Canonical calculi with (n,k)-ary quantifiers
CANONICAL CALCULI WITH ( n, k ) -ARY QUANTIFIERS ARNON AVRON AND ANNA ZAMANSKYTel Aviv University e-mail address : { aa,annaz } @post.tau.ac.il Abstract.
Propositional canonical Gentzen-type systems, introduced in 2001 by Avronand Lev, are systems which in addition to the standard axioms and structural rules haveonly logical rules in which exactly one occurrence of a connective is introduced and noother connective is mentioned. A constructive coherence criterion for the non-trivialityof such systems was defined and it was shown that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valuednon-deterministic matrices (2Nmatrices). In 2005 Zamansky and Avron extended theseresults to systems with unary quantifiers of a very restricted form. In this paper wesubstantially extend the characterization of canonical systems to ( n, k )-ary quantifiers,which bind k distinct variables and connect n formulas, and show that the coherencecriterion remains constructive for such systems. Then we focus on the case of k ∈ { , } and for a canonical calculus G show that it is coherent precisely when it has a stronglycharacteristic 2Nmatrix, which in turn is equivalent to admitting strong cut-elimination. Introduction
An ( n, k ) -ary quantifier (for n > k ≥
0) is a generalized logical connective, whichbinds k variables and connects n formulas. Any n -ary propositional connective can bethought of as an ( n, ∧ connective binds novariables and connects two formulas: ∧ ( ψ , ψ ). The standard first-order quantifiers ∃ and ∀ are (1 , ∀ xψ, ∃ xψ .Bounded universal and existential quantifiers used in syllogistic reasoning ( ∀ x ( p ( x ) → q ( x ))and ∃ x ( p ( x ) ∧ q ( x ))) can be represented as (2,1)-ary quantifiers ∀ and ∃ , binding one variableand connecting two formulas: ∀ x ( p ( x ) , q ( x )) and ∃ x ( p ( x ) , q ( x )). An example of ( n, k )-aryquantifiers for k > ([15, 17]). The simplest Henkin quantifier Q H F.4.1.
Key words and phrases:
Proof Theory, Automated Deduction, Cut Elimination, Gentzen-type Systems,Quantifiers, Many-valued Logic, Non-deterministic Matrices. Generalized quantifiers of this kind have been first considered in [16]. In [22] Natural Deduction calculiare provided for n -place connectives and quantifiers and it is shown that derivations in such calculi arenormalizable. It should be noted that the semantic interpretation of quantifiers used in this paper is not sufficient fortreating such quantifiers.
LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.2168/LMCS-4 (3:2) 2008 c (cid:13)
A. Avron and A. Zamansky CC (cid:13) Creative Commons
A. AVRON AND A. ZAMANSKY binds 4 variables and connects one formula: Q H x x y y ψ ( x , x , y , y ) := ∀ x ∃ y ∀ x ∃ y ψ ( x , x , y , y )In this way of recording combinations of quantifiers, dependency relations between variablesare expressed as follows: an existentially quantified variable depends on those universallyquantified variables which are on the left of it in the same row.According to a long tradition in the philosophy of logic, established by Gentzen in hisclassical paper Investigations Into Logical Deduction ([13]), an “ideal” set of introductionrules for a logical connective should determine the meaning of the connective (see, e.g. ,[29, 30], and also [10] for a general discussion). In [2, 3] the notion of a “canonical proposi-tional Gentzen-type rule” was defined in precise terms. A constructive coherence criterionfor the non-triviality of systems consisting of such rules was provided, and it was shown thata system of this kind admits cut-elimination iff it is coherent. It was further proved thatthe semantics of such systems is provided by two-valued non-deterministic matrices (2Nma-trices), which form a natural generalization of the classical matrix. In fact, a characteristic2Nmatrix was constructed for every coherent canonical propositional system.In [28] the results were extended to systems (of a restricted form) with unary quan-tifiers. A characterization of a “canonical unary quantificational rule” in such calculi wasproposed (the standard Gentzen-type rules for ∀ and ∃ are canonical according to it), anda constructive extension of the coherence criterion from [2, 3] for canonical systems of thistype was given. 2Nmatrices were extended to languages with unary quantifiers, using a dis-tributional interpretation of quantifiers ([20, 7]). Then it was proved that again a canonicalGentzen-type system of this type admits cut-elimination iff it is coherent, and that it iscoherent iff it has a characteristic 2Nmatrix.In this paper we make the intuitive notion of a “well-behaved” introduction rule for( n, k )-ary quantifiers formally precise. We considerably extend the scope of the charac-terizations of [2, 3, 28] to “canonical ( n, k )-ary quantificational rules”, so that both thepropositional systems of [2, 3] and the restricted quantificational systems of [28] are specificinstances of the proposed definition. We show that the coherence criterion for the definedsystems remains decidable. Then we focus on the case of k ∈ { , } and show that thefollowing statements concerning a canonical calculus G are equivalent: (i) G is coherent,(ii) G has a strongly characteristic 2Nmatrix, and (iii) G admits strong cut-elimination.We show that coherence is not a necessary condition for standard cut-elimination, and thencharacterize a subclass of canonical systems for which this property does hold.1. Preliminaries
For any n > k ≥
0, if a quantifier Q is of arity ( n, k ), then Q x ...x k ( ψ , ..., ψ n ) isa formula whenever x , ..., x k are distinct variables and ψ , ..., ψ n are formulas of L .For interpretation of quantifiers, we use a generalized notion of distributions (see, e.g [20, 7]).Given a set S , P + ( S ) is the set of all the nonempty subsets of S . We note that by ‘cut-elimination’ we mean here just the existence of proofs without (certain forms of)cuts, rather than an algorithm to transform a given proof to a cut-free one (for the assumptions-free casethe term “cut-admissibility” is sometimes used, but this notion is too weak for our purposes). See, e.g. , [6]for a resolution-based algorithm for cut-elimination in LK.
ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 3
Definition 1.1.
Given a set of truth value V , a distribution of a (1,1)-ary quantifier Q isa function λ Q : P + ( V ) → V .(1,1)-ary distribution quantifiers have been extensively studied and axiomatized inmany-valued logic. See, e.g. , [7, 21, 14].In what follows, L is a language with ( n, k )-ary quantifiers, that is with quantifiers Q , ..., Q m with arities ( n , k ), ..., ( n m , k m ) respectively. Denote by F rm clL the set of closed L -formulas and by T rm clL the set of closed L -terms. V ar = { v , v , ..., } is the set of variablesof L . We use the metavariables x, y, z to range over elements of V ar . ≡ α is the α -equivalence relation between formulas, i.e identity up to the renaming ofbound variables. Lemma 1.2.
Let Q be an ( n, k ) -ary quantifier of L and z , ..., z k fresh variables which donot occur in Q x ..x k ( ψ , ..., ψ n ) . Then: Q x ...x k ( ψ , ..., ψ n ) ≡ α Q y ...y k ( ψ ′ , ..., ψ ′ n ) iff ψ i { z /x , ..., z k /x k } ≡ α ψ ′ i { z /y , ..., z k /y k } for every ≤ i ≤ n . The proof is not hard and is left to the reader.We use [ ] for application of functions in the meta-language, leaving the use of ( ) tothe object language. A { t /x } denotes the formula obtained from A by substituting t for x . Given an L -formula A , F v [ A ] is the set of variables occurring free in A . We denote Q x ...x k A by Q−→ x A , and A ( x , ..., x k ) by A ( −→ x ).A set of sequents S satisfies the free-variable condition if the set of variables occurringbound in S is disjoint from the set of variables occurring free in S .2. Canonical Systems with (n,k)-ary quantifiers
In this section we propose a precise characterization of a “canonical ( n, k )-ary quantifi-cational Gentzen-type rule”.Using an introduction rule for an ( n, k )-ary quantifier Q , we should be able to derive asequent of the form Γ ⇒ Q x ...x k ( ψ , ..., ψ n ) , ∆ or of the form Γ , Qx ...x k ( ψ , ..., ψ n ) ⇒ ∆,based on some information about the subformulas of Q x ...x k ( ψ , ..., ψ n ) contained in thepremises of the rule. For instance, consider the following standard rules for the (1,1)-aryquantifier ∀ : 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 does not occur free in the conclusion. Our key observationis that the internal structure of A , as well as the exact term t or variable w used, areimmaterial for the meaning of ∀ . What is important here is the sequent on which A appears, as well as whether a term variable t or an eigenvariable z is used.It follows that the internal structure of the formulas of L used in the description of arule can be abstracted by using a simplified first-order language, i.e., the formulas of L inan introduction rule of a ( n, k )-ary quantifier, can be represented by atomic formulas withpredicate symbols of arity k . The case when the substituted term is any L -term, will besignified by a constant, and the case when it is a variable satisfying the above conditions- by a variable. In other words, constants serve as term variables, while variables areeigenvariables.Thus in addition to our original language L with ( n, k )-ary quantifiers we define another,simplified language. A. AVRON AND A. ZAMANSKY
Definition 2.1.
For k ≥ n ≥ Con , L nk ( Con ) is the (first-order)language with n k -ary predicate symbols p , ..., p n and the set of constants Con (and noquantifiers). The set of variables of L nk ( Con ) is
V ar = { v , v , ..., } .Note that L nk ( Con ) and L share the same set of variables. Furthermore, henceforthwe assume that for every ( n, k )-ary quantifier Q of L , L nk ( Con ) is a subset of L . Thisassumption is not necessary, but it makes the presentation easier, as will be explained inthe sequel.Next we formalize the notion of a canonical rule and its application. Definition 2.2.
Let
Con be some set of constants. A canonical quantificational rule ofarity ( n, k ) is an expression of the form { Π i ⇒ Σ i } ≤ i ≤ m /C , where m ≥ C is either ⇒ Q v ...v k ( p ( v , ..., v k ) , ..., p n ( v , ..., v k )) or Q v ...v k ( p ( v , ..., v k ) , ..., p n ( v , ..., v k )) ⇒ forsome ( n, k )-ary quantifier Q of L and for every 1 ≤ i ≤ m : Π i ⇒ Σ i is a clause over L nk ( Con ).Henceforth, in cases where the set of constants
Con is clear from the context (it is theset of all constants occurring in a canonical rule), we will write L nk instead of L nk ( Con ).A canonical rule is a schematic representation, while for an actual application we need toinstantiate the schematic variables by the terms and formulas of L . This is done using amapping function, defined as follows. Definition 2.3.
Let R = Θ /C be an ( n, k )-ary canonical rule, where C is of one of the forms( Q−→ v ( p ( −→ v ) , ..., p n ( −→ v )) ⇒ ) or ( ⇒ Q−→ v ( p ( −→ v ) , ..., p n ( −→ v ))). Let Γ be a set of L -formulasand z , ..., z k - distinct variables of L . An h R, Γ , z , ..., z k i -mapping is any function χ fromthe predicate symbols, terms and formulas of L nk to formulas and terms of L , satisfying thefollowing conditions: • For every 1 ≤ i ≤ n , χ [ p i ] is an L -formula. • χ [ y ] is a variable of L . • χ [ x ] = χ [ y ] for every two variables x = y . • χ [ c ] is an L -term, such that χ [ x ] does not occur in χ [ c ] for any variable x occurring in Θ. • For every 1 ≤ i ≤ n , whenever p i ( t , ..., t k ) occurs in Θ, for every 1 ≤ j ≤ k : χ [ t j ]is a term free for z j in χ [ p i ], and if t j is a variable, then χ [ t j ] does not occur free inΓ ∪ {Q z ...z k ( χ [ p ] , ..., χ [ p n ]) } . • χ [ p i ( t , ..., t k )] = χ [ p i ] { χ [ t ] /z , ..., χ [ t k ] /z k } .We extend χ to sets of L nk ( Con Θ )-formulas as follows: χ [∆] = { χ [ ψ ] | ψ ∈ ∆ } Given a schematic representation of a rule and an instantiation mapping, we can definean application of a rule as follows.
Definition 2.4. An application of a canonical rule of arity ( n, k ) R = { Π i ⇒ Σ i } ≤ i ≤ m / Q−→ v ( p ( −→ v ) , ..., p n ( −→ v )) ⇒ is any inference step of the form: { Γ , χ [Π i ] ⇒ ∆ , χ [Σ i ] } ≤ i ≤ m Γ , Q z ...z k ( χ [ p ] , ..., χ [ p n ]) ⇒ ∆where z , ..., z k are variables, Γ , ∆ are any sets of L -formulas and χ is some h R, Γ ∪ ∆ , z , ..., z k i -mapping. By a clause we mean a sequent containing only atomic formulas.
ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 5
An application of a canonical quantificational rule of the form { Π i ⇒ Σ i } ≤ i ≤ m / ⇒ Q−→ v ( p ( −→ v ) , ..., p n ( −→ v ))is defined similarly.Below we demonstrate the above definition by a number of examples. Examples 2.5. (1) The standard right introduction rule for ∧ , which can be thought ofas a (2 , {⇒ p , ⇒ p } / ⇒ p ∧ p . Its application is of the form:Γ ⇒ ψ , ∆ Γ ⇒ ψ , ∆Γ ⇒ ψ ∧ ψ , ∆(2) The standard introduction rules for the (1 , ∀ and ∃ can be formulatedas follows: { p ( c ) ⇒} / ∀ v p ( v ) ⇒ {⇒ p ( v ) } / ⇒ ∀ v p ( v ) {⇒ p ( d ) } / ⇒ ∃ v p ( v ) { p ( v ) ⇒} / ∃ v p ( v ) ⇒ Applications of these rules have 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 free for w in ψ , z is not free in Γ ∪ ∆ ∪ {∀ wψ } , and t is any term free for w in ψ .(3) Consider the bounded existential and universal (2 , ∀ and ∃ (corre-sponding to ∀ x.p ( x ) → p ( x ) and ∃ x.p ( x ) ∧ p ( x ) used in syllogistic reasoning). Theircorresponding rules can be formulated as follows: { p ( c ) ⇒ , ⇒ p ( c ) } / ∀ v ( p ( v ) , p ( v )) ⇒{ p ( v ) ⇒ p ( v ) } / ⇒ ∀ v ( p ( v ) , p ( v )) { p ( v ) , p ( v ) ⇒} / ∃ v ( p ( v ) , p ( v )) ⇒{⇒ p ( c ) , ⇒ p ( c ) } / ⇒ ∃ v ( p ( v ) , p ( v ))Applications of these rules are of the form:Γ , ψ { t /z } ⇒ ∆ Γ ⇒ ψ { t /z } , ∆Γ , ∀ z ( ψ , ψ ) ⇒ ∆ Γ , ψ { y/z } ⇒ ψ { y/z } , ∆Γ ⇒ ∀ z ( ψ , ψ ) , ∆Γ , ψ { y/z } , ψ { y/z } ⇒ ∆Γ , ∃ z ( ψ , ψ ) ⇒ ∆ Γ ⇒ ψ { t /x } , ∆ Γ ⇒ ψ { t /x } , ∆Γ ⇒ ∃ z ( ψ , ψ ) , ∆where t and y are free for z in ψ and ψ , y does not occur free in Γ ∪ ∆ ∪ {∃ z ( ψ , ψ ) } .(4) Consider the (2,2)-ary rule { p ( v , v ) ⇒ , p ( v , d ) ⇒ p ( c, d ) } / ⇒ Q v v ( p ( v , v ) , p ( v , v ))Its application is of the form:Γ , ψ { w /z , w /z } ⇒ ∆ Γ , ψ { w /z , t /z } ⇒ ∆ , ψ { t /z , t /z } Γ ⇒ ∆ , Q z z ( ψ , ψ )where w , w , w , t , t satisfy the appropriate conditions. A. AVRON AND A. ZAMANSKY
Note that although the derivability of the α -axiom is essential for any logical system, itis not guaranteed to be derivable in a canonical system. What natural syntactic conditionsguarantee its derivability is still a question for further research. For now we explicitly addthe α -axiom to the canonical calculi. Notation. (Following [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 may be empty) denote ite ( s, Φ ∪ { A } , Φ). Forinstance, the sequents A ⇒ and ⇒ A are denoted by A − s ⇒ A s for s = f and s = t respectively. According to this notation, a ( n, k )-ary canonical rule is of the form: { Σ j ⇒ Π j } ≤ j ≤ m / Q−→ v ( p ( −→ v ) , ..., p n ( −→ v )) − s ⇒ Q−→ v ( p ( −→ v ) , ..., p n ( −→ v )) s for s ∈ { t, f } . For further abbreviation, we denote such rule by { Σ j ⇒ Π j } ≤ j ≤ m / Q ( s ). Definition 2.6.
A Gentzen-type calculus G is canonical if in addition to the α -axiom A ⇒ A ′ for A ≡ α A ′ and the standard structural rules, G has only canonical rules. Definition 2.7.
Two ( n, k )-ary canonical introduction rules Θ /C and Θ /C for Q are dual if for some s ∈ { t, f } : C = A − s ⇒ A s and C = A s ⇒ A − s , where A = Q v ...v k ( p ( v , ..., v k ) , ..., p n ( v , ..., v k )).Although we can define arbitrary canonical systems using our simplified language L nk ,our quest is for systems, the syntactic rules of which define the semantic meaning of logicalconnectives/quantifiers. Thus we are interested in calculi with a “reasonable” or “non-contradictory” set of rules, which allows for defining a sound and complete semantics forthe system. This can be captured syntactically by the following extension of the coherence criterion of [2, 28]. Definition 2.8.
For two sets of clauses Θ , Θ over L nk , Rnm (Θ ∪ Θ ) is a set Θ ∪ Θ ′ ,where Θ ′ is obtained from Θ by a fresh renaming of constants and variables which occurin Θ .Henceforth it will be convenient (but not essential) to assume that the fresh constantsused for the renaming are in L . Definition 2.9. (Coherence) A canonical calculus G is coherent if for every two dualcanonical rules Θ / ⇒ A and Θ /A ⇒ , the set of clauses Rnm (Θ ∪ Θ ) is classicallyinconsistent.Note that the principle of renaming of clashing constants and variables is similar tothe one used in first-order resolution. The importance of this principle for the definition ofcoherence will be explained in the sequel. Proposition 2.10. (Decidability of coherence)
The coherence of a canonical calculus G is decidable.Proof. The question of classical consistency of a finite set of clauses without function sym-bols (over L nk ) can be shown to be equivalent to satisfiability of a finite set of universalformulas with no function symbols. This is decidable (by an obvious application of Her-brand’s theorem). A strongly related coherence criterion is defined in [19], where linear logic is used to reason aboutvarious sequent systems. Our coherence criterion is also equivalent in the context of canonical calculi to thereductivity condition in [9], as will be explained in the sequel.
ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 7 The semantic framework
Non-deterministic matrices.
Our main semantic tool are non-deterministic matri-ces (Nmatrices), first introduced in [2, 3] and extended in [27, 28]. These structures are ageneralization of the standard concept of a many-valued matrix, in which the truth-valueof a formula is chosen non-deterministically from a given non-empty set of truth-values.Thus, given a set of truth-values V , we can generalize the notion of a distribution functionof an ( n, k )-ary quantifier Q (from Definition. 1.1) to a function λ Q : P + ( V n ) → P + ( V ). Inother words, given some distribution Y of n-ary vectors of truth values, the interpretationfunction non-deterministically chooses the truth value assigned to Q−→ z ( ψ , ..., ψ n ) out from λ Q [ Y ] . Definition 3.1. (Non-deterministic matrix)
A non-deterministic matrix(henceforth Nmatrix) for L is a tuple M = < V , G , O > , where: • V is a non-empty set of truth values. • G (designated truth values) is a non-empty proper subset of V . • O is a set of interpretation functions: for every ( n, k )-ary quantifier Q of L, O includesthe corresponding distribution function ˜ Q M : P + ( V n ) → P + ( V ).Note the special treatment of propositional connectives in the definition above. In[2, 28], an Nmatrix includes an interpretation function ˜ ⋄ : V n → P + ( V ) for every n -aryconnective of the language; given a valuation v , the truth value v [ ⋄ ( ψ , ..., ψ n )] is chosennon-deterministically from ˜ ⋄ [ h v [ ψ ] , ..., v [ ψ n ] i ]. In the definition above, the interpretation ofa propositional connective ⋄ is a function of another type: ˜ ⋄ : P + ( V n ) → P + ( V ). This can bethought as a generalization of the previous definition, identifying the tuple h v [ ψ ] , ..., v [ ψ n ] i with the singleton {h v [ ψ ] , ..., v [ ψ n ] i} . The advantage of this generalization is that it allowsfor a uniform treatment of both quantifiers and propositional connectives. Definition 3.2. (L-structure)
Let M be an Nmatrix for L . An L-structure for M is apair S = h D, I i where D is a (non-empty) domain and I is a function interpreting constants,predicate symbols and function symbols of L , satisfying the following 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 function. I is extended to interpret closed terms of L as follows: I [ f ( t , ..., t n )] = I [ f ][ I [ t ] , ..., I [ t n ]]Here a note on our treatment of quantification in the framework of Nmatrices is inorder. The standard approach to interpreting quantified formulas is by using objectual (orreferential) semantics, where the variable is thought of as ranging over a set of objects fromthe domain (see, e.g. , [11, 12]). An alternative approach is substitutional quantification([18]), where quantifiers are interpreted substitutionally, i.e. a universal (an existential)quantification is true if and only if every one (at least one) of its substitution instances istrue (see, e.g. , [24, 26]). [27] explains the motivation behind choosing the substitutionalapproach for the framework of Nmatrices, and points out the problems of the objectualapproach in this context. The substitutional approach assumes that every element of thedomain has a closed term referring to it. Thus given a structure S = h D, I i , we extend thelanguage L with individual constants , one for each element of D . Definition 3.3. ( L(D) )
Let S= h D, I i be an L -structure for an Nmatrix M . L ( D ) isthe language obtained from L by adding to it the set of individual constants { a | a ∈ D } . S ′ = h D, I ′ i is the L ( D )-structure, such that I ′ is an extension of I satisfying: I ′ [ a ] = a . A. AVRON AND A. ZAMANSKY
Given an L -structure S = h D, I i , we shall refer to the extended L ( D )-structure h D, I ′ i as S and to I ′ as I when the meaning is clear from the context. Definition 3.4. ( S -substitution) Given an L -structure S = h D, I i for an Nmatrix M for L , an S -substitution is a function σ : V ar → T rm clL ( D ) . It is extended to σ : T rm L ∪ F rm L → T rm clL ( D ) ∪ F rm clL ( D ) as follows: for a term t of L ( D ), σ [ t ] is the closed term obtained from t by replacing every x ∈ F v [ t ] by σ [ x ]. For a formula ϕ , σ [ ϕ ] is the sentence obtained from ϕ by replacing every x ∈ F v [ ϕ ] by σ [ x ].Given a set Γ of formulas, we denote the set { σ [ ψ ] | ψ ∈ Γ } by σ [Γ].The motivation for the following definition is purely technical and is related to extendingthe language with the set of individual constants { a | a ∈ D } . Suppose we have a closedterm t , such that I [ t ] = a ∈ D . But a also has an individual constant a referring to it. Wewould like to be able to substitute t for a in every context. Definition 3.5. (Congruence of terms and formulas)
Let S be an L -structure for anNmatrix M . The relation ∼ S between terms of L ( D ) is defined inductively as follows: • x ∼ S x • For closed terms t , t ′ of L ( D ): t ∼ S t ′ when I [ t ] = I [ t ′ ]. • If t ∼ S t ′ , ..., t n ∼ S t ′ n , then f ( t , ..., t n ) ∼ S f ( t ′ , ..., t ′ n ).The relation ∼ S between formulas of L ( D ) is defined as follows: • If t ∼ S t ′ , t ∼ S t ′ , ..., t n ∼ S t ′ n , then p ( t , ..., t n ) ∼ S p ( t ′ , ..., t ′ n ). • If ψ {−→ z / −→ x } ∼ S ϕ {−→ z / −→ y } , ..., ψ n {−→ z / −→ x } ∼ S ϕ n {−→ z / −→ y } , where −→ x = x ...x k and −→ y = y ...y k are distinct variables and −→ z = z ...z k are new distinct variables, then Q−→ x ( ψ , ..., ψ n ) ∼ S Q−→ y ( ϕ , ..., ϕ n ) for any ( n, k )-ary quantifier Q of L .Intuitively, ψ ∼ S ψ ′ if ψ ′ can be obtained from ψ by possibly renaming bound variablesand by any number of substitutions of a closed term t for another closed term s , so that I [ t ] = I [ s ]. ≡ α ⊆∼ S . Lemma 3.6. ( [27] ) Let S be an L -structure for an Nmatrix M . Let ψ, ψ ′ be formulas of L ( D ) . Let t , t ′ be closed terms of L ( D ) , such 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 valuation)
Let S = h D, I i be an L -structure for an Nmatrix M .An S -valuation v : F rm clL ( D ) → V is legal in M if it satisfies the following conditions:(1) v [ ψ ] = v [ ψ ′ ] for every two sentences ψ, ψ ′ of L ( D ), such that ψ ∼ S ψ ′ .(2) v [ p ( t , ..., t n )] = I [ p ][ I [ t ] , ..., I [ t n ]].(3) For every ( n, k )-ary quantifier Q of L , v [ Q x , ..., x k ( ψ , ..., ψ n )] should be an element of˜ Q M [ {h v [ ψ { a /x , ..., a k /x k } ] , ..., v [ ψ n { a /x , ..., a k /x k } ] i | a , ..., a k ∈ D } ].Note that in case Q is a propositional connective (for k = 0), the function ˜ Q M is appliedto a singleton, as was explained above. Notation.
For a set of sequents S , we shall write S ⊢ G Γ ⇒ ∆ if a sequent Γ ⇒ ∆ has aproof from S in G . Definition 3.8.
Let S = h D, I i be an L -structure for an Nmatrix M . ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 9 (1) An M -legal S -valuation v is a model of a sentence ψ in M , denoted by S, v | = M ψ , if v [ ψ ] ∈ G .(2) Let v be an M -legal S -valuation. A sequent Γ ⇒ ∆ is M -valid in h S, v i if for every S -substitution σ : if S, v | = M σ [ ψ ] for every ψ ∈ Γ, then there is some ϕ ∈ ∆, such that S, v | = M σ [ ϕ ].(3) A sequent Γ ⇒ ∆ is M -valid, denoted by ⊢ M Γ ⇒ ∆, if for every L -structure S andevery M -legal S -valuation v , Γ ⇒ ∆ is M -valid in h S, v i .(4) For a set of sequents S , S ⊢ M Γ ⇒ ∆ if for every L -structure S and every M -legal S -valuation v : whenever the sequents of S are M -valid in h S, v i , Γ ⇒ ∆ is also M -validin h S, v i . Definition 3.9.
A system G is strongly sound for an Nmatrix M if for every set S ofsequents closed under substitution: S ⊢ G Γ ⇒ ∆ entails S ⊢ M Γ ⇒ ∆. A system G is strongly complete for an Nmatrix M if for every set S of sequents closed under substitution: S ⊢ M Γ ⇒ ∆ entails S ⊢ G Γ ⇒ ∆. An Nmatrix M is strongly characteristic for G if G isstrongly sound and strongly complete for M .Note that since the empty set of sequents is closed under substitutions, strong soundnessimplies (weak) soundness . A similar remark applies to completeness and a characteristicNmatrix.3.2. Semantics for simplified languages L nk . In addition to L -structures for languageswith ( n, k )-ary quantifiers, we also use L nk -structures for the simplified languages L nk , used forformulating the canonical rules. To make the distinction clearer, we shall use the metavari-able S for the former and N for the latter. Since the formulas of L nk are always atomic,the specific 2Nmatrix for which N is defined is immaterial, and can be omitted. We mayeven speak of classical validity of sequents over L nk . Thus henceforth instead of speaking of M -validity of a set of clauses Θ over L nk , we will speak simply of validity.Next we define the notion of a distribution of L nk -structures. Definition 3.10.
Let N = h D, I i be a structure for L nk . Dist N , the distribution of N isdefined as follows: Dist N = {h I [ p ][ a , ..., a k ] , ..., I [ p n ][ a , ..., a k ] i | a , ..., a k ∈ D } We say that an L nk -structure N is E -characteristic if Dist N = E .Note that the distribution of an L n -structure N is Dist N = {h I [ p ] , ..., I [ p n ] i} and soit is always a singleton. Furthermore, the validity of a set of clauses over L n can be reducedto propositional satisfiability as stated in the following lemma which can be easily proved: Lemma 3.11.
Let N be a L n -structure. Assume that Dist N = {h s , ..., s n i} for some s , ..., s n ∈ { t, f } . Let v Dist N be any propositional valuation satisfying v [ p i ] = s i for every ≤ i ≤ n . A set of clauses Θ is valid in N iff v Dist N propositionally satisfies Θ . Now we turn to the case k = 1. In this case it is convenient to define a special kind of L n -structures which we call canonical structures. These structures are sufficient to reflectthe behavior of all possible L n -structures. A more general definition would be without the restriction concerning the closure of S under substitution.However, in this case we would need to add substitution as a structural rule to canonical calculi. A system G is (weakly) sound for an Nmatrix M if ⊢ G Γ ⇒ ∆ entails ⊢ M Γ ⇒ ∆. Definition 3.12.
Let
E ∈ P + ( { t, f } n ). A L n -structure N = h D, I i is E -canonical if D = E and for every b = h s , ..., s n i ∈ D and every 1 ≤ i ≤ n : I [ p i ][ b ] = s i .Clearly, every E -canonical L n -structure is E -characteristic. Lemma 3.13.
Let Θ be a set of clauses over L n , which is valid in some structure N = h D, I i . Then there exists a Dist N -canonical structure N ′ in which Θ is valid.Proof. Suppose that Θ is valid in a structure N = h D, I i . Define the L n -structure N ′ = h I ′ , D ′ i as follows: • D ′ = Dist N . • I ′ [ c ] = h I [ p ][ I [ c ]] , ..., I [ p n ][ I [ c ]] i for every constant c occurring in Θ. • For every 1 ≤ i ≤ n : I ′ [ p i ][ h s , ..., s n i ] = t iff s i = t .Clearly, N ′ is Dist N -canonical. It is easy to verify that Θ is valid in N ′ . Corollary 3.14.
Let
E ∈ P + ( { t, f } n ) . For a finite set of clauses Θ over L n , the questionwhether Θ is valid in a E -characteristic structure is decidable.Proof. Follows from Lemma 3.13 and the fact that for any
E ∈ P + ( { t, f } n ), there are finitelymany E -canonical structures to check.4. Canonical systems with ( n, k ) -ary quantifiers for k ∈ { , } Now we turn to the class of canonical systems with ( n, k )-ary quantifiers for the case of k ∈ { , } and n ≥
1. Henceforth, unless stated otherwise, we assume that k ∈ { , } .4.1. Semantics for canonical systems for k ∈ { , } . In this section we explore theconnection between the coherence of a canonical calculus G , the existence for it of a stronglycharacteristic 2Nmatrix, and strong cut-elimination (in a sense explained below.) We startby defining the notion of suitability for G . Definition 4.1. (Suitability for G ) Let G be a canonical calculus over L . A 2Nmatrix M is suitable for G if for every ( n, k )-ary canonical rule Θ / Q ( s ) of G (where s ∈ { t, f } ), itholds that for every L nk -structure N in which Θ is valid: ˜ Q M [ Dist N ] = { s } . Theorem 4.2.
Let G be a canonical calculus and M - a 2Nmatrix suitable for G . Then G is strongly sound for M .Proof. see Appendix A.Now we come to the construction of a characteristic 2Nmatrix for every coherent canon-ical calculus. Definition 4.3.
Let G be a coherent canonical calculus. The Nmatrix M G for L is definedas follows for every ( n, k )-ary quantifier Q of L , every s ∈ { t, f } and every E ∈ P + ( { t, f } n ):˜ Q M G [ E ] = { s } if Θ / Q ( s ) ∈ G andΘ is valid in some
E − canonical L nk − structure { t, f } otherwise ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 11
First of all, note that by corollary 3.14, the above definition is constructive. Next, letus show that M G is well-defined. Assume by contradiction that there are two dual rulesΘ / ⇒ A and Θ /A ⇒ , such that both Θ and Θ are valid in some E -canonical structures N , N respectively. Obtain Θ ′ from Θ by renaming of constants and variables which occurin Θ . Then clearly Θ ′ is also valid in some E -canonical structure N . If k = 0, by Lemma3.11, the set of clauses Θ ∪ Θ ′ is satisfiable by a (classical) propositional valuation v E andis thus classically consistent, in contradiction to the coherence of G (see defn. 2.9).Otherwise, k = 1. The only difference between different E -canonical structures is in theinterpretation of constants, and since the sets of constants occurring in Θ and Θ ′ aredisjoint, an E -canonical structure N ′ = h D ′ , I ′ i (for the extended language containing theconstants of both Θ and Θ ) can be constructed, in which Θ ∪ Θ ′ are valid. Thus the setΘ ∪ Θ ′ = Rnm (Θ ∪ Θ ) is classically consistent, in contradiction to the coherence of G . Remark:
The construction of M G above is much simpler than the constructions car-ried out in [2, 28]: a canonical calculus there is first transformed into an equivalent normalform calculus, which is then used to construct the characteristic Nmatrix. The idea is totransform the calculus so that each rule dictates the interpretation for only one E . How-ever, the above definitions show that the transformation into normal form is actually notnecessary and we can construct M G directly from G .Next we demonstrate the construction of a characteristic 2Nmatrix for some coherentcanonical calculi. Examples 4.4. (1) It is easy to see that for any canonical coherent calculus G includingthe 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 following three (1 , { p ( v ) ⇒ p ( v ) } / ⇒ ∀ v ( p ( v ) , p ( v )) { p ( c ) ⇒ , ⇒ p ( c ) } / ∀ v ( p ( v ) , p ( v )) ⇒{⇒ p ( c ) , ⇒ p ( c ) } / ⇒ ∃ v ( p ( v ) , p ( v )) G ′ is obviously coherent. The 2Nmatrix M G ′ is defined as follows for every H ∈ P + ( { t, f } ):˜ ∀ [ H ] = ( { t } if h t , f i 6∈ H { f } otherwise ˜ ∃ [ H ] = ( { t } if h t , t i ∈ H { t, f } otherwise The first rule dictates the condition that ∀ [ H ] = { t } for the case of h t, f i 6∈ H . Thesecond rule dictates the condition that ∀ [ H ] = { f } for the case that h t, f i ∈ H . Since G ′ is coherent, these conditions are non-contradictory. The third rule dictates the conditionthat ∃ [ H ] = { t } in the case that h t, t i ∈ H . There is no rule which dictates conditionsfor the case of h t, t i 6∈ H , and so the interpretation in this case is non-deterministic.(3) Consider the canonical calculus G ′′ consisting of the following (1 , { p ( v ) , p ( v ) ⇒} / Q v ( p ( v ) , p ( v ) , p ( v )) ⇒ Of course, G ′′ is coherent. The 2Nmatrix M G ′′ is defined as follows for every H ∈ P + ( { t, f } ):˜ ∀ [ 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 Now we come to the main theorem, establishing a connection between the coherence of acanonical calculus G , the existence of a strongly characteristic 2Nmatrix for G and strongcut-elimination in G in the sense of [1]. Definition 4.5.
Let G be a canonical calculus and let S be a set of sequents closed undersubstitution. A proof P of Γ ⇒ ∆ from S in G is simple if all cuts in P are on formulasfrom S . Definition 4.6.
A calculus G admits strong cut-elimination if for every set of sequents S closed under substitution and every sequent Γ ⇒ ∆, such that S ∪ { Γ ⇒ ∆ } satisfies thefree-variable condition : if S ⊢ G Γ ⇒ ∆, then Γ ⇒ ∆ has a simple proof in G .Note that strong cut-elimination implies standard cut-elimination (which correspondsto the case of an empty set S ). Theorem 4.7.
Let G be a canonical calculus. Then the following statements concerning G are equivalent: (1) G is coherent. (2) G has a strongly characteristic 2Nmatrix. (3) G admits strong cut-elimination.Proof. First we prove that (2) implies (1). Suppose that G has a strongly characteristic2Nmatrix M . Assume by contradiction that G is not coherent. Then there exist two dual( n, k )-ary rules R = Θ / ⇒ A and R = Θ /A ⇒ in G , such that Rnm (Θ ∪ Θ ) isclassically consistent. Suppose that k = 1. Then A = Q v ( p ( v ) , ..., p n ( v )). Recall that Rnm (Θ ∪ Θ ) = Θ ∪ Θ ′ , where Θ ′ is obtained from Θ by renaming constants and variablesthat occur also in Θ (see defn. 2.8). For simplicity we assume that the fresh constantsused for renaming are all in L . Let Θ = { Σ j ⇒ Π j } ≤ j ≤ m and Θ ′ = { Σ j ⇒ Π j } ≤ j ≤ r .Since Θ ∪ Θ ′ is classically consistent, there exists an L nk -structure N = h D, I i , in whichboth Θ and Θ ′ are valid. Recall that we also assume that L nk is a subset of L and so thefollowing are applications of R and R respectively: { Σ j ⇒ Π j } ≤ j ≤ m ⇒ Q v ( p ( v ) , ..., p n ( v )) { Σ j ⇒ Π j } ≤ j ≤ m Q v ( p ( v ) , ..., p n ( v )) ⇒ Let S be any extension of N to L and v - any M -legal S -valuation. It is easy to see thatthe premises of the applications above are M -valid in h S, v i (since the premises containatomic formulas). Since G is strongly sound for M , both ⇒ Q v ( p ( v ) , ..., p n ( v )) and [1] does not assume that S is closed under substitution. Instead, a structural substitution rule is addedand the allowed cuts are on substitution instances of formulas from S . See section 1. This assumption is not necessary and is used only for simplification of presentation, since we caninstantiate the constants by any L -terms. This assumption is again not essential for the proof, but it simplifies the presentation.
ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 13 Q v ( p ( v ) , ..., p n ( v )) ⇒ should also be M -valid in h S, v i , which is of course impossible.The proof for the case of k = 0 is simpler and is left to the reader.Next, we prove that (3) implies (1). Let G be a canonical calculus which admits strongcut-elimination. Suppose by contradiction that G is not coherent. Then there are twodual rules of G : Θ / ⇒ A and Θ /A ⇒ , such that Rnm (Θ ∪ Θ ) is classically consistent.Let Θ be the minimal set of clauses, such that Rnm (Θ ∪ Θ ) ⊆ Θ and Θ is closed undersubstitutions. Θ ∪ {⇒} satisfy the free-variable condition, since only atomic formulas areinvolved and no variables are bound there. It is easy to see that Θ ⊢ G ⇒ A and Θ ⊢ G A ⇒ .By using cut, Θ ⊢ G ⇒ . But ⇒ has no simple proof in G from Θ (since Rnm (Θ ∪ Θ ) isconsistent and Θ is its closure under substitutions), in contradiction to the fact that G admits strong cut-elimination.To show that (1) implies both (2) and (3), we need the following proposition: Proposition 4.8.
Let G be a coherent calculus. Let S be a set of sequents closed undersubstitution and Γ ⇒ ∆ - a sequent, such that S ∪ { Γ ⇒ ∆ } satisfies the free-variablecondition. If Γ ⇒ ∆ has no simple proof from S in G , then S6⊢ M Γ ⇒ ∆ .Proof. see Appendix A.To prove that (1) implies (2), suppose that G is coherent. Let us show that M G is astrongly characteristic 2Nmatrix for G . By definition of M G , it is suitable for G (see defn.4.1). By theorem 4.2, G is strongly sound for M G .For strong completeness, let S be a set of sequents closed under substitution. Suppose thata sequent Γ ⇒ ∆ has no proof from S in G . If S ∪{ Γ ⇒ ∆ } does not satisfy the free-variablecondition, obtain S ′ ∪ { Γ ′ ⇒ ∆ ′ } by renaming the bound variables, so that S ′ ∪ { Γ ′ ⇒ ∆ ′ } satisfies the condition (otherwise, take Γ ′ ⇒ ∆ ′ and S ′ to be Γ ⇒ ∆ and S respectively).Then Γ ′ ⇒ ∆ ′ has no proof from S ′ in G (otherwise we could obtain a proof of Γ ⇒ ∆ from S by using cuts on logical axioms), and so it also has no simple proof from S ′ in G . ByProposition 4.8, S ′ M Γ ′ ⇒ ∆ ′ . That is, there is an L -structure S and an M -legal valuation v , such that the sequents in S ′ are M -valid in h S, v i , while Γ ′ ⇒ ∆ ′ is not. Since v respectsthe ≡ α -relation, the sequents of S are also M -valid in h S, v i , while Γ ⇒ ∆ is not. And so S6⊢ M Γ ⇒ ∆. We have shown that G is strongly complete (and strongly sound) for M G .Thus M G is a strongly characteristic 2Nmatrix for G .Finally, we prove that (1) implies (3). Let G be a coherent calculus. Let S be a set ofsequents closed under substitution, and let Γ ⇒ ∆ be a sequent, such that S ∪ { Γ ⇒ ∆ } satisfies the free-variable condition. Suppose that S ⊢ G Γ ⇒ ∆. We have already shownabove that M G is a strongly characteristic 2Nmatrix for G . Thus S ⊢ M Γ ⇒ ∆, andby Proposition 4.8, Γ ⇒ ∆ has a simple proof from S in G . Thus G admits strong cut-elimination. Remark.
At this point it should be noted that the renaming of clashing constants inthe definition of coherence (see defn. 2.9) is crucial. Consider, for instance, a canonicalcalculus G consisting of the introduction rules { p ( c ) ⇒ ; ⇒ p ( c ′ ) } / ⇒ Q v p ( v ) and { p ( c ′′ ) ⇒ ; ⇒ p ( c ) } / Q v p ( v ) ⇒ for a (1,1)-ary quantifier Q . Without renaming ofclashing constants, we would conclude that the set { p ( c ) ⇒ ; ⇒ p ( c ′ ) ; p ( c ′′ ) ⇒ , ⇒ p ( c ) } is classically inconsistent. However, G obviously has no strongly characteristic 2Nmatrix,since the rules dictate contradicting requirements for ˜ Q [ { t, f } ]. But if we perform renamingfirst, obtaining the set Rnm (Θ ∪ Θ ) = { p ( c ) ⇒ , ⇒ p ( c ′ ) , p ( c ′′ ) ⇒ , ⇒ p ( c ′′′ ) } , weshall see that Rnm (Θ ∪ Θ ) is classically consistent and so G is not coherent. Hence, bythe above theorem, G has no strongly characteristic 2Nmatrix. Corollary 4.9.
The existence of a strongly characteristic 2Nmatrix for a canonical calculus G is decidable.Proof. By theorem 4.7, the question whether G has a strongly characteristic 2Nmatrix isequivalent to the question whether G is coherent, and this, by Proposition 2.10, is decidable. Remark:
The above results are related to the results in [9], where a general class ofsequent calculi with ( n, k )-ary quantifiers and a (not necessarily standard) set of structuralrules called standard calculi are defined. A canonical calculus is a particular instance of astandard calculus which includes all of the standard structural rules. [9] formulate syntacticnecessary and sufficient conditions for a slightly generalized version of cut-elimination withnon-logical axioms. Unlike in this paper, the non-logical axioms must consist of atomic formulas (and must be closed under cuts and substitutions). But the results of [9] apply toa much wider class of calculi (since different combinations of structural rules are allowed).In addition, a constructive modular cut-elimination procedure is provided. The reductivitycondition of [9] can be shown to be equivalent to our coherence criterion in the context ofcanonical systems .4.2. Coherence and standard cut-elimination.
In the previous subsection we havestudied the connection between coherence and strong cut-elimination. In this subsectionwe focus on standard cut-elimination in canonical calculi. It easily follows from theorem4.7 that coherence implies cut-elimination:
Corollary 4.10.
Let G be a canonical calculus. If G is coherent, then for every sequent Γ ⇒ ∆ satisfying the free-variable condition: if Γ ⇒ ∆ is provable in G , then it has acut-free proof in G . Thus coherence is a sufficient condition for cut-elimination in a canonical calculus. Inthe more restricted canonical systems of [2, 28] it also is a necessary condition. However,things get more complicated with the more general canonical rules studied in this paper.
Example 4.11.
Consider, for instance, the following canonical calculus G consisting ofthe following two inference rules: Θ / ⇒ Q v ( p ( v ) , p ( v )) and Θ / Q v ( p ( v ) , p ( v )) ⇒ ,where:Θ = Θ = { p ( v ) ⇒ p ( v ) ; ⇒ p ( c ) ; ⇒ p ( c ) ; p ( c ) ⇒ ; p ( c ) ⇒ ; p ( c ) ⇒ ; ⇒ p ( c ) } Clearly, G is not coherent. We now sketch a proof that the only sequents provable in G are logical axioms. This immediately implies that G admits cut-elimination.To prove this it suffices to show that for every rule of G : if its premises are logicalaxioms, then its conclusion is a logical axiom. Suppose by contradiction that we can applyone of the rules on logical axioms and obtain a conclusion which is not a logical axiom.Suppose, without loss of generality, that it is the first rule. Then the application would beof the form:Γ , χ [ p ] { χ [ v ] /w } ⇒ ∆ , χ [ p ] { χ [ v ] /w . . . Γ ⇒ χ [ p ] { χ [ c ] /w } , ∆ Γ ⇒ χ [ p ] { χ [ c ] /w } , ∆Γ ⇒ Q w ( χ [ p ] , χ [ p ]) , ∆ We wish to thank Agata Ciabattoni for pointing out these facts to us in a personal correspondence.
ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 15
Since the proved sequent is not a logical axiom, (*) there are no A ∈ Γ and B ∈ ∆, such that A ≡ α B . Moreover, since Γ , χ [ p ] { χ [ v ] /w } ⇒ ∆ , χ [ p ] { χ [ y ] /w } is a logical axiom, either(i) there is some C ∈ ∆, such that C ≡ α χ [ p ] { χ [ v ] /w } , (ii) there is some C ∈ Γ, suchthat C ≡ α χ [ p ] { χ [ v ] /w } , or (iii) χ [ p ]( χ [ v ] /w ) ≡ α χ [ p ] { χ [ v ] /w } . Suppose (i) holds, i.e.there is some some C ∈ ∆, such that C ≡ α χ [ p ] { χ [ v ] /w } . Then since χ [ v ] cannot occurfree in ∆, w F v [ C ], and so w F v [ χ [ p ]]. Hence, χ [ p ] { χ [ c ] /w } = χ [ p ] { χ [ v ] /w } = χ [ p ]. Now since Γ ⇒ χ [ p ] { χ [ c ] /w } , ∆ is a logical axiom, and due to (*), there is some D ∈ Γ, such that D ≡ α χ [ p ] { χ [ c ] /w } . But since χ [ p ] { χ [ c ] /w } = χ [ p ] { χ [ v ] /w } , C ≡ α D , C ∈ ∆ and D ∈ Γ, in contradiction to (*). The case (ii) is treated similarly using theconstant c . The case (iii) is handled using the constant c .Thus, only logical axioms are provable in G and so it admits standard cut-elimination,although it is not coherent.Hence coherence is not a necessary condition for cut-elimination in general. However,below we characterize a more restricted subclass of canonical systems, for which this prop-erty does hold. Definition 4.12.
A canonical calculus G is simple if for every two dual ( n, k )-ary canonicalrules Θ / ⇒ A and Θ /A ⇒ one of the following properties holds:(1) k = 0, i.e. Θ / ⇒ A and Θ /A ⇒ are propositional rules.(2) k = 1 and one of the following holds for each variable y occurring in Rnm (Θ ∪ Θ ): • There is at most one 1 ≤ i ≤ n , such that y occurs in p i ( y ) in Rnm (Θ ∪ Θ ) andthere is at most one constant c , such that p i ( c ) also occurs in Rnm (Θ ∪ Θ ). • There are two different 1 ≤ i, j ≤ n , such that y occurs in p i ( y ) and p j ( y ) in Rnm (Θ ∪ Θ ) and for every constant c , there is no such 1 ≤ k ≤ n , that both p k ( y ) and p k ( c )occur in Rnm (Θ ∪ Θ ). Examples 4.13. (1) All the canonical calculi from examples 2.5 are simple.(2) Consider the canonical calculus G , consisting of the following two rules for a (3 , Q : { p ( v ) ⇒ ; p ( c ) , p ( c ) ⇒ } / ⇒ Q v ( p ( v ) , p ( v ) , p ( v )) {⇒ p ( v ) ; ⇒ p ( e ) } / Q v ( p ( v ) , p ( v ) , p ( v )) ⇒ It is easy to see that G is a simple coherent calculus.(3) If we modify the first rule of G as follows: { p ( v ) ⇒ ; p ( c ) , p ( c ) ⇒ ; p ( d ) ⇒ p ( d ) } / ⇒ Q v ( p ( v ) , p ( v ) , p ( v ))the resulting calculus is not simple, since both p ( c ) and p ( d ) occur in the premises ofthe rule, together with p ( v ).(4) The calculus G from example 4.11 is not simple, since for instance p ( v ), p ( c ) and p ( c ) occur in the premises (after renaming). Proposition 4.14.
If a simple canonical calculus G admits cut-elimination, then it iscoherent.Proof. see Appendix A. Summary and further research
In this paper we have considerably extended the characterization of canonical calculi of[2, 28] to ( n, k )-ary quantifiers. Focusing on the case of k ∈ { , } , we have shown that thefollowing statements concerning a canonical calculus G are equivalent: (i) G is coherent, (ii) G has a strongly characteristic 2Nmatrix, and (iii) G admits strong cut-elimination. We havealso shown that coherence is not a necessary condition for standard cut-elimination, andcharacterized a subclass of canonical systems called simple calculi, for which this propertydoes hold.In addition to these proof-theoretical results for a natural type of multiple conclusionGentzen-type systems with ( n, n, k ∈{ , } , we still need to characterize the most general subclass of canonical calculi, for whichcoherence is both a necessary and sufficient condition for standard cut-elimination (it is notclear whether the characterization of simple calculi can be further extended).Extending these results to the case of k > , G , consisting of the following two (1,2)-ary rules: { p ( c, x ) ⇒} / ⇒ Q z z p ( z , z ) {⇒ p ( y, d ) } / Q z z p ( z , z ) ⇒ G is coherent, but it is easy to see that M G is not well-defined in this case. And even if a2Nmatrix M suitable for G does exist, it is not necessarily sound for G . It is clear that thedistributional interpretation of quantifiers is no longer adequate for the case of k >
1, sinceit cannot capture any kind of dependencies between elements of the domain. Thus a moregeneral interpretation of quantifiers is needed.Another important research direction is extending canonical systems with equality. Thiswill allow us to treat counting ( n, k )-ary quantifiers, like “there are at most two elements a, b ,such that p ( a, b ) holds”. Clearly, equality must be incorporated also into the representationlanguage L nk . Standard and strong cut-elimination and its connection to the coherence ofcanonical systems are yet to be investigated for canonical systems with equality. Acknowledgement
This research was supported by the
Israel Science Foundation founded by the Israel Acad-emy of Sciences and Humanities (grant No 809/06).
ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 17
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Appendix A. Proofs of selected propositions
Proof of Theorem 4.2 : Suppose that M is suitable for G . Let S = h D, I i be some L -structure and v - an M -legal S -valuation. Let S be any set of sequents closed undersubstitution. We will show that if the sequents of S are M -valid in h S, v i , then any sequentprovable from S in G is M -valid in h S, v i . Obviously, the axioms of G are M -valid, andthe structural rules, including cut, are strongly sound. It remains to show that for everyapplication of a canonical rule R of G : if the premises of R are M -valid in h S, v i , then itsconclusion is M -valid in h S, v i . We will show this for the case of k = 1, leaving the easiercase of k = 0 to the reader.Let R be an ( n, G : R = Θ R / Q v ( p ( v ) , ..., p n ( v )) − r ⇒ Q v ( p ( v ) , ..., p n ( v )) r where r ∈ { t, f } and Θ R = { Σ j ⇒ Π j } ≤ j ≤ m . An application of R is of the form: { Γ , χ [Σ j ] ⇒ χ [Π j ] , ∆ } ≤ j ≤ m Γ , Q z ( χ [ p ] , ..., χ [ p n ]) − r ⇒ ∆ , Q z ( χ [ p ] , ..., χ [ p n ]) r where χ is some h R, Γ ∪ ∆ , z i -mapping. Suppose that { Γ , χ [Σ j ] ⇒ χ [Π j ] , ∆ } ≤ j ≤ m is M -valid in h S, v i . We will now show that Γ , Q z ( χ [ p ] , ..., χ [ p n ]) − r ⇒ ∆ , Q z ( χ [ p ] , ..., χ [ p n ]) r isalso M -valid in h S, v i . (a) Let σ be an S -substitution, such that S, v | = M σ [Γ] and for every ψ ∈ ∆: S, v = M σ [ ψ ].Denote by e ψ the L -formula obtained from a formula ψ by substituting every free occurrenceof w ∈ F v [ ψ ] − { z } for σ [ w ].Let E = {h v [ ] χ [ p ] { a/z } ] , ..., v [ ] χ [ p n ] { a/z } ] i | a ∈ D } . We will show that ˜ Q [ E ] = { r } , andso v [ σ [ Qz ( χ [ p ] , ..., χ [ p n ])]] = r . From ( a ) it will follow that Γ , Q z ( χ [ p ] , ..., χ [ p n ]) − r ⇒ ∆ , Q z ( χ [ p ] , ..., χ [ p n ]) r is M -valid in h S, v i .We prove this by showing that Θ R is valid in some E -characteristic L nk -structure. Then, bysuitability of M , we shall conclude that ˜ Q M [ E ] = r .Construct the L nk -structure N = h D ′ , I ′ i as follows: • D ′ = D . • For every a ∈ D : I ′ [ p i ][ a ] = v [ g χ [ p i ] { a/z } ]. • For every constant c , I ′ [ c ] = I [ σ [ χ [ c ]]].We will now show that Θ R = { Σ j ⇒ Π j } ≤ j ≤ m is valid in N . Suppose for contradictionthat it is not so. Then there exists some 1 ≤ j ≤ m , for which Σ j ⇒ Π j is not valid in N .Thus there is some N -substitution η , such that:( b ) whenever p i ( t ) ∈ Π j ∪ Σ j : p i ( t ) ∈ ite ( I ′ [ p i ][ I ′ [ η [ t ]]] , Σ j , Π j ). ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 19
We show now that Γ , χ [Σ j ] ⇒ χ [Π j ] , ∆ is not M -valid in h S, v i , in contradiction to ourassumption about the premises of the above application.Let ψ ∈ ite ( s, χ [Σ j ] , χ [Π j ]) for s ∈ { t, f } . Let σ ′ be the S -substitution similar to σ exceptthat σ ′ [ χ [ y ]] = a y , where a y = I ′ [ η [ y ]] for every variable y occurring in Θ R . Note that σ ′ iswell-defined, since for every two different variables x, y : χ [ x ] = χ [ y ] (recall defn. 2.3). Thenone of the following 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 ],such that for any variable y occurring in Θ R , χ [ y ] does not occur in χ [ c ]. Recall that by (b) , I ′ [ p i ][ I ′ [ η [ c ]]] = s . And 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 every variable y occurring in Θ R prevents χ [ y ] from occurring freely in Q z ( χ [ p ] , ..., χ [ p n ]), and that σ, σ ′ only differ for variables χ [ z ] where z occurs in Θ R .)By Lemma 3.6-2 and the legality of v : v [ g χ [ p i ] { σ [ χ [ c ]] /z } ] = v [ g χ [ p i ] { I [ σ [ χ [ c ]]] /z } ]By definition 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 } , where p i ( y ) ∈ ite ( s, Σ j , Π j ) and χ [ y ] does not occur in Γ ∪ ∆ ∪{Q z ( ψ , ..., ψ n ) } and is free for z in χ [ p i ]. Then I ′ [ p i ][ I ′ [ η [ y ]]] = s .Let a = I ′ [ η [ y ]]. Then, σ ′ [ χ [ y ]] = a and 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 Thus we have shown that v [ σ ′ [ ψ ]] = s whenever ψ ∈ ite ( s, χ [Σ j ] , χ [Π j ]). Also, there isno variable y occurring in Θ R , such that χ [ y ] occurs in Γ ∪ ∆, and so σ [Γ] = σ ′ [Γ] and σ [∆] = σ ′ [∆]. Thus for every ψ ∈ Γ ∪ χ [Σ j ], v [ σ ′ [ ψ ]] = t while for every ϕ ∈ ∆ ∪ χ [Π j ], v [ σ ′ [ ϕ ]] = f . Hence, Γ , χ [Σ j ] ⇒ ∆ , χ [Π j ] is not M -valid in h S, v i , in contradiction to ourassumption on the validity of the premises of the application above.We have shown that { Σ j ⇒ Π j } ≤ j ≤ m is valid in N . Obviously , Dist N = E . Since M is suitable for G : ˜ Q M [ E ] = { r } and so v [ σ [ Q z ( χ [ p ] , ..., χ [ p n ])]] = r . From this fact andassumption (a) it follows that Γ , Q z ( χ [ p ] , ..., χ [ p n ]) − r ⇒ ∆ , Q z ( χ [ p ] , ..., χ [ p n ]) r is M -validin h S, v i . Proof of Proposition 4.8 :Let S be a set of sequents closed under substitution and Γ ⇒ ∆ - a sequent, such that S ∪ { Γ ⇒ ∆ } satisfies the free-variable condition. Suppose that Γ ⇒ ∆ has no simple prooffrom S in G . To show that S6⊢ M Γ ⇒ ∆, we will construct a structure S and an M -legalvaluation v , such that the sequents of S are M -valid in h S, v i , while Γ ⇒ ∆ is not.It is easy to see that we can limit ourselves to the language L ∗ , which is a subset of L ,consisting of all the constants and predicate and function symbols, occurring in S ∪{ Γ ⇒ ∆ } .Let T be the set of all the terms in L ∗ which do not contain variables occurring bound inΓ ⇒ ∆ and S . It is a standard matter to show that Γ , ∆ can be extended to two (possiblyinfinite) sets Γ ′ , ∆ ′ (where Γ ⊆ Γ ′ and ∆ ⊆ ∆ ′ ), satisfying the following properties: Recall that E = {h v [ ] χ [ p ] { 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 . (1) For every finite Γ ⊆ Γ ′ and ∆ ⊆ ∆ ′ , Γ ⇒ ∆ has no simple proof in G .(2) There are no ψ ∈ Γ ′ and ϕ ∈ ∆ ′ , such that ψ ≡ α ϕ .(3) If { Σ j ⇒ Π j } ≤ j ≤ m / Q ( r ) is an ( n, G and Q ( ψ , ..., ψ n ) ∈ ite ( r, ∆ ′ , Γ ′ ),then there is some 1 ≤ j ≤ m , such that whenever p i ∈ ite ( s, Σ j , Π j ), ψ i ∈ ite ( s, Γ ′ , ∆ ′ )for s ∈ { t, f } .(4) If { Σ j ⇒ Π j } ≤ j ≤ m / Q ( r ) is an ( n, G and Q z ( ψ , ..., ψ n ) ∈ ite ( r, ∆ ′ , Γ ′ ),then there is some 1 ≤ j ≤ m , such that: • For every constant c , whenever p i ( c ) ∈ ite ( s, Σ j , Π j ) for some 1 ≤ i ≤ n , then ψ i { t /z } ∈ ite ( s, Γ ′ , ∆ ′ ) for every term t ∈ T . • For each variable y , there exists some t y ∈ T , such that whenever p i ( y ) ∈ ite ( s, Σ j , Π j )for some 1 ≤ i ≤ n , then ψ i { t y /z } ∈ ite ( s, Γ ′ , ∆ ′ ).Note that every t ∈ T is free for z in ψ i for every 1 ≤ i ≤ n .(5) For every formula ψ occurring in S , ψ ∈ Γ ′ ∪ ∆ ′ .Note that the last condition can be satisfied because cuts on formulas from S are allowedin a simple proof.Let S = h D, I i be the L ∗ -structure defined as follows: • D = T . • I [ c ] = c for every constant c of L ∗ . • I [ f ][ t , ..., t n ] = f ( t , ..., t n ) for every n -ary function symbol f . • I [ p ][ t , ..., t n ] = t iff p ( t , ..., t n ) ∈ Γ ′ for every n -ary predicate symbol p .Let σ ∗ be any S -substitution satisfying σ ∗ [ x ] = x for every x ∈ T . (Note that every x ∈ T is also a member of the domain and thus has an individual constant referring to itin L ∗ ( D ).)For an L ( D )-formula ψ (an L ( D )-term t ), we will denote by b ψ ( b t ) the L -formula ( L -term) obtained from ψ ( t ) by replacing every individual constant of the form s for some s ∈ T by the term s . More formally, b t and b ψ are defined as follows: • b x = x for any variable x of L . • b c = c for any constant c of L . • b t = t for any t ∈ T . • \ f ( t , ..., t n ) = f ( b t , ..., b t n ). • \ p ( t , ..., t n ) = p ( b t , ..., b t n ). • \ Q ( ψ , ..., ψ n ) = Q ( b ψ , ..., b ψ n ). • \ Q x ( ψ , ..., ψ n ) = Q x ( b ψ , ..., b ψ n ). Lemma A.1.
Let t be an L ( D ) -term and ψ - an L ( D ) -formula. (1) For 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) For every ψ ∈ Γ ′ ∪ ∆ ′ : [ σ ∗ [ ψ ] = ψ .Proof. The lemma is proved by a tedious induction on t and ψ .Define the S -valuation v as follows: • v [ p ( t , ..., t n )] = I [ p ][ I [ t ] , ..., I [ t n ]]. • For every ( n, Q of L , if there is some C ∈ Γ ′ ∪ ∆ ′ , such that C ≡ α \ Q ( ψ , ..., ψ n ), then v [ Q ( ψ , ..., ψ n )] = t iff C ∈ Γ ′ . Otherwise v [ Q ( ψ , ..., ψ n )] = t iff˜ Q [ {h v [ ψ ] , ..., v [ ψ n ] i} ] = { t } . ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 21 • For every ( n, Q of L , if there is some C ∈ Γ ′ ∪ ∆ ′ , such that C ≡ α \ Q x ( ψ , ..., ψ n ), then v [ Q x ( ψ , ..., ψ n )] = t iff C ∈ Γ ′ . Otherwise v [ Q x ( ψ , ..., ψ n )] = t iff˜ Q [ {h v [ ψ { a/x } ] , ..., v [ ψ n { a/x } ] i | a ∈ D } ] = { t } . Lemma A.2. (1) I ∗ [ σ ∗ [ t ]] = t for every t ∈ T . (2) For every two L ( D ) -formulas ψ, ψ ′ : if ψ ≡ α ψ ′ , then σ ∗ [ ψ ] ≡ α σ ∗ [ ψ ′ ] . (3) For every two L ( D ) -sentences ψ, ψ ′ : if ψ ∼ S ψ ′ , then b ψ ≡ α b ψ ′ .Proof. The claims are proven by induction on t in the first case, and on ψ and ψ ′ in thesecond and third cases. Lemma A.3.
For every ψ ∈ Γ ′ ∪ ∆ ′ : v ( σ ∗ [ ψ ]) = t iff ψ ∈ Γ ′ .Proof. If ψ = p ( t , ..., t n ), then v [ σ ∗ [ ψ ]] = I [ p ][ I [ σ ∗ [ t ]] , ..., I [ σ ∗ [ t n ]]]. Note that for every1 ≤ i ≤ n , t i ∈ T . By Lemma A.2-1, I [ σ ∗ [ t i ]] = t i , and by the definition of I , v [ σ ∗ [ ψ ]] = t iff p ( t , ..., t n ) ∈ Γ ′ .Otherwise ψ = Q ( ψ , ..., ψ n ) or ψ = Q ′ x ( ψ , ..., ψ n ). If ψ ∈ Γ ′ , then by Lemma A.1-3 [ σ ∗ [ ψ ] = ψ ∈ Γ ′ and so v [ σ ∗ [ ψ ]] = t . If ψ ∈ ∆ ′ then by property 2 of Γ ′ ∪ ∆ ′ it cannot bethe case that there is some C ∈ Γ ′ , such that C ≡ α [ σ ∗ [ ψ ] = ψ and so v [ σ ∗ [ ψ ]] = f .. Lemma A.4. v is legal in M G .Proof. First we need to show that v respects the ∼ S -relation. We prove by induction on L ∗ ( D )-sentences ψ, ψ ′ : if ψ ∼ S ψ ′ , then v [ ψ ] = v [ ψ ′ ]. • ψ = p ( t , ..., t n ), ψ ′ = p ( s , ..., 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 , ..., t n )] = I [ p ][ I [ t ] , ..., I [ t n ]] = I [ p ][ I [ s ] , ..., I [ s n ]]= v [ p ( s , ..., s n )]. • ψ = Q x ( ψ , ..., ψ n ), ψ ′ = Q y ( ψ ′ , ..., ψ ′ n ) and for every 1 ≤ i ≤ n : ψ i { z/x } ∼ S ψ ′ i { z/y } fora fresh variable z . Then by Lemma 3.6-2 for every 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 induction hypothesis, {h v [ ψ { a/x } ] , ..., v [ ψ n { a/x } ] i | a ∈ D } = {h v [ ψ ′ { a/x } ] , ..., v [ ψ ′ n { a/x } ] i | a ∈ D } . One ofthe following cases holds: − There is no C ∈ Γ ′ ∪ ∆ ′ , such that C ≡ α b ψ or C ≡ α b ψ ′ . Then v [ Q x ( ψ , ..., ψ n )] = t iff {h v [ ψ { a/x } ] , ..., v [ ψ n { a/x } ] i | a ∈ D } = t iff {h v [ ψ ′ { a/x } ] , ..., v [ ψ ′ n { a/x } ] i | a ∈ D } = t iff v [ Q y ( ψ ′ , ..., 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 ′ ∪ ∆ ′ , such that C ≡ α b ψ ′ . Similarly to the previous case, v [ ψ ] = v [ ψ ′ ] = t iff C ∈ Γ. • The case of ψ = Q ( ψ , ..., ψ n ), ψ ′ = Q ( ψ ′ , ..., ψ ′ n ) is treated similarly.It remains to show that v respects the interpretations of the ( n, k )-ary quantifiers in M G . The case of k = 0 is not hard and is left to the reader. We will show the prooffor the case of k = 1. Suppose by contradiction that there is some L ∗ ( D )-sentence A = This is obvious if t i does not occur in the set { Γ ⇒ ∆ } ∪ S . If it occurs in this set, then by thefree-variable condition t i does not contain variables bound in this set and so t i ∈ T by definition of T . Q z ( ψ , ..., ψ n ), such that v [ A ] ˜ Q [ H A ], where H A = {h v [ ψ { a/z } ] , ..., v [ ψ n { a/z } ] i | a ∈ D } .From the definition of v , it must be the case that :( a ) there is some L -formula C ∈ Γ ′ ∪ ∆ ′ , such that C ≡ α b A , and v [ A ] = t iff C ∈ Γ ′ .Suppose that ˜ Q [ H A ] = { t } and v [ A ] = f . By definition of M G and the fact that ˜ Q [ H A ]is a singleton, it must be the case that there is some canonical rule { Σ k ⇒ Π k } ≤ k ≤ m / ⇒Q v ( p ( v ) , ..., p n ( v )) in G , such that:( b ) { Σ k ⇒ Π k } ≤ k ≤ m is valid in a H A -characteristic structure N = h D N , I N i . A = Q z ( ψ , ..., ψ n ) and C ≡ α b A , so C is of the form Q w ( ϕ , ..., 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 be the formula obtained from ϕ i by substituting every x ∈ F v [ ϕ i ] = { w } for σ ∗ [ x ]. By Lemma 3.6-2, φ i { a/w } ∼ S ψ i { a/z } for every a ∈ D . Wehave already shown that v respects the ∼ S -relation, and so v [ φ i { a/w } ] = v [ ψ i { a/z } ]. Thus H A = {h v [ φ { a/w } ] , ..., v [ φ n { a/w } ] i | a ∈ D } .Since v [ A ] = f , it follows from (a) that C = Q w ( ϕ , ..., ϕ n ) ∈ ∆ ′ . Then by property3 of Γ ′ ∪ ∆ ′ , there is some 1 ≤ j ≤ m , such that whenever p i ( y ) ∈ ite ( r, Σ j , Π j ), thereis 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 . Since N is H A -characteristic, there is some a y ∈ D N , such that I N [ p i ][ a y ] = v [ φ i { σ ∗ [ t y ] /w } ] = r .Let us now show that Σ j ⇒ Π j is not valid in N (in contradiction to ( b )). Let µ beany N -substitution, such that µ [ y ] = a y for every variable y occurring in Σ j ∪ Π j . We nowshow that whenever 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 variable y , then I N [ p i ][ µ [ y ]] = I N [ p i ][ I N [ a y ]] = I N [ p i ][ a y ] = s . Otherwise t is some constant c . By property 3 of Γ ′ ∪ ∆ ′ , for every 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 .Thus for every t ∈ T : v [ φ i { σ ∗ [ t ] /w } ] = v [ φ i { t /w } ] = s . Since N is H A -characteristic, I N [ p c ][ I N [ c ]] = s . And so we have shown that Σ j ⇒ Π j is not valid in N , in contradictionto ( b ).The proof for the case of ˜ Q [ H A ] = { f } and v [ A ] = t is symmetric. Lemma A.5.
For every sequent Σ ⇒ Π ∈ S , Σ ⇒ Π is M -valid in h S, v i .Proof. Suppose by contradiction that there is some Σ ⇒ Π ∈ S , which is not M -valid in h S, v i . Then there exists some S -substitution µ , such that for every ψ ∈ Σ: S, v | = M µ [ ψ ],and for every ϕ ∈ Π: S, v = M µ [ ϕ ]. Note that for every φ ∈ Σ ∪ Π, d µ [ φ ] is a substitutioninstance of φ . Since S is closed under substitution, d µ [ φ ] also occurs in S , and thus byproperty 5 of Γ ′ ∪ ∆ ′ : d µ [ φ ] ∈ Γ ′ ∪ ∆ ′ . By Lemma A.3, if d µ [ φ ] ∈ Γ ′ then v [ σ ∗ [ d µ [ φ ]]] = t , andif d µ [ φ ] ∈ ∆ ′ then v [ σ ∗ [ d µ [ φ ]]] = f . By Lemma A.1-2, µ [ φ ] ∼ S σ ∗ [ d µ [ φ ]]. Since v is M -legal, itrespects the ∼ S -relation and so for every 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] has a simple proof from S in G , in contradiction toproperty 1 of Γ ′ ∪ ∆ ′ . If there is no L -formula C ∈ Γ ′ ∪ ∆ ′ , such that C ≡ α b A , then by definition of v , v [ A ] is always in˜ Q [ H A ], so this case is not possible. ANONICAL CALCULI WITH ( n, k )-ARY QUANTIFIERS 23
We have shown 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 -valid in h S, v i . From (ii) it follows that Γ ⇒ ∆ is not M -valid in h S, v i , which completes the proof. Proof of Proposition 4.14:
For a set of clauses Θ, denote by Θ { c/x } the set { Γ { c/x } ⇒ ∆ { c/x } | Γ ⇒ ∆ ∈ Θ } . Thenthe following lemma is easily proved: Lemma A.6.
Let Θ be a classically consistent set of clauses. Then for any constant c , Θ { c/x } is also classically consistent. Now suppose that a simple canonical calculus G is not coherent. Then there is a pairof ( n, k )-ary dual rules R = Θ / ⇒ A and R = Θ /A ⇒ , such that Rnm (Θ ∪ Θ ) isclassically consistent. If k = 0, then the proof is similar to the proof of theorem 4.7 in [2].Otherwise, k = 1, A = Q v ( p ( v ) , ..., p n ( v )) and whenever p i ( y ) occurs in Rnm (Θ ∪ Θ )for some variable y and some 1 ≤ i ≤ n , there is at most one constant c , such that p i ( c ) alsooccurs in Rnm (Θ ∪ Θ ). Recall that Rnm (Θ ∪ Θ ) = Θ ∪ Θ ′ , where Θ ′ is obtained fromΘ by renaming of constants and variables which occur in Θ (see defn. 2.8). We assumethat the new constants in Θ ′ are in L (this assumption is not necessary but it simplifiesthe presentation).Obtain the sets Υ , Υ from Θ , Θ ′ respectively as follows. For every 1 ≤ i ≤ n , if p i ( c ) occurs in Θ ∪ Θ ′ for some constant c , replace all variables y , such that p i ( y ) occursin Θ ∪ Θ ′ by c (note that this is well-defined due to the special property of simple calculi).Otherwise, replace all variables y , such that p i ( y ) occurs in Θ ∪ Θ ′ by a fresh constant d i of L . Then Υ = Υ ∪ Υ is obtained from Θ ∪ Θ ′ by replacing all variables by constants. SinceΘ ∪ Θ ′ is classically consistent, by repeated application of Lemma A.6, Υ is also classicallyconsistent. Then there exists some L -structure S in which the set of clauses Υ is (classically)valid. Since Υ consists of closed atomic formulas, there also exists a (classical) propositionalvaluation v S , which satisfies Υ. Let Φ = { A | v S [ A ] = t, A ∈ Γ ∪ ∆ , Γ ⇒ ∆ ∈ Υ } andΨ = { 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 and B are sets of standard axioms. (Since v S satisfies Π ⇒ Σ, there issome A ∈ Π, such that v S [ A ] = f , or some A ∈ ∆, such that v S [ A ] = t . In the former case, A ∈ Ψ and in the latter case, A ∈ Φ.)Let x be a fresh variable of L . Define the h R , Ψ ∪ Φ , x i -mapping χ (see defn. 2.3) asfollows. For every 1 ≤ i ≤ n , χ [ p i ] = p i ( x ) if there is some constant c , such that p i ( c ) occursin Θ ∪ Θ ′ . Otherwise, χ [ p i ] = p i ( d i ) (where d i is the fresh constant of L chosen above). Forevery constant c and variable y occurring in Θ ∪ Θ ′ : χ [ c ] = c and χ [ y ] = y . It is easy tosee that Υ = { χ [Σ ′ ] ⇒ χ [Π ′ ] | Σ ′ ⇒ Π ′ ∈ Θ } and Υ = { χ [Σ ′ ] ⇒ χ [Π ′ ] | Σ ′ ⇒ Π ′ ∈ Θ ′ } .Thus the following is an application of R : B Φ , Q x ( χ [ p ] , ..., χ [ p n ]) ⇒ ΨIt is easy to check that χ is also an h R , Ψ ∪ Φ , x i -mapping and so the following is also anapplication of R : B Φ ⇒ Ψ , Q x ( χ [ p ] , ..., χ [ p n ])By cut, Φ ⇒ Ψ is provable, but Φ and Ψ are disjoint sets of atomic formulas, thus theyhave no cut-free proof in G , in contradiction to our assumption. This work is licensed under the Creative Commons Attribution-NoDerivs License. To viewa copy of this license, visit http://creativecommons.org/licenses/by-nd/2.0/http://creativecommons.org/licenses/by-nd/2.0/