Canonical calculi with (n,k)-ary quantifiers

Logical Methods in Computer Science V ol. 4 (3:2) 2008, pp. 1–23 www .lmcs-online.org Submitted Feb . 1, 2007 Published Aug. 6, 2008 CANONIC AL CALCULI WITH ( n, k ) -AR Y QUANTIFI ERS ARNON A VR ON AND ANNA Z A MANSKY T el Aviv Universit y e-mail addr ess : { aa,annaz } @post.tau.ac.il Abstra ct. Propositional canonical Gentzen-t yp e sy stems, introd uced in 2001 by Av ron and Lev, are systems which in addition to the standard axioms and structu ral rules hav e only logical rules in which exactly one occu rrence of a connective is in trodu ced and no other connective is men tioned. A constructive coherence criterion for the non- trivialit y of suc h systems w as defined and it w as shown that a system of this k ind admits cut- elimination iff it is coherent. The semantics of such systems is provided using tw o-val ued non-deterministic matrices (2Nmatrices). In 2005 Zamansky and A vron extended th ese results to systems with unary qu antifiers of a very restricted form. In this pap er we substantial ly ex t end the characterizati on of canonical systems t o ( n, k ) -ary quantifiers, whic h bind k distinct v ariables and connect n formulas , and show th at the coherence criterion remains constructive for such systems. Then w e focus on the case of k ∈ { 0 , 1 } and for a canonical calculus G sho w that it is coherent precisely when it has a strongly chara cteristic 2Nmatrix, which in turn is equiv alen t to admitting strong cut-elimination. Introduction An ( n , k ) -ary quantifier 1 (for n > 0, k ≥ 0) is a generalized logical connectiv e, w hic h binds k v ariables and connects n f orm ulas. Any n -ary p rop ositional connectiv e can b e though t of as an ( n , 0)-ary quan tifier. F or instance, the standard ∧ conn ectiv e b inds no v ariables and connects tw o formulas: ∧ ( ψ 1 , ψ 2 ). The standard fir st-order quantifiers ∃ and ∀ are (1 , 1)-quantifiers, as they b ind one v ariable and connect one form u la: ∀ xψ , ∃ xψ . Bounded univ ersal and existen tial q u an tifiers used in syllogistic reasoning ( ∀ x ( p ( x ) → q ( x )) and ∃ x ( p ( x ) ∧ q ( x ))) can b e represen ted as (2,1)- ary quantifiers ∀ and ∃ , bindin g one v ariable and connecting t w o formulas: ∀ x ( p ( x ) , q ( x )) and ∃ x ( p ( x ) , q ( x )). An example of ( n, k )-ary quan tifiers for k > 1 are Henkin quanti fiers 2 ([15, 17]). The simp lest Henkin q u an tifier Q H 1998 ACM Subje ct Classific ation: F.4.1. Key wor ds and phr ases: Pro of Theory , Automated Deduction, Cut Elimination, Gentzen-type Systems, Quantifiers, Many-v alued Logic, Non-d et erministic Matrices. 1 Generalized qu anti fiers of this kind hav e b een first considered in [16 ]. In [22] Natu ral Dedu ction calculi are provided for n -place connectives and q uantifiers and it is shown t hat d eriv ations in such calculi are normalizable. 2 It should b e n oted that the seman tic interpretation of quantifiers used in this pap er is n ot sufficient for treating such quantifiers. LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.216 8/LMCS-4 (3:2) 2008 c  A. A vron and A. Zamansky CC  Crea tive Commons 2 A. A VR ON AN D A. ZAMA N SKY binds 4 v ariables an d connects one form u la: Q H x 1 x 2 y 1 y 2 ψ ( x 1 , x 2 , y 1 , y 2 ) := ∀ x 1 ∃ y 1 ∀ x 2 ∃ y 2 ψ ( x 1 , x 2 , y 1 , y 2 ) In this w a y of recordin g combinations of quan tifiers, dep endency relations b et ween v ariables are expressed as follo ws: an existen tially q u an tified v ariable dep end s on those universally quan tified v ariables whic h are on the left of it in the same r ow. According to a long tradition in th e philosoph y of logic, established b y Gentz en in his classical pap er Investigations Into L o gic al De duction ([13]), an “ideal” set of in tro duction rules for a logical connectiv e should determine the me aning of the connectiv e (see, e.g . , [29, 30], and also [10] for a general discussion). In [2, 3] the notion of a “canonica l p rop osi- tional Gen tzen-type rule” w as defined in precise terms. A constructiv e c oher enc e criterion for the non-trivialit y of sy s tems consisting of s uc h rules wa s pro vided, and it was sh o w n that a system of this kind admits cut-elimination iff it is coherent. It wa s furth er prov ed that the s eman tics of such systems is pr o v id ed b y t w o-v alued non-deterministic matrices (2Nma- trices), which form a natural generalization of the classical matrix. In fact, a c haracteristic 2Nmatrix was constructed for ev ery coherent canonical prop ositional system. In [28] the results were extended to sy s tems (of a restricted form) with u nary quan- tifiers. A charac terizatio n of a “canonical unary qu an tificatio nal ru le” in such calculi wa s prop osed (the s tandard Gen tzen-t yp e ru les f or ∀ and ∃ are canonical according to it), and a constru ctiv e extension of the coherence criterio n from [2, 3] for canonical s y s tems of this t y p e was giv en. 2Nmatrices were extended to languages with unary qu an tifiers, using a dis- tributional interpretation of quant ifiers ([20, 7]). T hen it was pro v ed that again a canonical Gen tzen-t yp e system of this t yp e admits cut-elimination 3 iff it is coherent, and that it is coheren t iff it has a c haracteristic 2Nmat rix. In this pap er w e mak e the in tuitiv e notion of a “w ell-b eha ved” intro d uction ru le f or ( n, k )-ary quan tifiers formally pr ecise. W e considerably extend the scop e of the charac- terizatio ns of [2, 3, 28] to “canonical ( n, k )-ary qu an tificatio nal r ules”, so that b oth th e prop ositional systems of [2, 3] and the r estricted quan tificatio nal systems of [28] are sp ecific instances of the prop osed definition. W e show th at the coherence criterion for th e defi n ed systems remains decidable. Then we fo cus on the case of k ∈ { 0 , 1 } and sho w th at the follo wing statemen ts concerning a canonical calculus G are equiv alen t: (i) G is coherent, (ii) G has a strongly c haracteristic 2Nmatrix, an d (iii) G admits strong cut-elimination. W e sh o w that coherence is not a necessary condition f or stand ard cut-elimination, and then c h aracterize a sub class of canonical systems for whic h this prop ert y do es h old. 1. Preliminaries F or any n > 0 and k ≥ 0, if a quant ifier Q is of arity ( n, k ), th en Q x 1 ...x k ( ψ 1 , ..., ψ n ) is a formula wheneve r x 1 , ..., x k are d istinct v ariables and ψ 1 , ..., ψ n are formulas of L . F or int erpretation of quant ifiers, we use a generalized notion of distributions (see, e.g [20, 7]). Giv en a set S , P + ( S ) is the set of all the nonempty subsets of S . 3 W e note that by ‘cut-elimination’ we mean here just the existenc e of proofs without (certain forms of ) cuts, rather than an alg orithm to t ransform a given proof to a cu t -free one (for the assumptions-free case the term “cut-admissibilit y” is sometimes used, bu t t his n otion is t oo weak for our purp oses). See, e. g. , [6 ] for a resolution-based algorithm for cut- elimination in LK. CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 3 Definition 1.1. Given a set of truth v alue V , a distribution of a (1,1)-a ry quantifier Q is a fu nction λ Q : P + ( V ) → V . (1,1)-a ry distribution quantifiers ha v e b een extensiv ely studied and a xiomatized in man y-v alued logic. S ee, e.g. , [7, 21, 14]. In w hat follo ws, L is a language w ith ( n, k )-ary quantifiers, that is with quan tifiers Q 1 , ..., Q m with arities ( n 1 , k 1 ), ..., ( n m , k m ) r esp ectiv ely . Denote by F r m cl L the s et of closed L -form ulas and b y T r m cl L the set of closed L -terms. V ar = { v 1 , v 2 , ..., } is the set of v ariables of L . W e use the meta v ariables x, y , z to range o ver elemen ts of V ar . ≡ α is the α -equiv alence relation b et ween formulas, i.e identit y up to the r enaming of b ound v ariables. Lemma 1.2. L et Q b e an ( n, k ) -ary quantifier of L and z 1 , ..., z k fr esh variables which do not o c cur in Q x 1 ..x k ( ψ 1 , ..., ψ n ) . Then: Q x 1 ...x k ( ψ 1 , ..., ψ n ) ≡ α Q y 1 ...y k ( ψ ′ 1 , ..., ψ ′ n ) iff ψ i { z 1 /x 1 , ..., z k /x k } ≡ α ψ ′ i { z 1 /y 1 , ..., z k /y k } for every 1 ≤ i ≤ n . The p r o of is not hard an d is left to the reader. W e use [ ] for application of fu nctions in the meta-language , leavi ng the use of ( ) to the ob ject language. A { t / x } d enotes the f orm ula obtained from A by su bstituting t for x . Given an L -formula A , F v [ A ] is the set of v ariables o ccur ring fr ee in A . W e d enote Q x 1 ...x k A by Q − → x A , and A ( x 1 , ..., x k ) by A ( − → x ). A set of sequents S satisfies th e fr e e-variable c ondition if the set of v ariables o ccurring b ound in S is d isjoin t from the set of v ariables occur r ing free in S . 2. Canonical Syste ms with (n,k)-ar y q uantifiers In this section we prop ose a p r ecise c haracterizat ion of a “canonical ( n, k )-ary quanti fi- cational Gent zen-t yp e rule”. Using an in tro duction rule for an ( n, k )-ary quantifier Q , we should b e able to derive a sequen t of the form Γ ⇒ Q x 1 ...x k ( ψ 1 , ..., ψ n ) , ∆ or of the form Γ , Qx 1 ...x k ( ψ 1 , ..., ψ n ) ⇒ ∆, based on s ome information ab out the subformulas of Q x 1 ...x k ( ψ 1 , ..., ψ n ) con tained in the premises of the rule. F or instance, consider the follo win g s tand ard rules for the (1,1)-ary quan tifier ∀ : Γ , A { t /w } ⇒ ∆ Γ , ∀ w A ⇒ ∆ ( ∀ ⇒ ) Γ ⇒ A { z /w } , ∆ Γ ⇒ ∀ w A, ∆ ( ⇒ ∀ ) where t , z are free for w in A and z do es not o ccur free in the conclusion. Our k ey observ ation is that the internal str u cture of A , as w ell as the exact term t or v ariable w used, are immaterial for the meaning of ∀ . What is imp ortan t here is the sequen t on which A app ears, as w ell as whether a term v ariable t or an eigen v ariable z is used. It follo ws th at the internal stru cture of the formulas of L us ed in the description of a rule can b e abstracted by using a simplified fir st-order language , i.e., the formulas of L in an in tro d uction r ule of a ( n, k )-ary quan tifier, can b e repr esented by atomic form ulas with predicate sym b ols of arit y k . The case when the s ubstituted term is an y L -term, will b e signified b y a constan t, and th e case when it is a v ariable satisfying the ab ov e conditions - by a v ariable. In other words, constan ts serve as term v ariables, while v ariables are eigen v ariables. Th us in addition to our original language L with ( n, k )-ary qu an tifiers we define another, simplified language . 4 A. A VR ON AN D A. ZAMA N SKY Definition 2.1. F or k ≥ 0, n ≥ 1 and a set of constant s C on , L n k ( C on ) is the (first-order) language with n k -ary p redicate symbols p 1 , ..., p n and the set of constan ts C on (and n o quan tifiers). T he set of v ariables of L n k ( C on ) is V ar = { v 1 , v 2 , ..., } . Note th at L n k ( C on ) and L sh are th e same set of v ariables. F urthermore, henceforth w e assume that for eve ry ( n, k )-ary quant ifier Q of L , L n k ( C on ) is a su bset of L . This assumption is n ot necessary , but it mak es the presenta tion easier, as will b e explained in the s equ el. Next we formalize the notion of a canonical rule and its application. Definition 2.2. L et C on b e some set of constants. A c anonic al q uantific ational rule of arity ( n, k ) is an exp r ession of the form { Π i ⇒ Σ i } 1 ≤ i ≤ m /C , where m ≥ 0, C is either ⇒ Q v 1 ...v k ( p 1 ( v 1 , ..., v k ) , ..., p n ( v 1 , ..., v k )) or Q v 1 ...v k ( p 1 ( v 1 , ..., v k ) , ..., p n ( v 1 , ..., v k )) ⇒ for some ( n, k )-ary qu an tifier Q of L and for every 1 ≤ i ≤ m : Π i ⇒ Σ i is a clause 4 o ver L n k ( C on ). Henceforth, in cases where the set of constants C on is clea r from the conte xt (it is th e set of all constant s occurr ing in a canonical rule), we will write L n k instead of L n k ( C on ). A canonical rule is a schematic repr esen tatio n, while for an actual app licatio n we need to instan tiate the s c h ematic v ariables by the terms and f orm ulas of L . This is done us ing a mapping f u nction, defin ed as f ollo ws. Definition 2.3. L et R = Θ /C b e an ( n, k )-ary canonical rule, wh ere C is of one of the forms ( Q − → v ( p 1 ( − → v ) , ..., p n ( − → v )) ⇒ ) or ( ⇒ Q − → v ( p 1 ( − → v ) , ..., p n ( − → v ))). Let Γ b e a set of L -form ulas and z 1 , ..., z k - distinct v ariables of L . An h R , Γ , z 1 , ..., z k i -mapping is any function χ from the predicate sym b ols, terms an d formulas of L n k to f orm ulas and term s of L , satisfying the follo wing conditions: • F or every 1 ≤ i ≤ n , χ [ p i ] is an L -formula. • χ [ y ] is a v ariable of L . • χ [ x ] 6 = χ [ y ] for ev ery t w o v ariables x 6 = y . • χ [ c ] is an L -term, such that χ [ x ] do es n ot o ccur in χ [ c ] f or an y v ariable x o ccur ring in Θ. • F or every 1 ≤ i ≤ n , wh enev er p i ( t 1 , ..., t k ) o ccur s in Θ, for ev ery 1 ≤ j ≤ k : χ [ t j ] is a term free for z j in χ [ p i ], and if t j is a v ariable, then χ [ t j ] do es not o ccur free in Γ ∪ {Q z 1 ...z k ( χ [ p 1 ] , ..., χ [ p n ]) } . • χ [ p i ( t 1 , ..., t k )] = χ [ p i ] { χ [ t 1 ] /z 1 , ..., χ [ t k ] /z k } . W e extend χ to sets of L n k ( C on Θ )-form ulas as follo ws: χ [∆] = { χ [ ψ ] | ψ ∈ ∆ } Giv en a schemati c rep r esen tatio n of a rule and an instan tiation mapping, we can defin e an application of a rule as follo ws. Definition 2.4. An app lic ation of a canonical r ule of arit y ( n, k ) R = { Π i ⇒ Σ i } 1 ≤ i ≤ m / Q − → v ( p 1 ( − → v ) , ..., p n ( − → v )) ⇒ is an y in f erence step of the form: { Γ , χ [Π i ] ⇒ ∆ , χ [Σ i ] } 1 ≤ i ≤ m Γ , Q z 1 ...z k ( χ [ p 1 ] , ..., χ [ p n ]) ⇒ ∆ where z 1 , ..., z k are v ariables, Γ , ∆ are any sets of L -formulas and χ is some h R, Γ ∪ ∆ , z 1 , ..., z k i - mapping. 4 By a clause we mean a seq uent con taining only atomic formulas . CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 5 An application of a canonical quantificat ional rule of the form { Π i ⇒ Σ i } 1 ≤ i ≤ m / ⇒ Q − → v ( p 1 ( − → v ) , ..., p n ( − → v )) is defin ed similarly . Belo w we demonstrate th e ab o v e d efinition by a num b er of examples. Examples 2.5. (1) The standard r igh t in tro duction rule for ∧ , wh ich can b e though t of as a (2 , 0)-ary quan tifier is {⇒ p 1 , ⇒ p 2 } / ⇒ p 1 ∧ p 2 . Its application is of the form: Γ ⇒ ψ 1 , ∆ Γ ⇒ ψ 2 , ∆ Γ ⇒ ψ 1 ∧ ψ 2 , ∆ (2) The standard introdu ction ru les f or th e (1 , 1)-ary qu an tifiers ∀ and ∃ can b e f ormulated as follo ws: { p 1 ( c ) ⇒} / ∀ v 1 p 1 ( v 1 ) ⇒ {⇒ p 1 ( v 1 ) } / ⇒ ∀ v 1 p 1 ( v 1 ) {⇒ p 1 ( d ) } / ⇒ ∃ v 1 p 1 ( v 1 ) { p 1 ( v 1 ) ⇒} / ∃ v 1 p 1 ( v 1 ) ⇒ Applications of these rules ha ve the forms: Γ , ψ { t /w } ⇒ ∆ Γ , ∀ w ψ ⇒ ∆ ( ∀ ⇒ ) Γ ⇒ ψ { z /w } , ∆ Γ ⇒ ∀ w ψ , ∆ ( ⇒ ∀ ) Γ ⇒ ψ { t /w } , ∆ Γ ⇒ ∃ w A, ∆ ( ⇒ ∃ ) Γ , ψ { z /w } ⇒ ∆ Γ , ∃ w ψ ⇒ ∆ ( ∃ ⇒ ) where z is f ree for w in ψ , z is not free in Γ ∪ ∆ ∪ {∀ w ψ } , and t is any term fr ee for w in ψ . (3) Consider the b ound ed existen tial and unive rsal (2 , 1)-ary qu an tifiers ∀ and ∃ (corre- sp ondin g to ∀ x.p 1 ( x ) → p 2 ( x ) and ∃ x.p 1 ( x ) ∧ p 2 ( x ) used in syllogistic reasoning). Their corresp ondin g ru les can b e form ulated as follo ws : { p 2 ( c ) ⇒ , ⇒ p 1 ( c ) } / ∀ v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) ⇒ { p 1 ( v 1 ) ⇒ p 2 ( v 1 ) } / ⇒ ∀ v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) { p 1 ( v 1 ) , p 2 ( v 1 ) ⇒} / ∃ v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) ⇒ {⇒ p 1 ( c ) , ⇒ p 2 ( c ) } / ⇒ ∃ v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) Applications of these r ules are of the form: Γ , ψ 2 { t /z } ⇒ ∆ Γ ⇒ ψ 1 { t /z } , ∆ Γ , ∀ z ( ψ 1 , ψ 2 ) ⇒ ∆ Γ , ψ 1 { y /z } ⇒ ψ 2 { y /z } , ∆ Γ ⇒ ∀ z ( ψ 1 , ψ 2 ) , ∆ Γ , ψ 1 { y /z } , ψ 2 { y /z } ⇒ ∆ Γ , ∃ z ( ψ 1 , ψ 2 ) ⇒ ∆ Γ ⇒ ψ 1 { t /x } , ∆ Γ ⇒ ψ 2 { t /x } , ∆ Γ ⇒ ∃ z ( ψ 1 , ψ 2 ) , ∆ where t and y are free for z in ψ 1 and ψ 2 , y do es not o ccur free in Γ ∪ ∆ ∪ { ∃ z ( ψ 1 , ψ 2 ) } . (4) Consider the (2,2)-ary r ule { p 1 ( v 1 , v 2 ) ⇒ , p 1 ( v 3 , d ) ⇒ p 2 ( c, d ) } / ⇒ Q v 1 v 2 ( p 1 ( v 1 , v 2 ) , p 2 ( v 1 , v 2 )) Its application is of the form: Γ , ψ 1 { w 1 /z 1 , w 2 /z 2 } ⇒ ∆ Γ , ψ 1 { w 3 /z 1 , t 1 /z 2 } ⇒ ∆ , ψ 2 { t 2 /z 1 , t 1 /z 2 } Γ ⇒ ∆ , Q z 1 z 2 ( ψ 1 , ψ 2 ) where w 1 , w 2 , w 3 , t 1 , t 2 satisfy th e appropriate cond itions. 6 A. A VR ON AN D A. ZAMA N SKY Note that although the deriv abilit y of the α -axiom is essential for an y logical system, it is not guarantee d to b e deriv able in a canonical system. What natural syn tactic cond itions guaran tee its deriv ability is still a question for further researc h . F or no w we explicitly add the α -axiom to th e canonical calc uli. Notation. ( F ollo wing [2 ], notations 3-5.) Let − t = f , − f = t and ite ( t, A, B ) = A , ite ( f , A, B ) = B . Let Φ , A s (where Φ ma y b e emp t y ) denote ite ( s, Φ ∪ { A } , Φ). F or instance, the sequen ts A ⇒ and ⇒ A are d enoted by A − s ⇒ A s for s = f and s = t resp ectiv ely . According to th is notation, a ( n, k )-ary canonica l rule is of the form: { Σ j ⇒ Π j } 1 ≤ j ≤ m / Q − → v ( p 1 ( − → v ) , ..., p n ( − → v )) − s ⇒ Q − → v ( p 1 ( − → v ) , ..., p n ( − → v )) s for s ∈ { t, f } . F or fur ther abbreviation, w e denote su c h rule by { Σ j ⇒ Π j } 1 ≤ j ≤ m / Q ( s ). Definition 2.6. A Gen tzen-t yp e calculus G is c anonic al if in addition to the α -axiom A ⇒ A ′ for A ≡ α A ′ and the standard stru ctural rules, G has on ly canonical rules. Definition 2.7. Two ( n, k )-ary canonical in tro duction rules Θ 1 /C 1 and Θ 2 /C 2 for Q are dual if for some s ∈ { t, f } : C 1 = A − s ⇒ A s and C 2 = A s ⇒ A − s , where A = Q v 1 ...v k ( p 1 ( v 1 , ..., v k ) , ..., p n ( v 1 , ..., v k )). Although we can define arbitrary canonical systems us ing our simplified language L n k , our quest is for systems, the sy ntactic ru les of whic h define the seman tic meaning of logical connectiv es/quantifiers. Th us we are in terested in calculi with a “reasonable” or “non- con tr adictory” set of rules, which allo ws for defining a sound and complete semantics for the system. This can b e captured s y ntactic ally b y the follo wing extension of the c oher enc e criterion of [2, 28]. Definition 2.8. F or tw o sets of clauses Θ 1 , Θ 2 o ver L n k , Rnm (Θ 1 ∪ Θ 2 ) is a s et Θ 1 ∪ Θ ′ 2 , where Θ ′ 2 is obtained from Θ 2 b y a fresh renaming of constan ts and v ariables whic h occur in Θ 1 . Henceforth it will b e con v en ien t (but not essen tial ) to assume that the fresh constant s used f or the r enaming are in L . Definition 2.9. (Coherence) 5 A canonical calculus G is c oher e nt if for every t w o dual canonical rules Θ 1 / ⇒ A and Θ 2 / A ⇒ , the set of clauses Rnm (Θ 1 ∪ Θ 2 ) is classically inconsisten t. Note th at the principle of renaming of clashing constants and v ariables is similar to the one u sed in first-order resolution. The imp ortance of this p rinciple for the d efinition of coherence will b e explained in the sequel. Prop osition 2.10. ( Decidabilit y of coherence) The c oher enc e of a c anonic al c alculus G is de cidable. Pr o of. The question of classical consistency of a fin ite set of clauses without fu nction sym- b ols (o v er L n k ) can b e shown to b e equiv alen t to satisfiabilit y of a finite set of un iv ersal form ulas with no function symb ols. This is decidable (by an obvious application of Her- brand’s theorem). 5 A strongly related coherence criterion is d efined in [19], where linear logic is used to reason ab out v arious sequent systems. Ou r coherence criterion is also equiv alen t in th e con text of canonical calculi to the reductivity condition in [9], as will b e ex plained in the sequel. CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 7 3. The seman tic frame w ork 3.1. Non-deterministic matrices. Our main seman tic tool are non-deterministic matri- ces (Nmatrices), first int ro duced in [2, 3] and extended in [27, 28 ]. T hese structures are a generalizat ion of the standard concept of a m an y-v alued matrix, in whic h th e truth-v alue of a formula is chosen n on-deterministically from a give n n on-empt y set of truth-v alues. Th us, giv en a set of truth-v alues V , w e can generaliz e the notion of a distribution function of an ( n, k )-ary quan tifier Q (from Definition. 1.1) to a function λ Q : P + ( V n ) → P + ( V ). I n other w ords, giv en some distribution Y of n-ary v ectors of truth v alues, the int erpretation function non-deterministically c ho oses the truth v alue assigned to Q − → z ( ψ 1 , ..., ψ n ) out fr om λ Q [ Y ] . Definition 3.1. (Non-dete rminist ic matrix) A non-deterministic matrix (henceforth Nmatrix) for L is a tuple M = < V , G , O > , where: • V is a n on -emp t y set of truth v alues. • G (designated tru th v alues) is a non-emp ty prop er su bset of V . • O is a set of inte rpretation functions: for ev ery ( n, k )-ary quantifier Q of L, O includ es the corresp onding distr ibution fu nction ˜ Q M : P + ( V n ) → P + ( V ). Note the sp ecial tr eatmen t of prop ositional connectiv es in the defin ition ab o ve. In [2, 28], an Nmatrix includes an interpretatio n function ˜ ⋄ : V n → P + ( V ) for every n -ary connectiv e of the language; giv en a v aluation v , the tru th v alue v [ ⋄ ( ψ 1 , ..., ψ n )] is c h osen non-deterministically from ˜ ⋄ [ h v [ ψ 1 ] , ..., v [ ψ n ] i ]. In the d efinition ab o ve, the inte rpretation of a p rop ositional conn ectiv e ⋄ is a f unction of another t yp e: ˜ ⋄ : P + ( V n ) → P + ( V ). This can b e though t as a generalization of the previous definition, id en tifying th e tup le h v [ ψ 1 ] , ..., v [ ψ n ] i with the singleton {h v [ ψ 1 ] , ..., v [ ψ n ] i} . The adv an tage of this generalization is that it allo ws for a uniform treatmen t of b oth quan tifiers and prop ositional connecti v es. Definition 3.2. (L-structure) Let M b e an Nmatrix for L . An L-stru ctur e for M is a pair S = h D , I i wh ere D is a (non-empty) domain and I is a fun ction inte rpreting constants, predicate symbols and f unction symb ols of L , satisfying the follo wing conditions: I [ c ] ∈ D , I [ p n ] : D n → V is an n-ary predicate, and I [ f n ] : D n → D is an n-ary fun ction. I is extended to interpret closed terms of L as follo ws: I [ f ( t 1 , ..., t n )] = I [ f ][ I [ t 1 ] , ..., I [ t n ]] Here a note on ou r treatmen t of qu an tificatio n in th e fr amew ork of Nmatrices is in order. Th e standard approac h to in terpreting quantified formulas is by u sing obje ctual (or referen tial) semantic s, where the v ariable is thought of as ranging o v er a set of ob jects from the d omain (see, e . g. , [11, 12]). An alternativ e appr oac h is substitutional qu an tificatio n ([18]), w here quant ifiers are in terpreted su bstitutionally , i.e. a unive rsal (an existen tial) quan tification is true if and only if every one (at least on e) of its sub s titution instances is true (see, e. g. , [24, 26]). [2 7] explains the motiv ation b ehind c ho osing the sub stitutional approac h for the framework of Nmatrices, and p oin ts out the p roblems of the ob jectual approac h in this context . The substitutional appr oac h assumes that ev ery element of the domain has a closed term referring to it. Thus giv en a structur e S = h D , I i , w e extend the language L with individual c onstants , one for eac h elemen t of D . Definition 3.3. ( L(D) ) Let S= h D , I i b e an L -structur e f or an Nmatrix M . L ( D ) is the language ob tained from L by adding to it the set of individual c onstants { a | a ∈ D } . S ′ = h D , I ′ i is the L ( D )-stru cture, su c h that I ′ is an extension of I satisfying: I ′ [ a ] = a . 8 A. A VR ON AN D A. ZAMA N SKY Giv en an L -structure S = h D , I i , we s hall refer to the extended L ( D )-structure h D , I ′ i as S and to I ′ as I when the meaning is clear fr om the context . Definition 3.4. ( S -substitution) Give n an L -structur e S = h D , I i for an Nmatrix M for L , an S -substitution is a function σ : V ar → T r m cl L ( D ) . It is extended to σ : T r m L ∪ F r m L → T r m cl L ( D ) ∪ F r m cl L ( D ) as follo ws: for a term t of L ( D ), σ [ t ] is th e closed term ob tained from t b y replacing ev ery x ∈ F v [ t ] by σ [ x ]. F or a formula ϕ , σ [ ϕ ] is the sente nce obtained from ϕ by replacing every x ∈ F v [ ϕ ] b y σ [ x ]. Giv en a set Γ of formulas, we denote the set { σ [ ψ ] | ψ ∈ Γ } by σ [Γ]. The motiv ation for the follo wing defin ition is purely technical and is related to extending the language with the set of ind ividual constan ts { a | a ∈ D } . Supp ose we hav e a closed term t , su ch that I [ t ] = a ∈ D . Bu t a also has an individual constan t a referrin g to it. W e w ould lik e to b e able to substitute t f or a in ev ery con text. Definition 3.5. (Congruence of terms and form ulas) Let S b e an L -structure for an Nmatrix M . Th e relation ∼ S b et w een terms of L ( D ) is defin ed inductiv ely as follo ws: • x ∼ S x • F or closed terms t , t ′ of L ( D ): t ∼ S t ′ when I [ t ] = I [ t ′ ]. • If t 1 ∼ S t ′ 1 , ..., t n ∼ S t ′ n , then f ( t 1 , ..., t n ) ∼ S f ( t ′ 1 , ..., t ′ n ). The relation ∼ S b et w een formulas of L ( D ) is defined as follo ws: • If t 1 ∼ S t ′ 1 , t 2 ∼ S t ′ 2 , ..., t n ∼ S t ′ n , th en p ( t 1 , ..., t n ) ∼ S p ( t ′ 1 , ..., t ′ n ). • If ψ 1 { − → z / − → x } ∼ S ϕ 1 { − → z / − → y } , ..., ψ n { − → z / − → x } ∼ S ϕ n { − → z / − → y } , where − → x = x 1 ...x k and − → y = y 1 ...y k are distinct v ariables and − → z = z 1 ...z k are new distinct v ariables, then Q − → x ( ψ 1 , ..., ψ n ) ∼ S Q − → y ( ϕ 1 , ..., ϕ n ) for an y ( n, k )-ary quantifier Q of L . In tuitiv ely , ψ ∼ S ψ ′ if ψ ′ can b e obtained from ψ by p ossibly renaming b ound v ariables and b y an y num b er of su b stitutions of a closed term t for another closed term s , so that I [ t ] = I [ s ]. ≡ α ⊆∼ S . Lemma 3.6. ( [27 ] ) L e t S b e an L - structur e for an Nmatrix M . L et ψ , ψ ′ b e formulas of L ( D ) . L et t , t ′ b e close d terms of L ( D ) , su c h that t ∼ S t ′ . (1) If ψ ≡ α ψ ′ , then ψ ∼ S ψ ′ . (2) If ψ ∼ S ψ ′ , then ψ { t /x } ∼ S ψ ′ { t ′ /x } . Definition 3.7. (Legal v aluat ion) Let S = h D , I i b e an L -structure for an Nmatrix M . An S -v aluation v : F r m cl L ( D ) → V is le g al in M if it satisfies the follo wing conditions: (1) v [ ψ ] = v [ ψ ′ ] for ev ery t wo sentences ψ , ψ ′ of L ( D ), suc h that ψ ∼ S ψ ′ . (2) v [ p ( t 1 , ..., t n )] = I [ p ][ I [ t 1 ] , ..., I [ t n ]]. (3) F or ev ery ( n , k )-ary quantifier Q of L , v [ Q x 1 , ..., x k ( ψ 1 , ..., ψ n )] shou ld b e an elemen t of ˜ Q M [ {h v [ ψ 1 { a 1 /x 1 , ..., a k /x k } ] , ..., v [ ψ n { a 1 /x 1 , ..., a k /x k } ] i | a 1 , ..., a k ∈ D } ]. Note that in case Q is a prop ositional connectiv e (for k = 0), the fu nction ˜ Q M is applied to a singleton, as w as explained ab o ve. Notation. F or a set of sequen ts S , w e shall write S ⊢ G Γ ⇒ ∆ if a sequen t Γ ⇒ ∆ h as a pro of f rom S in G . Definition 3.8. Let S = h D , I i b e an L -stru cture for an Nmatrix M . CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 9 (1) An M -legal S -v aluation v is a mo del of a sentenc e ψ in M , denoted b y S, v | = M ψ , if v [ ψ ] ∈ G . (2) Let v b e an M -legal S -v aluatio n. A sequent Γ ⇒ ∆ is M -valid in h S, v i if f or ev ery S -substitution σ : if S, v | = M σ [ ψ ] for eve ry ψ ∈ Γ , th en there is some ϕ ∈ ∆, such that S, v | = M σ [ ϕ ]. (3) A sequent Γ ⇒ ∆ is M -v alid, d enoted b y ⊢ M Γ ⇒ ∆ , if for every L -stru cture S and ev er y M -legal S -v aluation v , Γ ⇒ ∆ is M -v alid in h S, v i . (4) F or a set of sequen ts S , S ⊢ M Γ ⇒ ∆ if for ev ery L -structur e S and every M -legal S -v aluation v : wheneve r the sequent s of S are M -v alid in h S, v i , Γ ⇒ ∆ is also M -v alid in h S, v i . Definition 3.9. A system G is str ongly sound 6 for an Nmatrix M if f or eve ry set S of sequen ts closed u nder sub stitution: S ⊢ G Γ ⇒ ∆ en tails S ⊢ M Γ ⇒ ∆ . A system G is str ongly c omplete for an Nmatrix M if f or every set S of s equ en ts closed und er substitution: S ⊢ M Γ ⇒ ∆ en tails S ⊢ G Γ ⇒ ∆. An Nmatrix M is str ongly char acteristic for G if G is strongly s ou n d and strongly complete for M . Note that sin ce th e empt y set of sequ en ts is closed un der s ubstitutions, strong soun dness implies (weak) soun dness 7 . A similar remark applies to completeness and a c haracteristic Nmatrix. 3.2. Seman tics for simplified languages L n k . In addition to L -stru ctures for languages with ( n, k )-ary quantifiers, we also use L n k -structures f or the simp lified languages L n k , us ed for form ulating th e canonical rules. T o mak e the distinction clearer, we sh all use the meta v ari- able S for th e f ormer and N for the latter. Since the formulas of L n k are alw ays atomic, the sp ecific 2Nmatrix for wh ic h N is defi ned is imm aterial, an d can b e omitted. W e ma y ev en sp eak of classica l v alidity of sequen ts ov er L n k . Thus hen ceforth in stead of sp eaking of M -v alidit y of a set of clauses Θ o v er L n k , we will sp eak simp ly of v alidit y . Next we define the notion of a distribution of L n k -structures. Definition 3.10. Let N = h D , I i b e a structure for L n k . D ist N , the d istribution of N is defined as follo ws: D ist N = {h I [ p 1 ][ a 1 , ..., a k ] , ..., I [ p n ][ a 1 , ..., a k ] i | a 1 , ..., a k ∈ D } W e say that an L n k -structure N is E -charact eristic if D ist N = E . Note that the distribution of an L n 0 -structure N is D ist N = {h I [ p 1 ] , ..., I [ p n ] i} and so it is alw a ys a singleton. F urthermore, th e v alidit y of a set of clauses o v er L n 0 can b e redu ced to p rop ositional satisfiabilit y as stated in the follo wing lemma wh ic h can b e easily prov ed: Lemma 3.11. L et N b e a L n 0 -structur e. Assume that D ist N = {h s 1 , ..., s n i} for some s 1 , ..., s n ∈ { t, f } . L et v D ist N b e any pr op ositional valuation satisfying v [ p i ] = s i for every 1 ≤ i ≤ n . A set of clauses Θ is v alid in N iff v D ist N pr op ositional ly satisfies Θ . No w we turn to the case k = 1. In this case it is con venien t to define a sp ecial kind of L n 1 -structures wh ic h we call c anonic al structur es. These s tructures are su fficien t to r eflect the b eha vior of all p ossib le L n 1 -structures. 6 A more general defi nition would b e without the restriction concerning the closure of S under substitution. How ever, in this case w e would need to add su bstitution as a structu ral rule to canonical calculi. 7 A system G is (wea kly) soun d for an Nmatrix M if ⊢ G Γ ⇒ ∆ entails ⊢ M Γ ⇒ ∆. 10 A. A VR ON AN D A. ZAMA N SKY Definition 3.12. Let E ∈ P + ( { t, f } n ). A L n 1 -structure N = h D , I i is E -canonical if D = E and for ev ery b = h s 1 , ..., s n i ∈ D and ev ery 1 ≤ i ≤ n : I [ p i ][ b ] = s i . Clearly , every E -c anonical L n 1 -structure is E -c haracteristic . Lemma 3.13. L et Θ b e a set of clauses over L n 1 , which is v alid in some structur e N = h D , I i . Then ther e exists a D ist N -c anonic al structur e N ′ in which Θ is valid. Pr o of. Supp ose that Θ is v alid in a structure N = h D , I i . Define the L n 1 -structure N ′ = h I ′ , D ′ i as follo ws : • D ′ = D is t N . • I ′ [ c ] = h I [ p 1 ][ I [ c ]] , ..., I [ p n ][ I [ c ]] i f or ev ery constan t c o ccurring in Θ. • F or every 1 ≤ i ≤ n : I ′ [ p i ][ h s 1 , ..., s n i ] = t iff s i = t . Clearly , N ′ is D ist N -canonical. It is easy to v erify that Θ is v alid in N ′ . Corollary 3.14. L et E ∈ P + ( { t, f } n ) . F or a finite set of clauses Θ over L n 1 , the q uestion whether Θ is valid in a E -char acteristic structur e is de cidable. Pr o of. F ollo ws from L emma 3.13 and the fact that for an y E ∈ P + ( { t, f } n ), there are finitely man y E -canonical structur es to c hec k. 4. Canonical s ystems with ( n, k ) -ar y q uantifiers for k ∈ { 0 , 1 } No w w e tur n to the class of canonical sys tems with ( n, k )-ary q u an tifiers for the case of k ∈ { 0 , 1 } and n ≥ 1. Henceforth, unless stated otherwise, we assume that k ∈ { 0 , 1 } . 4.1. Seman tics for canonical systems for k ∈ { 0 , 1 } . In this section we explore the connection b et w een th e coherence of a canonical calculus G , the existence for it of a strongly c h aracteristic 2Nmatrix, and strong cut-elimination (in a sense explained b elo w.) W e start b y defin in g the n otion of suitability for G . Definition 4.1. (Suit abilit y for G ) Let G b e a canonical calculus o v er L . A 2Nmatrix M is su itable for G if for ev ery ( n, k )-ary canonical rule Θ / Q ( s ) of G (where s ∈ { t, f } ), it holds that for eve ry L n k -structure N in which Θ is v alid: ˜ Q M [ D ist N ] = { s } . Theorem 4.2. L et G b e a c anonic al c alculus and M - a 2Nmatrix suitable for G . Then G is str ongly sound for M . Pr o of. see App endix A. No w we come to the construction of a characte ristic 2Nmatrix for ev ery coherent canon- ical calculus. Definition 4.3. Let G b e a coherent canonical calculus. The Nmatrix M G for L is defin ed as follo w s for every ( n , k )-ary quan tifier Q of L , ev ery s ∈ { t, f } and ev ery E ∈ P + ( { t, f } n ): ˜ Q M G [ E ] =      { s } if Θ / Q ( s ) ∈ G and Θ is v alid in some E − c anonic al L n k − structur e { t, f } otherw ise CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 11 First of all, note that by corollary 3.14, the ab ov e definition is constru ctiv e. Next, let us show that M G is well-defined. Assu me by con tradiction that there are t w o dual r u les Θ 1 / ⇒ A and Θ 2 / A ⇒ , su c h that b oth Θ 1 and Θ 2 are v alid in some E -canonical stru ctures N 1 , N 2 resp ectiv ely . Ob tain Θ ′ 2 from Θ 2 b y renamin g of constan ts and v ariables whic h o ccur in Θ 1 . Then cle arly Θ ′ 2 is also v alid in some E -canonical structur e N 3 . If k = 0, by Lemma 3.11, the set of clauses Θ 1 ∪ Θ ′ 2 is satisfiable by a (classical) prop ositional v aluation v E and is thus classically consisten t, in contradicti on to the coherence of G (see defn. 2.9). Otherwise, k = 1. The only difference b etw een different E -canonica l structures is in the in terpretation of constants, and sin ce the s ets of constant s o ccurring in Θ 1 and Θ ′ 2 are disjoin t, an E -canonical structur e N ′ = h D ′ , I ′ i (for the extended language contai ning the constan ts of b oth Θ 1 and Θ 2 ) can b e constructed, in whic h Θ 1 ∪ Θ ′ 2 are v alid. Thus the s et Θ 1 ∪ Θ ′ 2 = Rnm (Θ 1 ∪ Θ 2 ) is classica lly consisten t, in con tradictio n to the coherence of G . Remark: The construction of M G ab o v e is muc h simpler than the constructions car- ried out in [2, 28]: a canonical calculus there is firs t tran s formed in to an equiv alen t normal form calculus, whic h is then used to construct the c haracteristic Nmatrix. The idea is to transform the calculus so that eac h rule dictates the int erpretation f or only one E . Ho w- ev er , the ab o ve definitions show that the transformation in to normal form is actually not necessary and w e can construct M G directly fr om G . Next we d emonstrate the constru ction of a characte ristic 2Nmatrix f or some coherent canonical calculi. Examples 4.4. (1) It is easy to see that for an y canonical coheren t calculus G includin g the standard (1,1)- ary rules for ∀ and ∃ from Example 2.5-2: ˜ ∀ M G [ { t, f } ] = ˜ ∀ M G [ { f } ] = ˜ ∃ M G [ { f } ] = { f } ˜ ∀ M G [ { t } ] = ˜ ∃ M G [ { t, f } ] = ˜ ∃ M G [ { t } ] = { t } (2) Consider the canonical calculus G ′ consisting of the f ollo wing th ree (1 , 2)-ary r ules from Example 2.5-3: { p 1 ( v 1 ) ⇒ p 2 ( v 1 ) } / ⇒ ∀ v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) { p 2 ( c ) ⇒ , ⇒ p 1 ( c ) } / ∀ v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) ⇒ {⇒ p 1 ( c ) , ⇒ p 2 ( c ) } / ⇒ ∃ v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) G ′ is obviously coherent. The 2Nmatrix M G ′ is d efined as follo ws for eve ry H ∈ P + ( { t, f } 2 ): ˜ ∀ [ H ] = ( { t } if h t , f i 6∈ H { f } other w ise ˜ ∃ [ H ] = ( { t } if h t , t i ∈ H { t, f } other wis e The fir s t ru le dictates the condition that ∀ [ H ] = { t } for the case of h t, f i 6∈ H . T h e second rule dictates the condition that ∀ [ H ] = { f } for the case that h t, f i ∈ H . S ince G ′ is coheren t, th ese conditions are n on-con trad ictory . Th e third rule dictates th e condition that ∃ [ H ] = { t } in the case that h t, t i ∈ H . There is no rule whic h dictates conditions for the case of h t, t i 6∈ H , and so the interpretation in this case is non-deterministic. (3) Consider the canonical calc ulus G ′′ consisting of the follo wing (1 , 3)-ary rule: { p 2 ( v 1 ) , p 3 ( v 1 ) ⇒} / Q v 1 ( p 1 ( v 1 ) , p 2 ( v 1 ) , p 3 ( v 1 )) ⇒ 12 A. A VR ON AN D A. ZAMA N SKY Of course, G ′′ is coherent . T he 2Nmatrix M G ′′ is defin ed as follo ws for eve ry H ∈ P + ( { t, f } 2 ): ˜ ∀ [ H ] = ( { f } if H ⊆ {h t , t , f i , h t , f , t i , h t , f , f i , h f , t , f i , h f , f , t i , h f , f , f i} { t, f } if h f , t , t i ∈ H or h t , t , t i ∈ H No w w e come to the main th eorem, establishin g a connection b et w een the coherence of a canonical calculus G , th e existence of a stron gly c haracteristic 2Nmatrix for G and str ong cut-elimination in G in the sense of [1]. Definition 4.5. Let G b e a canonical calculus and let S b e a set of sequen ts closed und er substitution. A pro of P of Γ ⇒ ∆ from S in G is simple if all cuts in P are on formula s from S . Definition 4.6. A calculus G admits str ong cut-elimination 8 if for eve ry set of sequen ts S closed u nder sub stitution and every sequent Γ ⇒ ∆, su c h that S ∪ { Γ ⇒ ∆ } satisfies the free-v ariable condition 9 : if S ⊢ G Γ ⇒ ∆, then Γ ⇒ ∆ has a simple pro of in G . Note that strong cut-elimination implies standard cu t-eliminatio n (which corresp onds to the case of an empty set S ). Theorem 4.7. L et G b e a c anonic al c alculus. Then the fol lowing statements c onc erning G ar e e quivalent: (1) G is c oher ent. (2) G has a str ongly char acteristic 2Nmatrix. (3) G admits str ong cut-elimination. Pr o of. First we p ro v e that (2) implies (1). Supp ose that G has a strongly characte ristic 2Nmatrix M . Assu me by contradicti on that G is not coheren t. Th en there exist t w o dual ( n, k )-ary rules R 1 = Θ 1 / ⇒ A and R 2 = Θ 2 / A ⇒ in G , such that R nm (Θ 1 ∪ Θ 2 ) is classically consisten t. S upp ose that k = 1. Then A = Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )). Recall that Rnm (Θ 1 ∪ Θ 2 ) = Θ 1 ∪ Θ ′ 2 , where Θ ′ 2 is obtained from Θ 2 b y renamin g constan ts and v ariables that o ccur also in Θ 1 (see defn. 2.8). F or simplicit y 10 w e assume th at the fr esh constan ts used for renaming are all in L . Let Θ 1 = { Σ 1 j ⇒ Π 1 j } 1 ≤ j ≤ m and Θ ′ 2 = { Σ 2 j ⇒ Π 2 j } 1 ≤ j ≤ r . Since Θ 1 ∪ Θ ′ 2 is classically consistent , there exists an L n k -structure N = h D , I i , in whic h b oth Θ 1 and Θ ′ 2 are v alid. Recall that w e also assume that L n k is a su b set of L 11 and so the follo wing are applications of R 1 and R 2 resp ectiv ely: { Σ 1 j ⇒ Π 1 j } 1 ≤ j ≤ m ⇒ Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )) { Σ 2 j ⇒ Π 2 j } 1 ≤ j ≤ m Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )) ⇒ Let S b e any extension of N to L and v - an y M -legal S -v aluation. I t is easy to see that the premises of the ap p lications ab o v e are M -v alid in h S, v i (since the p remises con tain atomic formulas). S ince G is strongly soun d f or M , b oth ⇒ Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )) and 8 [1] do es n ot assume that S is closed u nder substitution. Instead, a struct u ral substitution rule is add ed and th e allow ed cuts are on substitution instances of formulas from S . 9 See section 1. 10 This assumption is not necessary and is u sed only for simplification of presentation, since w e can instantia te t he constants by any L -terms. 11 This assumption is again not essentia l for the p roof, bu t it simplifies the presentatio n. CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 13 Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )) ⇒ sh ou ld also b e M -v alid in h S, v i , which is of course imp ossible. The p r o of for the case of k = 0 is simpler and is left to the reader. Next, we prov e that (3) implies (1). Let G b e a canonical calculus which adm its strong cut-elimination. Supp ose by con tradiction that G is not coherent. Then th ere are tw o dual rules of G : Θ 1 / ⇒ A and Θ 2 / A ⇒ , suc h that Rnm (Θ 1 ∪ Θ 2 ) is classically consisten t. Let Θ b e the minimal set of clauses, such that R nm (Θ 1 ∪ Θ 2 ) ⊆ Θ and Θ is closed u n der substitutions. Θ ∪ {⇒} satisfy the free-v ariable condition, since only atomic form ulas are in v olved and n o v ariables are b ound there. It is easy to see that Θ ⊢ G ⇒ A and Θ ⊢ G A ⇒ . By usin g cut, Θ ⊢ G ⇒ . But ⇒ has no simple pro of in G from Θ (sin ce Rn m (Θ 1 ∪ Θ 2 ) is consisten t and Θ is its closure und er substitutions), in con tr adiction to the fact that G admits strong cut-elimination. T o sh ow that (1) implies b oth (2) and (3), w e need the follo wing prop osition: Prop osition 4.8. L et G b e a c oher ent c alculus. L e t S b e a set of se que nts close d under substitution and Γ ⇒ ∆ - a se que nt, such that S ∪ { Γ ⇒ ∆ } satisfies the fr e e-v ariable c ondition. If Γ ⇒ ∆ has no simple pr o of fr om S in G , then S 6⊢ M Γ ⇒ ∆ . Pr o of. see App endix A. T o prov e that (1) implies (2), supp ose that G is coherent. Let us sh o w that M G is a strongly c haracteristic 2Nmatrix for G . By defin ition of M G , it is suitable for G (see defn. 4.1). By theorem 4.2, G is strongly sound for M G . F or strong completeness, let S b e a set of sequents closed u nder sub s titution. Supp ose that a sequen t Γ ⇒ ∆ has no pro of from S in G . If S ∪ { Γ ⇒ ∆ } do es not satisfy the free-v ariable condition, obtain S ′ ∪ { Γ ′ ⇒ ∆ ′ } by renaming the b ound v ariables, so that S ′ ∪ { Γ ′ ⇒ ∆ ′ } satisfies the condition (otherwise, tak e Γ ′ ⇒ ∆ ′ and S ′ to b e Γ ⇒ ∆ and S resp ectiv ely). Then Γ ′ ⇒ ∆ ′ has no p ro of from S ′ in G (otherwise w e could obtain a pro of of Γ ⇒ ∆ from S by usin g cuts on logical axioms), and so it also has no simple pro of from S ′ in G . By Prop osition 4.8, S ′ 6⊢ M Γ ′ ⇒ ∆ ′ . That is, there is an L -structure S and an M -legal v aluation v , suc h that the sequ en ts in S ′ are M -v alid in h S, v i , while Γ ′ ⇒ ∆ ′ is not. S in ce v resp ects the ≡ α -relation, the sequents of S are also M -v alid in h S, v i , while Γ ⇒ ∆ is not. And so S 6⊢ M Γ ⇒ ∆. W e hav e sho wn that G is strongly complete (and strongly s ou n d) for M G . Th us M G is a strongly charac teristic 2Nmatrix for G . Finally , w e pro v e th at (1) implies (3). Let G b e a coheren t cal culus. Let S b e a set of sequen ts closed under substitution, and let Γ ⇒ ∆ b e a sequent, su c h th at S ∪ { Γ ⇒ ∆ } satisfies the free-v ariable condition. Sup p ose that S ⊢ G Γ ⇒ ∆. W e ha v e already shown ab o v e that M G is a str on gly charac teristic 2Nmatrix for G . Th u s S ⊢ M Γ ⇒ ∆, an d b y Prop osition 4.8, Γ ⇒ ∆ h as a simple pro of from S in G . Thus G adm its strong cut- elimination. Remark. At this p oin t it should b e noted that th e renaming of clashing constant s in the defin ition of coherence (see defn. 2.9) is cru cial. Consider, for in stance, a canonical calculus G consisting of th e introdu ction r ules { p 1 ( c ) ⇒ ; ⇒ p 1 ( c ′ ) } / ⇒ Q v 1 p 1 ( v 1 ) and { p 1 ( c ′′ ) ⇒ ; ⇒ p 1 ( c ) } / Q v 1 p ( v 1 ) ⇒ for a (1,1)-ary q u an tifier Q . Without renaming of clashing constant s, we wo uld conclude that the set { p 1 ( c ) ⇒ ; ⇒ p 1 ( c ′ ) ; p 1 ( c ′′ ) ⇒ , ⇒ p 1 ( c ) } is classically inconsistent. Ho w ever, G obviously h as no strongly c haracteristic 2Nmatrix, since th e r u les dictate con tradicting requirements for ˜ Q [ { t, f } ]. But if w e p erform ren amin g first, ob taining th e set R nm (Θ 1 ∪ Θ 2 ) = { p 1 ( c ) ⇒ , ⇒ p 1 ( c ′ ) , p 1 ( c ′′ ) ⇒ , ⇒ p 1 ( c ′′′ ) } , we shall see that Rnm (Θ 1 ∪ Θ 2 ) is classically consisten t and so G is n ot coheren t. Hence, by the ab o v e theorem, G has no strongly c haracteristic 2Nmatrix. 14 A. A VR ON AN D A. ZAMA N SKY Corollary 4.9. The existenc e of a str ongly char acteristic 2Nmatrix for a c anonic al c alculus G is de cidable. Pr o of. By theorem 4.7, the qu estion whether G has a strongly charact eristic 2Nmatrix is equiv alen t to th e question whether G is coherent, and this, by Prop osition 2.10, is d ecidable. Remark: The ab ov e results are related to the results in [9], where a general class of sequen t calculi with ( n, k )-ary qu an tifiers and a (not necessarily standard) set of structural rules called standar d calculi are defined. A canonical calculus is a particular instance of a standard calculus which includ es all of the stand ard structural rules. [9 ] formulate syn tactic necessary and sufficien t cond itions for a slightl y generalized ve rsion of cut-elimination with non-logical axioms. Unlik e in this pap er, the non -logical axioms must consist of atomic form ulas (and must b e closed u nder cu ts and substitutions). But the results of [9] apply to a m u c h wider class of calculi (since different combinatio ns of structur al ru les are allo wed). In addition, a constructiv e mo d ular cut-eliminatio n pr o cedure is provided. T he r eductivit y condition of [9] can b e sho wn to b e equiv alent to our coherence criterion in th e con text of canonical sys tems 12 . 4.2. Coherence and standard cut-elimination. In the previous sub section w e ha v e studied the connection b et w een coherence and strong cu t-elimination. In this sub s ection w e fo cus on standard cut-elimination in canonical calculi. It easily follo w s from theorem 4.7 that coherence imp lies cut-elimination: Corollary 4.10. L et G b e a c anonic al c alculus. If G is c oher ent, then for e v ery se quent Γ ⇒ ∆ satisfying the fr e e- variable c ondition: if Γ ⇒ ∆ is pr ovable in G , then it has a cut-fr e e pr o of in G . Th us coherence is a sufficient cond ition for cut-elimination in a canonical calculus. In the more restricted canonical systems of [2, 28] it also is a necessary condition. Ho w ev er, things get more complicated with the more general canonical ru les studied in this pap er. Example 4.11. Consid er, for instance, the follo w in g canonical calculus G 0 consisting of the follo win g t w o inference rules: Θ 1 / ⇒ Q v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) and Θ 2 / Q v 1 ( p 1 ( v 1 ) , p 2 ( v 1 )) ⇒ , where: Θ 1 = Θ 2 = { p 1 ( v 1 ) ⇒ p 2 ( v 1 ) ; ⇒ p 1 ( c 1 ) ; ⇒ p 2 ( c 1 ) ; p 1 ( c 2 ) ⇒ ; p 2 ( c 2 ) ⇒ ; p 1 ( c 3 ) ⇒ ; ⇒ p 2 ( c 3 ) } Clearly , G 0 is not coherent. W e no w sket c h a pr o of that the only sequents p r o v able in G 0 are logical axioms. Th is immediately implies that G 0 admits cut-el imination. T o p ro v e this it su ffices to s ho w th at for ev ery rule of G 0 : if its p remises are logical axioms, then its conclusion is a logic al axio m. Supp ose by con tradiction that w e can apply one of the ru les on logical axioms and obtain a conclusion whic h is n ot a logical axiom. Supp ose, without loss of generalit y , that it is the first r ule. Then the application would b e of th e form: Γ , χ [ p 1 ] { χ [ v 1 ] /w } ⇒ ∆ , χ [ p 2 ] { χ [ v 1 ] /w . . . Γ ⇒ χ [ p 1 ] { χ [ c 1 ] /w } , ∆ Γ ⇒ χ [ p 2 ] { χ [ c 1 ] /w } , ∆ Γ ⇒ Q w ( χ [ p 1 ] , χ [ p 2 ]) , ∆ 12 W e wish to thank Agata Ciabattoni for p ointing out these facts to us in a p ersonal corresp ondence. CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 15 Since the p ro v ed s equ en t is n ot a logical axiom, (*) th ere are no A ∈ Γ and B ∈ ∆, suc h that A ≡ α B . Moreo v er, since Γ , χ [ p 1 ] { χ [ v 1 ] /w } ⇒ ∆ , χ [ p 2 ] { χ [ y ] /w } is a logical axiom, either (i) th er e is some C ∈ ∆, such that C ≡ α χ [ p 1 ] { χ [ v 1 ] /w } , (ii) there is some C ∈ Γ, su c h that C ≡ α χ [ p 2 ] { χ [ v 1 ] /w } , or (iii) χ [ p 1 ]( χ [ v 1 ] /w ) ≡ α χ [ p 2 ] { χ [ v 1 ] /w } . Sup p ose (i) holds, i.e. there is some some C ∈ ∆, such that C ≡ α χ [ p 1 ] { χ [ v 1 ] /w } . Then since χ [ v 1 ] cann ot o ccur free in ∆, w 6∈ F v [ C ], an d so w 6∈ F v [ χ [ p 1 ]]. Hence, χ [ p 1 ] { χ [ c 1 ] /w } = χ [ p 1 ] { χ [ v 1 ] /w } = χ [ p 1 ]. No w sin ce Γ ⇒ χ [ p 1 ] { χ [ c 1 ] /w } , ∆ is a logical axiom, and due to (*), there is some D ∈ Γ, suc h that D ≡ α χ [ p 1 ] { χ [ c 1 ] /w } . But since χ [ p 1 ] { χ [ c 1 ] /w } = χ [ p 1 ] { χ [ v 1 ] /w } , C ≡ α D , C ∈ ∆ and D ∈ Γ, in con tradictio n to (*). The case (ii) is treated similarly using the constan t c 2 . T he case (iii) is handled using the constan t c 3 . Th us, only logical axioms are prov able in G 0 and so it admits standard cut-elimination, although it is not coheren t. Hence coherence is n ot a necessary condition for cut-elimination in general. How ev er, b elo w we charact erize a m ore restricted su b class of canonical systems, for wh ic h this prop- ert y do es hold. Definition 4.12. A canonical calculus G is si mple if for eve ry t w o dual ( n, k )-ary canonical rules Θ 1 / ⇒ A and Θ 2 / A ⇒ one of the follo w ing prop erties h olds: (1) k = 0, i.e. Θ 1 / ⇒ A and Θ 2 / A ⇒ are prop ositional ru les. (2) k = 1 and one of the follo wing holds for eac h v ariable y o ccurring in Rnm (Θ 1 ∪ Θ 2 ): • There is at most one 1 ≤ i ≤ n , suc h that y o ccurs in p i ( y ) in Rnm (Θ 1 ∪ Θ 2 ) and there is at most one constan t c , suc h that p i ( c ) also occurs in Rnm (Θ 1 ∪ Θ 2 ). • There are tw o d ifferen t 1 ≤ i, j ≤ n , suc h that y o ccurs in p i ( y ) and p j ( y ) in Rnm (Θ 1 ∪ Θ 2 ) and for ev ery constant c , there is n o suc h 1 ≤ k ≤ n , that b oth p k ( y ) and p k ( c ) o ccur in Rnm (Θ 1 ∪ Θ 2 ). Examples 4.13. (1) All the canonica l calculi from examples 2.5 are simple. (2) Consider the canonical calculus G 1 , consisting of the f ollo wing t w o rules for a (3 , 1)-ary quan tifier Q 1 : { p 1 ( v 1 ) ⇒ ; p 1 ( c ) , p 2 ( c ) ⇒ } / ⇒ Q 1 v 1 ( p 1 ( v 1 ) , p 2 ( v 1 ) , p 3 ( v 1 )) {⇒ p 1 ( v 1 ) ; ⇒ p 2 ( e ) } / Q 1 v 1 ( p 1 ( v 1 ) , p 2 ( v 1 ) , p 3 ( v 1 )) ⇒ It is easy to see that G 1 is a simple coheren t calculus. (3) If w e mod ify the first rule of G 1 as follo ws: { p 1 ( v 1 ) ⇒ ; p 1 ( c ) , p 2 ( c ) ⇒ ; p 1 ( d ) ⇒ p 3 ( d ) } / ⇒ Q 1 v 1 ( p 1 ( v 1 ) , p 2 ( v 1 ) , p 3 ( v 1 )) the resulting calculus is not simple, since b oth p 1 ( c ) and p 1 ( d ) o ccur in the pr emises of the rule, together with p 1 ( v 1 ). (4) The calculus G 0 from example 4.11 is n ot simple, since for instance p 1 ( v 1 ), p 1 ( c 1 ) and p 1 ( c 5 ) o ccur in the premises (after renaming). Prop osition 4.14. If a simple c anonic al c alculus G admits cut-elimination, then it is c oher ent. Pr o of. see App endix A. 16 A. A VR ON AN D A. ZAMA N SKY 5. Summar y and fur the r rese arch In this pap er we h a ve considerably extended the c h aracteriza tion of canonical calculi of [2, 28] to ( n , k )-ary quan tifiers. F o cusing on the case of k ∈ { 0 , 1 } , w e ha v e sho wn that th e follo wing statemen ts concerning a canonical calculus G are equiv alen t: (i) G is coheren t, (ii) G h as a strongly c h aracteristic 2Nmatrix, and (iii) G admits strong cut-elimination. W e ha v e also sho wn that coherence is not a necessary condition f or standard cut-elimination, and c h aracterized a sub class of canonical systems called simple cal culi, for whic h this prop ert y do es h old. In addition to these pro of-theoretical r esults for a n atural type of multi ple conclusion Gen tzen-t yp e systems w ith ( n , 1)-ary and ( n, 0)-ary quan tifiers, this work also pro vides further evidence for th e thesis th at the meaning of a logica l constan t is given by its in- tro duction (and “elimination”) rules . W e h a ve sho wn that at least in the framework of m ultiple-conclusion consequence relations, an y “reasonable” set of canonical qu an tificatio nal rules completely determines the seman tics of the qu an tifier. This p ap er also demons tr ates the imp ortan t role of the seman tic framework of Nmatri- ces ([2, 27]), whic h sub stan tially contributes to the unders tand ing of th e connection b etw een syn tactic rules and seman tic in terpretations of qu an tifiers. Due to the mo d ularit y of th e framew ork, we w ere able to detect the semantic effect of eac h of the canonical rules, wh ic h of cours e is n ot p ossible usin g deterministic matrices. Some of the most immediate research d irections are as f ollo ws. I n the case of k ∈ { 0 , 1 } , w e still need to c haracterize the most general sub class of canonical calculi, f or whic h coherence is b oth a necessary and sufficien t condition for standard cut-elimination (it is not clear whether the charact erization of simp le calculi can b e fu rther extend ed ). Extending these resu lts to the case of k > 1 might lead to n ew insights on Henkin quan tifiers and other imp ortant generalized qu an tifiers. How ev er, ev en for the simp lest case of (1 , 2)-ary qu an tifiers the extension is far from straigh tforward. Consider, for instance, the calculus G , consisting of the follo wing t w o (1,2)-ary rules: { p ( c, x ) ⇒} / ⇒ Q z 1 z 2 p ( z 1 , z 2 ) {⇒ p ( y , d ) } / Q z 1 z 2 p ( z 1 , z 2 ) ⇒ G is coherent, but it is easy to see that M G is n ot w ell-defin ed in this case. An d even if a 2Nmatrix M suitable for G d o es exist, it is not necessarily sound f or G . It is clear that the distributional in terpretation of quantifiers is no longer adequate for the case of k > 1, since it cannot capture an y kind of dep endencies b et w een elements of the d omain. T hus a more general interpretati on of qu an tifiers is n eeded. Another imp ortant researc h direction is extending canonical systems with equalit y . This will allo w us to treat counting ( n, k )-ary quan tifiers, like “there are at most t w o elemen ts a, b , suc h that p ( a, b ) h olds”. Clearly , equalit y must b e incorp orated also into the r epresen tation language L n k . S tandard and stron g cut-elimination and its connection to the coherence of canonical sys tems are yet to b e in v estigated for canonical systems with equalit y . A cknowledgement This researc h w as supp orted by the Isr ael Scienc e F oundation founded by th e Israel Acad- em y of Sciences and Humanities (gran t No 809/06) . 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Proofs of selec ted propositions Pro of of Theorem 4.2 : Supp ose that M is s u itable for G . Let S = h D , I i b e some L -structure and v - an M -legal S -v aluation. Let S b e an y set of sequent s closed u nder substitution. W e will s h o w that if the sequents of S are M -v alid in h S, v i , then any sequent pro v able from S in G is M -v alid in h S, v i . Obvio usly , the axioms of G are M -v alid, and the stru ctural rules, including cut, are strongly sou n d. It remains to sho w that for ev ery application of a canonical ru le R of G : if the premises of R are M -v alid in h S, v i , then its conclusion is M -v alid in h S, v i . W e will sho w this for the case of k = 1, lea vin g the easier case of k = 0 to the r eader. Let R b e an ( n, 1)-ary rule of G : R = Θ R / Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )) − r ⇒ Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )) r where r ∈ { t, f } and Θ R = { Σ j ⇒ Π j } 1 ≤ j ≤ m . An application of R is of the form: { Γ , χ [Σ j ] ⇒ χ [Π j ] , ∆ } 1 ≤ j ≤ m Γ , Q z ( χ [ p 1 ] , ..., χ [ p n ]) − r ⇒ ∆ , Q z ( χ [ p 1 ] , ..., χ [ p n ]) r where χ is some h R, Γ ∪ ∆ , z i -mapping. Supp ose that { Γ , χ [Σ j ] ⇒ χ [Π j ] , ∆ } 1 ≤ j ≤ m is M - v alid in h S, v i . W e will no w sho w that Γ , Q z ( χ [ p 1 ] , ..., χ [ p n ]) − r ⇒ ∆ , Q z ( χ [ p 1 ] , ..., χ [ p n ]) r is also M -v alid in h S, v i . (a) L et σ b e an S -substitution, suc h that S , v | = M σ [Γ] and for ev ery ψ ∈ ∆: S, v 6| = M σ [ ψ ]. Denote by e ψ the L -formula obtained f rom a f orm ula ψ b y sub stituting ev ery free o ccurren ce of w ∈ F v [ ψ ] − { z } for σ [ w ]. Let E = {h v [ ] χ [ p 1 ] { a/z } ] , ..., v [ ] χ [ p n ] { a/z } ] i | a ∈ D } . W e will sho w that ˜ Q [ E ] = { r } , and so v [ σ [ Qz ( χ [ p 1 ] , ..., χ [ p n ])]] = r . F rom ( a ) it w ill follo w that Γ , Q z ( χ [ p 1 ] , ..., χ [ p n ]) − r ⇒ ∆ , Q z ( χ [ p 1 ] , ..., χ [ p n ]) r is M -v alid in h S, v i . W e pro ve this by s ho wing that Θ R is v alid in some E -charac teristic L n k -structure. Th en, by suitabilit y of M , we shall conclud e that ˜ Q M [ E ] = r . Construct the L n k -structure N = h D ′ , I ′ i as follo ws: • D ′ = D . • F or every a ∈ D : I ′ [ p i ][ a ] = v [ g χ [ p i ] { a/z } ]. • F or every constan t c , I ′ [ c ] = I [ σ [ χ [ c ]]]. W e will no w show that Θ R = { Σ j ⇒ Π j } 1 ≤ j ≤ m is v alid in N . Supp ose for con tradiction that it is not so. Th en there exists some 1 ≤ j ≤ m , for which Σ j ⇒ Π j is not v alid in N . Th us there is some N -su bstitution η , suc h that: ( b ) w henev er p i ( t ) ∈ Π j ∪ Σ j : p i ( t ) ∈ ite ( I ′ [ p i ][ I ′ [ η [ t ]]] , Σ j , Π j ). CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 19 W e sho w now that Γ , χ [Σ j ] ⇒ χ [Π j ] , ∆ is not M -v alid in h S, v i , in con tradiction to our assumption ab out the premises of the ab o ve application. Let ψ ∈ ite ( s, χ [Σ j ] , χ [Π j ]) for s ∈ { t, f } . Let σ ′ b e the S -substitution similar to σ except that σ ′ [ χ [ y ]] = a y , w here a y = I ′ [ η [ y ]] for ev ery v ariable y o ccurring in Θ R . Note that σ ′ is w ell-defined, since for eve ry t w o different v ariables x, y : χ [ x ] 6 = χ [ y ] (recall defn . 2.3). Th en one of the follo wing holds: • ψ = χ [ p i ] { χ [ c ] /z } , where p i ( c ) ∈ ite ( s, Σ j , Π j ) and χ [ c ] is some term free for z in χ [ p i ], suc h that for any v ariable y o ccur r ing in Θ R , χ [ y ] do es not occur in χ [ c ]. Recall that b y (b) , I ′ [ p i ][ I ′ [ η [ c ]]] = s . An d so: v [ σ ′ [ ψ ]] = v [ σ ′ [ χ [ p i ] { χ [ c ] /z } ]] = v [ g χ [ p i ] { σ ′ [ χ [ c ]] /z } ] = v [ g χ [ p i ] { σ [ χ [ c ]] /z } ] (Recall that ev ery v ariable y o ccurr ing in Θ R prev en ts χ [ y ] from o ccurring freely in Q z ( χ [ p 1 ] , ..., χ [ p n ]), and that σ, σ ′ only d iffer for v ariables χ [ z ] where z o ccurs in Θ R .) By Lemma 3.6-2 and the legalit y of v : v [ g χ [ p i ] { σ [ χ [ c ]] /z } ] = v [ g χ [ p i ] { I [ σ [ χ [ c ]]] /z } ] By d efinition of I ′ , I ′ [ c ] = I [ σ [ χ [ c ]]] and so: v [ g χ [ p i ] { I [ σ [ χ [ c ]]] /z } ] = v [ g χ [ p i ] { I ′ [ c ] /z } ] = I ′ [ p i ][ I ′ [ c ]] = I ′ [ p i ][ I ′ [ η [ c ]]] = s • ψ = χ [ p i ] { χ [ y ] /z } , wh ere p i ( y ) ∈ ite ( s, Σ j , Π j ) and χ [ y ] do es not o ccur in Γ ∪ ∆ ∪ {Q z ( ψ 1 , ..., ψ n ) } and is free f or z in χ [ p i ]. T h en I ′ [ p i ][ I ′ [ η [ y ]]] = s . Let a = I ′ [ η [ y ]]. T h en, σ ′ [ χ [ y ]] = a an d so: v [ σ ′ [ ψ ]] = v [ σ ′ [ χ [ p i ] { χ [ y ] /z } ] = v [ g χ [ p i ] { σ ′ [ χ [ y ]] /z } ] = = v [ g χ [ p i ] { a/z } ] = I ′ [ p i ][ a ] = I ′ [ p i ][ I ′ [ µ [ y ]]] = s Th us we ha v e sh own that v [ σ ′ [ ψ ]] = s whenever ψ ∈ ite ( s, χ [Σ j ] , χ [Π j ]). Also, there is no v ariable y o ccurring in Θ R , such that χ [ y ] o ccurs in Γ ∪ ∆ , and so σ [Γ] = σ ′ [Γ] and σ [∆] = σ ′ [∆]. Thus for eve ry ψ ∈ Γ ∪ χ [Σ j ], v [ σ ′ [ ψ ]] = t while for ev ery ϕ ∈ ∆ ∪ χ [Π j ], v [ σ ′ [ ϕ ]] = f . Hence, Γ , χ [Σ j ] ⇒ ∆ , χ [Π j ] is not M -v alid in h S, v i , in con tr ad iction to our assumption on the v alidit y of the premises of the application ab ov e. W e ha v e shown that { Σ j ⇒ Π j } 1 ≤ j ≤ m is v alid in N . Ob viously 13 , D ist N = E . Since M is suitable for G : ˜ Q M [ E ] = { r } and so v [ σ [ Q z ( χ [ p 1 ] , ..., χ [ p n ])]] = r . F rom this fact and assumption (a) it follo ws that Γ , Q z ( χ [ p 1 ] , ..., χ [ p n ]) − r ⇒ ∆ , Q z ( χ [ p 1 ] , ..., χ [ p n ]) r is M -v alid in h S, v i . Pro of of Prop osition 4.8 : Let S b e a set of sequents closed under substitution and Γ ⇒ ∆ - a sequent, such that S ∪ { Γ ⇒ ∆ } satisfies the free-v ariable condition. Sup p ose that Γ ⇒ ∆ has no simple p ro of from S in G . T o sh ow that S 6⊢ M Γ ⇒ ∆, we will construct a str u cture S and an M -legal v aluation v , su c h that the sequents of S are M -v alid in h S, v i , while Γ ⇒ ∆ is not. It is easy to see that we can limit our selv es to the language L ∗ , w hic h is a s ubset of L , consisting of all th e constan ts and p r edicate and function sym b ols, o ccurrin g in S ∪{ Γ ⇒ ∆ } . Let T b e the set of all the terms in L ∗ whic h do not con tain v ariables o ccurrin g b ound in Γ ⇒ ∆ and S . It is a standard matter to sho w that Γ , ∆ can b e extended to t w o (p ossibly infinite) sets Γ ′ , ∆ ′ (where Γ ⊆ Γ ′ and ∆ ⊆ ∆ ′ ), satisfying the follo wing prop erties: 13 Recall that E = {h v [ ] χ [ p 1 ] { a/z } ] , ..., v [ ] χ [ p n ] { a/z } ] i | a ∈ D } and I ′ [ p i ][ a ] = v [ ] χ [ p i ] { a/z } ] for every a ∈ D and every 1 ≤ i ≤ n . 20 A. A V RON AND A. ZAM ANSKY (1) F or ev ery finite Γ 1 ⊆ Γ ′ and ∆ 1 ⊆ ∆ ′ , Γ 1 ⇒ ∆ 1 has no simple pr o of in G . (2) There are no ψ ∈ Γ ′ and ϕ ∈ ∆ ′ , su c h that ψ ≡ α ϕ . (3) If { Σ j ⇒ Π j } 1 ≤ j ≤ m / Q ( r ) is an ( n, 0)-ary rule of G and Q ( ψ 1 , ..., ψ n ) ∈ ite ( r , ∆ ′ , Γ ′ ), then there is some 1 ≤ j ≤ m , such that whenev er p i ∈ ite ( s, Σ j , Π j ), ψ i ∈ ite ( s, Γ ′ , ∆ ′ ) for s ∈ { t, f } . (4) If { Σ j ⇒ Π j } 1 ≤ j ≤ m / Q ( r ) is an ( n , 1)-ary r ule of G and Q z ( ψ 1 , ..., ψ n ) ∈ ite ( r , ∆ ′ , Γ ′ ), then there is some 1 ≤ j ≤ m , suc h that: • F or eve ry constan t c , whenev er p i ( c ) ∈ ite ( s, Σ j , Π j ) for some 1 ≤ i ≤ n , th en ψ i { t /z } ∈ ite ( s, Γ ′ , ∆ ′ ) for ev ery term t ∈ T . • F or eac h v ariable y , there exists some t y ∈ T , such that w h enev er p i ( y ) ∈ ite ( s, Σ j , Π j ) for some 1 ≤ i ≤ n , th en ψ i { t y /z } ∈ ite ( s, Γ ′ , ∆ ′ ). Note that ev ery t ∈ T is free f or z in ψ i for eve ry 1 ≤ i ≤ n . (5) F or ev ery form ula ψ o ccurring in S , ψ ∈ Γ ′ ∪ ∆ ′ . Note that the last condition can b e satisfied b ecause cuts on formulas from S are allo w ed in a simple pro of. Let S = h D , I i b e the L ∗ -structure d efined as follo ws: • D = T . • I [ c ] = c for eve ry constan t c of L ∗ . • I [ f ][ t 1 , ..., t n ] = f ( t 1 , ..., t n ) for ev ery n -ary f unction symb ol f . • I [ p ][ t 1 , ..., t n ] = t iff p ( t 1 , ..., t n ) ∈ Γ ′ for eve ry n -ary pr edicate symbol p . Let σ ∗ b e any S -substitution s atisfying σ ∗ [ x ] = x for ev ery x ∈ T . (Note that ev er y x ∈ T is also a mem b er of th e domain and thus has an in dividual constan t r eferr ing to it in L ∗ ( D ).) F or an L ( D )-form ula ψ (an L ( D )-term t ), we will denote b y b ψ ( b t ) the L -form ula ( L - term) obtained from ψ ( t ) by replacing ev ery in dividual constan t of the form s for s ome s ∈ T by the term s . More formally , b t and b ψ are defined as follo ws: • b x = x for an y v ariable x of L . • b c = c f or an y constan t c of L . • b t = t for an y t ∈ T . • \ f ( t 1 , ..., t n ) = f ( b t 1 , ..., b t n ). • \ p ( t 1 , ..., t n ) = p ( b t 1 , ..., b t n ). • \ Q ( ψ 1 , ..., ψ n ) = Q ( b ψ 1 , ..., b ψ n ). • \ Q x ( ψ 1 , ..., ψ n ) = Q x ( b ψ 1 , ..., b ψ n ). Lemma A.1. L et t b e an L ( D ) - term and ψ - an L ( D ) - formula. (1) F or any z , x : b t { z /x } = \ t { z /x } and b ψ { z /x } = \ ψ { z /x } . (2) ψ ∼ S σ ∗ [ b ψ ] . (3) F or every ψ ∈ Γ ′ ∪ ∆ ′ : [ σ ∗ [ ψ ] = ψ . Pr o of. The lemma is prov ed b y a tedious induction on t an d ψ . Define th e S -v aluation v as follo ws: • v [ p ( t 1 , ..., t n )] = I [ p ][ I [ t 1 ] , ..., I [ t n ]]. • F or every ( n, 0)-ary quantifier Q of L , if there is some C ∈ Γ ′ ∪ ∆ ′ , su c h th at C ≡ α \ Q ( ψ 1 , ..., ψ n ), then v [ Q ( ψ 1 , ..., ψ n )] = t iff C ∈ Γ ′ . Otherwise v [ Q ( ψ 1 , ..., ψ n )] = t iff ˜ Q [ {h v [ ψ 1 ] , ..., v [ ψ n ] i} ] = { t } . CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 21 • F or every ( n, 1)-ary quantifier Q of L , if there is some C ∈ Γ ′ ∪ ∆ ′ , su c h th at C ≡ α \ Q x ( ψ 1 , ..., ψ n ), then v [ Q x ( ψ 1 , ..., ψ n )] = t iff C ∈ Γ ′ . Otherwise v [ Q x ( ψ 1 , ..., ψ n )] = t iff ˜ Q [ {h v [ ψ 1 { a/x } ] , ..., v [ ψ n { a/x } ] i | a ∈ D } ] = { t } . Lemma A.2. (1) I ∗ [ σ ∗ [ t ]] = t for eve ry t ∈ T . (2) F or every two L ( D ) -formulas ψ , ψ ′ : if ψ ≡ α ψ ′ , then σ ∗ [ ψ ] ≡ α σ ∗ [ ψ ′ ] . (3) F or every two L ( D ) -sentenc es ψ , ψ ′ : if ψ ∼ S ψ ′ , then b ψ ≡ α b ψ ′ . Pr o of. The claims are prov en by in duction on t in th e first case, and on ψ and ψ ′ in the second and third cases. Lemma A.3. F or eve ry ψ ∈ Γ ′ ∪ ∆ ′ : v ( σ ∗ [ ψ ]) = t iff ψ ∈ Γ ′ . Pr o of. If ψ = p ( t 1 , ..., t n ), then v [ σ ∗ [ ψ ]] = I [ p ][ I [ σ ∗ [ t 1 ]] , ..., I [ σ ∗ [ t n ]]]. Note 14 that for ev ery 1 ≤ i ≤ n , t i ∈ T . By Lemma A.2-1, I [ σ ∗ [ t i ]] = t i , and b y the definition of I , v [ σ ∗ [ ψ ]] = t iff p ( t 1 , ..., t n ) ∈ Γ ′ . Otherwise ψ = Q ( ψ 1 , ..., ψ n ) or ψ = Q ′ x ( ψ 1 , ..., ψ n ). If ψ ∈ Γ ′ , then by Lemma A.1-3 [ σ ∗ [ ψ ] = ψ ∈ Γ ′ and so v [ σ ∗ [ ψ ]] = t . I f ψ ∈ ∆ ′ then by prop ert y 2 of Γ ′ ∪ ∆ ′ it cannot b e the case that there is some C ∈ Γ ′ , su c h that C ≡ α [ σ ∗ [ ψ ] = ψ and so v [ σ ∗ [ ψ ]] = f . . Lemma A.4. v is le gal in M G . Pr o of. First we need to sh o w that v resp ects the ∼ S -relation. W e pr o ve by induction on L ∗ ( D )-sen tences ψ, ψ ′ : if ψ ∼ S ψ ′ , th en v [ ψ ] = v [ ψ ′ ]. • ψ = p ( t 1 , ..., t n ), ψ ′ = p ( s 1 , ..., s n ) and t i ∼ S s i for every 1 ≤ i ≤ n . Then I [ t i ] = I [ s i ] and by definition of v : v [ p ( t 1 , ..., t n )] = I [ p ][ I [ t 1 ] , ..., I [ t n ]] = I [ p ][ I [ s 1 ] , ..., I [ s n ]] = v [ p ( s 1 , ..., s n )]. • ψ = Q x ( ψ 1 , ..., ψ n ), ψ ′ = Q y ( ψ ′ 1 , ..., ψ ′ n ) and f or every 1 ≤ i ≤ n : ψ i { z /x } ∼ S ψ ′ i { z /y } for a fresh v ariable z . Th en by Lemm a 3.6-2 for ev ery a ∈ D : ψ i { z /x }{ a/z } = ψ i { a/x } ∼ S ψ ′ i { a/y } = ψ i { z /y }{ a /z } . By the indu ction hypothesis, {h v [ ψ 1 { a/x } ] , ..., v [ ψ n { a/x } ] i | a ∈ D } = {h v [ ψ ′ 1 { a/x } ] , ..., v [ ψ ′ n { a/x } ] i | a ∈ D } . One of the f ollo wing cases holds: − There is no C ∈ Γ ′ ∪ ∆ ′ , such that C ≡ α b ψ or C ≡ α b ψ ′ . T hen v [ Q x ( ψ 1 , ..., ψ n )] = t iff {h v [ ψ 1 { a/x } ] , ..., v [ ψ n { a/x } ] i | a ∈ D } = t iff {h v [ ψ ′ 1 { a/x } ] , ..., v [ ψ ′ n { a/x } ] i | a ∈ D } = t iff v [ Q y ( ψ ′ 1 , ..., ψ ′ n )] = t . − There is some C ∈ Γ ′ ∪ ∆ ′ , such that C ≡ α b ψ . By Lemma A.2-3, b ψ ≡ α b ψ ′ , and so v [ ψ ] = v [ ψ ′ ] = t iff C ∈ Γ. − There is some C ∈ Γ ′ ∪ ∆ ′ , suc h that C ≡ α b ψ ′ . Similarly to the previous case, v [ ψ ] = v [ ψ ′ ] = t iff C ∈ Γ. • The case of ψ = Q ( ψ 1 , ..., ψ n ), ψ ′ = Q ( ψ ′ 1 , ..., ψ ′ n ) is treated sim ilarly . It remains to sh o w that v resp ects the interpretatio ns of the ( n, k )-ary qu an tifiers in M G . The case of k = 0 is n ot hard and is left to the reader. W e w ill sh o w the pro of for the case of k = 1. Su p p ose by contradict ion th at there is some L ∗ ( D )-sen tence A = 14 This is obvious if t i does not o ccur in t he set { Γ ⇒ ∆ } ∪ S . If it o ccurs in this set, then by the free-v ariable condition t i does not contain va riables b ound in this set and so t i ∈ T by defi n ition of T . 22 A. A V RON AND A. ZAM ANSKY Q z ( ψ 1 , ..., ψ n ), suc h that v [ A ] 6∈ ˜ Q [ H A ], wh ere H A = {h v [ ψ 1 { a/z } ] , ..., v [ ψ n { a/z } ] i | a ∈ D } . F r om the d efi nition of v , it m ust b e the case that 15 : ( a ) there is some L -form u la C ∈ Γ ′ ∪ ∆ ′ , su c h that C ≡ α b A , and v [ A ] = t iff C ∈ Γ ′ . Supp ose that ˜ Q [ H A ] = { t } and v [ A ] = f . By d efinition of M G and the fact that ˜ Q [ H A ] is a singleton, it m ust b e the case that there is some canonical rule { Σ k ⇒ Π k } 1 ≤ k ≤ m / ⇒ Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )) in G , such that: ( b ) { Σ k ⇒ Π k } 1 ≤ k ≤ m is v alid in a H A -c h aracteristic structure N = h D N , I N i . A = Q z ( ψ 1 , ..., ψ n ) and C ≡ α b A , so C is of the form Q w ( ϕ 1 , ..., ϕ n ). By Lemma A.2- 2, σ ∗ [ C ] ≡ α σ ∗ [ b A ]. By Lemma 3.6-1, σ ∗ [ C ] ∼ S σ ∗ [ b A ]. By Lemma A.1-2, σ ∗ [ b A ] ∼ S A , and thus σ ∗ [ C ] ∼ S A . Let φ i b e the formula obtained from ϕ i b y su bstituting ev ery x ∈ F v [ ϕ i ] = { w } for σ ∗ [ x ]. By Lemm a 3.6-2, φ i { a/w } ∼ S ψ i { a/z } for ev ery a ∈ D . W e ha v e already sh own that v resp ects the ∼ S -relation, and so v [ φ i { a/w } ] = v [ ψ i { a/z } ]. Thus H A = {h v [ φ 1 { a/w } ] , ..., v [ φ n { a/w } ] i | a ∈ D } . Since v [ A ] = f , it follo ws fr om (a) that C = Q w ( ϕ 1 , ..., ϕ n ) ∈ ∆ ′ . Then b y pr op ert y 3 of Γ ′ ∪ ∆ ′ , th ere is s ome 1 ≤ j ≤ m , su c h that whenev er p i ( y ) ∈ ite ( r , Σ j , Π j ), there is some t y ∈ T , such that ϕ i { t y /w } ∈ ite ( r, Γ ′ , ∆ ′ ). By Lemma A.3, v [ σ ∗ [ ϕ i { t y /w } ]] = v [ φ i { σ ∗ [ t y ] /w } ] = r . Sin ce N is H A -c h aracteristic, there is some a y ∈ D N , suc h that I N [ p i ][ a y ] = v [ φ i { σ ∗ [ t y ] /w } ] = r . Let us no w sho w that Σ j ⇒ Π j is n ot v alid in N (in con tradiction to ( b )). Let µ b e an y N -substitution, su c h that µ [ y ] = a y for eve ry v ariable y o ccurring in Σ j ∪ Π j . W e n o w sho w that whenev er p ( t ) ∈ ite ( s, Σ j , Π j ), I [ p ][ I [ µ [ t ]]] = s . Let p ( t ) ∈ ite ( s, Σ j , Π j ). If t is some v ariable y , then I N [ p i ][ µ [ y ]] = I N [ p i ][ I N [ a y ]] = I N [ p i ][ a y ] = s . Oth erwise t is some constant c . By prop ert y 3 of Γ ′ ∪ ∆ ′ , for ev ery t ∈ T : ϕ i { t /x } ∈ ite ( s, Σ j , Π j ). By Lemma A.3 , v [ σ ∗ [ ϕ i { t /w } ]] = v [ φ i { σ ∗ [ t ] /w } ] = s . Th us for eve ry t ∈ T : v [ φ i { σ ∗ [ t ] /w } ] = v [ φ i { t /w } ] = s . Since N is H A -c h aracteristic, I N [ p c ][ I N [ c ]] = s . And so we h a ve sh o w n that Σ j ⇒ Π j is n ot v alid in N , in cont radiction to ( b ). The p r o of for the case of ˜ Q [ H A ] = { f } and v [ A ] = t is symmetric. Lemma A.5. F or eve ry se quent Σ ⇒ Π ∈ S , Σ ⇒ Π is M -valid in h S, v i . Pr o of. Supp ose by con tr adiction that there is some Σ ⇒ Π ∈ S , wh ic h is not M -v alid in h S, v i . Then there exists some S -substitution µ , su ch that for ev ery ψ ∈ Σ: S, v | = M µ [ ψ ], and for ev ery ϕ ∈ Π: S, v 6| = M µ [ ϕ ]. Note that for every φ ∈ Σ ∪ Π, d µ [ φ ] is a s ubstitution instance of φ . Since S is closed und er su bstitution, d µ [ φ ] also o ccurs in S , and thus by prop erty 5 of Γ ′ ∪ ∆ ′ : d µ [ φ ] ∈ Γ ′ ∪ ∆ ′ . By Lemma A.3, if d µ [ φ ] ∈ Γ ′ then v [ σ ∗ [ d µ [ φ ]]] = t , and if d µ [ φ ] ∈ ∆ ′ then v [ σ ∗ [ d µ [ φ ]]] = f . By Lemma A.1-2, µ [ φ ] ∼ S σ ∗ [ d µ [ φ ]]. S in ce v is M -legal, it resp ects the ∼ S -relation and so for ev ery φ ∈ Σ ∪ Π: v [ µ [ φ ]] = v [ σ ∗ [ d µ [ φ ]]]. Thus d µ [Σ] ⊆ Γ ′ and d µ [Π] ⊆ ∆ ′ . But d µ [Σ] ⇒ d µ [Π] h as a simple p ro of from S in G , in con tradiction to prop erty 1 of Γ ′ ∪ ∆ ′ . 15 If there is no L -formula C ∈ Γ ′ ∪ ∆ ′ , such that C ≡ α b A , th en b y definition of v , v [ A ] is alw a ys in ˜ Q [ H A ], so this case is not p ossible. CANONICAL CALCU LI WITH ( n, k )-AR Y Q UANTIFIERS 23 W e ha v e sho wn that (i) v is legal in M , (ii) for every ψ ∈ Γ ′ ∪ ∆ ′ : v [ σ ∗ [ ψ ]] = t iff ψ ∈ Γ ′ , and (iii) the sequents in S are M -v alid in h S, v i . F rom (ii) it follo w s that Γ ⇒ ∆ is n ot M -v alid in h S, v i , which completes th e p ro of. Pro of of Prop osition 4.14: F or a s et of clauses Θ, denote by Θ { c/x } the set { Γ { c/x } ⇒ ∆ { c/x } | Γ ⇒ ∆ ∈ Θ } . Then the f ollo wing lemma is easily pr o ved: Lemma A.6. L et Θ b e a classic al ly c onsistent set of clauses. Then for any c onstant c , Θ { c/x } is also classic al ly c onsistent. No w supp ose that a simple canonical calculus G is not coherent. Then there is a pair of ( n, k )-ary du al rules R 1 = Θ 1 / ⇒ A and R 2 = Θ 2 / A ⇒ , su c h that Rnm (Θ 1 ∪ Θ 2 ) is classically consisten t. If k = 0, then the p ro of is similar to the pro of of theorem 4.7 in [2]. Otherwise, k = 1, A = Q v 1 ( p 1 ( v 1 ) , ..., p n ( v 1 )) and whenev er p i ( y ) o ccurs in Rnm (Θ 1 ∪ Θ 2 ) for some v ariable y and some 1 ≤ i ≤ n , there is at most one constant c , su c h that p i ( c ) also o ccurs in Rnm (Θ 1 ∪ Θ 2 ). Recall that Rnm (Θ 1 ∪ Θ 2 ) = Θ 1 ∪ Θ ′ 2 , where Θ ′ 2 is obtained from Θ 2 b y renaming of constan ts and v ariables whic h o ccur in Θ 1 (see defn. 2.8). W e assume that the new constan ts in Θ ′ 2 are in L (this assu mption is not n ecessary b ut it sim p lifies the p resen tation). Obtain th e sets Υ 1 , Υ 2 from Θ 1 , Θ ′ 2 resp ectiv ely as follo ws. F or ev ery 1 ≤ i ≤ n , if p i ( c ) o ccurs in Θ 1 ∪ Θ ′ 2 for some constan t c , r eplace all v ariables y , such that p i ( y ) o ccurs in Θ 1 ∪ Θ ′ 2 b y c (note that this is well-defined due to the s p ecial prop er ty of simple calculi). Otherwise, replace all v ariables y , such that p i ( y ) o ccurs in Θ 1 ∪ Θ ′ 2 b y a fr esh constant d i of L . Th en Υ = Υ 1 ∪ Υ 2 is obtained fr om Θ 1 ∪ Θ ′ 2 b y replacing all v ariables by constan ts. Since Θ 1 ∪ Θ ′ 2 is classicall y consisten t, by rep eated application of Lemma A.6, Υ is also classically consisten t. T hen there exists some L -structure S in which the set of clauses Υ is (classica lly) v alid. Since Υ consists of closed atomic f ormulas, there also exists a (classical) prop ositional v aluation v S , w hic h satisfies Υ. Let Φ = { A | v S [ A ] = t, A ∈ Γ ∪ ∆ , Γ ⇒ ∆ ∈ Υ } and Ψ = { A | v S [ A ] = f , A ∈ Γ ∪ ∆ , Γ ⇒ ∆ ∈ Υ } . Let B j = { Π , Φ ⇒ Σ , Ψ | Π ⇒ Σ ∈ Υ j } for j = 1 , 2. Then B 1 and B 2 are sets of stand ard axioms. (Since v S satisfies Π ⇒ Σ, there is some A ∈ Π, su c h that v S [ A ] = f , or some A ∈ ∆, suc h that v S [ A ] = t . I n the former case, A ∈ Ψ and in the latter case, A ∈ Φ.) Let x b e a fresh v ariable of L . Define the h R 1 , Ψ ∪ Φ , x i -mapping χ (see defn . 2.3) as follo ws. F or ev ery 1 ≤ i ≤ n , χ [ p i ] = p i ( x ) if there is some constan t c , such that p i ( c ) o ccurs in Θ 1 ∪ Θ ′ 2 . Otherw ise, χ [ p i ] = p i ( d i ) (where d i is the fresh constan t of L c hosen ab o v e). F or ev er y constan t c and v ariable y o ccurrin g in Θ 1 ∪ Θ ′ 2 : χ [ c ] = c and χ [ y ] = y . I t is easy to see that Υ 1 = { χ [Σ ′ ] ⇒ χ [Π ′ ] | Σ ′ ⇒ Π ′ ∈ Θ 1 } and Υ 2 = { χ [Σ ′ ] ⇒ χ [Π ′ ] | Σ ′ ⇒ Π ′ ∈ Θ ′ 2 } . Th us the follo wing is an application of R 1 : B 1 Φ , Q x ( χ [ p 1 ] , ..., χ [ p n ]) ⇒ Ψ It is easy to c h ec k that χ is also an h R 2 , Ψ ∪ Φ , x i -mapp ing and so the follo win g is also an application of R 2 : B 2 Φ ⇒ Ψ , Q x ( χ [ p 1 ] , ..., χ [ p n ]) By cut, Φ ⇒ Ψ is pro v able, b ut Φ and Ψ are disjoint sets of atomic form ulas, th us they ha v e no cut-free pro of in G , in con tradictio n to our assumption. This work is licensed under t he Cr eative Commons Attr ibution-NoDer ivs L icense. 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