Canonical calculi with (n,k)-ary quantifiers

Propositional canonical Gentzen-type systems, introduced in 2001 by Avron and Lev, are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. A constructive coherence criterion for the non-triviality of 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-valued non-deterministic matrices (2Nmatrices). In 2005 Zamansky and Avron extended these results to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n,k)-ary quantifiers, which bind k distinct variables and connect n formulas, and show that the coherence criterion remains constructive for such systems. Then we focus on the case of k∈{0,1} and for a canonical calculus G show that it is coherent precisely when it has a strongly characteristic 2Nmatrix, which in turn is equivalent to admitting strong cut-elimination.


binds 4 variables and connects one formula:
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 way of recording combinations of quantifiers, dependency relations between variables are expressed as follows: an existentially quantified variable depends on those universally quantified 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 his classical paper Investigations Into Logical Deduction ( [13]), an "ideal" set of introduction rules 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 propositional Gentzen-type rule" was defined in precise terms. A constructive coherence criterion for the non-triviality of systems consisting of such rules was provided, and it was shown that a system of this kind admits cut-elimination iff it is coherent. It was further proved that the semantics of such systems is provided by two-valued non-deterministic matrices (2Nmatrices), which form a natural generalization of the classical matrix. In fact, a characteristic 2Nmatrix was constructed for every coherent canonical propositional system.
In [28] the results were extended to systems (of a restricted form) with unary quantifiers. A characterization of a "canonical unary quantificational rule" in such calculi was proposed (the standard Gentzen-type rules for ∀ and ∃ are canonical according to it), and a constructive extension of the coherence criterion from [2,3] for canonical systems of this type was given. 2Nmatrices were extended to languages with unary quantifiers, using a distributional interpretation of quantifiers ( [20,7]). Then it was proved that again a canonical Gentzen-type system of this type admits cut-elimination 3 iff it is coherent, and that it is coherent 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 characterizations of [2,3,28] to "canonical (n, k)-ary quantificational rules", so that both the propositional systems of [2,3] and the restricted quantificational systems of [28] are specific instances of the proposed definition. We show that the coherence criterion for the defined systems remains decidable. Then we focus on the case of k ∈ {0, 1} and show that the following 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 then characterize a subclass of canonical systems for which this property does hold.

Preliminaries
For any n > 0 and k ≥ 0, if a quantifier Q is of arity (n, k), then Qx 1 ...x k (ψ 1 , ..., ψ n ) is a formula whenever x 1 , ..., x k are distinct variables and ψ 1 , ..., 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. Definition 1.1. Given a set of truth value V, a distribution of a (1,1)-ary quantifier Q is a function λ Q : P + (V) → V.
In what follows, L is a language with (n, k)-ary quantifiers, that is with quantifiers Q 1 , ..., Q m with arities (n 1 , k 1 ), ..., (n m , k m ) respectively. Denote by F rm cl L the set of closed L-formulas and by T rm cl L the set of closed L-terms. V ar = {v 1 , v 2 , ..., } is the set of variables of L. We use the metavariables x, y, z to range over elements of V ar.
We use [ ] for application of functions in the meta-language, leaving the use of ( ) to the 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 A set of sequents S satisfies the free-variable condition if the set of variables occurring bound in S is disjoint from the set of variables occurring free in S.

Canonical Systems with (n,k)-ary quantifiers
In this section we propose a precise characterization of a "canonical (n, k)-ary quantificational Gentzen-type rule".
Using an introduction rule for an (n, k)-ary quantifier Q, we should be able to derive a sequent of the form Γ ⇒ Qx 1 ...x k (ψ 1 , ..., ψ n ), ∆ or of the form Γ, Qx 1 ...x k (ψ 1 , ..., ψ n ) ⇒ ∆, based on some information about the subformulas of Qx 1 ...x k (ψ 1 , ..., ψ n ) contained in the premises of the rule. For instance, consider the following standard rules for the (1,1)-ary quantifier ∀: where t, z are free for w in A and z does not occur free in the conclusion. Our key observation is that the internal structure of A, as well as the exact term t or variable w used, are immaterial 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 a rule can be abstracted by using a simplified first-order language, i.e., the formulas of L in an introduction rule of a (n, k)-ary quantifier, can be represented by atomic formulas with predicate symbols of arity k. The case when the substituted term is any L-term, will be signified 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 are eigenvariables.
Thus in addition to our original language L with (n, k)-ary quantifiers we define another, simplified language. For k ≥ 0, n ≥ 1 and a set of constants Con, L n k (Con) is the (first-order) language with n k-ary predicate symbols p 1 , ..., p n and the set of constants Con (and no quantifiers). The set of variables of L n k (Con) is V ar = {v 1 , v 2 , ..., }. Note that L n k (Con) and L share the same set of variables. Furthermore, henceforth we assume that for every (n, k)-ary quantifier Q of L, L n k (Con) is a subset of L. This assumption is not necessary, but it makes the presentation easier, as will be explained in the 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 of arity (n, k) is an expression of the form Henceforth, in cases where the set of constants Con is clear from the context (it is the set of all constants occurring in a canonical rule), we will write L n k instead of L n k (Con). A canonical rule is a schematic representation, while for an actual application we need to instantiate the schematic variables by the terms and formulas of L. This is done using a mapping function, defined as follows.
). Let Γ be a set of L-formulas and z 1 , ..., z k -distinct variables of L. An R, Γ, z 1 , ..., z k -mapping is any function χ from the predicate symbols, terms and formulas of L n k to formulas and terms of L, satisfying the following conditions: does not occur in χ[c] for any variable x occurring in Θ.
• For every 1 ≤ i ≤ n, whenever p i (t 1 , ..., t k ) occurs in Θ, for every 1 ≤ j ≤ k: 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 in We extend χ to sets of L n k (Con Θ )-formulas as follows: Given a schematic representation of a rule and an instantiation mapping, we can define an application of a rule as follows.
Definition 2.4. An application of a canonical rule of arity (n, k) where z 1 , ..., z k are variables, Γ, ∆ are any sets of L-formulas and χ is some R, Γ∪∆, z 1 , ..., z kmapping. 4 By a clause we mean a sequent containing only atomic formulas.
CANONICAL CALCULI WITH (n, k)-ARY QUANTIFIERS 5 An application of a canonical quantificational rule of the form 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 of as a (2, 0)-ary quantifier is {⇒ p 1 , ⇒ p 2 }/ ⇒ p 1 ∧ p 2 . Its application is of the form: The standard introduction rules for the (1, 1)-ary quantifiers ∀ and ∃ can be formulated as follows: Applications of these rules have the forms: 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, 1)-ary quantifiers ∀ and ∃ (corresponding to ∀x.p 1 (x) → p 2 (x) and ∃x.p 1 (x) ∧ p 2 (x) used in syllogistic reasoning). Their corresponding rules can be formulated as follows: ) Applications of these rules are of the form: , ∆ where t and y are free for z in ψ 1 and ψ 2 , y does not occur free in Γ ∪ ∆ ∪ {∃z(ψ 1 , ψ 2 )}. (4) Consider the (2,2)-ary rule Its application is of the form: where w 1 , w 2 , w 3 , t 1 , t 2 satisfy the appropriate conditions.
Note that although the derivability of the α-axiom is essential for any logical system, it is not guaranteed to be derivable in a canonical system. What natural syntactic conditions guarantee its derivability is still a question for further research. For now we explicitly add the 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). For instance, 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: For further abbreviation, we denote such rule by {Σ j ⇒ Π j } 1≤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 1 /C 1 and Θ 2 /C 2 for Q are dual if for some s ∈ {t, f }: Although we can define arbitrary canonical systems using our simplified language L n k , our quest is for systems, the syntactic rules of which define the semantic meaning of logical connectives/quantifiers. Thus we are interested in calculi with a "reasonable" or "noncontradictory" set of rules, which allows for defining a sound and complete semantics for the system. This can be captured syntactically by the following extension of the coherence criterion of [2,28].
is obtained from Θ 2 by a fresh renaming of constants and variables which occur in Θ 1 .
Henceforth it will be convenient (but not essential) to assume that the fresh constants used for the renaming are in L. Definition 2.9. (Coherence) 5 A canonical calculus G is coherent if for every two dual canonical rules Θ 1 / ⇒ A and Θ 2 /A ⇒, the set of clauses Rnm(Θ 1 ∪ Θ 2 ) is classically inconsistent.
Note that the principle of renaming of clashing constants and variables is similar to the one used in first-order resolution. The importance of this principle for the definition of coherence will be explained in the sequel. Proof. The question of classical consistency of a finite set of clauses without function symbols (over L n k ) can be shown to be equivalent to satisfiability of a finite set of universal formulas with no function symbols. This is decidable (by an obvious application of Herbrand's theorem).

The semantic framework
3.1. Non-deterministic matrices. Our main semantic tool are non-deterministic matrices (Nmatrices), first introduced in [2, 3] and extended in [27,28]. These structures are a generalization of the standard concept of a many-valued matrix, in which the truth-value of 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 function of an (n, k)-ary quantifier Q (from Definition. 1.1) to a function λ Q : P + (V n ) → P + (V). In other words, given some distribution Y of n-ary vectors of truth values, the interpretation function non-deterministically chooses the truth value assigned to 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-ary connective of the language; given a valuation v, In the definition above, the interpretation of a propositional connective ⋄ is a function of another type:⋄ : P + (V n ) → P + (V). This can be thought as a generalization of the previous definition, identifying the tuple v[ The advantage of this generalization is that it allows for a uniform treatment of both quantifiers and propositional connectives. I is extended to interpret closed terms of L as follows: Here a note on our treatment of quantification in the framework of Nmatrices is in order. The standard approach to interpreting quantified formulas is by using objectual (or referential) semantics, where the variable is thought of as ranging over a set of objects from the 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 is true (see, e.g., [24,26]). [27] explains the motivation behind choosing the substitutional approach for the framework of Nmatrices, and points out the problems of the objectual approach in this context. The substitutional approach assumes that every element of the domain has a closed term referring to it. Thus given a structure S = D, I , we extend the language L with individual constants, one for each element of D.
Given a set Γ of formulas, we denote the set The motivation for the following definition is purely technical and is related to extending the language with the set of individual constants {a | a ∈ D}. Suppose we have a closed term t, such that I[t] = a ∈ D. But a also has an individual constant a referring to it. We would 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 an Nmatrix M. The relation ∼ S between terms of L(D) is defined inductively as follows: . The relation ∼ S between formulas of L(D) is defined as follows: ..x k and − → y = y 1 ...y k are distinct variables and − → z = z 1 ...z k are new distinct variables, then .., 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 variables and by any number of substitutions of a closed term t for another closed term s, so that Note that in case Q is a propositional connective (for k = 0), the functionQ M is applied to a singleton, as was explained above.
Notation. For a set of sequents S, we shall write S ⊢ G Γ ⇒ ∆ if a sequent Γ ⇒ ∆ has a proof from S in G.
Definition 3.8. Let S = D, I be an L-structure for an Nmatrix M.
Definition 3.9. A system G is strongly sound 6 for an Nmatrix M if for every set S of sequents closed under substitution: is strongly sound and strongly complete for M.
Note that since the empty set of sequents is closed under substitutions, strong soundness implies (weak) soundness 7 . A similar remark applies to completeness and a characteristic Nmatrix.

3.2.
Semantics for simplified languages L n k . In addition to L-structures for languages with (n, k)-ary quantifiers, we also use L n k -structures for the simplified languages L n k , used for formulating the canonical rules. To make the distinction clearer, we shall use the metavariable S for the former and N for the latter. Since the formulas of L n k are always atomic, the specific 2Nmatrix for which N is defined is immaterial, and can be omitted. We may even speak of classical validity of sequents over L n k . Thus henceforth instead of speaking of M-validity of a set of clauses Θ over L n k , we will speak simply of validity. Next we define the notion of a distribution of L n k -structures.
Now we turn to the case k = 1. In this case it is convenient to define a special kind of L n 1 -structures which we call canonical structures. These structures are sufficient to reflect the behavior of all possible L n 1 -structures. 6 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. 7 A system G is (weakly) sound for an Nmatrix M if ⊢G Γ ⇒ ∆ entails ⊢M Γ ⇒ ∆. 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 1 , the question whether 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 finitely many E-canonical structures to check.

Canonical systems with (n, k)-ary quantifiers for k ∈ {0, 1}
Now we turn to the class of canonical systems with (n, k)-ary quantifiers for the case of k ∈ {0, 1} and n ≥ 1. Henceforth, unless stated otherwise, we assume that k ∈ {0, 1}. 4.1. Semantics for canonical systems for k ∈ {0, 1}. In this section we explore the connection between the coherence of a canonical calculus G, the existence for it of a strongly characteristic 2Nmatrix, and strong cut-elimination (in a sense explained below.) We start by defining the notion of suitability for G.
First of all, note that by corollary 3.14, the above definition is constructive. Next, let us show that M G is well-defined. Assume by contradiction that there are two dual rules Θ 1 / ⇒ A and Θ 2 /A ⇒, such that both Θ 1 and Θ 2 are valid in some E-canonical structures N 1 , N 2 respectively. Obtain Θ ′ 2 from Θ 2 by renaming of constants and variables which occur in Θ 1 . Then clearly Θ ′ 2 is also valid in some E-canonical structure N 3 . If k = 0, by Lemma 3.11, the set of clauses Θ 1 ∪ Θ ′ 2 is satisfiable by a (classical) propositional valuation v E and is 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 the interpretation of constants, and since the sets of constants occurring in Θ 1 and Θ ′ 2 are disjoint, an E-canonical structure N ′ = D ′ , I ′ (for the extended language containing the constants of both Θ 1 and Θ 2 ) can be constructed, in which Θ 1 ∪ Θ ′ 2 are valid. Thus the set is classically consistent, in contradiction to the coherence of G.
Remark: The construction of M G above is much simpler than the constructions carried out in [2,28]: a canonical calculus there is first transformed into an equivalent normal form calculus, which is then used to construct the characteristic Nmatrix. The idea is to transform the calculus so that each rule dictates the interpretation for only one E. However, the above definitions show that the transformation into normal form is actually not necessary and we can construct M G directly from G. Next we demonstrate the construction of a characteristic 2Nmatrix for some coherent canonical calculi. (1) It is easy to see that for any canonical coherent calculus G including the standard (1,1)-ary rules for ∀ and ∃ from Example 2.5-2: Consider the canonical calculus G ′ consisting of the following three (1, 2)-ary rules from Example 2.5-3: The first rule dictates the condition that ∀[H] = {t} for the case of t, f ∈ H. The second rule dictates the condition that ∀[H] = {f } for the case that t, f ∈ H. Since G ′ is coherent, these conditions are non-contradictory. The third rule dictates the condition that ∃[H] = {t} in the case that t, t ∈ H. There is no rule which dictates conditions for the case of t, t ∈ H, and so the interpretation in this case is non-deterministic.

A. AVRON AND A. ZAMANSKY
Of course, G ′′ is coherent. The 2Nmatrix M G ′′ is defined as follows for every H ∈ P + ({t, f } 2 ): Now we come to the main theorem, establishing a connection between the coherence of a canonical calculus G, the existence of a strongly characteristic 2Nmatrix for G and strong cut-elimination in G in the sense of [1].  Note that strong cut-elimination implies standard cut-elimination (which corresponds to 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 is classically consistent, there exists an L n k -structure N = D, I , in which both Θ 1 and Θ ′ 2 are valid. Recall that we also assume that L n k is a subset of L 11 and so the following are applications of R 1 and R 2 respectively: Let S be any extension of N to L and v -any M-legal S-valuation. It is easy to see that the premises of the applications above are M-valid in S, v (since the premises contain atomic formulas). Since G is strongly sound for M, both ⇒ Qv 1 (p 1 (v 1 ), ..., p n (v 1 )) and 8 [1] does not assume that S is closed under substitution. Instead, a structural substitution rule is added and the allowed cuts are on substitution instances of formulas from S. 9 See section 1. 10 This assumption is not necessary and is used only for simplification of presentation, since we can instantiate the constants by any L-terms. 11 This assumption is again not essential for the proof, but it simplifies the presentation.
Qv 1 (p 1 (v 1 ), ..., p n (v 1 )) ⇒ should also be M-valid in S, v , 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 strong cut-elimination. Suppose by contradiction that G is not coherent. Then there are two dual rules of G: Θ 1 / ⇒ A and Θ 2 /A ⇒, such that Rnm(Θ 1 ∪ Θ 2 ) is classically consistent. Let Θ be the minimal set of clauses, such that Rnm(Θ 1 ∪ Θ 2 ) ⊆ Θ and Θ is closed under substitutions. Θ ∪ {⇒} satisfy the free-variable condition, since only atomic formulas are involved 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 1 ∪ Θ 2 ) is consistent 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: Proof. see Appendix A.
To prove that (1) implies (2), suppose that G is coherent. Let us show that M G is a strongly 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 that a sequent Γ ⇒ ∆ has no proof from S in G. If S ∪{Γ ⇒ ∆} does not satisfy the free-variable condition, 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. By Proposition 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 S, v , while Γ ′ ⇒ ∆ ′ is not. Since v respects the ≡ α -relation, the sequents of S are also M-valid in S, v , while Γ ⇒ ∆ is not. And so S ⊢ 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 of sequents 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 shown above that M G is a strongly characteristic 2Nmatrix for G. Thus S ⊢ M Γ ⇒ ∆, and by Proposition 4.8, Γ ⇒ ∆ has a simple proof from S in G. Thus G admits strong cutelimination.
Proof. By theorem 4.7, the question whether G has a strongly characteristic 2Nmatrix is equivalent 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 of sequent calculi with (n, k)-ary quantifiers and a (not necessarily standard) set of structural rules called standard calculi are defined. A canonical calculus is a particular instance of a standard calculus which includes all of the standard structural rules. [9] formulate syntactic necessary and sufficient conditions for a slightly generalized version of cut-elimination with non-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 to a much wider class of calculi (since different combinations of structural rules are allowed). In addition, a constructive modular cut-elimination procedure is provided. The reductivity condition of [9] can be shown to be equivalent to our coherence criterion in the context of canonical systems 12 .

4.2.
Coherence and standard cut-elimination. In the previous subsection we have studied the connection between coherence and strong cut-elimination. In this subsection we focus on standard cut-elimination in canonical calculi. It easily follows from theorem 4.7 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 a cut-free proof in G.
Thus coherence is a sufficient condition for cut-elimination in a canonical calculus. In the 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.
To prove this it suffices to show that for every rule of G 0 : if its premises are logical axioms, then its conclusion is a logical axiom. Suppose by contradiction that we can apply one 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 be of the form:  Thus, only logical axioms are provable in G 0 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 property does hold.
(2) k = 1 and one of the following holds for each variable y occurring in Rnm(Θ 1 ∪ Θ 2 ): • There is at most one 1 ≤ i ≤ n, such that y occurs in p i (y) in Rnm(Θ 1 ∪ Θ 2 ) and there is at most one constant c, such that p i (c) also occurs in Rnm(Θ 1 ∪ Θ 2 ). • There are two different 1 ≤ i, j ≤ n, such that y occurs in p i (y) and p j (y) in Rnm(Θ 1 ∪ Θ 2 ) 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 1 ∪ Θ 2 ). (2) Consider the canonical calculus G 1 , consisting of the following two rules for a (3, 1)-ary quantifier Q 1 : ) ⇒ It is easy to see that G 1 is a simple coherent calculus.
(3) If we modify the first rule of G 1 as follows: the resulting calculus is not simple, since both p 1 (c) and p 1 (d) occur in the premises of the rule, together with p 1 (v 1 ). (4) The calculus G 0 from example 4.11 is not simple, since for instance p 1 (v 1 ), p 1 (c 1 ) and p 1 (c 5 ) occur in the premises (after renaming).
Proposition 4.14. If a simple canonical calculus G admits cut-elimination, then it is coherent.
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 ∈ {0, 1}, we have shown that the following 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 have also shown that coherence is not a necessary condition for standard cut-elimination, and characterized a subclass of canonical systems called simple calculi, for which this property does hold.
In addition to these proof-theoretical results for a natural type of multiple conclusion Gentzen-type systems with (n, 1)-ary and (n, 0)-ary quantifiers, this work also provides further evidence for the thesis that the meaning of a logical constant is given by its introduction (and "elimination") rules . We have shown that at least in the framework of multiple-conclusion consequence relations, any "reasonable" set of canonical quantificational rules completely determines the semantics of the quantifier.
This paper also demonstrates the important role of the semantic framework of Nmatrices ( [2,27]), which substantially contributes to the understanding of the connection between syntactic rules and semantic interpretations of quantifiers. Due to the modularity of the framework, we were able to detect the semantic effect of each of the canonical rules, which of course is not possible using deterministic matrices.
Some of the most immediate research directions are as follows. In the case of k ∈ {0, 1}, we still need to characterize the most general subclass of canonical calculi, for which coherence is both a necessary and sufficient condition for standard cut-elimination (it is not clear whether the characterization of simple calculi can be further extended).
Extending these results to the case of k > 1 might lead to new insights on Henkin quantifiers and other important generalized quantifiers. However, even for the simplest case of (1, 2)-ary quantifiers the extension is far from straightforward. Consider, for instance, the calculus G, consisting of the following two (1,2)-ary rules: {p(c, x) ⇒}/ ⇒ Qz 1 z 2 p(z 1 , z 2 ) {⇒ p(y, d)}/Qz 1 z 2 p(z 1 , z 2 ) ⇒ G is coherent, but it is easy to see that M G is not well-defined in this case. And even if a 2Nmatrix M suitable for G does exist, it is not necessarily sound for G. It is clear that the distributional interpretation of quantifiers is no longer adequate for the case of k > 1, since it cannot capture any kind of dependencies between elements of the domain. Thus a more general interpretation of quantifiers is needed.
Another important research direction is extending canonical systems with equality. This will 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 representation language L n k . Standard and strong cut-elimination and its connection to the coherence of canonical systems are yet to be investigated for canonical systems with equality. We will now show that Θ R = {Σ j ⇒ Π j } 1≤j≤m is valid in N . Suppose for contradiction that 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: is not M-valid in S, v , in contradiction to our assumption on the validity of the premises of the application above. We have shown that {Σ j ⇒ Π j } 1≤j≤m is valid in N . Obviously 13  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 proof from S in G. To show that S ⊢ M Γ ⇒ ∆, we will construct a structure S and an M-legal valuation v, such that the sequents of S are M-valid in S, v , 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 in Γ ⇒ ∆ and S. It is a standard matter to show that Γ, ∆ can be extended to two (possibly infinite) sets Γ ′ , ∆ ′ (where Γ ⊆ Γ ′ and ∆ ⊆ ∆ ′ ), satisfying the following properties: (1) For every finite Γ 1 ⊆ Γ ′ and ∆ 1 ⊆ ∆ ′ , Γ 1 ⇒ ∆ 1 has no simple proof in G.
Let S = D, I be the L * -structure defined as follows: • D = T.
• I[c] = c for every constant c of L * .
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 it in L * (D).) For an L(D)-formula ψ (an L(D)-term t), we will denote by ψ ( t) the L-formula (Lterm) obtained from ψ (t) by replacing every individual constant of the form s for some s ∈ T by the term s. More formally, t and ψ are defined as follows: • x = x for any variable x of L.
• c = c for any constant c of L.
Define the S-valuation v as follows: Proof. The claims are proven by induction on t in the first case, and on ψ and ψ ′ in the second and third cases.