OOn inclusions between predicate provability logics
Taishi Kurahashi ∗† Abstract
We investigate several consequences of inclusion relations betweenpredicate provability logics. Moreover, we give a necessary and sufficientcondition for the inclusion relation between predicate provability logicswith respect to Σ arithmetical interpretations. The notion of provability is a kind of modality, and modal logical studies offormalized provability have been extensively proceeded by many authors. Suchstudies have had many successes, especially in the framework of propositionalmodal logic. Solovay’s arithmetical completeness theorem [13] is one of them.For every recursively enumerable extension T of Peano Arithmetic PA , letPr T ( x ) be a usual provability predicate of T . A T -arithmetical interpretation isa mapping f T from the set of all propositional modal formulas to the set of sen-tences of arithmetic such that f T commutes with each propositional connectiveand f T maps (cid:3) A to Pr T ( (cid:112) f T ( A ) (cid:113) ). Let PL ( T ) be the set of all propositionalmodal formulas A such that T (cid:96) f T ( A ) for every T -arithmetical interpreta-tion f T . This set is called the propositional provability logic of T . Solovay’sarithmetical completeness theorem states that if T is Σ -sound recursively enu-merable extension of PA , then PL ( T ) is exactly the propositional modal logic GL . Thus PL ( T ) is recursive, but does not contain any elements specific to thetheory T .The formalized provability is also studied in the framework of predicatemodal logic. The main target of this study is the predicate provability logic QPL ( T ) of T , which consists of predicate modal sentences verifiable in T underany T -arithmetical interpretation. Boolos [3] asked if QPL ( PA ) is recursivelyenumerable or not, and in contrast to the propositional case, Vardanyan [14]proved that QPL ( PA ) is Π -complete. Hence the analogue of Solovay’s arith-metical completeness theorem never holds in the case of predicate logic. More-over, Montagna [12] investigated that some results which hold in the case ofpropositional logic are not inherited in predicate case. Among other things, he ∗ Email: [email protected] † Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan. a r X i v : . [ m a t h . L O ] J a n roved that QPL ( PA ) is not a subset of QPL ( BG ), where BG is the Bernays–G¨odel set theory. Thus QPL ( T ) can vary depending on the theory T .Moreover, Artemov [1] showed that predicate provability logic QPL ( T ) of T can be different depending on the choice of a formula defining T . More precisely,we say that a formula τ ( v ) is a definition of a theory T if for any natural number n , τ ( n ) is true if and only if n is the G¨odel number of an axiom of T . For eachΣ definition τ ( v ) of T , we can construct a Σ provability predicate Pr τ ( x ) of T saying that “ x is (the G¨odel number of a formula) provable in the theory definedby τ ( v )”. The notion of τ -arithmetical interpretations is introduced as well byusing Pr τ ( x ) instead of Pr T ( x ). Then, the predicate provability logic QPL τ ( T )of τ ( v ) is defined to be the set of all predicate modal sentences provable in T under all τ -arithmetical interpretations. Artemov proved that for any Σ -soundrecursively enumerable extension T of PA and any Σ definition τ ( v ) of T ,there exists a Σ definition τ ( v ) of T such that QPL τ ( T ) (cid:42) QPL τ ( T ).The results of Montagna and Artemov seem to indicate that inclusion rela-tions between predicate provability logics are rarely established. Indeed, Kura-hashi [9] proved that for any natural numbers i and j with 0 < i < j , there existsa Σ definition σ i ( v ) of the theory IΣ i such that for all Σ definitions σ j ( v ) of IΣ j , QPL σ i ( IΣ i ) (cid:42) QPL σ j ( IΣ j ) and QPL σ j ( IΣ j ) (cid:42) QPL σ i ( IΣ i ). The situationof the inclusion relation between predicate provability logics is completely dif-ferent from that of propositional case: it is known that for any theories T and T , at least one of PL ( T ) ⊆ PL ( T ) and PL ( T ) ⊆ PL ( T ) holds (cf. Visser [16]).From this point of view, in the present paper, we investigate several conse-quences of the inclusion QPL τ ( T ) ⊆ QPL τ ( T ) between predicate provabilitylogics. Among other things, we prove that if QPL τ ( T ) ⊆ QPL τ ( T ), then1. T + Con τ is a subtheory of T + Con τ ;2. T is Σ -conservative over T ;3. Con τ and Con τ are provably equivalent over T ; and4. For any formula ϕ ( (cid:126)x ), T (cid:96) ∀ (cid:126)x (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ x ) (cid:113) ) (cid:17) . Thus from our results, we certify that inclusion relation between predicate prov-ability logics holds only under limited situations. Moreover, our these resultsalso show that the predicate provability logic
QPL τ ( T ) is not only complex,but also possesses much information about the theory T and the provabilitypredicate Pr τ ( x ).We also investigate provability logics with respect to Σ arithmetical in-terpretations. In the propositional case, a T -arithmetical interpretation f T iscalled Σ if for any propositional variable p , f T ( p ) is a Σ sentence. Let PL Σ ( T )be the set of all propositional modal formulas A such that T (cid:96) f T ( A ) for every T -arithmetical interpretation f T which is Σ . Visser [15] proved that PL Σ ( PA )2s also recursive and exactly the propositional modal logic GLV . In the pred-icate case, Berarducci [2] also proved that
QPL Σ ( PA ) is Π -complete. Thus,the situations of Σ provability logics do not seem to be different from those ofusual provability logics.On the other hand, there is an advantage to dealing with Σ arithmeticalinterpretations for our purposes, which allows us to improve Artemov’s Lemmaused in the proof of Vardanyan’s theorem. Then, we can give a necessary andsufficient condition for the inclusion relation between predicate provability log-ics with respect to Σ arithmetical interpretations. Namely, we prove that QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ) if and only if T is a subtheory of T and for anyformula ϕ ( (cid:126)x ), T (cid:96) ∀ (cid:126)x (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) )). Let L A = { , S, + , × , <, = } be the language of first-order arithmetic. We call aset of L A -sentences simply a theory . Peano Arithmetic PA is the theory con-sisting of basic axioms for L A and induction axioms for L A -formulas. IΣ isthe theory obtained from PA by restricting induction axioms to Σ formulas.Throughout the present paper, T , T and T always denote recursively enumer-able consistent extensions of IΣ . Let Th ( T ) be the set of all L A -sentencesprovable in T . Also, for each class Γ of formulas, let Th Γ ( T ) := Th (Γ) ∩ Γ. Thestandard model of arithmetic is denoted by N . We say that T is Σ -sound ifevery element of Th Σ ( T ) is true in N .For each natural number n , the numeral for n is denoted by n . We fix somenatural G¨odel numbering, and for each L A -formula ϕ , let (cid:112) ϕ (cid:113) be the numeralfor the G¨odel number of ϕ . We say a formula τ ( v ) is a definition of a theory T if for any natural number n , N | = τ ( n ) if and only if n is the G¨odel numberof some axiom of T . Hereafter, we assume that τ ( v ), τ ( v ) and τ ( v ) alwaysdenote Σ definitions of T , T and T , respectively. Then, we can construct aΣ formula Pr τ ( x ) saying that “ x is (the G¨odel number of a formula) provablein the theory defined by τ ( v )”. The following fact is well-known. Fact 2.1 (Derivability conditions (see Boolos [4] and Lindstr¨om [11])) . For anyformulas ϕ ( (cid:126)x ) and ψ ( (cid:126)x ) ,1. If T (cid:96) ϕ ( (cid:126)x ) , then IΣ (cid:96) Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ;2. IΣ (cid:96) Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) → ψ ( (cid:126) ˙ x ) (cid:113) ) → (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) → Pr τ ( (cid:112) ψ ( (cid:126) ˙ x ) (cid:113) )) ;3. If ϕ ( (cid:126)x ) is a Σ formula, then IΣ (cid:96) ϕ ( (cid:126)x ) → Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) . Here (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) is an abbreviation for (cid:112) ϕ ( ˙ x , . . . , ˙ x n ) (cid:113) that is a primitive recur-sive term corresponding to a primitive recursive function calculating the G¨odelnumber of ϕ ( k , . . . , k n ) from k , . . . , k n .3et Con τ be the Π sentence ¬ Pr τ ( (cid:112) (cid:113) ) stating that the theory definedby τ ( v ) is consistent. For each sentence ϕ , let ( τ + ϕ )( v ) be the Σ definition τ ( v ) ∨ v = (cid:112) ϕ (cid:113) of T + ϕ . Then it is known that the formalized version ofdeduction theorem holds: IΣ (cid:96) ∀ x (Pr τ + ϕ ( x ) ↔ Pr τ ( (cid:112) ϕ (cid:113) ˙ → x )). Here u ˙ → v is a primitive recursive term corresponding to a primitive recursive functioncalculating the G¨odel number of ϕ → ψ from the G¨odel numbers of ϕ and ψ .The language of predicate modal logic is the language of first-order predicatelogic without function and constant symbols equipped with the unary modaloperators (cid:3) and ♦ . We may assume that the languages of predicate modal logicand first-order arithmetic have the same variables. Definition 2.2.
A mapping f from the set of all atomic formulas of predi-cate modal logic to the set of L A -formulas satisfying the following condition iscalled an arithmetical interpretation : For each atomic formula P ( x , . . . , x n ), f ( P ( x , . . . , x n )) is an L A -formula ϕ ( x , . . . , x n ) with the same free variables,and moreover f ( P ( y , . . . , y n )) is ϕ ( y , . . . , y n ) for any variables y , . . . , y n . Definition 2.3.
Each arithmetical interpretation f is uniquely extended toa mapping f τ from the set of all predicate modal formulas to the set of L A -formulas inductively as follows:1. f τ ( ⊥ ) is 0 = 1;2. f τ commutes with each propositional connective and quantifier;3. f τ ( (cid:3) A ( x , . . . , x n )) is the formula Pr τ ( (cid:112) f τ ( A ( ˙ x , . . . , ˙ x n )) (cid:113) ).Notice that any predicate modal formula A has the same free variables as f τ ( A ). We are ready to introduce the predicate provability logic of τ ( v ). Definition 2.4.
The predicate provability logic
QPL τ ( T ) of τ ( v ) is the set { A | A is a sentence and for all arithmetical interpretations f, T (cid:96) f τ ( A ) } . The main purpose of the present paper is to investigate inclusion relation
QPL τ ( T ) ⊆ QPL τ ( T ) between predicate provability logics. For this purpose,we heavily use Artemov’s Lemma (Fact 2.8) that is used in the proof of Var-danyan’s theorem on the Π -completeness of the predicate provability logic of PA . To state Artemov’s Lemma, we prepare some definitions. Definition 2.5.
We prepare predicate symbols P Z ( x ), P S ( x, y ), P A ( x, y, z ), P M ( x, y, z ), P L ( x, y ) and P E ( x, y ) corresponding to members 0, S , +, × , < and= of L A , respectively. For each L A -formula ϕ , let ϕ ∗ be a logically equivalent L A -formula whose each atomic formula is one of the forms x = 0, S ( x ) = y , x + y = z , x × y = z , x < y and x = y . Let ϕ ◦ be a relational formula obtainedfrom ϕ ∗ by replacing each atomic formula with corresponding relation symbolin { P Z , P S , P A , P M , P L , P E } adequately. Then ϕ ◦ is a predicate modal formula.4et Seq( s ) be the formula naturally expressing that “ s is a finite sequence”.Also let lh ( s ) and ( s ) x be primitive recursive terms corresponding to primi-tive recursive functions calculating the length and x -th component of a finitesequence s , respectively. Definition 2.6.
For each arithmetical interpretation f , let R f ( x, y ) be theformula ∃ s (Seq( s ) ∧ lh ( s ) = x +1 ∧ ( s ) x = y ∧ f ( P Z (( s ) )) ∧∀ z < x f ( P S (( s ) z , ( s ) z +1 ))) . Let R f ( (cid:126)x, (cid:126)y ) denote a conjunction R f ( x , y ) ∧ R f ( x , y ) ∧ · · · ∧ R f ( x n , y n ).The formula R f ( x, y ) means that y represents x under the interpretation that f ( P Z ( x )) and f ( P S ( x, y )) say “ x represents 0” and “ y represents the successorof a number represented by x ”, respectively. Definition 2.7.
Let D be the modal sentence (cid:94) K ∈{ Z,S,A,M,L,E } (cid:16) ∀ (cid:126)x ( P K ( (cid:126)x ) → (cid:3) P K ( (cid:126)x )) ∧ ∀ (cid:126)x ( ¬ P K ( (cid:126)x ) → (cid:3) ¬ P K ( (cid:126)x )) (cid:17) . Fact 2.8 (Artemov’s Lemma (see [4, p.232])) . There exists an L A -sentence χ such that IΣ (cid:96) χ and for any arithmetical interpretation f and L A -formula ϕ ( (cid:126)x ) , IΣ (cid:96) Con τ ∧ f τ (D) ∧ f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → (cid:16) ϕ ( (cid:126)x ) ↔ f τ ( ϕ ◦ ( (cid:126)y )) (cid:17) . We give a short outline of a proof of Artemov’s Lemma based on the pre-sentation in [8]. Let M be a model of IΣ + Con τ ∧ f τ (D) ∧ f τ ( χ ◦ ). By theaid of f τ ( χ ◦ ), f τ ( P E ( x, y )) defines an equivalence relation ∼ on M . Let [ a ] bethe equivalence class of a ∈ M with respect to ∼ . Then, the relations on M defined by the formulas P K ( (cid:126)x ) for K ∈ { Z, S, A, M, L } induce an L A -structure M f with the domain { [ a ] | a ∈ M } . For instance, M f | = [ a ] + [ b ] = [ c ] ⇐⇒ M | = f τ ( P A ( a, b, c )). The sentence f τ ( χ ◦ ) guarantees that M f is well-definedand indeed an L A -structure satisfying a sufficiently strong fragment of IΣ , andthat for any (cid:126)a ∈ M , M f | = ϕ ( (cid:126) [ a ]) ⇐⇒ M | = f τ ( ϕ ◦ ( (cid:126)a )). Also M is isomorphicto an initial segment of M f via an embedding defined by the formula R f ( x, y ).Moreover, from the sentence Con τ ∧ f τ (D), we obtain the equivalences f τ ( P K ( (cid:126)x )) ↔ Pr τ ( (cid:112) f τ ( P K ( (cid:126) ˙ x )) (cid:113) ) and ¬ f τ ( P K ( (cid:126)x )) ↔ Pr τ ( (cid:112) ¬ f τ ( P K ( (cid:126) ˙ x )) (cid:113) )in M for each K ∈ { Z, S, A, M, L, E } . Then both f τ ( P K ( (cid:126)x )) and ¬ f τ ( P K ( (cid:126)x ))are equivalent to Σ formulas in M . By applying a proof of Tennenbaum’stheorem (see Kaye [7]), we obtain that M and M f are in fact isomorphic, andhence are elementarily equivalent. Therefore, if M | = R f ( (cid:126)a,(cid:126)b ), then M | = ϕ ( (cid:126)a )is equivalent to M f | = ϕ ( (cid:126) [ b ]). Hence M | = ϕ ( (cid:126)a ) ↔ f τ ( ϕ ◦ ( (cid:126)b )).In the proof of Artemov’s Lemma, the following facts are also used.5 act 2.9 (See Boolos [4, Lemma 6]) . For any Σ formula ϕ ( (cid:126)x ) and arithmeticalinterpretation f , IΣ (cid:96) f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → (cid:16) ϕ ( (cid:126)x ) → f τ ( ϕ ◦ ( (cid:126)y )) (cid:17) . Fact 2.10 (See Boolos [4, Lemma 8]) . For any arithmetical interpretation f , IΣ (cid:96) Con τ ∧ f τ (D) ∧ f τ ( χ ◦ ) → ∀ y ∃ xR f ( x, y ) . Facts 2.9 and 2.10 follow from the observations that M is isomorphic toan initial segment of M f and R f ( x, y ) defines a surjection from M onto M f ,respectively. In Boolos [4], these facts including Artemov’s Lemma are statedin the forms that the corresponding formulas are proved in PA , and de Jonge[5] proved that PA is replaced with IΣ (see also [8]). Definition 2.11.
An arithmetical interpretation f is natural if for each K ∈{ Z, S, A, M, L, E } , f maps P K ( (cid:126)x ) to the intended atomic formula (for example, f ( P A ( x, y, z )) is x + y = z ).For every predicate modal formula A , let (cid:0) A be an abbreviation for A ∧ (cid:3) A . Proposition 2.12.
Let f be any natural arithmetical interpretation.1. For any L A -formula ϕ ( (cid:126)x ) , IΣ (cid:96) ∀ (cid:126)x ( f τ ( ϕ ◦ ( (cid:126)x )) ↔ ϕ ( (cid:126)x )) ;2. IΣ (cid:96) f τ ( (cid:0) D) ∧ f τ ( (cid:0) χ ◦ ) .Proof.
1. By induction on the construction of ϕ ( (cid:126)x ).2. For each K ∈ { Z, S, A, M, L, E } , since f τ ( P K ( (cid:126)x )) is ∆ , it follows fromFact 2.1.3 that IΣ proves f τ ( P K ( (cid:126)x )) → Pr τ ( (cid:112) f τ ( P K ( (cid:126) ˙ x )) (cid:113) ) and ¬ f τ ( P K ( (cid:126)x )) → Pr τ ( (cid:112) ¬ f τ ( P K ( (cid:126) ˙ x )) (cid:113) ). Thus IΣ (cid:96) f τ (D). By Fact 2.1.1, IΣ (cid:96) Pr τ ( (cid:112) f τ (D) (cid:113) ),and hence IΣ (cid:96) f τ ( (cid:0) D).Also by clause 1, IΣ (cid:96) f τ ( χ ◦ ) ↔ χ . Since IΣ proves χ , IΣ (cid:96) f τ ( χ ◦ ). Asabove, IΣ (cid:96) f τ ( (cid:0) χ ◦ ) also holds.Artemov’s Lemma is used to prove Vardanyan’s theorem, but what is im-portant to us is the following observation by Visser and de Jonge. Fact 2.13 (Visser and de Jonge [17, Theorem 3]) . For any L A -sentence ϕ , thefollowing are equivalent:1. T + Con τ (cid:96) ϕ .2. ♦ (cid:62) ∧ D ∧ χ ◦ → ϕ ◦ ∈ QPL τ ( T ) . We give a proof of Visser and de Jonge’s fact.6 roof. (1 ⇒ T + Con τ (cid:96) ϕ . By Artemov’s Lemma, for any arith-metical interpretation f , T (cid:96) Con τ ∧ f τ (D) ∧ f τ ( χ ◦ ) → f τ ( ϕ ◦ ) . Thus T (cid:96) f τ ( ♦ (cid:62) ∧ D ∧ χ ◦ → ϕ ◦ ). Hence ♦ (cid:62) ∧ D ∧ χ ◦ → ϕ ◦ ∈ QPL τ ( T ).(2 ⇒ ♦ (cid:62) ∧ D ∧ χ ◦ → ϕ ◦ ∈ QPL τ ( T ). For a natural arithmeticalinterpretation f , T (cid:96) Con τ ∧ f τ (D) ∧ f τ ( χ ◦ ) → f τ ( ϕ ◦ ) . By Proposition 2.12, T + Con τ (cid:96) ϕ .Visser and de Jonge’s fact states that QPL τ ( T ) has the complete informationabout Th ( T + Con τ ). Then we obtain some corollaries concerning inclusionsbetween predicate provability logics. Corollary 2.14.
1. If
QPL τ ( T ) ⊆ QPL τ ( T ) , then Th ( T + Con τ ) ⊆ Th ( T + Con τ ) ;2. If QPL τ ( T ) = QPL τ ( T ) , then Th ( T + Con τ ) = Th ( T + Con τ ) .Proof.
1. Suppose
QPL τ ( T ) ⊆ QPL τ ( T ). Let ϕ be any L A -sentence with T + Con τ (cid:96) ϕ . Then from Fact 2.13, ♦ (cid:62) ∧ D ∧ χ ◦ → ϕ ◦ ∈ QPL τ ( T ).By the supposition, ♦ (cid:62) ∧ D ∧ χ ◦ → ϕ ◦ ∈ QPL τ ( T ). From Fact 2.13 again, T + Con τ (cid:96) ϕ . Therefore Th ( T + Con τ ) ⊆ Th ( T + Con τ ).2 follows from clause 1.The following corollary is an immediate consequence of Corollary 2.14.2. Corollary 2.15. If QPL τ ( T ) = QPL τ ( T ) and Th ( T ) ⊆ Th ( T ) , then T (cid:96) Con τ ↔ Con τ . Inspired by Visser and de Jonge’s fact, we explore further consequences of in-clusion relationships between predicate provability logics that result from Arte-mov’s Lemma.
In this subsection, we prove variations of Visser and de Jonge’s Fact 2.13 and itsconsequences. The following proposition is a variation of Fact 2.13 with respectto Σ sentences. Proposition 3.1.
For any Σ sentence ϕ , the following are equivalent: . T (cid:96) ϕ .2. χ ◦ → ϕ ◦ ∈ QPL τ ( T ) .Proof. (1 ⇒ T (cid:96) ϕ . By Fact 2.9, for any arithmetical interpretation f , IΣ (cid:96) f τ ( χ ◦ ) ∧ ϕ → f τ ( ϕ ◦ ). Hence T (cid:96) f τ ( χ ◦ → ϕ ◦ ). We have χ ◦ → ϕ ◦ ∈ QPL τ ( T ).(2 ⇒ Corollary 3.2.
1. If
QPL τ ( T ) ⊆ QPL τ ( T ) , then Th Σ ( T ) ⊆ Th Σ ( T ) ;2. If QPL τ ( T ) = QPL τ ( T ) , then Th Σ ( T ) = Th Σ ( T ) . By applying Fact 2.9, Corollary 2.15 is strengthened as follows.
Proposition 3.3. If QPL τ ( T ) ⊆ QPL τ ( T ) , then T (cid:96) Con τ ↔ Con τ .Proof. Suppose
QPL τ ( T ) ⊆ QPL τ ( T ). Then, T (cid:96) Con τ → Con τ by Corol-lary 2.15.1, and so it suffices to prove T (cid:96) Con τ → Con τ . Let f be anyarithmetical interpretation. Since ¬ Con τ is a Σ sentence, by Fact 2.9, IΣ (cid:96) f τ ( χ ◦ ) → ( ¬ Con τ → f τ ( ¬ Con ◦ τ )) . Hence T (cid:96) f τ ( χ ◦ ∧ (cid:3) ⊥ → ¬ Con ◦ τ ), and thus χ ◦ ∧ (cid:3) ⊥ → ¬ Con ◦ τ is in QPL τ ( T ). From the supposition, χ ◦ ∧ (cid:3) ⊥ → ¬ Con ◦ τ ∈ QPL τ ( T ). Byconsidering a natural arithmetical interpretation, we obtain that T proves ¬ Con τ → ¬ Con τ . Therefore T (cid:96) Con τ → Con τ . Corollary 3.4. If T (cid:96) Con τ , then QPL τ ( T ) (cid:42) QPL τ ( T ) .Proof. Assume T (cid:96) Con τ . If QPL τ ( T ) ⊆ QPL τ ( T ), then by Proposition3.3, T (cid:96) Con τ ↔ Con τ . From the supposition, T (cid:96) Con τ and this contra-dicts G¨odel’s second incompleteness theorem. Therefore we get QPL τ ( T ) (cid:42) QPL τ ( T ).The following corollary is a refinement of the result of Artemov [1]. Corollary 3.5.
Suppose that T is Σ -sound. Then, for any Σ definition τ ( v ) of T , there exists a Σ definition τ (cid:48) ( v ) of T such that QPL τ ( T ) (cid:42) QPL τ (cid:48) ( T ) and QPL τ (cid:48) ( T ) (cid:42) QPL τ ( T ) . roof. Let τ ( v ) be any Σ definition of T . Since ¬ Con τ is Σ , by Fact 2.1.3, T (cid:96) ¬ Con τ → Pr τ ( (cid:112) ¬ Con τ (cid:113) ). Equivalently, T (cid:96) Con τ +Con τ → Con τ . Since T is Σ -sound, Con τ +Con τ is a true Π sentence. Then, it is known that there existsa Σ definition τ (cid:48) ( v ) of T such that T (cid:96) Con τ (cid:48) ↔ Con τ +Con τ (cf. Lindstr¨om[11, Theorem 2.8.(b)]).Suppose, towards a contradiction, T (cid:96) Con τ → Con τ (cid:48) . Then, T provesCon τ → Con τ +Con τ and Pr τ ( (cid:112) ¬ Con τ (cid:113) ) → ¬ Con τ . By L¨ob’s theorem, T alsoproves ¬ Con τ . This contradicts the Σ -soundness of T . Thus T (cid:48) Con τ → Con τ (cid:48) .Moreover, T (cid:48) Con τ ↔ Con τ (cid:48) . It follows from Proposition 3.3 that QPL τ ( T ) (cid:42) QPL τ (cid:48) ( T ) and QPL τ (cid:48) ( T ) (cid:42) QPL τ ( T ). In this subsection, we investigate further consequences of inclusions betweenpredicate provability logics via Artemov’s Lemma. In particular, we show thatsome provable equivalences of provability predicates are derived from inclusion.First, we prepare the following lemma.
Lemma 3.6.
Let f be any arithmetical interpretation.1. PA (cid:96) f τ (D) → ( R f ( x, y ) → Pr τ ( (cid:112) R f ( ˙ x, ˙ y ) (cid:113) )) ;2. If f ( P Z ( x )) and f ( P S ( x, y )) are Σ formulas, then IΣ (cid:96) R f ( x, y ) → Pr τ ( (cid:112) R f ( ˙ x, ˙ y ) (cid:113) ) .Proof.
1. By the definition of D, f τ ( P Z ( x )) → Pr τ ( (cid:112) f τ ( P Z ( ˙ x )) (cid:113) ) and f τ ( P S ( x, y )) → Pr τ ( (cid:112) f τ ( P S ( ˙ x, ˙ y )) (cid:113) ) are provable in PA + f τ (D). Also if PA (cid:96) ϕ → Pr τ ( (cid:112) ϕ (cid:113) )and PA (cid:96) ϕ → Pr τ ( (cid:112) ϕ (cid:113) ), then PA (cid:96) ϕ ∧ ϕ → Pr τ ( (cid:112) ϕ ∧ ϕ (cid:113) ) and T (cid:96) ∃ sϕ → Pr τ ( (cid:112) ∃ sϕ (cid:113) ). Thus it suffices to show that PA + f τ (D) proves ∀ z < x f τ ( P S (( s ) z , ( s ) z +1 )) → Pr τ ( (cid:112) ∀ z < ˙ x f τ ( P S (( ˙ s ) z , ( ˙ s ) z +1 )) (cid:113) ) . Let ψ ( x ) denote this formula. Since T (cid:96) ∀ z < f τ ( P S (( s ) z , ( s ) z +1 )), by Fact2.1.1, PA (cid:96) Pr τ ( (cid:112) ∀ z < f τ ( P S (( ˙ s ) z , ( ˙ s ) z +1 )) (cid:113) ). Thus PA (cid:96) ψ (0). Also PA + f τ (D) proves ψ ( x ) ∧ ∀ z < S ( x ) f τ ( P S (( s ) z , ( s ) z +1 )) → ∀ z < x f τ ( P S (( s ) z , ( s ) z +1 )) ∧ f τ ( P S (( s ) x , ( s ) x +1 )) , → Pr τ ( (cid:112) ∀ z < ˙ x f τ ( P S (( ˙ s ) z , ( ˙ s ) z +1 )) ∧ f τ ( P S (( ˙ s ) ˙ x , ( ˙ s ) ˙ x +1 )) (cid:113) ) , → Pr τ ( (cid:112) ∀ z < S ( ˙ x ) f τ ( P S (( ˙ s ) z , ( ˙ s ) z +1 )) (cid:113) ) . Hence PA + f τ (D) (cid:96) ψ ( x ) → ψ ( S ( x )), and by the induction axiom, we conclude PA + f τ (D) (cid:96) ∀ xψ ( x ).2. If f ( P Z ( x )) and f ( P S ( x, y )) are Σ formulas, then R f ( x, y ) is also a Σ formula. Then the statement follows from Fact 2.1.3.We are ready to prove one of our main theorem of this subsection.9 heorem 3.7. Suppose Th ( PA ) ⊆ Th ( T ) . If QPL τ ( T ) ⊆ QPL τ ( T ) , thenfor any L A -formula ϕ ( (cid:126)y ) , T (cid:96) ∀ (cid:126)y (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ↔ Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) (cid:17) . Proof.
Suppose Th ( PA ) ⊆ Th ( T ) and QPL τ ( T ) ⊆ QPL τ ( T ). Let f be anyarithmetical interpretation. By Artemov’s Lemma, IΣ (cid:96) Con τ ∧ f τ (D) ∧ f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → ( ϕ ( (cid:126)x ) ↔ f τ ( ϕ ◦ ( (cid:126)y ))) . Then T proves f τ (D) ∧ f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → ((Con τ → ϕ ( (cid:126)x )) ↔ (Con τ → f τ ( ϕ ◦ ( (cid:126)y )))) . By Fact 2.1, we have IΣ (cid:96) f τ ( (cid:3) D) ∧ f τ ( (cid:3) χ ◦ ) ∧ Pr τ ( (cid:112) R f ( (cid:126) ˙ x, (cid:126) ˙ y ) (cid:113) ) → (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ f τ ( (cid:3) ( ♦ (cid:62) → ϕ ◦ ( (cid:126)y ))) (cid:17) . (1)By Artemov’s Lemma again, IΣ (cid:96) Con τ ∧ f τ (D) ∧ f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ f τ (Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) (cid:17) . (2)From Lemma 3.6.1, PA + f τ (D) (cid:96) R f ( (cid:126)x, (cid:126)y ) → Pr τ ( (cid:112) R f ( (cid:126) ˙ x, (cid:126) ˙ y ) (cid:113) ). By combiningthis with (1) and (2), we obtain PA (cid:96) Con τ ∧ f τ ( (cid:0) D) ∧ f τ ( (cid:0) χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → (cid:16) f τ (Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) ↔ f τ ( (cid:3) ( ♦ (cid:62) → ϕ ◦ ( (cid:126)y ))) (cid:17) . Since (cid:126)x does not appear in the consequent of the formula, PA (cid:96) Con τ ∧ f τ ( (cid:0) D) ∧ f τ ( (cid:0) χ ◦ ) ∧ ∃ (cid:126)xR f ( (cid:126)x, (cid:126)y ) → (cid:16) f τ (Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) ↔ f τ ( (cid:3) ( ♦ (cid:62) → ϕ ◦ ( (cid:126)y ))) (cid:17) . From Fact 2.10, IΣ (cid:96) Con τ ∧ f τ (D) ∧ f τ ( χ ◦ ) → ∀ (cid:126)y ∃ (cid:126)xR f ( (cid:126)x, (cid:126)y ). Hence PA (cid:96) Con τ ∧ f τ ( (cid:0) D) ∧ f τ ( (cid:0) χ ◦ ) → (cid:16) f τ (Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) ↔ f τ ( (cid:3) ( ♦ (cid:62) → ϕ ◦ ( (cid:126)y ))) (cid:17) . Since Th ( PA ) ⊆ Th ( T ), we obtain that the sentence ∀ (cid:126)y (cid:16) ♦ (cid:62) ∧ (cid:0) D ∧ (cid:0) χ ◦ → (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ↔ (cid:3) ( ♦ (cid:62) → ϕ ◦ ( (cid:126)y )) (cid:17)(cid:17)
10s contained in
QPL τ ( T ). By the supposition, this sentence is also in QPL τ ( T ).By considering a natural arithmetical interpretation and by Proposition 2.12, T + Con τ (cid:96) ∀ (cid:126)y (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ↔ Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) (cid:17) . By Proposition 3.3, T (cid:96) Con τ → Con τ . Thus T + ¬ Con τ (cid:96) ¬ Con τ , andhence T + ¬ Con τ (cid:96) ∀ (cid:126)y (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ↔ Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) (cid:17) . Therefore we conclude T (cid:96) ∀ (cid:126)y (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) ↔ Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ y ) (cid:113) ) (cid:17) . In our proof of Theorem 3.7, Lemma 3.6 is used to replace the formulaPr τ ( (cid:112) R f ( (cid:126) ˙ x, (cid:126) ˙ y ) (cid:113) ) with R f ( (cid:126)x, (cid:126)y ) in the antecedent of a formula. If ϕ is a sentence,then this procedure is no longer needed, and so the proof proceeds without usingLemma 3.6. Then other parts of our proof of Theorem 3.7 work within IΣ .Thus we also obtain the following theorem. Theorem 3.8. If QPL τ ( T ) ⊆ QPL τ ( T ) , then for any L A -sentence ϕ , T (cid:96) Pr τ ( (cid:112) Con τ → ϕ (cid:113) ) ↔ Pr τ ( (cid:112) Con τ → ϕ (cid:113) ) . Using Fact 2.9, we prove a variation of Theorem 3.7 with respect to Π formulas. Theorem 3.9.
Suppose Th ( PA ) ⊆ Th ( T ) . If QPL τ ( T ) ⊆ QPL τ ( T ) , thenfor any Π formula ϕ ( (cid:126)y ) , T (cid:96) ∀ (cid:126)y (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) → Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) )) . Proof.
Suppose Th ( PA ) ⊆ Th ( T ) and QPL τ ( T ) ⊆ QPL τ ( T ). Let f beany arithmetical interpretation and let ϕ ( (cid:126)y ) be any Π formula. Since ¬ ϕ ( (cid:126)y )is Σ , by Fact 2.9, IΣ (cid:96) f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) ∧ ¬ ϕ ( (cid:126)x ) → f τ ( ¬ ϕ ◦ ( (cid:126)y )). Then, T (cid:96) f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) ∧ f τ ( ϕ ◦ ( (cid:126)y )) → ϕ ( (cid:126)x ). By Fact 2.1, IΣ (cid:96) f τ ( (cid:3) χ ◦ ) ∧ Pr τ ( (cid:112) R f ( (cid:126) ˙ x, (cid:126) ˙ y ) (cid:113) ) ∧ f τ ( (cid:3) ϕ ◦ ( (cid:126)y )) → Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) . (3)By Artemov’s Lemma, IΣ provesCon τ ∧ f τ (D) ∧ f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) ∧ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) → f τ (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) . (4)By combining Lemma 3.6 with (3) and (4), PA provesCon τ ∧ f τ (D) ∧ f τ ( (cid:0) χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) ∧ f τ ( (cid:3) ϕ ◦ ( (cid:126)y )) → f τ (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) .
11s in the proof of Theorem 3.7, R f ( (cid:126)x, (cid:126)y ) is removed from the antecedent of theformula, that is, PA (cid:96) Con τ ∧ f τ (D) ∧ f τ ( (cid:0) χ ◦ ) ∧ f τ ( (cid:3) ϕ ◦ ( (cid:126)y )) → f τ (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) . Since Th ( PA ) ⊆ Th ( T ), ∀ (cid:126)y (cid:16) ♦ (cid:62) ∧ D ∧ (cid:0) χ ◦ ∧ (cid:3) ϕ ◦ ( (cid:126)y ) → Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ (cid:17) ∈ QPL τ ( T ) ⊆ QPL τ ( T ) . By considering a natural arithmetical interpretation, we obtain T + Con τ (cid:96) ∀ (cid:126)y (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) → Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) )) . By Proposition 3.3, T + ¬ Con τ (cid:96) ¬ Con τ , and in particular, T + ¬ Con τ proves ∀ (cid:126)y Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ). Therefore we conclude T (cid:96) ∀ (cid:126)y (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) → Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) )) . As above, we also obtain the following theorem.
Theorem 3.10. If QPL τ ( T ) ⊆ QPL τ ( T ) , then for any Π sentence ϕ , T (cid:96) Pr τ ( (cid:112) ϕ (cid:113) ) → Pr τ ( (cid:112) ϕ (cid:113) ) . As consequences of theorems proved in this subsection, we obtain severalcorollaries.
Corollary 3.11. If QPL τ ( T ) ⊆ QPL τ ( T ) and T is Σ -sound, then1. Th ( T + Con τ ) = Th ( T + Con τ ) ; and2. Th Π ( T ) ⊆ Th Π ( T ) .Proof. Suppose
QPL τ ( T ) ⊆ QPL τ ( T ) and T is Σ -sound.1. By Corollary 2.14, Th ( T + Con τ ) ⊆ Th ( T + Con τ ). On the other hand,let ϕ be any L A -sentence ϕ with T +Con τ (cid:96) ϕ . Then, T (cid:96) Pr τ ( (cid:112) Con τ → ϕ (cid:113) )by Fact 2.1.1. By Theorem 3.8, T (cid:96) Pr τ ( (cid:112) Con τ → ϕ (cid:113) ) ↔ Pr τ ( (cid:112) Con τ → ϕ (cid:113) ) , and hence T (cid:96) Pr τ ( (cid:112) Con τ → ϕ (cid:113) ). Then, Pr τ ( (cid:112) Con τ → ϕ (cid:113) ) is true in N because T is Σ -sound. This means T (cid:96) Con τ → ϕ . Therefore we conclude Th ( T + Con τ ) ⊆ Th ( T + Con τ ).2. Let ϕ be any Π sentence such that T (cid:96) ϕ . Then T (cid:96) Pr τ ( (cid:112) ϕ (cid:113) )by Fact 2.1.1. By Theorem 3.10, T (cid:96) Pr τ ( (cid:112) ϕ (cid:113) ) → Pr τ ( (cid:112) ϕ (cid:113) ), and hence T (cid:96) Pr τ ( (cid:112) ϕ (cid:113) ). Since T is Σ -sound, T (cid:96) ϕ . Thus Th Π ( T ) ⊆ Th Π ( T ).12n the next subsection, we will prove that the assumption of the Σ -soundnessof T in the statement of Corollary 3.11 cannot be removed (see Propositions3.21 and 3.22). Remark 3.12.
We say that a theory T is faithfully interpretable in a the-ory T if there exists an interpretation I of T in T such that for any L A -sentence ϕ , T (cid:96) ϕ if and only if T (cid:96) I ( ϕ ). Lindstr¨om [10] proved thatif T and T are consistent recursively enumerable extensions of PA , then T is faithfully interpretable in T if and only if Th Π ( T ) ⊆ Th Π ( T ) and Th Σ ( T ) ⊆ Th Σ ( T ). Therefore from Corollary 3.11 and Proposition 3.2, weobtain that if Th ( PA ) ⊆ Th ( T ) ∩ Th ( T ), QPL τ ( T ) ⊆ QPL τ ( T ) and T isΣ -sound, then T is faithfully interpretable in T .We show that if T is Σ -sound and proves the Σ -soundness of T , then QPL τ ( T ) and QPL τ ( T ) are incomparable in the following strong sense. Corollary 3.13.
Suppose that T is Σ -sound and for some Σ definition σ ( v ) of T , for all Σ sentences ϕ , T (cid:96) Pr σ ( (cid:112) ϕ (cid:113) ) → ϕ . Then, for any respec-tive Σ definitions τ ( v ) and τ ( v ) of T and T , QPL τ ( T ) (cid:42) QPL τ ( T ) and QPL τ ( T ) (cid:42) QPL τ ( T ) .Proof. First, we show
QPL τ ( T ) (cid:42) QPL τ ( T ). By the supposition, T provesPr σ ( (cid:112) (cid:113) ) → σ . On the other hand, T (cid:48) Con σ by the second incompleteness theorem. Since Con σ is a Π sentence, Th Π ( T ) (cid:42) Th Π ( T ). Therefore QPL τ ( T ) (cid:42) QPL τ ( T ) by Corollary 3.11because T is Σ -sound.Secondly, we show QPL τ ( T ) (cid:42) QPL τ ( T ). Since ¬ Con τ is Σ , T provesPr σ ( (cid:112) ¬ Con τ (cid:113) ) → ¬ Con τ , and also proves Con τ → Con σ +Con τ . On theother hand, assume, towards a contradiction, Con τ → Con σ +Con τ is provablein T . Then, by the second incompleteness theorem, T + Con τ is inconsistent,and so T (cid:96) ¬ Con τ . By Fact 2.1.1, T (cid:96) Pr σ ( (cid:112) ¬ Con τ (cid:113) ). Hence T (cid:96) ¬ Con τ ,and this contradicts the Σ -soundness of T . We obtain T + Con τ (cid:48) Con τ → Con σ +Con τ . Therefore Th ( T ) (cid:42) Th ( T + Con τ ). By Corollary 2.14, weconclude QPL τ ( T ) (cid:42) QPL τ ( T ). Remark 3.14.
Let i and j be any natural numbers with 0 < i < j . Then,the theory IΣ j is Σ -sound and proves Pr IΣ i ( (cid:112) ϕ (cid:113) ) → ϕ for all Σ sentences ϕ (cf. H´ajek and Pudl´ak [6, Corollary I.4.34]). From Corollary 3.13, for any respec-tive Σ definitions σ i ( v ) and σ j ( v ) of IΣ i and IΣ j , QPL σ i ( IΣ i ) (cid:42) QPL σ j ( IΣ j )and QPL σ j ( IΣ j ) (cid:42) QPL σ i ( IΣ i ). This is a refinement of a result of Kurahashi[9]. Lemma 3.15.
Let σ ( v ) be any Σ definition of some theory. Suppose that forall L A -formulas ϕ ( (cid:126)x ) , T (cid:96) ∀ (cid:126)x (Pr σ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) )) . Then, for anypredicate modal formula A and any arithmetical interpretation f , T (cid:96) f σ ( A ) ↔ f τ ( A ) .Proof. We prove the lemma by induction on the construction of A . We onlygive a proof of the case that A is of the form (cid:3) B . Assume that T proves13 σ ( B ) ↔ f τ ( B ). Then, by Fact 2.1, IΣ (cid:96) Pr τ ( (cid:112) f σ ( B ) (cid:113) ) ↔ f τ ( (cid:3) B ). Since T (cid:96) f σ ( (cid:3) B ) ↔ Pr τ ( (cid:112) f σ ( B ) (cid:113) ) by the supposition, we obtain that f σ ( (cid:3) B ) ↔ f τ ( (cid:3) B ) is provable in T . Corollary 3.16. If Th ( PA ) ⊆ Th ( T ) and QPL τ ( T ) ⊆ QPL τ ( T ) , then QPL τ +Con τ ( T + Con τ ) ⊆ QPL τ +Con τ ( T + Con τ ) .Proof. Suppose Th ( PA ) ⊆ Th ( T ) and QPL τ ( T ) ⊆ QPL τ ( T ). Let A be anyelement of QPL τ +Con τ ( T + Con τ ) and f be arbitrary arithmetical interpreta-tion. Then, T + Con τ (cid:96) f τ +Con τ ( A ). Since Th ( T + Con τ ) ⊆ Th ( T + Con τ )by Corollary 2.14, T + Con τ (cid:96) f τ +Con τ ( A ). By Theorem 3.7, for any L A -formula ϕ ( (cid:126)x ), T (cid:96) ∀ (cid:126)x (cid:16) Pr τ +Con τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ +Con τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) (cid:17) . Thus by Lemma 3.15, T + Con τ (cid:96) f τ +Con τ ( A ) ↔ f τ +Con τ ( A ), and hence T +Con τ (cid:96) f τ +Con τ ( A ). Since f is arbitrary, A is contained in QPL τ +Con τ ( T +Con τ ).Moreover, we strengthen Proposition 3.3 and Corollary 3.16. Definition 3.17.
We define a sequence (Con nτ ) n ∈ N of Π consistency statementsof T inductively as follows:1. Con τ : ≡ n +1 τ : ≡ Con τ +Con nτ .Since ¬ Con nτ is a Σ sentence, IΣ (cid:96) ¬ Con nτ → Pr τ ( (cid:112) ¬ Con nτ (cid:113) ) by Fact 2.1.3.Equivalently, IΣ (cid:96) Con n +1 τ → Con nτ . Thus Con nτ ∧ Con τ +Con nτ is provablyequivalent to Con n +1 τ over IΣ . Corollary 3.18. If Th ( PA ) ⊆ Th ( T ) and QPL τ ( T ) ⊆ QPL τ ( T ) , then forany natural number n ≥ ,1. QPL τ +Con nτ ( T + Con nτ ) ⊆ QPL τ +Con nτ ( T + Con nτ ) ; and2. T (cid:96) Con nτ ↔ Con nτ .Proof. Suppose Th ( PA ) ⊆ Th ( T ) and QPL τ ( T ) ⊆ QPL τ ( T ).1. By induction on n ≥
1. For n = 1, the statement is exactly Corollary 3.16.Suppose QPL τ +Con nτ ( T + Con nτ ) ⊆ QPL τ +Con nτ ( T + Con nτ ). As commentedabove, Con nτ i ∧ Con τ i +Con nτi is equivalent to Con n +1 τ i for i ∈ { , } , and hence byCorollary 3.16, QPL τ +Con n +1 τ ( T + Con n +1 τ ) ⊆ QPL τ +Con n +1 τ ( T + Con n +1 τ ) .
2. By induction on n ≥
1. For n = 1, the statement is exactly Propo-sition 3.3. Suppose T (cid:96) Con nτ ↔ Con nτ . By clause 1, QPL τ +Con nτ ( T +14on nτ ) ⊆ QPL τ +Con nτ ( T +Con nτ ). Then by Proposition 3.3, T +Con nτ provesCon τ +Con nτ ↔ Con τ +Con nτ . This means T + Con nτ (cid:96) Con n +1 τ ↔ Con n +1 τ . (5)We prove T (cid:96) Con n +1 τ ↔ Con n +1 τ . Since T + Con n +1 τ (cid:96) Con nτ , it followsfrom (5) that T + Con n +1 τ (cid:96) Con n +1 τ . Conversely, since T + Con n +1 τ (cid:96) Con nτ , T + Con n +1 τ (cid:96) Con nτ by induction hypothesis. Then, T + Con n +1 τ (cid:96) Con n +1 τ from (5).Under certain suppositions, we give the following necessary and sufficientcondition for QPL τ ( T ) ⊆ QPL τ ( T ). Corollary 3.19.
Suppose that Th ( T ) ⊆ Th ( T ) and there exists a Π sentence π satisfying the following two conditions: • T (cid:96) Con τ → ¬ Pr τ ( (cid:112) π (cid:113) ) ; • T (cid:96) Pr τ ( (cid:112) π (cid:113) ) .Then, QPL τ ( T ) ⊆ QPL τ ( T ) if and only if T (cid:96) ¬ Con τ ∧ ¬ Con τ .Proof. ( ⇒ ): Suppose QPL τ ( T ) ⊆ QPL τ ( T ). Let π be a Π sentence satisfy-ing two conditions stated above. By Theorem 3.10, Pr τ ( (cid:112) π (cid:113) ) → Pr τ ( (cid:112) π (cid:113) )is provable in T , and hence T (cid:96) Pr τ ( (cid:112) π (cid:113) ) by the choice of π . On theother hand, by Corollary 2.14, Th ( T + Con τ ) ⊆ Th ( T + Con τ ), and thus T + Con τ (cid:96) ¬ Pr τ ( (cid:112) π (cid:113) ). Therefore T + Con τ is inconsistent, and we obtain T (cid:96) ¬ Con τ . By Proposition 3.3, T (cid:96) Con τ → Con τ . Hence T (cid:96) ¬ Con τ .( ⇐ ): Assume that T proves ¬ Con τ and ¬ Con τ . Then, for any L A -formula ϕ ( (cid:126)x ), T (cid:96) ∀ (cid:126)x (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) )). Let A be any element of QPL τ ( T ) and f be any arithmetical interpretation. Then, T (cid:96) f τ ( A ). Since Th ( T ) ⊆ Th ( T ), T (cid:96) f τ ( A ). By Lemma 3.15, f τ ( A ) ↔ f τ ( A ) is provablein T , and hence T (cid:96) f τ ( A ). Therefore A ∈ QPL τ ( T ). We have proved QPL τ ( T ) ⊆ QPL τ ( T ). In this subsection, we give some counterexamples to several statements. Beforegiving them, we prepare a lemma.
Lemma 3.20.
For any L A -sentence ϕ with T (cid:96) ϕ → Pr τ ( (cid:112) ϕ (cid:113) ) , QPL τ ( T ) ⊆ QPL τ + ϕ ( T + ϕ ) . Proof.
Suppose T (cid:96) ϕ → Pr τ ( (cid:112) ϕ (cid:113) ). Let A be any element of QPL τ ( T ) and f beany arithmetical interpretation. Then, T (cid:96) f τ ( A ). Since T + ϕ proves Pr τ ( (cid:112) ϕ (cid:113) ),for any L A -formula ψ ( (cid:126)x ), it follows from Fact 2.1.2 that T + ϕ (cid:96) Pr τ + ϕ ( (cid:112) ψ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) ϕ → ψ ( (cid:126) ˙ x ) (cid:113) ) , ↔ Pr τ ( (cid:112) ψ ( (cid:126) ˙ x ) (cid:113) ) . T + ϕ (cid:96) f τ ( A ) ↔ f τ + ϕ ( A ). Hence T + ϕ (cid:96) f τ + ϕ ( A ). Weconclude QPL τ ( T ) ⊆ QPL τ + ϕ ( T + ϕ ).The following two propositions show that in the statement of Corollary 3.11,the assumption of the Σ -soundness of T cannot be omitted. Proposition 3.21.
There exist consistent recursively enumerable extensions T and T of IΣ and respective Σ definitions τ ( v ) and τ ( v ) of T and T satisfying the following conditions:1. QPL τ ( T ) ⊆ QPL τ ( T ) ;2. T + Con τ and T + Con τ are consistent; and3. Th ( T + Con τ ) (cid:42) Th ( T + Con τ ) .Proof. Let T be any Σ -sound recursively enumerable extension of IΣ and τ ( v ) be any Σ definition of T . Also let ϕ be the Σ sentence ¬ Con τ . Then N | = ¬ ϕ . Let T := T + ϕ and τ ( v ) be ( τ + ϕ )( v ).1. Since ϕ is a Σ sentence, T (cid:96) ϕ → Pr τ ( (cid:112) ϕ (cid:113) ) by Fact 2.1.3. Then byLemma 3.20, QPL τ ( T ) ⊆ QPL τ ( T ).2. Since T is Σ -sound, T + Con τ is consistent. Suppose, towards acontradiction, that T + Con τ is inconsistent. Then T + ϕ (cid:96) ¬ Con τ + ϕ , andhence T (cid:96) ϕ → Pr τ ( (cid:112) ¬ ϕ (cid:113) ). Since T (cid:96) ϕ → Pr τ ( (cid:112) ϕ (cid:113) ), we have T (cid:96) ϕ →¬ Con τ . It follows T (cid:96) Pr τ ( (cid:112) ¬ Con τ (cid:113) ) → ¬ Con τ . By L¨ob’s theorem, T (cid:96)¬ Con τ . This contradicts the Σ -soundness of T . Therefore T + Con τ isconsistent.3. Since T + Con τ is also Σ -sound, T + Con τ (cid:48) ϕ . On the other hand, T + Con τ (cid:96) ϕ , and hence Th ( T + Con τ ) (cid:42) Th ( T + Con τ ). Proposition 3.22.
There exist consistent recursively enumerable extensions T and T of IΣ and respective Σ definitions τ ( v ) and τ ( v ) of T and T satisfying the following conditions:1. QPL τ ( T ) ⊆ QPL τ ( T ) ; and2. Th Π ( T ) (cid:42) Th Π ( T ) .Proof. Let T be arbitrary consistent recursively enumerable extension of IΣ and τ ( v ) be any Σ definition of T . Let ρ be a Π Rosser sentence of T ,and let T := T + ¬ ρ and τ ( v ) be ( τ + ¬ ρ )( v ). By Rosser’s theorem, T is consistent. Since ¬ ρ is Σ , by Lemma 3.20, QPL τ ( T ) ⊆ QPL τ ( T ). It isknown Th Π ( T ) (cid:42) Th Π ( T ) (see Lindstr¨om [11, Chapter 5 Exercise 1]).The following proposition shows that the converse implications of Proposi-tion 3.3, Theorem 3.7 and Corollary 3.11 do not hold. Proposition 3.23.
There exist consistent recursively enumerable extensions T and T of IΣ and respective Σ definitions τ ( v ) and τ ( v ) of T and T satisfying the following conditions: . IΣ (cid:96) Con τ ↔ Con τ ;2. T is Σ -sound and Th ( T + Con τ ) = Th ( T + Con τ ) ;3. For any L A -formula ϕ ( (cid:126)x ) , IΣ (cid:96) ∀ (cid:126)x (cid:16) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ x ) (cid:113) ) (cid:17) ; QPL τ ( T ) (cid:42) QPL τ ( T ) .Proof. Let T be any Σ -sound recursively enumerable extension of IΣ and τ ( v ) be any Σ definition of T . Let ρ be a Π Rosser sentence of T definedby using τ ( v ). Also let T := T + ρ and τ ( v ) be ( τ + ρ )( v ).1. Since IΣ (cid:96) Con τ ↔ ¬ Pr τ ( (cid:112) ¬ ρ (cid:113) ), IΣ (cid:96) Con τ ↔ Con τ .2. Let ψ be any Σ sentence with T (cid:96) ψ . Then T (cid:96) ¬ ρ ∨ ψ . Since T isΣ -sound, N | = ¬ ρ ∨ ψ . Since N | = ρ , N | = ψ . Hence T is Σ -sound.Moreover, since IΣ (cid:96) Con τ → ρ , T + Con τ is deductively equivalent to T + ρ + Con τ , and to T + Con τ .3. For any L A -formula ϕ ( (cid:126)x ), IΣ (cid:96) Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ +Con τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) , ↔ Pr τ + ρ +Con τ ρ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) , ↔ Pr τ ( (cid:112) Con τ → ϕ ( (cid:126) ˙ x ) (cid:113) ) .
4. Since T (cid:96) ρ , T (cid:96) Pr τ ( (cid:112) ρ (cid:113) ). Also T (cid:96) Con τ → ¬ Pr τ ( (cid:112) ρ (cid:113) ). Suppose,towards a contradiction, T (cid:96) ¬ Con τ . Then T + ρ (cid:96) ¬ Con τ . Since T + ¬ ρ (cid:96)¬ Con τ , we get T (cid:96) ¬ Con τ . This contradicts the Σ -soundness of T . Thus T (cid:48) ¬ Con τ . It follows from Corollary 3.19 that QPL τ ( T ) (cid:42) QPL τ ( T ). Σ arithmetical interpretations In this section, we investigate inclusions between predicate provability logicswith respect to Σ arithmetical interpretations. The main goal of this sectionis to give a necessary and sufficient condition for inclusion relation betweenpredicate provability logics with respect to Σ arithmetical interpretations. Definition 4.1.
An arithmetical interpretation f is Σ n if for any atomic formula P ( (cid:126)x ) of predicate modal logic, f ( P ( (cid:126)x )) is a Σ n formula.Notice that there are natural Σ arithmetical interpretations. We introducethe predicate provability logics with respect to Σ n arithmetical interpretations. Definition 4.2.
QPL Σ n τ ( T ) := { ϕ | ϕ is a sentence and for all Σ n arithmeticalinterpretations f , T (cid:96) f τ ( ϕ ) } . 17erarducci [2] proved that restricting arithmetical interpretations to Σ n doesnot change the complexity of predicate provability logics, that is, for each n ≥ PA with respect to Σ n arithmetical interpretations is also Π -complete.On the other hand, it is beneficial to deal with Σ arithmetical interpreta-tions in our study. In the proof of Artemov’s Lemma, the assumption Con τ ∧ f τ (D) is prepared to make the formulas f ( P K ( x )) and ¬ f ( P K ( x, y )) equiva-lent to Σ formulas for each K ∈ { Z, S, A, M, L, E } . In the case that f is aΣ arithmetical interpretation, the same result holds without the assumptionCon τ ∧ f τ (D) by adding sufficiently many theorems of IΣ to the sentence χ asconjuncts. This is guaranteed by the following equivalences: • ¬ P Z ( x ) ↔ ∃ yP S ( y, x ); • ¬ P S ( x, y ) ↔ ∃ z ( P S ( x, z ) ∧ ( P L ( z, y ) ∨ P L ( y, z ))); • ¬ P A ( x, y, z ) ↔ ∃ w ( P A ( x, y, w ) ∧ ( P L ( w, z ) ∨ P L ( z, w ))); • ¬ P M ( x, y, z ) ↔ ∃ w ( P M ( x, y, w ) ∧ ( P L ( w, z ) ∨ P L ( z, w ))); • ¬ P L ( x, y ) ↔ P E ( x, y ) ∨ P L ( y, x ); • ¬ P E ( x, y ) ↔ P L ( x, y ) ∨ P L ( y, x ).Thus we obtain the following variation of Artemov’s Lemma with respect to Σ arithmetical interpretations. Theorem 4.3 (Σ -Artemov’s Lemma) . There exists an L A -sentence χ suchthat IΣ (cid:96) χ and for any Σ arithmetical interpretation f and any L A -formula ϕ ( (cid:126)x ) , IΣ (cid:96) f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → ( ϕ ( (cid:126)x ) ↔ f τ ( ϕ ◦ ( (cid:126)y ))) . We also obtain a variation of Fact 2.10 with respect to Σ arithmeticalinterpretations. Proposition 4.4.
For any Σ arithmetical interpretation f , IΣ (cid:96) f τ ( χ ◦ ) → ∀ y ∃ xR f ( x, y ) . The following proposition is a variation of Fact 2.13 with respect to Σ arithmetical interpretations. Proposition 4.5.
For any L A -sentence ϕ , the following are equivalent:1. T (cid:96) ϕ .2. χ ◦ → ϕ ◦ ∈ QPL Σ τ ( T ) . roof. (1 ⇒ T (cid:96) ϕ . By Σ -Artemov’s Lemma, for any Σ arith-metical interpretation f , IΣ (cid:96) f τ ( χ ◦ ) → ( ϕ ↔ f τ ( ϕ ◦ )). Then T proves f τ ( χ ◦ → ϕ ◦ ). Thus χ ◦ → ϕ ◦ ∈ QPL Σ τ ( T ).(2 ⇒ χ ◦ → ϕ ◦ ∈ QPL Σ τ ( T ). By considering a natural Σ arithmetical interpretation, we obtain T (cid:96) ϕ .We prove the following main theorem of this section. Theorem 4.6.
The following are equivalent:1.
QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ) .2. Th ( T ) ⊆ Th ( T ) and for any L A -formula ϕ ( (cid:126)x ) , T (cid:96) ∀ (cid:126)x (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) )) .Proof. (1 ⇒ QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ).First, we prove Th ( T ) ⊆ Th ( T ). Let ϕ be any sentence with T (cid:96) ϕ . Thenby Proposition 4.5, χ ◦ → ϕ ◦ ∈ QPL Σ τ ( T ). By the supposition, this sentence isalso in QPL Σ τ ( T ). Then by Proposition 4.5 again, we obtain T (cid:96) ϕ . Therefore Th ( T ) ⊆ Th ( T ).Secondly, we prove the T -provable equivalence of two provability predicates.Let ϕ ( (cid:126)y ) be any L A -formula. By Σ -Artemov’s Lemma, for any Σ arithmeticalinterpretation f , IΣ (cid:96) f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → ( ϕ ( (cid:126)x ) ↔ f τ ( ϕ ◦ ( (cid:126)y ))) . By Fact 2.1, IΣ (cid:96) f τ ( (cid:3) χ ◦ ) ∧ Pr τ ( (cid:112) R f ( (cid:126) ˙ x, (cid:126) ˙ y ) (cid:113) ) → (cid:16) Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ f τ ( (cid:3) ϕ ◦ ( (cid:126)y )) (cid:17) . (6)By Σ -Artemov’s Lemma again, IΣ (cid:96) f τ ( χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → (cid:16) f τ (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) ↔ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) (cid:17) . (7)By Lemma 3.6.2, IΣ (cid:96) R f ( (cid:126)x, (cid:126)y ) → Pr τ ( (cid:112) R f ( (cid:126) ˙ x, (cid:126) ˙ y ) (cid:113) ). By combining thiswith (6) and (7), IΣ (cid:96) f τ ( (cid:0) χ ◦ ) ∧ R f ( (cid:126)x, (cid:126)y ) → (cid:16) f τ (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) ↔ f τ ( (cid:3) ϕ ◦ ( (cid:126)y )) (cid:17) . Since (cid:126)x does not appear in the consequent of the formula, IΣ (cid:96) f τ ( (cid:0) χ ◦ ) ∧ ∃ (cid:126)xR f ( (cid:126)x, (cid:126)y ) → (cid:16) f τ (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) ↔ f τ ( (cid:3) ϕ ◦ ( (cid:126)y )) (cid:17) . By Proposition 4.4, IΣ (cid:96) f τ ( χ ◦ ) → ∀ (cid:126)y ∃ (cid:126)xR f ( (cid:126)x, (cid:126)y ). Then, IΣ (cid:96) f τ ( (cid:0) χ ◦ ) → (cid:16) f τ (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ) ↔ f τ ( (cid:3) ϕ ◦ ( (cid:126)y )) (cid:17) .
19e obtain ∀ (cid:126)y (cid:16) (cid:0) χ ◦ → (cid:16) Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ◦ ↔ (cid:3) ϕ ◦ ( (cid:126)y ) (cid:17)(cid:17) ∈ QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ) . By considering a natural Σ arithmetical interpretation, we conclude T (cid:96) ∀ (cid:126)y (cid:16) Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) ↔ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ y ) (cid:113) ) (cid:17) . (2 ⇒ A be any element of QPL Σ τ ( T ) and f be any Σ arithmetical interpretation. Then, T (cid:96) f τ ( A ).Since Th ( T ) ⊆ Th ( T ), T (cid:96) f τ ( A ). By the assumption and Lemma 3.15, wehave T (cid:96) f τ ( A ) ↔ f τ ( A ), and thus T (cid:96) f τ ( A ). Therefore A is in QPL Σ τ ( T ).We have proved QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ).Similar to the proof of (2 ⇒
1) of Theorem 4.6, it can be proved that clause2 in the statement of Theorem 4.6 implies
QPL τ ( T ) ⊆ QPL τ ( T ). Corollary 4.7. If QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ) , then QPL τ ( T ) ⊆ QPL τ ( T ) . We propose the following proposition.
Problem 4.8.
Does the converse implication of Corollary 4.7 hold?
We close this section with the following corollary.
Corollary 4.9. If QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ) and T is Σ -sound, then QPL Σ τ ( T ) = QPL Σ τ ( T ) .Proof. Suppose
QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ) and T is Σ -sound. By Theorem4.6, Th ( T ) ⊆ Th ( T ) and T (cid:96) ∀ (cid:126)x (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) )) for any L A -formula ϕ ( (cid:126)x ). Let ψ be any L A -sentence with T (cid:96) ψ . Since T provesPr τ ( (cid:112) ψ (cid:113) ) by Fact 2.1.1, we have T (cid:96) Pr τ ( (cid:112) ψ (cid:113) ). Since T is Σ -sound, T (cid:96) ψ . We have shown Th ( T ) ⊆ Th ( T ). Then, for any L A -formula ϕ ( (cid:126)x ), T (cid:96)∀ (cid:126)x (Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) ) ↔ Pr τ ( (cid:112) ϕ ( (cid:126) ˙ x ) (cid:113) )). By Theorem 4.6, we conclude QPL Σ τ ( T ) ⊆ QPL Σ τ ( T ). References [1] Sergei N. Artemov. Numerically correct logics of provability (in Russian).
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