3 Hilbert Calculi

intuitionistic logic

Marcelo E. Coniglio¹ Cristina Sernadas²

1GTAL, Departamento de Filosofia, Universidade Estadual de Campinas, Brazil

2CLC/CMA, Departamento de Matem´atica, IST, Portugal

Abstract

Two Hilbert calculi for higher-order logic (or theory of types) are intro- duced. The first is defined in a language that uses just exponential types of power type, and it is obtained by adapting the sequent calculus for local set theory introduced by Bell in [3]. The second one, originally introduced in [5], is defined in a language with arbitrary functional types. Using usual topos semantics we show that both systems are sound and complete.

Introduction

Higher-order logic (a.k.a. hol, or theory of types) is defined in a very rich language which permits to express most of mathematics reasoning. Theory of types was introduced by Russell in 1908 as a solution to paradoxes in set theory (see [19]) and was reformulated by Church in [4]. By G¨odel’s second theorem, it is immediate to show that there is no (reasonable) proof system complete for the standard (set) semantics of hol. On the other hand, Henkin proves in [8] that it is possible to give an axiomatization of hol sound and complete w.r.t. a wider class of models, calledgeneral models, in which types of the form (θ→θ⁰) are interpreted as subsets of the set of maps from (the carrier of)θ to (the carrier of)θ⁰. From the works of Lawvere (see for example [14, 15]) it was proved that the usual axiomatizations of hol are sound and complete w.r.t. an extremely elegant topos semantics (see, for instance, [12, 3, 16, 9, 17]).

In this research report we introduce two very simple Hilbert-style axiom- atizations of hol, which are sound and complete w.r.t. topos semantics. The first one is obtained by adapting the sequent calculus-style axiomatization of hol calledlocal set theory, introduced by Bell in [3], defined in a language with power types but without arbitrary functional types. The second one, originally introduced in [5], is a simple extension of the former to a language with arbi- trary functional types. The notions of signature with schema variables and of Hilbert calculus with provisos, as well as the notion of local and global entail- ment used here, are taken and adapted from the forthcoming paper [5], and this technical report should be seen as a companion to that paper.

1 Higher-Order Languages

In this section we recall the notion of signature introduced in [5]. This is a simplified version, enough for our purposes.

Definition 1.1 Given a setS with distinguished element1, we denote by Θ(S) the set inductively defined as follows: (i)S ⊆Θ(S); (ii) ifθ₁, . . . , θ_n∈Θ(S) for integern≥2 then (θ₁× · · · ×θ_n)∈Θ(S); (iii) ifθ∈Θ(S) thenP(θ)∈Θ(S).4 As usual, we write θⁿ for the n-th power of θ (the product of θ with itself n times) and by conventionθ⁰ is1 and θ¹ isθ.

Definition 1.2 Asignatureis a tuple Σ =hS,1,Ξ, X, Fiwhere:

• S is a set with distinguished element 1;

• Ξ ={Ξ_θ}_θ∈Θ(S) where each Ξ_θ is a denumerable set Ξ_θ={ξ_k^θ |k∈N};

• X={X_θ}_θ∈Θ(S) where eachX_θ is a denumerable set X_θ ={x^θ_k |k∈N};

• F ={F_θθ⁰}_θ,θ⁰_∈Θ(S) where eachF_θθ⁰ is a set. 4 The elements ofSare known assortsorground types. The elements of Θ(S) are known as types over S. Ground type 1 is called the unit sort. The type P(1), denoted by Ω, is called the truth value type. The elements of each Ξ_θ and X_θ are called schema variables and variables of type θ, respectively. The elements of eachF_θθ⁰ are called function symbols of typeθθ⁰.

Definition 1.3 The family ST(Σ) ={ST(Σ)_θ}_θ∈Θ(S) is inductively defined as follows:

• Ξ_θ∪X_θ ⊆ST(Σ)_θ;

• ifx∈X_θ⁰,ξ⁰ ∈Ξ_θ⁰ and ξ∈Ξ_θ then^x_ξ0ξ ∈ST(Σ)_θ;

• iff ∈F_θθ⁰ andt∈ST(Σ)_θ then (f t)∈ST(Σ)_θ⁰;

• hi ∈ST(Σ)₁;

• ift_i∈ST(Σ)_θifor 1≤i≤nwithn≥2 thenht₁, . . . , t_ni ∈ST(Σ)_{(θ1×···×θn)};

• ift∈ST(Σ)_{(θ1×···×θn)},n≥2 and 1≤i≤nthen (t)_i ∈ST(Σ)_θi;

• ift₁, t₂∈ST(Σ)_θ then (=_θht₁, t₂i)∈ST(Σ)_Ω;

• ift₁ ∈ST(Σ)_θ and t₂ ∈ST(Σ)_P(θ) then (∈_θht₁, t₂i)∈ST(Σ)_Ω;

• ifx∈X_θ and t∈ST(Σ)_Ω then (set_θx t)∈ST(Σ)_P(θ). 4

The elements of each ST(Σ)_θ are called schema terms of type θ. Schema terms of type Ω are also known as schema formulae. Schema terms without occurrences of schema variables are called terms: T(Σ)_θ denotes the set of terms of typeθ. Note that schema terms with occurrences of^x_ξ0ξ are not terms.

Schema formulae without schema variables are calledformulae. We writeSL(Σ) and L(Σ) for ST(Σ)_Ω and ST(Σ)_Ω, respectively. As we shall see in Section 3, schema variables are used in Hilbert calculi to express arbitrary terms within rules. Thus, with respect to semantics we are just interested in terms, and schema terms will be useful just as a tool for Hilbert calculi.

Every occurrence of a variablexin a schema term (set_θx δ) or in ^x_ξ0ξ, inside a schema termt, is said to beboundint. Any other occurrence ofxin a schema termt is said to befree int. In particular, the unique bound occurrences of a variablexin a termtare in the scope of a term (set_θx ϕ) occurring int. Ift, t⁰ are schema terms and x is a variable of the same type that t then t^x_t0 denotes the schema term obtained fromt by substituting every free occurrence of x in tby t⁰. We say that a termt⁰ ∈ST(Σ)_θ is free for a variable x∈X_θ in a term t if, for every variable y occurring free in t⁰, every occurrence of y in t^x_t0 not already int is free.

Frequently we will omit the types attached to the symbols. As usual, we will adopt infix notation, writing for example (t₁ =t₂) instead of (=_θ ht₁, t₂i).

We also write {x : γ} for (set_θx γ), and t₁ ∈_θ t₂ (or even t₁ ∈ t₂) instead of (∈_θht₁, t₂i).

Other logical operations can be introduced through abbreviations (cf. [3]):

• Equivalence: (δ₁⇔δ₂) for (δ₁ =_Ωδ₂).

• True: tfor (hi=₁hi).

• Conjunction: (δ₁∧δ₂) for (hδ₁, δ₂i=_(Ω×Ω)ht,ti).

• Implication: (δ₁⇒δ₂) for ((δ₁∧δ₂)⇔δ₁).

• Universal quantification: (∀_θx^θ_kδ) for ({x^θ_k : δ}=_P(θ) {x^θ_k : t}).

• False: f for (∀_Ωx^Ω₁ x^Ω₁).

• Negation: (¬δ) for (δ⇒f).

• Disjunction: (δ₁∨δ₂) for

(∀_Ωx^Ω_i (((δ₁⇒x^Ω_i )∧(δ₂⇒x^Ω_i ))⇒x^Ω_i )),

wherex^Ω_i is the first variable of type Ω not occurring free inhδ₁, δ₂i.

• Existential quantification: (∃_θx^θ_kδ) for

(∀_Ωx^Ω_i (∀_θx^θ_k((δ⇒x^Ω_i )⇒x^Ω_i ))),

wherex^Ω_i is the first variable of type Ω not occurring free inδ.

2 Topos Semantics

Higher-order languages can be interpreted in any topos (see, for instance, [12, 3, 16, 9, 17]). In order to interpret Σ-terms in a given topos we need to introduce the notion of context.

By a Σ-contextwe mean a finite sequence~x=x₁. . . x_nof distinct variables.

We denote by [] the empty context. Given a context ~x = x₁. . . x_n where the variablesx₁. . . x_nare of typeθ₁, . . . , θ_n, respectively, we writeθ_~xforθ₁×· · ·×θ_n and say thatθ_~x is the type of the context~x. By definitionθ_[] is 1.

The setST(Σ, ~x)_θ is composed by all Σ-schema termst of typeθsuch that every variable occurring free int appears in the context~x. The sets ST(Σ, ~x), SL(Σ, ~x),T(Σ, ~x) andL(Σ, ~x) are defined analogously.

Given a finite set Γ of terms we may refer to its canonical context formed exclusively by the variables occurring free in some term tof Γ (this canonical context is unique once we fix a total ordering of the variables).

Definition 2.1 Let Σ be a signature. A Σ-structureis a pairM =hE,·_Misuch thatE is a (non-degenerate) topos and·_M is a map such that:

• θ_M is an object of E for all θ ∈ Θ(S) such that 1_M is terminal 1, (θ₁×

· · · ×θ_n)_M is θ_1M × · · · ×θ_nM and P(θ)_M is the exponential Ω^θM (thus, Ω_M is identified with the subobject classifier Ω);

• iff ∈F_θθ⁰ thenf_M :θ_M →θ⁰_M in E. 4 Given a Σ-structureM and a context~xof typeθ_~x letθ_~xM beθ_1M× · · · ×θ_nM. Definition 2.2 If t∈ T(Σ, ~x)_θ and M is a Σ-structure then we define induc- tively a morphism [[t]]^M_~x :θ_~xM →θ_M as follows:

• [[x_i]]_~x^M is the canonical projection over θ_iM;

• [[hi]]^M_~x is the unique map fromθ_~xM to 1;

• [[(f t)]]_~x^M is the composite f_M ◦[[t]]^M_~x ;

• [[ht₁, . . . , t_ni]]^M_~x is ([[t₁]]^M_~x , . . . ,[[t_n]]_~x^M);

• [[(t)_i]]^M_~x isp_i◦[[t]]^M_~x , wheretis of type (θ₁⁰×· · ·×θ_m⁰ ) andp_iis the canonical projection overθ⁰_iM;

• [[(t₁ =_θ t₂)]]_~x^M is the characteristic map of m : dom(m) ,→ θ_~xM, the monomorphism obtained from the equalizer of{[[t_i]]^M_~x :θ_~xM →θ_M}_i=1,2;

• [[(t₁ ∈_θ t₂)]]^M_~x is eval◦([[t₂]]^M_~x ,[[t₁]]^M_~x ), where eval: Ω^θM ×θ_M → Ω is the evaluation map associated to the exponential Ω^θM.

• [[{x : ϕ}]]^M_~x is the exponential transpose of [[ϕ^x_y]]^M_~xy:θ_~xM×θ_M →Ω with respect to θ_M, wherey is the first variable free for x in ϕ not occurring

in~x. 4

Definition 2.3 An interpretation system is a pair S = hΣ,Mi where Σ is a

signature andM is a class of Σ-structures. 4

Using our abbreviations, it can be proved that quantifiers and connectives are interpreted in any topos in the usual way (see for instance [16, 17]). As mentioned in Section 1, schema variables are just used for express rules in Hilbert calculi and we are not interested in interpret them.

In order to define semantic entailment we need to introduce the following notation: given an object A in a topos E, then true_A : A → Ω is the char- acteristic map of the monomorphism id_A : A → A. Recall that Sub(A) is the collection of equivalence classes [m] of monomorphismsm :dom(m),→A, wherem ∼n iff there exists an isomorphismf :dom(m)→ dom(n) such that m = n◦f. Given [m_i] ∈ Sub(A) (i= 1,2) we say that [m₁] ≤[m₂] iff there exists a morphism f : dom(m₁) → dom(m₂) such that m₁ = m₂ ◦f. Then hSub(A),≤i is a Heyting algebra. If X is a finite subset of Sub(A) then ^

X will denote the infimum ofX w.r.t. the Heyting algebra-structure of Sub(A).

Usually, monomorphisms are identified with their equivalence classes (see, for instance, [16]).

Definition 2.4 LetS be an interpretation system. Given a finite subset Ψ∪ {ϕ} of L(Σ, ~x) we say:

• Ψglobally entailsϕwithinSand~x, written Ψ²^S_p~x ϕ, iff, for everyM ∈ M:

^

ψ∈Ψ

[[ψ]]_~x^M =true_θ~xM implies [[ϕ]]^M_~x =true_θ~xM;

• Ψ^locally entailsϕwithinS and~x, written Ψ²^S_d~x ϕ, iff, for everyM ∈ M,

ψ∈Ψ

[[ψ]]_~x^M ≤[[ϕ]]^M_~x . 4

If Ψ∪ {ϕ} ⊆L(Σ, ~x) is finite and~y is the canonical context of Ψ∪ {ϕ}then we write Ψ ²^S_o ϕ instead of Ψ ²^S_o~y ϕ, for o ∈ {p,d}. It is easy to prove that Ψ²^S_o ϕimplies Ψ²^S_o~x ϕ (and the converse is not necessarily true, because the possibly empty domains used in the interpretation of types of Σ).

The usual notion of semantic entailment considered in categorial logic is the local one. On the other hand, we will define two different notions of syntactical inference, one for each concept of semantic entailment. In several contexts it is useful to maintain the distinction between the two notions of (semantic and syntactical) inferences (cf. [21, 22, 5]).

3 Hilbert Calculi

In this section we recall (a simplified version of) the notion of Hilbert calculus introduced in [5]. In order to represent arbitrary terms in rules of Hilbert calculi we will use schema variables. Moreover, some rules will have provisos which control their range of application. Thus we need to introduce the following concepts.

By a Σ-substitution ρ we mean a Θ(S)-indexed family of maps from Ξ_θ to T(Σ)_θ. As usual we write ξρ instead of ρ_θ(ξ). Any Σ-substitution ρ induces a map ρb: ST(Σ)−→ T(Σ) defined inductively as usual, with: ρ(b^x_ξ0ξ) = (ξρ)^x_ξ0ρ, where the right-side expression is the Σ-term obtained fromξρ by substituting every free occurrence of x by ξ⁰ρ. Note that ρ(δ)b ∈T(Σ)_θ ifδ ∈ST(Σ)_θ. We denoteρ(δ) byb δρ. LetSbs(Σ) be the set of all Σ-substitutions.

By a Σ-provisowe mean a map π:Sbs(Σ)→2. Intuitively,π(ρ) = 1 iff the Σ-substitutionρis allowed. (In [5] it is introduced a different notion of proviso which is suitable to perform fibring of deduction systems.) Provisos are very common in rules of logics. For instance, it is well known that a substitution instanceξρ⇒∀x ξρof the schema formulaξ⇒∀x ξis valid in first-order predicate logicprovided thatx is not free inξρ; in this case we haveπ(ρ) = 1 iffx is not free inξρ. We denote by Prov(Σ) the set of all Σ-provisos. Theunit provisou maps every Σ-substitution to 1. Binary product of provisosπuπ⁰ is defined as expected: (πuπ⁰)(ρ) =π(ρ)uπ⁰(ρ).

Definition 3.1 A Σ-ruleis a triplehΓ, δ, πi where Γ∪ {δ} ⊆SL(Σ) and π is a

Σ-proviso. 4

When Γ =∅ the conclusionδ of the rule is also known as anaxiom. When Γ is finite the rule is said to befinitary.

Definition 3.2 A deduction system is a triple D = hΣ,R_d,R_pi where Σ is a signature and bothR_p and R_d are sets of finitary Σ-rules andR_d⊆ R_p. 4 The elements of R_p are called proof rules and those of R_d are known as derivation rules.

Definition 3.3 A ~x-proof within a deduction system D of ϕ ∈ L(Σ, ~x) from Ψ⊆L(Σ, ~x) is a sequence ϕ₁, . . . , ϕ_n of formulae in L(Σ, ~x) such that ϕ_n isϕ and for eachi= 1, . . . , n:

• eitherϕ_i∈Ψ;

• or there is a rule h{γ₁, . . . , γ_k}, δ, πi ∈ R_p and a Σ-substitution ρ such that:

1. π(ρ) = 1;

2. ϕ_i =δρ;

3. for eachj = 1, . . . , k, there is ai_j ∈ {1, . . . , i−1}such thatϕ_ij =γ_jρ.

When there is such a~x-proof in Dof ϕ from Ψ, we write Ψ`<sup>D</sup><sub>p~x</sub> ϕ. And when there is a context~xsuch that Ψ`^D_p~x ϕwe write Ψ`^D_p ϕ. 4 Definition 3.4 A ~x-derivation within a deduction system D of ϕ ∈ L(Σ, ~x) from Ψ⊆L(Σ, ~x) is a sequenceϕ₁, . . . , ϕ_n of formulae inL(Σ, ~x) such thatϕ_n isϕand for each i= 1, . . . , n:

• eitherϕ_i∈Ψ;

• or∅ `^D_p~x ϕ_i;

• or there is a rule h{γ₁, . . . , γ_k}, δ, πi ∈ R_d and a Σ-substitution ρ such that:

1. π(ρ) = 1;

2. ϕ_i =δρ;

3. for eachj = 1, . . . , k, there is ai_j ∈ {1, . . . , i−1}such thatϕ_ij =γ_jρ.

When there is such a~x-derivation in D of ϕ from Ψ, we write Ψ `<sup>D</sup><sub>d~x</sub> ϕ. And when there is a context~xsuch that Ψ`^D_d~xϕ we write Ψ`<sup>D</sup><sub>d</sub> ϕ. 4 As usual, with respect to both proofs and derivations, we may drop the reference to the assumptions when Γ =∅. Note that `^D_p~x ϕiff`^D_d~x ϕ.

Definition 3.5 A logic system is a tuple L =hΣ,M,R_d,R_pi such that S = hΣ,Miis an interpretation system andD=hΣ,R_d,R_piis a deduction system.

4 Definition 3.6 A logic system L is said to be sound iff, for o ∈ {p,d}, any context~xand every finite Ψ∪ {ϕ} ⊆L(Σ, ~x):

• Ψ`^D_o~x ϕimplies Ψ²^S_o~xϕ.

A logic systemLis said to becompleteiff, for o∈ {p,d}and finite Ψ∪{ϕ} ⊆ L(Σ):

• Ψ²^S_o ϕimplies Ψ`^D_o ϕ. 4

We say that a Σ-structure M satisfies D if Ψ`^D_o~x ϕ implies Ψ ²^hΣ,{M}i_o~x ϕ for every Ψ, ϕ,~xand o∈ {p,d}.

Remark 3.7 We recall here the observation made in [5] about the strangeness of Definition 3.6. It is clear that the intended definition of soundness of a logic systemL is, for o∈ {p,d},

Ψ`^D_o ϕimplies Ψ²^S_o ϕ.

Unfortunately, this definition is not correct in the realm of logic systems, be- cause the (possibly) empty domains interpreting the types of Σ. In fact, from Ψ, ψ²^S_o ϕand Ψ²^S_o ψwe cannot infer Ψ²^S_o ϕ, for o∈ {p,d}(see, for instance, [3]). On the other hand, it is obvious that any deduction system D satisfies the following property: from Ψ, ψ `<sup>D</sup><sub>o</sub> ϕ and Ψ `^D_o ψ we infer Ψ `^D_o ϕ, for o ∈ {p,d}. Therefore the standard definition of soundness must be changed, and we must live with the fact that it is possible to have

Ψ`^D_o ϕbut Ψ6²^S_o ϕ

even in a sound logic systemL. 4

4 Local Set Theories

In general, the logic of topoi is expressed through a sequent calculus or in natural deduction-style (see, for instance, [3, 9]). One reason for this is probably related to the problems that the definition of soundness involves for Hilbert calculi (cf.

Remark 3.7 and [5]). In this section we recall the sequent calculus calledlocal set theory introduced by Bell in [3], which is sound (in the usual sense) and complete for topos semantics. And in Section 5 we will rewrite this calculus in a Hilbert-style.

Definition 4.1 Let Σ be a signature as in Definition 1.2 but allowing terms of the formhti, which are identified with t. A Σ-sequent (or simply asequent) is a pairhΨ, ϕi, where Ψ∪ {ϕ} is a finite set of formulae over Σ. 4 A sequent hΨ, ϕi will be denoted by (Ψ :ϕ) or simply Ψ :ϕ. If Ψ =∅then we will write (:ϕ) or :ϕ. As usual,

ϕ,Ψ :ψ Ψ, ϕ:ψ and Φ,Ψ :ψ

will stand for {ϕ} ∪Ψ : ψ and Φ∪Ψ : ψ, respectively. If Ψ is a finite set of formulae then Ψ^x_τ will stands for the finite set {ϕ^x_τ |ϕ∈Ψ}.

Definition 4.2 Local Set Theories (cf. [3]) A local set theory is a sequent calculus defined as follows

Tautology ϕ:ϕ Unity :x₁ =hi

Equality x=y, ϕ^z_x:ϕ^z_y (x andy free for z inϕ) Product1 : (hx₁, . . . , x_ni)_i =x_i (1≤i≤n) Product2 x=h(x)₁, . . . ,(x)_ni (n≥1) Comprehension :x∈ {x : ϕ}

Thinning

Ψ :ϕ ψ,Ψ :ϕ Cut Ψ :ϕ ϕ,Ψ :ψ

Ψ :ψ (any free variable of ϕfree in Ψ or ψ) Substitution

Ψ :ϕ

Ψ^x_τ :ϕ^x_τ (τ free for xin Ψ and ϕ) Extensionality

Ψ :x∈σ ⇔x∈τ

Ψ :σ=τ (x not free in Ψ, σ, τ)

Equivalence

ϕ,Ψ :ψ ψ,Ψ :ϕ Ψ :ϕ⇔ψ

4 In [3], the term hti is defined to be t, for every term t. This allows us to prove the sequent (:∀x(x=_θ x)) for all θ. Recall from [3] the following: letS be a set of sequents. Then the local set theoryS generated by S is defined as follows: (Ψ :ϕ)∈S iff Ψ`<sub>S</sub> ϕiff there exists a proof of (Ψ :ϕ) possibly using sequents ofS as assumptions in the rules. IfS =∅then we write Ψ`ϕinstead of Ψ`_Sϕ. The following result was proved in [3].

Proposition 4.3 The following is true in any local set theory:

1. ϕ,Ψ :ψ

Ψ :ϕ⇒ψ and Ψ :ϕ⇒ψ ϕ,Ψ :ψ

2. Ψ :ψ

Ψ :∀x ψ

provided either (i) x is not free in Ψ or (ii) xis not free inψ.

3.

∀x ψ`ψ providedx is free inψ.

4. ϕ^x_τ,Ψ :ψ

∀x ϕ,Ψ :ψ

provided thatτ is free forx inϕ,xis free inϕand any free variable ofτ is free in∀x ϕ, Ψ or ψ.

In Section 5 we will define a deduction system called HOL and prove that inferences in local set theories correspond to deductions in HOL, obtaining the theorem of adequacy ofHOL w.r.t. topos semantics. Because the different definition of soundness we state in 3.6, we need to extend the notion of inference in local set theories (cf. Definition 4.5).

Remark 4.4 For convenience, we adopt the following notation. Let Ψ be a finite subset ofL(Σ); then (V

Ψ) denotes a formula obtained from Ψ by taking the conjunction of all the formulae in Ψ in an arbitrary order and association (if Ψ =∅ then we take (V

Ψ) to be t). It is easy to prove that, if Ψ∪ {ψ}is a finite subset ofL(Σ) and (V

Ψ)₁, (V

Ψ)₂ are two conjunctions defined as above then{(V

Ψ)₁} `(V

Ψ)₂, therefore {(V

Ψ)₁⇒ψ} `(V

Ψ)₂⇒ψ. 4

Definition 4.5 LetS be a set of sequents formed by formulae inL(Σ) and let Ψ∪{ϕ}be a finite subset ofL(Σ, ~x) for some context~x=x₁. . . x_n. Let (~x=~x) be a formula (V

{(x₁ =x₁), . . . ,(x_n =x_n)}) obtained as in Remark 4.4. Then

Ψ`<sub>S~x</sub>ϕwill stands for Ψ,(~x=~x)`_Sϕ. 4

As usual we omit the reference to the theory S when S=∅.

Let S = hΣ,Mi be the interpretation system such that M is the class of all the Σ-structures M = hE,·_Mi. Using the soundness and completeness theorem of local set theory stated in [3] we obtain easily the following theorem of adequacy:

Theorem 4.6 Let~x be a context, Ψ∪ {ϕ} a finite subset of L(Σ, ~x) andS a set of sequents inL(Σ). Then Ψ `<sub>S~x</sub>ϕ iff Ψ²<sup>S</sup><sub>d~x</sub><sup>S</sup> ϕ, where S<sub>S</sub> =hΣ,M<sub>S</sub>i and M<sub>S</sub> is the subclass ofMformed by all the models ofS. In particular: Ψ`_~xϕ iff Ψ²^S_d~x ϕ, Ψ`<sub>S</sub> ϕiff Ψ²<sup>S</sup><sub>d</sub><sup>S</sup> ϕand Ψ`ϕiff Ψ²^S_d ϕ.

5 Hilbert-style axiomatization of Higher-order logic

In this section we will adapt the sequent calculus-style presentation of local set theory to a Hilbert-style one, defining a deduction system called HOL. The main results to be stated are the following:

• Ψ`<sup>HOL</sup><sub>d~x</sub> <sup>S</sup> ϕ implies Ψ`_S~xϕ;

• Ψ`<sub>S</sub>ϕ implies Ψ`^HOL_d ^S ϕ,

where HOL_S is obtained from HOL by adding the sequents of S as axioms (under an appropriate form). In order to do this, we begin with some notation.

For any schema formulaδ, any typeθ, any variablexof typeθand any schema termδ₁ of typeθ we define the following provisos:

• (δ₁Bx:δ)(ρ) = 1 iff δ₁ρ is free for x inδρ;

• (x≺δ)(ρ) = 1 iffx occurs free inδρ;

• (x6≺δ)(ρ) = 1 iffx does not occur free in δρ.

Definition 5.1 Hilbert calculus for intuitionistic hol.

We define the deduction system HOL = hΣ,R_p,R_di as follows (here i ∈ N, k≥2 and θ,θ₁, ...,θ_k are types):

• R_d is the set composed by:

taut1: h∅, ξ₁⇒(ξ₂⇒ξ₁),ui;

taut2: h∅,(ξ₁⇒(ξ₂⇒ξ₃))⇒((ξ₁⇒ξ₂)⇒(ξ₁⇒ξ₃)),ui;

taut3: h∅,(ξ₁⇒ξ₂)⇒((ξ₁⇒ξ₃)⇒(ξ₁⇒(ξ₂∧ξ₃))),ui;

taut4: h∅, ξ₁⇒(ξ₂⇒(ξ₁∧ξ₂)),ui;

uni: h∅,∀x₁(x₁ =hi),ui;

equa_i,θ: h∅,(ξ₁ =ξ₂)⇒(^x_ξ1ⁱξ₃ ⇒ ^x_ξ2ⁱξ₃),(ξ₁Bx_i:ξ₃)u(ξ₂Bx_i :ξ₃)i;

ref_θ: h∅,∀x₁(x₁ =x₁),ui;

proj_{k,θ1,...,θk,i}: h∅,∀x₁· · · ∀x_k((hx₁, . . . , x_ki)_i=x_i),ui for 1≤i≤k;

prod_{k,θ1,...,θk}: h∅,∀x₁(x₁ =h(x₁)₁, . . . ,(x₁)_ki),ui;

comph_θ: h∅,∀x₁(x₁ ∈ {x₁ :ξ₁} ⇔ξ₁),ui;

subs_i,θ: h∅,(∀x_iξ₂) ⇒ ^x_ξⁱ

1ξ₂,(ξ₁Bx_i:ξ₂)u(x_i≺ξ₂)i;

ext_θ: h∅,(∀x₁(x₁ ∈ξ₁⇔x₁∈ξ₂)⇒(ξ₁ =ξ₂),(x₁ 6≺ξ₁)u(x₁ 6≺ξ₂)i;

equiv: h∅,(ξ₁⇒ξ₂)⇒((ξ₂⇒ξ₁)⇒(ξ₁⇔ξ₂)),ui;

MP: h{ξ₁, ξ₁⇒ξ₂}, ξ₂,ui;

• R_p is obtained by adding toR_d the following rules:

GEN_i,θ: h{ξ₁⇒ξ₂}, ξ₁⇒(∀x_iξ₂),(x_i 6≺ξ₁)i. 4 Of course, rules with subscripts are in fact “schema-rules” (in the usual sense). Thus, eachi∈Nand each typeθdefine a particular instance ofequa_i,θ, and so on. Since, in contrast with the approach in [3], we do not define terms of the formhti(1-tupling), we need to include the axioms ref_θ. From now on, we will omit the subscripts in the name of the rules.

Remark 5.2 If Ψ∪{ψ}is a finite subset ofL(Σ, ~x) and (V

Ψ)₁, (V

Ψ)₂ are two conjunctions defined as in Remark 4.4 then{(V

Ψ)₁} `^HOL_d~x (V

Ψ)₂. Therefore {(V

Ψ)₁⇒ψ} `^HOL_d~x (V

Ψ)₂⇒ψ. 4

Given a set S of sequents we define the set of rules R_S={h∅,(^

Ψ)⇒ϕ,ui | (Ψ :ϕ)∈S}, where (V

Ψ) is defined as in Remark 4.4. The systemHOL_S is given by hΣ,R_p∪ R_S,R_d∪ R_Si.

Proposition 5.3 HOL_S satisfies the Metatheorem of Deduction with respect to~x-derivations: for every context~xand finite Ψ∪ {ϕ, ψ} ⊆L(Σ, ~x),

ϕ,Ψ`<sup>HOL</sup><sub>d~x</sub> <sup>S</sup> ψ iff Ψ`^HOL_d~x ^S ϕ⇒ψ.

Proof: By straightforward induction on the length of a~x-derivation of ψfrom Ψ∪ {ϕ} we get Ψ`^HOL_d~x ^S ϕ⇒ψ. The converse is immediate by MP. QED

The following useful properties ofHOL can be easily proved.

Lemma 5.4 Letϕ, ψ, ψ⁰∈L(Σ, ~x). The following holds in HOL.

1. `^HOL_d[] t.

2. {ϕ} `^HOL_d~x (t⇒ϕ).

3. {ϕ} `^HOL_p~x (∀xϕ).

4. {(ϕ⇒ψ),(ψ⇒ψ⁰)} `^HOL_d~x (ϕ⇒ψ⁰).

5. {(ϕ⇒ψ),(ϕ⇒ψ⁰)} `^HOL_d~x (ϕ⇒(ψ∧ψ⁰)).

6. {ϕ∧ψ} `^HOL_d~x ϕ.

7. {ϕ∧ψ} `^HOL_d~x ψ.

8. {ϕ⇒(ψ⇒ψ⁰)} `^HOL_d~x ((ϕ∧ψ)⇒ψ⁰).

Proof: (1) Consider the following []-derivation inHOL:

1. (∀x₁(x₁ =hi)) (uni)

2. ((∀x₁(x₁=hi))⇒t) (subs) 3. t(MP1,2).

(2) Consider the following~x-derivation:

1. ϕ(Hyp)

2. (ϕ⇒(t⇒ϕ)) (taut1) 3. (t⇒ϕ) (MP1,2).

(3) Consider the following~x-proof:

1. ϕ(Hyp)

2. (t⇒ϕ) (Item (2), 1) 3. (t⇒(∀xϕ)) (GEN2) 4. t(Item (1))

5. (∀xϕ) (MP4,3).

(4) Straightforward, bytaut1,taut2and MP.

(5) Straightforward, bytaut3and MP.

(6) It is easy to show that {hϕ₁, ϕ₂i = hψ₁, ψ₂i} `<sup>HOL</sup><sub>d~x</sub> (ϕ<sub>i</sub> = ψ<sub>i</sub>) for i = 1,2 and {ϕ<sub>1</sub>, ϕ<sub>2</sub>, ψ<sub>1</sub>, ψ<sub>2</sub>} ⊆ L(Σ, ~x). On the other hand, {ϕ = t} `^HOL_d~x ϕ, as the following~x-derivation shows.

1. (ϕ=t) (Hyp) 2. (t=ϕ) (1)

3. ((t=ϕ)⇒(x^x_t⇒x^x_ϕ)) (equa) 4. (t⇒ϕ) (MP2,3)

5. t(Item (1))

6. ϕ(MP5,4).

From the results above the desired result follows straightforwardly, because (ϕ∧ψ) ishϕ, ψi=ht,ti.

(7) Idem to item (6).

(8) An easy consequence of Items (6),(7) and Proposition 5.3. QED Lemma 5.5 Let ~xbe a context and let ϕ ∈ L(Σ, ~x). Then `^HOL_d~x ^S ϕ implies

`_S~xϕ.

Proof: By induction on the length n of a ~x-derivation of ϕ from ∅ within HOL_S. Ifn= 0 then we have the following cases:

(1) ϕ is an instance of (taut i) (i = 1, ...,4). The result follows from the completeness of pure local set theory andThinning.

(2) ϕ is an instance (∀x)(x ∈ σ ⇔x ∈ τ)⇒(σ = τ) of ext. Then x does not occur free in hσ, τi. Consider the following proof fromS in pure local set theory:

1. x∈σ⇔x∈τ :x∈σ⇔x∈τ Tautology

2. (∀x(x∈σ⇔x∈τ)) :x∈σ⇔x∈τ Proposition 4.3(3), 1 3. (∀x(x∈σ⇔x∈τ)) :σ =τ Extensionality, 2

4. (~x=~x),(∀x(x∈σ⇔x∈τ)) :σ=τ Thinning, 3

5. (~x=~x) : (∀x(x∈σ⇔x∈τ))⇒σ =τ Proposition 4.3(1), 4.

(3)ϕis an instance (∀xψ)⇒ψ_τ^x ofsubs. Thenτ is free forxinψandxoccurs free in ψ, and we can construct the following proof from S in pure local set theory:

1. ψ:ψ Tautology

2. (∀xψ) :ψ Proposition 4.3(3), 1 3. (∀xψ) :ψ_τ^x Substitution, 2 4. (~x=~x),(∀xψ) :ψ^x_τ Thinning, 3

5. (~x=~x) : (∀xψ)⇒ψ_τ^x Proposition 4.3(1), 4.

(4) The other cases forn= 0 are easy.

Suppose that the result is true for every ~x-derivation within HOL_S in k ≤ n steps, and letϕobtained from∅through a~x-derivation inn+ 1 steps. We have the following cases:

(a)ϕis an instance of an axiom. The proof is as above.

(b)ϕis obtained fromψandψ⇒ϕbyMP. Thus we can construct the following proof fromS in pure local set theory:

1. (~x=~x) :ψ (IH)

2. (~x=~x) :ψ⇒ϕ (IH)

3. ψ,(~x=~x) :ϕ Proposition 4.3(1), 2 4. (~x=~x) :ϕ Cut 3,1.

Note that the application of Cut is legitimated by the presence of a suitable (~x=~x) which captures all variable occurring free in ψ.

(c)ϕis ψ₁⇒(∀xψ₂) obtained fromψ₁⇒ψ₂ by GEN. Thusx does not occur free inψ₁. We have the following cases:

CASE 1: x does not occur free in ψ₂. Then we construct the following proof fromS in pure local set theory:

1. (~x=~x) :ψ₁⇒ψ₂ (IH)

2. ψ₁,(~x=~x) :ψ₂ Proposition 4.3(1), 1 3. ψ₁,(~x=~x) : (∀xψ₂) Proposition 4.3(2), 2 4. (~x=~x) :ψ₁⇒(∀xψ₂) Proposition 4.3(1), 3.

CASE 2: xoccurs free inψ₂. Then~xis, let’s say,~yx, and we can construct the following proof fromS in pure local set theory:

1. x=x,(~y=~y) :ψ₁⇒ψ₂ (IH)

2. (∀x(x=x)),(~y =~y) :ψ₁⇒ψ₂ Proposition 4.3(4), 1 3. : (∀x(x=x))

4. (~y=~y) :ψ₁⇒ψ₂ Cut 2,3

5. ψ₁,(~y=~y) :ψ₂ Proposition 4.3(1), 4 6. ψ₁,(~y=~y) : (∀xψ₂) Proposition 4.3(2), 5 7. (~y=~y) :ψ₁⇒(∀xψ₂) Proposition 4.3(1), 6 8. (~x=~x) :ψ₁⇒(∀xψ₂) Thinning, 7.

This concludes the proof. QED

Proposition 5.6 Let ~x be a context and let Ψ ∪ {ϕ} be a finite subset of L(Σ, ~x). Then Ψ`<sup>HOL</sup><sub>d~x</sub> <sup>S</sup> ϕ implies Ψ`_S~xϕ.

Proof: Is an immediate consequence of propositions 5.3 and 4.3(1), and Lemma

5.5. QED

Lemma 5.7 Let Ψ∪ {ϕ} be a finite subset of L(Σ). Then Ψ `_S ϕ implies

`^HOL_d ^S (V

Ψ)⇒ϕ.

Proof: Induction on the lengthnof the proof of the sequent (Ψ :ϕ) from S.

Ifn= 0 then we have two possibilities:

(1) (Ψ :ϕ) is an axiom of the pure local set theory. The result follows easily using the axioms ofHOL and Lemma 5.4. For example, any instance of axiom Equalitycan be derived in HOL as follows:

1. ((x=y)⇒(ψ_x^z⇒ψ_y^z)) (equa)

2. (((x=y)∧ψ^z_x)⇒ψ_y^z) (Lemma 5.4(8), 1), provided that bothx, y are free forz inψ.

(2) (Ψ :ϕ) is an instance of a sequent in S. The conclusion is immediate, by the definition ofR_S.

Assume that the result is true for any proof of length ≤n, and consider a sequent (Ψ :ϕ) which is proved from S inn+ 1 steps. We have the following new cases:

(a) (Ψ : ϕ) is of the form (ψ,Φ : ϕ), and it is obtained from (Φ : ϕ) by Thinning. Then there exists a context ~x and a ~x-derivation within HOL_S of ((V

Φ)⇒ϕ) from ∅, by induction hypothesis. From one of such~x-derivations we can construct the following~x-derivation in HOL_S:

1. ((V

Φ)⇒ϕ) (IH) 2. ((ψ∧(V

Φ))⇒(V

Φ)) (Lemma 5.4(7), Proposition 5.3) 3. ((ψ∧(V

Φ))⇒ϕ) (Lemma 5.4(4), 2,1).

(b) (Ψ : ϕ) is obtained from (Ψ : ψ) and (ψ,Ψ : ϕ) by Cut. There exists a context~xand~x-derivations inHOL_S of ((V

Ψ)⇒ψ) and ((ψ∧(V

Ψ))⇒ϕ), by induction hypothesis and Remark 5.2. From one of such~x-derivations we can construct the following~x-derivation in HOL_S:

1. ((V

Ψ)⇒ψ) (IH) 2. ((ψ∧(V

Ψ))⇒ϕ) (IH) 3. ((V

Ψ)⇒(V Ψ)) 4. ((V

Ψ)⇒(ψ∧(V

Ψ))) (Lemma 5.4(5) 1,3) 5. ((V

Ψ)⇒ϕ) (Lemma 5.4(4), 4,2).

(c) (Ψ : ϕ) is of the form (Φ^x_τ : ψ^x_τ), and it is obtained from (Φ : ψ) by Substitution. Note that τ is free forxin (V

Φ)⇒ψ. Ifxdoes not occur free in (V

Φ)⇒ψ then (Ψ :ϕ) is (Φ : ψ) and there exists a ~x-derivation in HOL_S of ((V

Ψ)⇒ϕ), by induction hypothesis. Ifx occurs free in (V

Φ)⇒ψthen there exists a~x-derivation in HOL_S of ((V

Φ)⇒ψ), by induction hypothesis. From one of such~x-derivations we can construct the following~y-derivation inHOL_S, for some context~y:

1. ((V

Φ)⇒ψ) (IH)

2. (∀x((V

Φ)⇒ψ)) (Lemma 5.4(3), 1) 3. (∀x((V

Φ)⇒ψ))⇒((V

Φ)⇒ψ)^x_τ (subs) 4. (V

Φ^x_τ)⇒ψ^x_τ (MP2,3).

(d) (Ψ :ϕ) is of the form (Ψ :σ=τ), and it is obtained from (Ψ :x∈σ⇔x∈τ) byExtensionality. Note thatxdoes not occur free inh(V

Ψ), σ, τi. There exists a~x-derivation in HOL_S of (V

Ψ)⇒(x∈σ⇔x∈τ), by induction hypothesis.

From one of such~x-derivations we can construct the following~x-derivation in HOL_S:

1. (V

Ψ)⇒(x∈σ⇔x∈τ) (IH) 2. (V

Ψ)⇒(∀x(x∈σ⇔x∈τ)) (GEN1) 3. (∀x(x∈σ⇔x∈τ))⇒(σ=τ) (ext) 4. (V

Ψ)⇒(σ =τ) (Lemma 5.4(4), 2,3).

(e) (Ψ :ϕ) is (Ψ :ψ₁⇔ψ₂), and it is obtained from (ψ₁,Ψ :ψ₂) and (ψ₂,Ψ :ψ₁) byEquivalence. Then there exists~x-derivations inHOL_S of ((ψ₁∧(V

Ψ))⇒ψ₂) and ((ψ₂∧(V

Ψ))⇒ψ₁), by induction hypothesis and Remark 5.2. From one of such~x-derivations we can construct the following~x-derivation inHOL_S:

1. ((ψ₁∧(V

Ψ))⇒ψ₂) (IH) 2. ((ψ₂∧(V

Ψ))⇒ψ₁) (IH) 3. ((V

Ψ)⇒(ψ₁⇒ψ₂)) (1) 4. ((V

Ψ)⇒(ψ₂⇒ψ₁)) (2)

5. ((ψ₁⇒ψ₂)⇒((ψ₂⇒ψ₁)⇒(ψ₁⇔ψ₂))) (equiv) 6. ((V

Ψ)⇒((ψ₂⇒ψ₁)⇒(ψ₁⇔ψ₂))) (Lemma 5.4(4), 3,5) 7. (((V

Ψ)⇒((ψ₂⇒ψ₁)⇒(ψ₁⇔ψ₂)))⇒(((V

Ψ)⇒(ψ₂⇒ψ₁))⇒((V Ψ)⇒ (ψ₁⇔ψ₂)))) (taut2)

8. (((V

Ψ)⇒(ψ₂⇒ψ₁))⇒((V

Ψ)⇒(ψ₁⇔ψ₂))) (MP6,7) 9. ((V

Ψ)⇒(ψ₁⇔ψ₂)) (MP 4,8).

QED Proposition 5.8 Let Ψ∪ {ϕ}be a finite subset ofL(Σ). Then Ψ`<sub>S</sub> ϕimplies Ψ`^HOL_d ^S ϕ.

Proof: Immediate by Lemma 5.7 and Proposition 5.3. QED From propositions 5.6 and 5.8 we obtain the desired result.

Theorem 5.9 (Adequacy of HOL) LetS_S =hΣ,M_Sisuch thatM_S is the class of all the Σ-structures satisfyingS. Then:

1. HOL_S is d-sound and d-complete w.r.t. S_S, that is, for every context ~x and finite Ψ∪ {ϕ} ⊆L(Σ, ~x):

• Ψ`^HOL_d~x ^S ϕ implies Ψ²^S_d~x^S ϕ.

• Ψ²^S_d^S ϕ implies Ψ`^HOL_d ^S ϕ.

2. HOL_S is p-sound and p-complete w.r.t. S_S, that is, for every context ~x and finite Ψ∪ {ϕ} ⊆L(Σ, ~x):

• Ψ`^HOL_p~x ^S ϕ implies Ψ²^S_p~x^S ϕ.

• Ψ²^S_p^S ϕ implies Ψ`^HOL_p ^S ϕ.

Proof: (1) It is an immediate consequence of Theorem 4.6 and propositions 5.6 and 5.8.

(2) Recall the notation introduced in Remark 4.4 and Definition 4.5. LetHOL^Ψ_S be the deduction system obtained fromHOL_S by adding the axiom

h∅,(^

Ψ)∧(~x=~x),ui,

and letS_S^Ψ =hΣ,M^Ψ_Si be the corresponding interpretation system. Then, by definition of proofs and derivations and by item (1):

Ψ`<sup>HOL</sup><sub>p~x</sub> <sup>S</sup> ϕ implies `^HOL_d~x ^ΨS ϕ implies [[ϕ]]^M_~x =true_θ~xM for every M ∈ M^Ψ_S. But the last affirmation implies the following: for everyM ∈ M_S,

if [[ψ]]^M_~x =true_θ~xM for every ψ∈Ψ then [[ϕ]]^M_~x =true_θ~xM.

This means that Ψ²^S_p~x^S ϕand thenHOL_Sis p-sound w.r.t. S_S. Finally, suppose that Ψ²^S_p^S ϕ. Then, for everyM ∈ M_S:

if [[ψ]]^M_~x =true_θ~xM for every ψ∈Ψ then [[ϕ]]^M_~x =true_θ~xM, where~xis the canonical context of Ψ∪ {ϕ}. Then

[[ϕ]]_~x^M = [[ϕ∧(~x=~x)]]^M_~x =true_θ~xM

for every M ∈ M^Ψ_S, that is, ²^S_d^SΨ (ϕ∧(~x =~x)). By item (1) we infer `<sup>HOL</sup><sub>d</sub> <sup>ΨS</sup> (ϕ∧(~x=~x)). Therefore Ψ`^HOL_p ^S ϕand then we obtain the desired result. QED

6 Extending the language

In [5] it is shown that it is possible to extend the set Θ(S) of Definition 1.1 to a wider collection allowing functional types instead of the particular cases P(θ). Of course the logic obtained is the same than HOL, because arbitrary exponentials can be expressed in any topos just using exponentials of the form Ω^A, finite limits and the properties of Ω (see for instance [16]).