This chapter examines global policing as it takes shape through the work of Interpol, the International Criminal Police Organization. Global policing emerges in the legal, political and technological amalgam through which transnational police cooperation is carried out, and includes the police practices inflected and made possible by this phenomenon. Interpol’s role is predominantly in the circulation of information, through which it enters into relationships and provides services that affect aspects of governance, from the local to national, regional and global. The (...) chapter describes this assemblage as a noteworthy experiment in developing what McKeon called a frame for common action. Drawing on Interpol publications, news stories, interviews with staff, and fieldwork at the General Secretariat in Lyon, France, the history, institutional structure, and daily practices are described. Three cases are analyzed, concerning Red Notices, national sovereignty, and terrorism, in order to explore some of the problems arising in Interpol’s political and technical operating arrangements. In conclusion, international and global policing are compared schematically, together with Interpol’s attempts to give institutional and procedural direction to the still-evolving form of global policing. (shrink)

A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov-Gentzen-Gödel style translation, and a new (...) translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a Kiriyama-Ono style translation with features of a Kolmogorov-Gentzen-Gödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics . (shrink)

In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi $ with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive (...) interpolation’ (see Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent. (shrink)

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting (...) to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants. (shrink)

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.

We analyse the structure of propositional proofs in the sequent calculus focusing on the well-known procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing the flow of formulas in it . We show some general facts about logical graphs such as acyclicity of cut-free proofs and acyclicity of contraction-free proofs , and we give (...) a proof of a strengthened version of the Craig Interpolation Theorem based on flows of formulas. We show that tautologies having minimal interpolants of non-linear size must have proofs with certain precise structural properties. We then show that given a proof Π and a cut-free form Π′ associated to it , certain subgraphs of Π′ which are logical graphs correspond to subgraphs of Π which are logical graphs for the same sequent. This locality property of cut elimination leads to new results on the complexity of interpolants, which cannot follow from the known constructions proving the Craig Interpolation Theorem. (shrink)

Arrow and turnstile interpolations are investigated in UCL [introduced by Sernadas et al. ], a logic that is a complete extension of classical propositional logic for reasoning about connectives that only behave as expected with a given probability. Arrow interpolation is shown to hold in general and turnstile interpolation is established under some provisos.

Decision theory requires agents to assign probabilities to states of the world and utilities to the possible outcomes of different actions. When agents commit to having the probabilities and/or utilities in a decision problem defined by objective features of the world, they may find themselves unable to decide which actions maximize expected utility. Decision theory has long recognized that work-around strategies are available in special cases; this is where dominance reasoning, minimax, and maximin play a role. Here we describe a (...) different work around, wherein a rational decision about one decision problem can be reached by ‘interpolating’ information from another problem that the agent believes has already been rationally solved. (shrink)

An interpolation theorem holds for many standard modal logics, but first order $S5$ is a prominent example of a logic for which it fails. In this paper it is shown that a first order $S5$ interpolation theorem can be proved provided the logic is extended to contain propositional quantifiers. A proper statement of the result involves some subtleties, but this is the essence of it.

Interpolation has become one of the standard properties that logicians investigate when designing a logic. In this paper, we provide strong evidence that the presence of interpolants is not only cogent for scientific reasoning but has also important practical implications in computer science. We illustrate that interpolation in general, and uniform splitting interpolants, in particular, play an important role in applications where formality and modularity are invoked. In recognition of the fact that common logical formalisms often lack uniform (...) interpolants, we advocate the need for developing general methods to engineer a specification logic so that some critical uniform interpolants become available. (shrink)

We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.

An interpolation Theorem is proved for first order logic with infinitary predicates. Our proof is algebraic via cylindric algebras.1.

In this paper we give a rather detailed algebraic investigation of interpolation and Beth’s property in propositional many-valued logics extending Hájek’s Basic Logic [P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, 1998], and we connect such properties with amalgamation and strong amalgamation in the corresponding varieties of algebras. It turns out that, while the most interesting extensions of in the language of have deductive interpolation, very few of them have Beth’s property or Craig interpolation. Thus in the last (...) part of the paper we look for conservative extensions of having such properties. (shrink)

A method is presented that connects the existence of uniform interpolants to the existence of certain sequent calculi. This method is applied to several modal logics and is shown to cover known results from the literature, such as the existence of uniform interpolants for the modal logic \. New is the result that \ has uniform interpolation. The results imply that for modal logics \ and \, which are known not to have uniform interpolation, certain sequent calculi cannot (...) exist. (shrink)

Preservation and interpolation results are obtained for L ∞ω and sublogics $\mathscr{L} \subseteq L_{\infty\omega}$ such that equivalence in L can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms.

We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness (...) and positive partial and linear orders. (shrink)

Craig's interpolation lemma (if φ → ψ is valid, then φ → θ and θ → ψ are valid, for θ a formula constructed using only primitive symbols which occur both in φ and ψ) fails for many propositional and first order modal logics. The interpolation property is often regarded as a sign of well-matched syntax and semantics. Hybrid logicians claim that modal logic is missing important syntactic machinery, namely tools for referring to worlds, and that adding such (...) machinery solves many technical problems. The paper presents strong evidence for this claim by defining interpolation algorithms for both propositional and first order hybrid logic. These algorithms produce interpolants for the hybrid logic of every elementary class of frames satisfying the property that a frame is in the class if and only if all its point-generated subframes are in the class. In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur. (shrink)

We study interpolation for elementary fragments of classical linear logic. Unlike in intuitionistic logic (see [Renardel de Lavalette, 1989]) there are fragments in linear logic for which interpolation does not hold. We prove interpolation for a lot of fragments and refute it for the multiplicative fragment (→, +), using proof nets and quantum graphs. We give a separate proof for the fragment with implication and product, but without the structural rule of permutation. This is nearly the Lambek (...) calculus. There is an appendix explaining what quantum graphs are and how they relate to proof nets. (shrink)

The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R-mingle with unit” logic that have the Craig interpolation property. This latter characterization is obtained using a model-theoretic quantifier elimination strategy to determine the varieties (...) of Sugihara monoids admitting the amalgamation property. (shrink)

A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (...) (1) Feasible interpolation theorems for the following proof systems: (a) resolution (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (α) (c) linear equational calculus (d) cutting planes. (2) New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]) (b) for the cutting planes proof system with coefficients written in unary ([4]). (3) An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory S 2 2 (α). In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle. (shrink)

The Interpolation Theorem, first formulated and proved by W. Craig fifty years ago for predicate logic, has been extended to many other logical frameworks and is being applied in several areas of computer science. We give a short overview, and focus on the theory of software systems and modules. An algebra of theories TA is presented, with a nonstandard interpretation of the existential quantifier . In TA, the interpolation property of the underlying logic corresponds with the quantifier combination (...) property . It is shown how the Modularization Theorem, the Factorization Lemma and the Normal Form Theorem for module expressions can be proved in TA. (shrink)

We investigate interpolation properties of many-valued propositional logics related to continuous t-norms. In case of failure of interpolation, we characterize the minimal interpolating extensions of the languages. For finite-valued logics, we count the number of interpolating extensions by Fibonacci sequences.

Continuing work initiated by Jónsson, Daigneault, Pigozzi and others; Maksimova proved that a normal modal logic (with a single unary modality) has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property (cf. [Mak 91], [Mak 79]). The aim of this paper is to extend the latter result to a large class of logics. We will prove that the characterization can be extended to all algebraizable logics containing Boolean fragment and having a certain kind of (...) local deduction property. We also extend this characterization of the interpolation property to arbitrary logics under the condition that their algebraic counterparts are discriminator varieties. We also extend Maksimova's result to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks smaller than 2, too.The problem of extending the above characterization result to no n-normal non-unary modal logics remains open. (shrink)

The splitting theorem says that any set of formulae has a finest representation as a family of letter-disjoint sets. Parikh formulated this for classical propositional logic, proved it in the finite case, used it to formulate a criterion for relevance in belief change, and showed that AGMpartial meet revision can fail the criterion. In this paper we make three further contributions. We begin by establishing a new version of the well-known interpolation theorem, which we call parallel interpolation, use (...) it to prove the splitting theorem in the infinite case, and show how AGM belief change operations may be modified, if desired, so as to ensure satisfaction of Parikh’s relevance criterion. (shrink)

Preservation and interpolation results are obtained for L ∞ω and sublogics $\mathscr{L} \subseteq L_{\infty\omega}$ such that equivalence in L can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms.

Craig interpolation theorem implies that the derivability of X,X′ Y′ implies existence of an interpolant I in the common language of X and X′ Y′ such that both X I and I,X′ Y′ are derivable. For classical logic this extends to X,X′ Y,Y′, but for intuitionistic logic there are counterexamples. We present a version true for intuitionistic propositional logic, and more complicated version for the predicate case.

Several extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language ℒ' is the least expressive extension of ℒ with interpolation. For instance, let ℳ be the extension of the basic modal language with a difference operator [7]. First-order logic is the least expressive extension of ℳ with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment.

We show in this paper that all fragments of intuitionistic propostional logic based on a subset of the connectives $\wedge, \vee, \rightarrow, \neg$ satisfy interpolation. Fragments containing $\leftrightarrow$ or $\neg\neg$ are briefly considered.

A semantical definition of abstract logics is given. It is shown that the Craig interpolation property implies the Beth definability property, and that the Souslin-Kleene interpolation property implies the weak Beth definability property. An example is given, showing that Beth does not imply Souslin-Kleene.

Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of first-order logic, there is an interpolant in which each relation symbol appears positively (negatively) only if it appears positively (negatively) in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some fixed tuple U of unary predicates U, all formulae under consideration have all quantifiers explicitly relativised to one (...) of the U. Under this stipulation, existential (universal) quantification over U contributes a positive (negative) occurrence of U. It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unifies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on positive vs. negative occurrences of predicates and on existentially vs. universally quantified sorts. (shrink)

A family of prepositional logics is considered to be intermediate between the intuitionistic and classical ones. The generalized interpolation property is defined and proved is the following.Theorem on interpolation. For every intermediate logic L the following statements are equivalent:(i) Craig's interpolation theorem holds in L, (ii) L possesses the generalized interpolation property, (iii) Robinson's consistency statement is true in L.

Given some links between Lyndon’s Interpolation Theorem, term distribution, and Sommers and Englebretsen’s logic, in this contribution we attempt to capture a sense of interpolation for Sommers and Englebretsen’s Term Functor Logic. In order to reach this goal we first expound the basics of Term Functor Logic, together with a sense of term distribution, and then we offer a proof of our main contribution.

We introduce a family of notions of interpolation for sentential logics. These concepts generalize the ones for substructural logics introduced in [5]. We show algebraic characterizations of these notions for the case of equivalential logics and study the relation between them and the usual concepts of Deductive, Robinson, and Maehara interpolation properties.

The problem of interpolation is a classical problem in logic. Given a consequence relation |~ and two formulas φ and ψ with φ |~ ψ we try to find a “simple" formula α such that φ |~ α |~ ψ. “Simple" is defined here as “expressed in the common language of φ and ψ". Non-monotonic logics like preferential logics are often a mixture of a non-monotonic part with classical logic. In such cases, it is natural examine also variants of (...) the interpolation problem, like: is there “simple" α such that φ ⊢ α |~ ψ where ⊢ is classical consequence? We translate the interpolation problem from the syntactic level to the semantic level. For example, the classical interpolation problem is now the question whether there is some “simple" model set X such that M(φ) ⫅ X ⫅ M(ψ). We can show that such X always exist for monotonic and antitonic logics. The case of non-monotonic logics is more complicated, there are several variants to consider, and we mostly have only partial results. (shrink)

The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the \ calculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. (...) In terms of the \ calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics. (shrink)

Model-theoretic methods are used to extend Craig's Interpolation Theorem to the loop-free portion of Pratt's dynamic logic of programs with simple assignments.

In this paper we will be concerned with the interpretability logic of PA and in particular with the fact that this logic, which is denoted by ILM, does not have the interpolation property. An example for this fact seems to emerge from the fact that ILM cannot express Σ₁-ness. This suggests a way to extend the expressive power of interpretability logic, namely, by an additional operator for Σ₁-ness, which might give us a logic with the interpolation property. We (...) will formulate this extension, give an axiomatization which is modally complete and arithmetically complete (although for proofs of these theorems we refer to an earlier paper), and investigate interpolation. We show that this logic still does not have the interpolation property. (shrink)

The Interpolation Theorem, first formulated and proved by W. Craig fifty years ago for predicate logic, has been extended to many other logical frameworks and is being applied in several areas of computer science. We give a short overview, and focus on the theory of software systems and modules. An algebra of theories TA is presented, with a nonstandard interpretation of the existential quantifier ∃. In TA, the interpolation property of the underlying logic corresponds with the quantifier combination (...) property ∃Σ ∃П S ≡ ∃(Σ ∪ П) S. It is shown how the Modularization Theorem, the Factorization Lemma and the Normal Form Theorem for module expressions can be proved in TA. (shrink)

The guarded fragment (GF) was introduced by Andréka, van Benthem and Németi as a fragment of first order logic which combines a great expressive power with nice, modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. Slightly generalizing the admissible relativizations yields the packed fragment (PF). In this paper we investigate interpolation and definability in these fragments. We first show that the interpolation property of first order logic fails (...) in restriction to GF and PF. However, each of these fragments turns out to have an alternative interpolation property that closely resembles the interpolation property usually studied in modal logic. These results are strong enough to entail the Beth definability property for GF and PF. Even better, every guarded or packed finite variable fragment has the Beth property. For interpolation, we characterize exactly which finite variable fragments of GF and PF enjoy this property. (shrink)

In a recent paper we have defined an analytic tableau calculus PL_16 for a functionally complete extension of Shramko and Wansing's logic based on the trilattice SIXTEEN_3. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic---such as the relations |=_t, |=_f, and |=_i that each correspond to a lattice order in SIXTEEN_3; and |=, the intersection of |=_t and |=_f,. -/- It turns out that our method of characterising these semantic relations---as (...) intersections of auxiliary relations that can be captured with the help of a single calculus---lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that |=, when restricted to L_{tf}, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano. (shrink)

In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.

Important positive as well as negative results on interpolation property in fragments of the intuitionistic propositional logic (INT) were obtained by J. I. Zucker in [6]. He proved that the interpolation theorem holds in purely implicational fragment of INT. He also gave an example of a fragment of INT for which interpolation fails. This fragment is determined by the constant falsum (), well known connectives: implication () and conjunction (), and by a ternary connective defined as follows: (...) (p, q, r)= df (pq)(pr).Extending this result of J. I. Zucker, G. R. Renardel de Lavalette proved in [5] that there are continuously many fragments of INT without the interpolation property. (shrink)

In this paper we deal with the logic which is the infinitary extension of propositional modal logic that has conjunctions and disjunctions only for countable sets of formulas. After introducing some basic concepts and tools from modal logic, we modify Makkai's generalization of the notion of consistency property to make it fit for modal purposes. Using this construction as a universal instrument, we prove, among other things, interpolation for as well as preservation results for universal, existential, and positive -formulas.

We show that proofs in the intuitionistic propositional logic factor through interpolants-in this way we prove a stronger interpolation property than the usual one which gives only the existence of interpolants.Translating that to categorical terms, we show that Pushouts (bipushouts) of bicartesian closed categories have the interpolation property (Theorem 3.2).

We introduce a notion of a real game (a generalisation of the Karchmer-Wigderson game (cf. [3]) and of real communication complexity, and relate this complexity to the size of monotone real formulas and circuits. We give an exponential lower bound for tree-like monotone protocols (defined in [4, Definition 2.2]) of small real communication complexity solving the monotone communication complexity problem associated with the bipartite perfect matching problem. This work is motivated by a research in interpolation theorems for prepositional logic (...) (by a problem posed in [5, Section 8], in particular). Our main objective is to extend the communication complexity approach of [4, 5] to a wider class of proof systems. In this direction we obtain an effective interpolation in a form of a protocol of small real communication complexity. Together with the above mentioned lower bound for tree-like protocols this yields as a corollary a lower bound on the number of steps for particular semantic derivations of Hall's theorem (these include tree-like cutting planes proofs for which an exponential lower bound was demonstrated in [2]). (shrink)

We prove several interpolation theorems for many-valued infinitary logic with quantifiers by studying expansions of MV-algebras in the spirit of polyadic and cylindric algebras. We prove for various reducts of polyadic MV-algebras of infinite dimensions that if is the free algebra in the given signature,, is in the subalgebra of generated by, is in the subalgebra of generated by and, then there exists an interpolant in the subalgebra generated by and such that. We call this a varying interpolation (...) property because the integer depends on the inequality. We also address cases where this interpolation property fails, but other weaker ones hold. One such interpolation theorem says that though an interpolant may not be found as above, an interpolant can always be found if finitely many universal quantifiers are applied to making it smaller and the same number of existential quantifiers are applied to making it bigger. This number of quantifiers also varies; it depends on the inequality. Se.. (shrink)

We define the interpolative fusion T∪∗ of a family i∈I of first-order theories over a common reduct T∩, a notion that generalizes many examples of random or generic structures in the model-theo...