorem, these properties can be reproven for IPC and IQC. Acknowledgements. The Translation Theorem for the old axiomatization of BQC has been proven in my Ph.D. thesis under supervision of Professor W. Ruitenburg. I would like to express my gratitude to him for his guidance. Thanks to M. Moniri, J. Simms and H. Vahid for helpful suggestions and discussions. REFERENCES [Ar95] M. Ardeshir, Aspects of Basic Logic, Ph.D. thesis, Marquette University, Milwaukee, 1995. [Ar97] M. Ardeshir, Robinson’s Consistency Theorem in Basic Predicate Logic, Technical Report, Institute for Studies in Theoretical Physics and Mathematics, IPM-97-186, Tehran, Iran. [AR95] M. Ardeshir, W. Ruitenburg, Basic Propositional Calculus, I, 1995, to appear. [Kl52] S. Kleene, Introduction to Metamathematics, North-Holland, 1952. [Ru91] W. Ruitenburg, Constructive Logic and the Paradoxes, Modern Logic 1, No. 4, 1991, pp. 271-301. [Ru93] W. Ruitenburg, Basic Logic and Fregean Set Theory, in: H. Barendregt, M. Bezem, J. W. Klop, eds., Dirk Van Dalen Festschrift, Quaes12

tiones Infinitae, Vol. 5, Department of Philosophy, Utrecht University, 1993, pp. 121-142. [Ru96] W. Ruitenburg, Basic Predicate Calculus, I, 1996, to appear. [Sc79] D. Scott, Identity and existence in intuitionistic logic, in: M. P. Fourman, C. J. mulvey and D. S. Scott, eds., Applications of Sheaves, Lecture notes in Mathematics, Springer-verlag, Vol. 753, 1979, pp 660-696. [Sm73] C. Smorynski, Investigations of intuitionistic formal systems by means of Kripke models, Ph.D. thesis, University of Chicago, 1973. [TD88] A. S. Troelstra, D. van Dalen, Constructivism in Mathematics, An Introduction, Vol. II, North-Holland, 1988. [Vi81] A. Visser, A Propositional logic with explicit fixed points, Studia Logica 40, 1981, 155-175. Department of Mathematics, Sharif University of Technology, P.O. Box 11365-1194, Tehran, Iran. Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

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