aa r X i v : . [ m a t h . L O ] F e b An Escape from Vardanyan’s Theorem
Ana de Almeida Borges ∗ Joost J. Joosten † Universitat de BarcelonaFebruary 26, 2021
Abstract
Vardanyan’s Theorem states that quantified provability logic is Π -complete,and in particular impossible to recursively axiomatize for consistent theories con-taining a minimum of arithmetic. However, the proof of this fact cannot beperformed in a strictly positive signature. The logic QRC was introduced in [3]as a candidate first-order provability logic. Here we show that QRC is indeedcomplete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As a corollary weconclude that QRC is also the quantified provability logic of HA . Keywords:
Provability logic, strictly positive logics, quantified modal logic, arith-metic interpretations, feasible fragments.
This document is a draft and as such is missing an introduction and literature review.For context, please refer to the introduction of [3]. Here we merely list the new obtainedresults: • Kripke models for
QRC don’t need to be inclusive in order to be sound. • QRC is complete with respect to constant domain Kripke models. • QRC is complete with respect to arithmetical semantics. • QRC is the quantified provability logic of HA and of all recursively axiomatizableextensions of IΣ . ∗ [email protected] † [email protected] Quantified Reflection Calculus with one modality
The
Quantified Reflection Calculus with one modality , or
QRC , is a sequent logic in astrictly positive predicate modal language introduced in [3].Given variables x, y, z, x i , . . . and a signature Σ fixing the constants c, d, c i , . . . andrelation symbols S, S i , . . . , the formulas of QRC are built up from ⊤ , n -ary relationsymbols applied to n terms (which are either variables or constants), the binary ∧ , theunary ♦ and the quantifier ∀ x .The free variables of a formula are defined as usual. The expression ϕ [ x ← t ] denotesthe formula ϕ with all free occurrences of the variable x simultaneously replaced by theterm t . We say that t is free for x in ϕ if no occurrence of a free variable in t becomesbound in ϕ [ x ← t ].The axioms and rules of QRC are listed in the following definition from [3]. Here weremoved the axiom ♦ ∀ x ϕ ⊢ ∀ x ♦ ϕ because it is an easy consequence of the calculuswithout it. Definition 2.1.
Let Σ be a signature and ϕ , ψ , and χ be any formulas in that language.The axioms and rules of QRC are the following:(i) ϕ ⊢ ⊤ and ϕ ⊢ ϕ ;(ii) ϕ ∧ ψ ⊢ ϕ and ϕ ∧ ψ ⊢ ψ ;(iii) if ϕ ⊢ ψ and ϕ ⊢ χ , then ϕ ⊢ ψ ∧ χ ;(iv) if ϕ ⊢ ψ and ψ ⊢ χ , then ϕ ⊢ χ ;(v) if ϕ ⊢ ψ , then ♦ ϕ ⊢ ♦ ψ ;(vi) ♦♦ ϕ ⊢ ♦ ϕ ; (vii) if ϕ ⊢ ψ , then ϕ ⊢ ∀ x ψ ( x / ∈ fv( ϕ ));(viii) if ϕ [ x ← t ] ⊢ ψ then ∀ x ϕ ⊢ ψ ( t free for x in ϕ );(ix) if ϕ ⊢ ψ , then ϕ [ x ← t ] ⊢ ψ [ x ← t ]( t free for x in ϕ and ψ );(x) if ϕ [ x ← c ] ⊢ ψ [ x ← c ], then ϕ ⊢ ψ ( c not in ϕ nor ψ ).If ϕ ⊢ ψ , we say that ψ follows from ϕ in QRC . When the signature is not clearfrom the context, we write ϕ ⊢ Σ ψ instead.We observe that our axioms do not include a universal quantifier elimination. How-ever, this and various other rules are readily available via the following easy lemma. Lemma 2.2.
The following are theorems (or derivable rules) of
QRC :(i) ♦ ∀ x ϕ ⊢ ∀ x ♦ ϕ ; (ii) ∀ x ∀ y ϕ ⊢ ∀ y ∀ x ϕ ;(iii) ∀ x ϕ ⊢ ϕ [ x ← t ] ( t free for x in ϕ );(iv) ∀ x ϕ ⊢ ∀ y ϕ [ x ← y ] ( y free for x in ϕ and y / ∈ fv( ϕ )); In [3] this was presented as an axiom of
QRC . ϕ ⊢ ψ , then ϕ ⊢ ψ [ x ← t ] ( x not free in ϕ and t free for x in ψ );(vi) if ϕ ⊢ ψ [ x ← c ], then ϕ ⊢ ∀ x ψ ( x not free in ϕ and c not in ϕ nor ψ ).The following are two useful complexity measures on the formulas of QRC definedin [3]. Definition 2.3.
Given a formula ϕ , its modal depth d ♦ ( ϕ ) is defined inductively asfollows: • d ♦ ( ⊤ ) := d ♦ ( S ( x , . . . , x n − )) := 0; • d ♦ ( ψ ∧ χ ) := max { d ♦ ( ψ ) , d ♦ ( χ ) } ; • d ♦ ( ∀ x ψ ) := d ♦ ( ψ ); • d ♦ ( ♦ ψ ) := d ♦ ( ψ ) + 1.Given a finite set of formulas Γ, its modal depth is d ♦ (Γ) := max ϕ ∈ Γ { d ♦ ( ϕ ) } .The definition of quantifier depth d ∀ is analogous except for: • d ∀ ( ∀ x ψ ) = d ∀ ( ψ ) + 1; and • d ∀ ( ♦ ψ ) = d ∀ ( ψ ).The modal depth provides a necessary condition for derivability, which in particularimplies irreflexivity. Lemma 2.4 ([3]) . • If ϕ ⊢ ψ , then d ♦ ( ϕ ) ≥ d ♦ ( ψ ). • For any formula ϕ , we have ϕ ♦ ϕ .Finally, the signature of QRC can be extended without strengthening the calculus. Lemma 2.5 ([3]) . Let Σ be a signature and let C be a collection of new constants notyet occurring in Σ. By Σ C we denote the signature obtained by including these newconstants C in Σ. Let ϕ, ψ be formulas in the language of Σ. Then, if ϕ ⊢ Σ C ψ , so does ϕ ⊢ Σ ψ . 3 Relational semantics
QRC was proven sound and complete with respect to relational semantics in [3]. Herewe extend both of those results in the following ways: we relax the requirement forthe adequateness of a frame, and we prove constant domain completeness, i.e., that if ϕ ψ then there exists a counter model that, in addition to being finite and irreflexive,also has a constant domain.We begin by slightly changing the definition of frame and relational model presentedin [3]. There, models for QRC were described as a number of first-order models (theworlds) connected through a transitive relation R . We additionally required inclusive-ness : that whenever w and u are worlds connected through R , the domain of w beincluded in the domain of u . We then used the inclusion (identity) function ι w,u to referto the element of the domain of u corresponding to an element in the domain of w .Here, we no longer have the inclusiveness restriction. In fact, as we’ll see bellow,any configuration of domains is sound as long as the functions η w,u relating the domainof w with the domain of u respect the transitivity of R . This is clearly the case if theframe is inclusive and η w,u = ι w,u , so the definitions presented in [3] are a particularcase of the ones presented here. Definition 3.1. A relational model M in a signature Σ is a tuple h W, R, { M w } w ∈ W , { η w,v } wRv , { I w } w ∈ W , { J w } w ∈ W i where: • W is a non-empty set (the set of worlds, where individual worlds are referred toas w, u, v , etc); • R is a binary relation on W (the accessibility relation); • each M w is a finite set (the domain of the world w , whose elements are referredto as d, d , d , etc); • if wRv , then η w,v is a function from M w to M v ; • for each w ∈ W , the interpretation I w assigns an element of the domain M w toeach constant c ∈ Σ, written c I w ; and • for each w ∈ W , the interpretation J w assigns a set of tuples S J w ⊆ ℘ (( M w ) n ) toeach n -ary relation symbol S ∈ Σ.The h W, R, { M w } w ∈ W , { η w,v } wRv i part of the model is called its frame . We say thatthe frame (or model) is finite if W is finite.The relevant frames and models will need to satisfy a number of requisites. Definition 3.2.
A frame F is adequate if: • R is transitive: if wRu and uRv , then wRv ; and4 the η functions respect transitivity: if wRu and uRv , then η w,v = η u,v ◦ η w,u .A model is adequate if it is based on an adequate frame and it is: • concordant: if wRu , then c I u = η w,u ( c I w ) for every constant c .Note that in an adequate and rooted model the interpretation of the constants is fullydetermined by the interpretation at the root.As in [3], we use assignments to define truth at a world in a first-order model. A w -assignment g is a function assigning a member of the domain M w to each variablein the language. Any w -assignment can be seen as a v -assignment as long as wRv , bycomposing it with η w,v on the left. We write g η to shorten η w,v ◦ g when w and v areclear from the context.Two w -assignments g and h are x -alternative , written g ∼ x h , if they coincide on allvariables other than x . A w -assignment g is extended to terms by defining g ( c ) := c I w for any constant c . Note that this meshes nicely with the concordant restriction of anadequate model: for any term t , if wRu then g η ( t ) = η w,u ( g ( t )).We are finally ready to define satisfaction at a world. This definition is a straight-forward adaptation of the one presented in [3] to our current definition of model. Theonly difference is in the case of ♦ ϕ , where we use g η instead of g ι . Definition 3.3.
Let M = h W, R, { M w } w ∈ W , { η w,u } wRu , { I w } w ∈ W , { J w } w ∈ W i be a rela-tional model in some signature Σ, and let w ∈ W be a world, g be a w -assignment, S be an n -ary relation symbol, and ϕ, ψ be formulas in the language of Σ.We define M , w (cid:13) g ϕ ( ϕ is true at w under g ) by induction on ϕ as follows. • M , w (cid:13) g ⊤ ; • M , w (cid:13) g S ( t , . . . , t n − ) iff h g ( t ) , . . . , g ( t n − ) i ∈ S J w ; • M , w (cid:13) g ϕ ∧ ψ iff both M , w (cid:13) g ϕ and M , w (cid:13) g ψ ; • M , w (cid:13) g ♦ ϕ iff there is v ∈ W such that wRv and M , v (cid:13) g η ϕ ; • M , w (cid:13) g ∀ x ϕ iff for all w -assignments h such that h ∼ x g , we have M , w (cid:13) h ϕ . Theorem 3.4 (Relational soundness) . If ϕ ⊢ ψ , then for any adequate model M , forany world w ∈ W , and for any w -assignment g : M , w (cid:13) g ϕ = ⇒ M , w (cid:13) g ψ. Proof.
By induction on the proof of ϕ ⊢ ψ , making the same arguments as in [3]. Herewe highlight only the transitivity axiom, where the transitivity of the η functions comesinto play, and also remark on the generalization on constants rule.The transitivity axiom is ♦♦ ϕ ⊢ ♦ ϕ , so assume that M , w (cid:13) g ♦♦ ϕ . Then there isa world v such that wRv and M , v (cid:13) η w,v ◦ g ♦ ϕ , and also a subsequent world u such that5 Ru and M , u (cid:13) η v,u ◦ ( η w,v ◦ g ) ϕ . Since R is transitive, we know that wRu and thus that η v,u ◦ ( η w,v ◦ g ) coincides η w,u ◦ g . Then M , u (cid:13) η w,u ◦ g ψ , and consequently M , w (cid:13) g ♦ ϕ ,as desired.The soundness of the generalization on constants rule, Rule 2.1.(x), is the mostinvolved part of the proof and depends on the construction of a model where theinterpretation of a constant is changed. Building that model in this context is a simplematter of taking care to propagate that change to all future worlds using the η functions. In [3] we proved relational completeness by building a term model that satisfies ϕ and doesn’t satisfy ψ when ϕ ψ . That construction provides a finite, irreflexive,and rooted model with increasing domains. Here we show that it is possible to build aconstant domain model instead, i.e., a model where the domain of every world is exactlythe same. This is extremely useful to prove the arithmetical completeness theorem inSection 6.Before starting the formal proof, we briefly describe the main idea. The term modelswe build are such that each world is a pair of sets of closed formulas p = h p + , p − i . Thefirst set, p + , is the set of formulas that will be satisfied at that world, or the positivepart . The second set, p − , is the set of formulas that will not be satisfied at that world,or the negative part . All worlds must be well-formed with respect to some set of closedformulas Φ, which means that: • p is closed: every formula in p + and p − is closed; • p is Φ-maximal: every formula of Φ is in either p + or p − (and there are no formulasin p but not in Φ); • p is consistent: if δ ∈ p − then V p + δ ; and • p is fully-witnessed: if ∀ x ϕ ∈ p − then there is some constant c such that ϕ [ x ← c ] ∈ p − .In that case we say that p is Φ-maximal consistent and fully witnessed, or Φ-MCW forshort (the closeness condition is included in the concept, although in practice almostevery formula in this section will be closed). If p and q are pairs, we write p ⊆ q when p + ⊆ q + and p − ⊆ q − . Furthermore, if Φ is a set of formulas we write p ⊆ Φ instead of p + ∪ p − ⊆ Φ.We want to have Φ-MCW pairs where Φ is closed under subformulas. However, thenaive subformulas of ∀ x ϕ would be open. In order to avoid that, we use the notionof closure with respect to a set of constants C defined in [3], where ϕ [ x ← c ] is a validsubformula of ∀ x ϕ as long as c ∈ C . 6 efinition 4.1. Given a set of constants C , the closure of a formula ϕ under C ,written C ℓ C ( ϕ ), is defined by induction on the formula as such: C ℓ C ( ⊤ ) := {⊤} ; C ℓ C ( S ( t , . . . , t n − )) := { S ( t , . . . , t n − )) , ⊤} ; C ℓ C ( ϕ ∧ ψ ) := { ϕ ∧ ψ } ∪ C ℓ C ( ϕ ) ∪ C ℓ C ( ψ ); C ℓ C ( ♦ ϕ ) := { ♦ ϕ } ∪ C ℓ C ( ϕ ); and C ℓ C ( ∀ x ϕ ) := {∀ x ϕ } ∪ [ c ∈ C C ℓ C ( ϕ [ x ← c ]) . The closure under C of a set of formulas Γ is the union of the closures under C ofeach of the formulas in Γ: C ℓ C (Γ) := [ γ ∈ Γ C ℓ C ( γ ) . The closure of a pair p is defined as the closure of p + ∪ p − .Going back to the overview of the completeness proof, suppose that ϕ ψ , (assum-ing for now that ϕ and ψ are closed). Defining p := h{ ϕ } , { ψ }i , the counter-model willbe rooted on a C ℓ C ( p )-MCW extension of p , where C is a set of constants to determine.The set of constants C will be used as the domain of the root. Note that p is alreadyclosed and consistent, so taking the step to maximality is as simple as deciding whetherto add each χ ∈ C ℓ C ( p ) to the positive or negative part of the root without ruining itsconsistency. The hard part is doing so in a way that guarantees that the resulting pairis fully witnessed.The completeness proof shown in [3] uses the observation that if V p +
6⊢ ∀ x χ , thenalso V p + χ [ x ← c ] where c does not appear in either p + or χ (Lemma 2.2.(vi)). Thissuggests a way of sorting the formulas of C ℓ C ( p ) into positive and negative: mark aformula as positive if and only if it is a consequence of p + . This guarantees that thereare witnesses for the negative universal formulas as long as there are enough constantsto go around. The way to make sure that there are enough constants is precisely thedifference between the proof presented in [3] and the proof presented here. To that end,we introduce the following definition. Definition 4.2.
The number of different constants in a formula ϕ is represented byd c ( ϕ ). The maximum number of different constants per formula in a set of formulas Γis defined as d c (Γ) := max ϕ ∈ Γ { d c ( ϕ ) } .We observe that the maximum number of distinct constants per formula in theclosure under C of a set of formulas can be upper bounded by a number that does notdepend on C . Remark 4.3.
For any formula ϕ , set of formulas Φ, and set of constants C : • d c ( ϕ ) ≤ d c ( ϕ [ x ← c ]) ≤ d c ( ϕ ) + 1 • d c ( ϕ ) ≤ d c ( C ℓ C ( ϕ )) ≤ d c ( ϕ ) + d ∀ ( ϕ ) • d c (Φ) ≤ d c ( C ℓ C (Φ)) ≤ d c (Φ) + d ∀ (Φ)7e are now ready to prove a Lindenbaum-like lemma. Lemma 4.4.
Given a finite signature Σ with constants C and a finite set of closedformulas Φ in the language of Σ such that | C | > c (Φ) + 2d ∀ (Φ), if p ⊆ C ℓ C (Φ) is aclosed consistent pair and p + is a singleton, there is a pair q ⊇ p in the language of Σsuch that q is C ℓ C (Φ)-MCW, and d ♦ ( q + ) = d ♦ ( p + ). Proof.
Like in [3], we start by defining a pair q ⊇ p such that for each χ ∈ C ℓ C (Φ), χ ∈ q + if and only if p + ⊢ χ (otherwise χ ∈ q − ). This pair q is C ℓ C (Φ)-maximalconsistent and d ♦ ( q ) = d ♦ ( p ) as shown in [3].It remains to show that q is fully witnessed. Let ∀ x ψ be a formula in q − . Weclaim that there is d ∈ C such that d does not appear either in p + or in ∀ x ψ . Forthis it is enough to see that | C | > d c ( V p + ) + d c ( ∀ x ψ ), which is the same as | C | > d c ( p + ) + d c ( ∀ x ψ ) because p + is a singleton by assumption. Since p + ⊆ C ℓ C (Φ), weknow that d c ( p + ) ≤ d c ( C ℓ C (Φ)), and similarly for ∀ x ψ . Then by Remark 4.3 we mayconclude that d c ( p + ) + d c ( ∀ x ψ ) ≤ c (Φ) + 2d c (Φ), which suffices by our assumptionon the size of C .Since d ∈ C does not appear in either p + or ∀ x ψ , we conclude that p + ψ [ x ← d ]by Lemma 2.2.(vi), and consequently ψ [ x ← d ] ∈ q − as desired.We now recall the definition of ˆ R from [3], which is the relation we use to connectthe worlds of the term model. Definition 4.5.
The relation ˆ R between pairs is such that p ˆ Rq if and only if both offollowing hold:1. for any formula ♦ ϕ ∈ p − we have ϕ, ♦ ϕ ∈ q − ; and2. there is some formula ♦ ψ ∈ p + ∩ q − . Lemma 4.6 ([3]) . The relation ˆ R restricted to consistent pairs is transitive and ir-reflexive.We now see that if w is a C ℓ C (Φ)-MCW pair with ♦ ϕ ∈ w + , we can find a C ℓ C (Φ)-MCW pair v with ϕ ∈ v + and w ˆ Rv . The proof is the same as in [3], except that wenow use Lemma 4.4 to obtain constant domains throughout the model. Lemma 4.7 (Pair existence) . Let Σ be a signature with a finite set of constants C , andΦ be a finite set of closed formulas in the language of Σ such that | C | > c (Φ)+2d ∀ (Φ).If p is a C ℓ C (Φ)-MCW pair and ♦ ϕ ∈ p + , then there is a C ℓ C (Φ)-MCW pair q such that p ˆ Rq , ϕ ∈ q + , and d ♦ ( q + ) < d ♦ ( p + ). Proof.
Consider the pair r = h{ ϕ } , { δ, ♦ δ | ♦ δ ∈ p − } ∪ { ♦ ϕ }i . It is easy to check that r is consistent and that p ˆ Rr (details can be found in [3]), and clearly r ⊆ C ℓ C (Φ). Wethen use Lemma 4.4 to obtain a C ℓ C (Φ)-MCW pair q ⊇ r such that d ♦ ( q + ) = d ♦ ( r + ) =d ♦ ( ϕ ) < d ♦ ( p + ). We obtain p ˆ Rq as a straightforward consequence of p ˆ Rr .8e can now define an adequate and constant domain model M [ p ] from any givenfinite and consistent pair p such that M [ p ] satisfies the formulas in p + and doesn’tsatisfy the formulas in p − . The idea is exactly the same as in [3]: build a term modelwhere each world w is a C ℓ M ( p )-MCW pair, and the worlds are related by (a sub-relationof) ˆ R . Definition 4.8.
Let Σ be a signature. Given a finite consistent pair p of closed formulasin Σ such that p + is a singleton, we define an adequate model M [ p ]. Here we will useΦ := p + ∪ p − .Let C be a set of at least 2d c (Φ) + 2d ∀ (Φ) + 1 different constants, including all ofthe ones appearing in Σ and adding more if necessary. The pairs of formulas we workwith are in the signature Σ extended by C .We start by defining the underlying frame in an iterative manner. The root is givenby Lemma 4.4 applied to C and p , obtaining the C ℓ C (Φ)-MCW pair q . Frame F isthen defined such that its set of worlds is W := { q } , its relation R is empty, and thedomain of q is M q := C .Assume now that we already have a frame F i , and we set out to define F i +1 as anextension of F i . For each leaf w of F i , i.e., each world such that there is no world v ∈ F i with wR i v , and for each formula ♦ ϕ ∈ w + , use Lemma 4.7 to obtain a C ℓ C (Φ)-MCWpair v such that w ˆ Rv , ϕ ∈ v + , and d ♦ ( v + ) < d ♦ ( w + ). Now add v to W i +1 , add h w, v i to R i +1 , define M i +1 v as C , and define η w,v as the identity.The process described above terminates because each pair is finite and the modaldepth of C ℓ C (Φ) (and consequently of w + , for any w ⊆ C ℓ C (Φ)) is also finite. Thus thereis a final frame F m . This frame is constant domain by construction, but not transitive.We obtain F [ p ] as the transitive closure of F m , which is clearly still constant domain.The η functions are all the identity in C , thus satisfying the transitivity condition. Weconclude that the frame F [ p ] is adequate.In order to obtain the model M [ p ] based on the frame F [ p ], let I q take constantsin Σ to their corresponding version as domain elements and if w is any other world,let I w coincide with I q . This is necessary to make sure that the model is concordant,because q sees every other world, and is sufficient to see that M [ p ] is adequate. Finally,given an n -ary predicate letter S and a world w , define S J w as the set of n -tuples h d , . . . , d n − i ⊆ ( M w ) n such that S ( d , . . . , d n − ) ∈ w + .Since everything up until now was meant for closed formulas, and furthermore weare potentially adding new constants to the signature of the formulas we care about,we provide a way of replacing the free variables of a formula with new constants. Definition 4.9 ([3]) . Given a formula ϕ in a signature Σ and a function g from the setof variables to a set of constants in some signature Σ ′ ⊇ Σ, we define the formula ϕ g inthe signature Σ ′ as ϕ with each free variable x simultaneously replaced by g ( x ).The constant domain model defined above coincides with the non-constant domaindefinition provided in [3] in everything other than that it refers to the stronger Lem-mas 4.4 and 4.7 which keep the domain constant. Thus the Truth Lemma holds withexactly the same proof. 9 emma 4.10 (Truth lemma, [3]) . Let Σ be a signature. For any finite non-emptyconsistent pair p of closed formulas in the language of Σ such that p + is a singleton, world w ∈ M [ p ], assignment g , and formula ϕ in the language of Σ such that ϕ g ∈ C ℓ M w ( p ),we have that M [ p ] , w (cid:13) g ϕ ⇐⇒ ϕ g ∈ w + . Theorem 4.11 (Constant domain completeness) . Let Σ be a signature and ϕ, ψ for-mulas in Σ. If ϕ ψ , then there is an adequate, finite, irreflexive, and constant domainmodel M , a world w ∈ M , and an assignment g such that: M , w (cid:13) g ϕ and M , w (cid:13) g ψ. Proof.
As in the original proof of relational completeness [3], define a new constant c x for each free variable x of ϕ and ψ and let Σ ′ be the signature Σ augmented withthese new constants. Let g be any assignment taking each x to c x . Note that ϕ g Σ ′ ψ g by Rule 2.1.(x) and Lemma 2.5. Build M := M [ h{ ϕ g } , { ψ g }i ] as described inDefinition 4.8, with root w . Then by Lemma 4.10 both M , w (cid:13) g ϕ and M , w (cid:13) g ψ , asdesired. Recall that the language of arithmetic is that of first-order logic together with the sym-bols { , , + , × , ≤ , = } with their usual arities (see [5] for details). It is also possible torefer to the provability and consistency of a formal theory T by using G¨odel’s prov-ability predicate (cid:3) τ and its dual ♦ τ . Roughly, (cid:3) τ ϕ is a Σ formula stating that thereis a Hilbert-style proof of ϕ , i.e. a finite sequence π , . . . , π n − such that π n − is ϕ andthat each π i is either (the G¨odel number of) an axiom of T or follows from previouselements of the sequence through a rule. The τ ( u ) is a formula axiomatizing T , suchthat N (cid:15) τ (¯ n ) if and only if n is the G¨odel number of an axiom of T .We wish to interpret the strictly positive formulas in the language of QRC asparametrized axiomatizations of arithmetical theories extending IΣ . Let τ IΣ ( u ) be astandard axiomatization of IΣ .A realization · ⋆ interprets each n -ary relation symbol S ( x , c ) as an ( n + 1)-ary Σ formula σ ( u, y , z ) in the language of arithmetic, where y matches with x and z matcheswith c . We then extend this notion to any formula as follows (we add τ IΣ ( u ) to theaxioms of the interpretation of any relation symbol to guarantee that every theory isan extension of IΣ ). Definition 5.1 ([3]) . • ⊤ ⋆ := τ IΣ ( u ); In the sections on arithmetic, we always use x for variables of QRC and y, z, u for variables ofarithmetic. Furthermore, u is reserved for the G¨odel numbers of theories. ( S ( x , c )) ⋆ := σ ( u, y , z ) ∨ τ IΣ ( u ); • ( ψ ∧ δ ) ⋆ := ψ ⋆ ∨ δ ⋆ ; • ( ♦ ψ ) ⋆ := τ IΣ ( u ) ∨ ( u = p♦ ψ ⋆ ⊤ q ); • ( ∀ x i ψ ) ⋆ := ∃ y i ψ ⋆ .Let T be a computably enumerable theory in the language of arithmetic extendingIΣ . QRC is defined as (recall that χ ⋆ will in general depend on y and z ): QRC ( T ) := { ϕ ( x , c ) ⊢ ψ ( x , c ) | ∀ · ⋆ T ⊢ ∀ θ ∀ y ∀ z ( (cid:3) ψ ⋆ θ → (cid:3) ϕ ⋆ θ ) } . We show that
QRC = QRC ( T ) , for any recursively axiomatizable and sound theory T extending IΣ . The left-to-rightinclusion was already proved in [3] (noting that QRC (IΣ ) ⊆ QRC ( T ) for any relevant T ). Theorem 5.2 (Arithmetical soundness, [3]) . QRC ⊆ QRC (IΣ ).The proof of the other inclusion is shown in the next section. The proof of arithmetical completeness closely follows the proof of Solovay’s Theoremfound in [2]. The arithmetical semantics used there is different from the one presentedin the previous section. This is not a problem as we show later that they are equivalentin this setting. We use · ⋆ when interpreting formulas as axiomatizations of formaltheories, and · ∗ when interpreting them as generic formulas. • ⊤ ∗ := ⊤ • ( S ( t )) ∗ := any arithmetical formula with the same arity as S • ( ϕ ∧ ψ ) ∗ := ϕ ∗ ∧ ψ ∗ • ( ♦ ϕ ) ∗ := ♦ τ ϕ ∗ • ( ∀ x k ϕ ) ∗ := ∀ y k ϕ ∗ A theory is computably enumerable if its axioms can be defined by a Σ formula. ϕ and ψ such that ϕ ψ in QRC , let M ϕ,ψ be a finite, irreflexive, and constant domain adequate model satisfying ϕ and not sat-isfying ψ at the root under an assignment g ϕ,ψ . This model and assignment exist byTheorem 4.11.We assume that the worlds of M ϕ,ψ are W = { , , . . . , N } , where 1 is the root. Wedefine a new model M that is a copy of M ϕ,ψ , except that it has an extra world 0 asthe new root. This world 0 is connected to all the other worlds through R ′ and has thesame domain, constant interpretation, and relational symbol interpretation as 1.Fix a recursively axiomatizable and sound theory T extending IΣ with axiomatiza-tion τ ( u ). Let the λ i be the Solovay sentences as defined in [2], satisfying the followingEmbedding Lemma. Lemma 6.1 (Embedding, [2]) . (1) N (cid:15) λ ;(2) T ⊢ W i ≤ N λ i ;(3) T ⊢ λ i → V j ≤ N & j = i ¬ λ j ;(4) T ⊢ λ i → V iRj ♦ τ λ j ;(5) T ⊢ λ i → (cid:3) τ (cid:0) W iRj λ j (cid:1) , with i = 0.The domain of every world is M . Let m be the size of M , and p · q be a bijectionbetween M and the set of natural numbers { , . . . , m − } .We now define for a given n -ary predicate symbol S and terms t = t , . . . , t n − the · ∗ interpretation as follows: ( S ( t )) ∗ := _ i ≤ N ( λ i ∧ Φ S ( t ) i )Φ S ( t ) i := _ h a ,...,a n − i∈ S Ji ^ ≤ l Given a QRC formula ϕ and a QRC variable x k , we have that x k ∈ fv( ϕ )if and only if y k ∈ fv( ϕ ∗ ). 12 emma 6.3. For any QRC formula ϕ and any T variable y : T ⊢ ϕ ∗ ↔ ϕ ∗ [ y ← y mod m ] . Proof. By external induction on the complexity of ϕ . The cases of ⊤ and ∧ are straight-forward.For the case of the relation symbols, consider without loss of generality S ( x ). Theonly free variable of ( S ( x )) ∗ is y , so the result is trivial when y is not y . Thus wecheck that T ⊢ ( S ( x )) ∗ ↔ ( S ( x )) ∗ [ y ← y mod m ]. We have( S ( x )) ∗ = _ j ∈ W (cid:16) λ j ∧ _ h a i∈ S Jj ( p a q = y mod m ) (cid:17) and hence( S ( x )) ∗ [ y ← y mod m ] = _ j ∈ W (cid:16) λ j ∧ _ h a i∈ S Jj ( p a q = ( y mod m ) mod m ) (cid:17) . These are equivalent because ( y mod m ) mod m = y mod m .Consider now the case of ∀ x ϕ . If y is y there is nothing to show ( y is not a freevariable of ( ∀ x ϕ ) ∗ ). Thus we may conclude that ( ∀ x ϕ ) ∗ [ y ← y mod m ] is the sameas ∀ y ϕ ∗ [ y ← y mod m ]. By the induction hypothesis, T ⊢ ϕ ∗ ↔ ϕ ∗ [ y ← y mod m ], fromwhich we obtain T ⊢ ∀ y ϕ ∗ ↔ ∀ y ϕ ∗ [ y ← y mod m ] as desired.Finally in the case of ♦ ϕ , note that ( ♦ ϕ ) ∗ [ y ← y mod m ] is ♦ τ ϕ ∗ [ y ← y mod m ]. Thusthe result is straightforward from the induction hypothesis under the box. Lemma 6.4 (Truth Lemma: positive) . For any QRC formula ϕ with free variables x , . . . , x n − , any world i ≤ N , and any assignment g : M , i (cid:13) g ϕ = ⇒ T ⊢ λ i → ϕ ∗ [ y ← p g ( x ) q ] · · · [ y n − ← p g ( x n − ) q ] . Proof. By external induction on the complexity of ϕ . The cases of ⊤ and ∧ are straight-forward.In the case of relational symbols, we assume without loss of generality that therelevant formula is P ( x , c ). If M , i (cid:13) g P ( x , c ), then h g ( x ) , ( c ) I i i ∈ P J i . Rea-son in T and assume λ i . It suffices to prove Φ P ( x ,c ) i [ y ← p g ( x ) q ], which implies( P ( x , c )) ∗ [ y ← p g ( x ) q ] under the assumption of λ i . We need to find h a , a i ∈ P J i such that p a q = p g ( x ) q mod m and p a q = p ( c ) I i q . Noting that p g ( x ) q mod m is p g ( x ) q , we can pick a := g ( x ) and a := ( c ) I i by assumption.For ∀ x ϕ , assume without loss of generality that the free variables of ϕ are x and x . If M , i (cid:13) g ∀ x ϕ then for every assignment h ∼ x g we have M , i (cid:13) h ϕ . We wish toshow T ⊢ λ i → ∀ y ϕ ∗ [ y ← p g ( x ) q ] . Reason in T , assume λ i , and let y be arbitrary. Our goal is ϕ ∗ [ y ← p g ( x ) q ][ y ← y ]. ByLemma 6.3, T proves the equivalence of ϕ ∗ and ϕ ∗ [ y ← y mod m ], so we can instead13how ϕ ∗ [ y ← y mod m ][ y ← p g ( x ) q ][ y ← y ], which is ϕ ∗ [ y ← y mod m ][ y ← p g ( x ) q ].Since p · q is a bijection, there is a ∈ M such that p a q = y mod m . We define anassignment h such that h ∼ x g and h ( x ) := a . By assumption, M , i (cid:13) h ϕ , so bythe induction hypothesis we obtain ϕ ∗ [ y ← p h ( x ) q ][ y ← p h ( x ) q ]. This concludes theargument because p h ( x ) q = y mod m and h ( x ) = g ( x ).Finally in the case of ♦ ϕ , assume without loss of generality that fv( ϕ ) = { x } . If M , i (cid:13) g ♦ ϕ , then there is j ∈ W such that iRj and M , j (cid:13) g ϕ . Reason in T andassume λ i . By Lemma 6.1.(4), we obtain ♦ τ λ j . Then the induction hypothesis underthe box gives us ♦ τ ( ϕ ∗ [ y ← p g ( x ) q ]), as desired. Lemma 6.5 (Truth Lemma: negative) . For any QRC formula ϕ with free variables x , . . . , x n − , any world i > 0, and any assignment g : M , i (cid:13) g ϕ = ⇒ T ⊢ λ i → ¬ ϕ ∗ [ y ← p g ( x ) q ] · · · [ y n − ← p g ( x n − ) q ] . Proof. By external induction on the complexity of ϕ . The cases of ⊤ and ∧ are straight-forward.For the relational symbols, consider P ( x , c ) without loss of generality. If M , i (cid:13) g P ( x , c ), then h g ( x ) , ( c ) I i i / ∈ P J i . Reason in T and assume λ i . By Lemma 6.1.(3),we have ¬ λ j for every j = i , and hence we only need to show ¬ Φ P ( x ,c ) i [ y ← p g ( x ) q ].In other words, we need to check that if h a , a i ∈ P J i , either p a q = p g ( x ) q mod m ,or p a q = p ( c ) I i q . But this follows from our assumption, taking into account that p g ( x ) q mod m is just p g ( x ) q and that p · q is injective.Consider now the case of ∀ x ϕ . We assume without loss of generality that fv( ϕ ) = { x , x } . If M , i (cid:13) g ∀ x ϕ , then there is an assignment h ∼ x g such that M , i (cid:13) h ϕ . Reason in T and assume λ i . By the induction hypothesis we may conclude that ¬ ϕ ∗ [ y ← p h ( x ) q ][ y ← p h ( x ) q ], which implies ¬ ∀ y ϕ ∗ [ y ← p h ( x ) q ]. But this is whatwe wanted, taking into account that h ( x ) = g ( x ).Finally, in the case of ♦ ϕ with x as the only free variable (without loss of gener-ality), assume that M , i (cid:13) g ♦ ϕ . Then for every j such that iRj , we have M , j (cid:13) g ϕ and thus for each such j the induction hypothesis gives us T ⊢ λ j → ¬ ϕ ∗ [ y ← p g ( x ) q ],which put together gives T ⊢ W iRj λ j → ¬ ϕ ∗ [ y ← p g ( x ) q ]. Reason in T and assume λ i . By Lemma 6.1.(5) and our assumption, we obtain (cid:3) T W iRj λ j . Taking the pre-vious observation under the box, we conclude (cid:3) T ¬ ϕ ∗ [ y ← p g ( x ) q ], which is precisely ¬ ( ♦ ϕ ) ∗ [ y ← p x q ]. Theorem 6.6. If ϕ, ψ are QRC formulas with free variables x such that ϕ ψ , wehave T ( ϕ ∗ → ψ ∗ )[ y ← p g ϕ,ψ ( x ) q ]. Proof. Build M as discussed at the beginning of this section, such that M satisfies ϕ and not ψ at world 1 under the assignment g := g ϕ,ψ . We will check that the · ∗ definedfrom M in this section is the right interpretation.Since M , (cid:13) g ϕ , we obtain from Lemma 6.4 that T ⊢ λ → ϕ ∗ [ y ← p g ( x ) q ].Since M , (cid:13) g ψ , we obtain from Lemma 6.5 that T ⊢ λ → ¬ ψ ∗ [ y ← p g ( x ) q ]. Thus T ⊢ λ → ¬ ( ϕ ∗ → ψ ∗ )[ y ← p g ( x ) q ]. 14y Lemma 6.1.(4) and the fact that 0 R 1, we obtain T ⊢ λ → ♦ τ λ , so putting thistogether with the previous observation, T ⊢ λ → ♦ τ ¬ ( ϕ ∗ → ψ ∗ )[ y ← p g ( x ) q ].By Lemma 6.1.(1), we know that N (cid:15) λ , and thus by the soundness of T , we knowthat N (cid:15) ♦ τ ¬ ( ϕ ∗ → ψ ∗ )[ y ← p g ( x ) q ]. Then T ( ϕ ∗ → ψ ∗ )[ y ← p g ( x ) q ].We now define an arithmetical realization · ⋆ in the style of Section 5 that behaveslike · ∗ . Since · ⋆ realizations take QRC constants to arithmetical variables, we definemodified version of Φ S ( t ) i as follows:Ψ S ( t ,...,t n − ) i := _ h a ,...,a n − i∈ S Ji ^ ≤ l For any strictly positive formula ϕ with free variables x and constants c : T ⊢ ∀ ψ ∀ y (cid:16) (cid:3) ϕ ⋆ [ z ← p c I q ] ψ ↔ (cid:3) τ ( ϕ ∗ → ψ ) (cid:17) . Proof. By external induction on ϕ . There is nothing to show for ⊤ .For the case of relational symbols, assume without loss of generality that the relevantformula is S ( x , c ). Note that( S ( x , c )) ⋆ = τ ( u ) ∨ ( u = p ^ i ≤ N ( λ i ∧ _ h a ,a i ( p a q = ˙ y mod m ∧ p a q = ˙ z mod m ) q )Reason in T and let ψ, y be arbitrary. By the formalized Deduction Theorem, wehave (cid:3) S ( x ,c ) ⋆ [ z ← ( c ) I ][ y ← y ] ψ if and only if (cid:3) τ (( S ( x , c )) ∗ ′ [ z ← ( c ) I ][ y ← y ] → ψ ),which is equivalent to our desired goal by the relationship between · ∗ ′ and · ∗ mentionedabove.In the case of ∧ , we take ϕ = ϕ ∧ ϕ .( → ) Reason in T and fix an arbitrary ψ assuming (cid:3) ( ϕ ⋆ ∨ ϕ ⋆ )[ z ← p c I q ] ψ . Thus, there is afinite sequence π = π , . . . , π n with π n = ψ that is a proof of ψ in the theory axiomatizedby ( ϕ ⋆ ∨ ϕ ⋆ )[ z ← p c I q ] = ϕ ⋆ [ z ← p c I q ] ∨ ϕ ⋆ [ z ← p c I q ]. Each axiom occurring in π iseither an axiom of T or a formula χ i which is not an axiom of T and which satisfieseither ϕ ⋆ [ z ← p c I q ] or ϕ ⋆ [ z ← p c I q ]. Thus, for each χ i we have in particular that either (cid:3) ϕ ⋆ [ z ← p c I q ] χ i or (cid:3) ϕ ⋆ [ z ← p c I q ] χ i , whence by the inductive hypothesis (cid:3) τ ( ϕ ∗ → χ i ) or (cid:3) τ ( ϕ ∗ → χ i ). In either case we have (cid:3) τ ( ϕ ∗ ∧ ϕ ∗ → χ i ), which yields the required resultsince ( ϕ ∧ ϕ ) ∗ = ϕ ∗ ∧ ϕ ∗ .( ← ) By the induction hypothesis (taking ψ to be ϕ ∗ ) we see that (cid:3) ϕ ⋆ [ z ← p c I q ] ϕ ∗ andlikewise we see that (cid:3) ϕ ⋆ [ z ← p c I q ] ϕ ∗ . Thus (cid:3) ( ϕ ⋆ ∨ ϕ ⋆ )[ z ← p c I q ] ϕ ∗ ∧ ϕ ∗ . By assumption we15ave (cid:3) τ ( ϕ ∗ ∧ ϕ ∗ → ψ ), and since ϕ ⋆ [ z ← p c I q ] extends T , we may conclude (cid:3) ϕ ⋆ [ z ← p c I q ] ψ as desired.The ♦ case follows easily from the (boxed) induction hypothesis applied to ⊥ andthe formalized Deduction Theorem.Finally, consider ∀ x ϕ . Note that x is represented by y in T , and this is alwaysa different variable from any z used to represent QRC constants.( → ) If (cid:3) ∃ y ϕ ⋆ [ z ← p c I q ] ψ , then there is a proof π = π , . . . , π n where π n = ψ and eachaxiom χ i in π that is not an axiom of T satisfies ϕ ⋆ [ z ← p c I q ][ y ← k i ] for some number k i , and consequently (cid:3) τ ( ϕ ∗ [ z ← p c I q ][ y ← k i ] → χ i ) by the induction hypothesis foreach i . Then by weakening we conclude (cid:3) τ ( ∀ y ϕ ∗ [ z ← p c I q ] → χ i ) for each i , and weare done.( ← ) By the induction hypothesis (with ψ as ϕ ∗ [ y ← k ] and y as k ), we obtain (cid:3) ϕ ⋆ [ z ← p g ( c I ) q ][ y ← k ] ϕ ∗ [ y ← k ] for any k , and thus (cid:3) ∃ y ϕ ⋆ [ z ← p g ( c I ) q ] ∀ y ϕ ∗ . By our as-sumption, (cid:3) τ ( ∀ y ϕ ∗ → ψ ), and since ∃ y ϕ ⋆ [ z ← p g ( c I ) q ] extends T , we conclude (cid:3) ∃ y ϕ ⋆ [ z ← p g ( c I ) q ] ψ , as desired. Theorem 6.8 (Arithmetical completeness) . QRC ⊇ QRC ( T ). Proof. Recall the definition of QRC ( T ): QRC ( T ) = { ϕ ( x , c ) ⊢ ψ ( x , c ) | ∀ · ⋆ T ⊢ ∀ θ ∀ y ∀ z ( (cid:3) ψ ⋆ θ → (cid:3) ϕ ⋆ θ ) } . We show that if ϕ ψ , then the · ⋆ defined above is such that T 6⊢ ∀ θ ∀ y ∀ z ( (cid:3) ψ ⋆ θ → (cid:3) ϕ ⋆ θ ) . Suppose towards a contradiction that T does prove this formula. Then we can instan-tiate the z variables by p c I q , and then by Lemma 6.7: T ⊢ ∀ θ ∀ y ( (cid:3) τ ( ψ ∗ → θ ) → (cid:3) τ ( ϕ ∗ → θ )) , so in particular T ⊢ ∀ y ( (cid:3) τ ( ϕ ∗ → ψ ∗ )) and hence T ⊢ (cid:3) τ ( ϕ ∗ → ψ ∗ )[ y ← p g ϕ,ψ ( x ) q ] . But this together with the soundness of T contradict Theorem 6.6.We conclude by remarking that QRC is also the provability logic of Heyting Arith-metic (the intuitionistic counterpart to PA ). It is well-known (see eg. [4]) that PA isΠ -conservative over HA . The definition of QRC ( PA ) only mentions provability of Π formulas, because (cid:3) ϕ ∗ θ is always Σ . Thus we conclude that QRC ( PA ) = QRC ( HA ).The same observation applied to the arithmetical completeness of RC [1] shows thatit is the propositional provability logic of HA .16 eferences [1] Beklemishev, L. D. (2014). Positive provability logic for uniform reflection principles. Annals of Pure and Applied Logic , (1), 82–105.[2] Boolos, G. S. (1993). The Logic of Provability . Cambridge: Cambridge UniversityPress.[3] Borges, A. A., & Joosten, J. J. (2020). Quantified reflection calculus with onemodality. In N. Olivetti, R. Verbrugge, S. Negri, & G. Sandu (Eds.) Advances inModal Logic 13 , (pp. 13–32). College Publications.[4] Friedman, H. (1978). Classically and intuitionistically provably recursive functions.In D. S. Scott, & G. H. Muller (Eds.) Higher Set Theory , vol. 699, (pp. 21–28).Springer Verlag.[5] H´ajek, P., & Pudl´ak, P. (1993).