We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic \(\textbf{IL}\). We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.

The sole purpose of this paper is to give an algebraic characterization, in terms of a superamalgamation property, of a local version of Craig interpolation theorem that has been introduced and studied in earlier papers. We continue ongoing research in abstract algebraic logic and use the framework developed by Andréka– Németi and Sain.

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)

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)

The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the (...) range of such logics is small. (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.

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.

The main result proves Lyndon’s and Craig’s interpolation properties for the logic of strict implication ${\textsf{F}}$, with a purely syntactical method. A cut-free G3-style sequent calculus $ {\textsf{GF}} $ and its single-succedent variant $ \textsf{GF}_{\textsf{s}} $ are introduced. $ {\textsf{GF}} $ can be extended to a G3-variant of the sequent calculus GBPC3 for Visser’s basic logic. Also a simple syntactic proof of known embedding result of $ {\textsf{F}} $ into $ {\textsf{K}} $ is provided. An extension of $ (...) {\textsf{F}} $, namely $ \textsf{FD}, $ is considered as well. (shrink)

A logic |$\mathcal{L}$| is said to satisfy the descending chain condition, DCC, if any descending chain of formulas in |$\mathcal{L}$| with ordering induced by |$\vdash _{\mathcal{L}};$| eventually stops. In this short note, we first establish a general theorem, which states that if a propositional logic |$\mathcal{L}$| satisfies both DCC and has the Craig Interpolation Property, CIP, then it satisfies the Uniform Interpolation Property, UIP, as well. As a result, by using the Nishimura lattice, we give (...) a new simply proof of uniform interpolation for |$\textbf{IPL}_2$|⁠, the two-variable fragment of Intuitionistic Propositional Logic; and one-variable uniform interpolation for |$\textbf{IPL}$|⁠. Also, we will see that the modal logics |$\textbf{S}_4$| and |$\textbf{K}_4$| do not satisfy atomic DCC. (shrink)

In this paper we introduce the modelwise interpolation property of a logic that states that whenever \(\models\phi\to\psi\) holds for two formulas \(\phi\) and \(\psi\), then for every model \(\mathfrak{M}\) there is an interpolant formula \(\chi\) formulated in the intersection of the vocabularies of \(\phi\) and \(\psi\), such that \(\mathfrak{M}\models\phi\to\chi\) and \(\mathfrak{M}\models\chi\to\psi\), that is, the interpolant formula in Craig interpolation may vary from model to model. We compare the modelwise interpolation property with the standard (...) class='Hi'>Craig interpolation and with the local interpolation property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic. (shrink)

We formulate a general institution-independent (i.e. independent of the details of the actual logic formalised as institution) version of the Craig Interpolation Theorem and prove it in dependence of Birkhoff-style axiomatizability properties of the actual logic.We formalise Birkhoff-style axiomatizability within the general abstract model theoretic framework of institution theory by the novel concept of Birkhoff institution.

LetL be one of the intuitionistic modal logics considered in [7] (or one of its extensions) and letM L be the “algebraic semantics” ofL. In this paper we will extend toL the equivalence, proved in the classical case (see [6]), among he weak Craig interpolation theorem, the Robinson theorem and the amalgamation property of varietyM L. We will also prove the equivalence between the Craig interpolation theorem and the super-amalgamation property of varietyM L. Then (...) we obtain the Craig interpolation theorem and Robinson theorem for two intuitionistic modal logics, one ofS 4-type and the other one ofS 5-type, showing the super-amalgamation property of the corresponding algebraic semantics. (shrink)

The idea that human beings have a distinct moral worth—a moral significance over and above any moral worth, such as that may be, possessed by other animals—has a long history and has traditionally been taken for granted by philosophers and theologians. However, in a variety of quarters in recent philosophy, this idea has come into disrepute, seeming to indicate a mere prejudice in favour of our own species. For example, Peter Singer has argued that such a position is mere speciesism, (...) a prejudice of a kind with racism and sexism in that it involves making moral distinctions between our own and other species that cannot be morally justified. What on such views is needed to justify any such distinction is a difference in terms of the morally relevant properties possessed by our own species as compared with other species. I will call this view the moral property view. Insofar as other species share with us morally relevant properties, for example the capacity to suffer cognitive ability and so on, it is mere prejudice not to accept moral requirements with respect to them as we do with respect to our own species. While on the surface such a view may seem morally enlightened, it indicates what will seem to many problematic moral judgments with respect to severely disabled human beings. In this paper, I will respond to these concerns by suggesting a different basis for the idea of human moral distinctiveness, one that draws on recent work by Wittgensteinian moral philosophers and which denies what I called above the property view. According to this view, while our shared life with other animals involves the recognition of their moral significance, our shared life with other human beings involves recognising that human beings as human beings have a distinctive moral value. (shrink)

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)

Though deceptively simple and plausible on the face of it, Craig's interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of (...) subsequent generalizations and applications, especially of many-sorted interpolation theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for "reasonable" logics extending first-order logic within the framework of abstract model theory. (shrink)

Many contemporary theorists charge that naïve realists are incapable of accounting for illusions. Various sophisticated proposals have been ventured to meet this charge. Here, we take a different approach and dispute whether the naïve realist owes any distinctive account of illusion. To this end, we begin with a simple, naïve account of veridical perception. We then examine the case that this account cannot be extended to illusions. By reconstructing an explicit version of this argument, we show that it depends critically (...) on the contention that perceptual experience is diaphanous, or more minimally and precisely, that there can be no difference in phenomenal properties between two experiences without a difference in the scenes presented in those experiences. Finding no good reason to accept this claim, we develop and defend a simple, naïve account of both veridical perception and illusion, here dubbed Simple, Austere Naïve Realism. (shrink)

The kalam cosmological argument-perhaps the most discussed philosophical argument for God's existence in recent decades-maintains that whatever begins to exist must have a cause. And since the universe began to exist, there must be a transcendent cause of its beginning, a conclusion which is confirmatory of theism. So this medieval argument for the finitude of the past has received fresh wind in its sails from recent scientific discoveries. This collection reviews and assesses the merits of the latest scientific evidences for (...) the universe's beginning. It ends with the kalam argument's conclusion that the universe has a cause-a personal cause with properties of theological significance. (shrink)

Though deceptively simple and plausible on the face of it, Craig's interpolation theorem has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, (...) especially of many-sorted interpolation theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for "reasonable" logics extending first-order logic within the framework of abstract model theory. (shrink)

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)

This article analyzes “farmers’ rights” as a strategy of resistance against the perceived inequities of intellectual property rights regimes for plant varieties. As commercial models of intellectual property have made their way into agriculture, farmers’ traditional seed-saving practices have been increasingly delegitimized. In response, farmers have adopted the language of farmers’ rights to demand greater material recognition of their contributions and better measures to protect their autonomy. This campaign has mixed implications. On one hand, farmers’rights are a unique (...) form of right that may help transform conventions of intellectual property in ways that are better suited for registering and materially encouraging alternative forms of innovation, such as those offered by farming communities. On the other hand, farmers’ rights have proved enormously difficult to enact. And by situating farmers’rights alongside easily enacted commercial breeders’rights, the campaign risks further legitimizing the inequities it is responding to. (shrink)

In the paper we give a sucient condition of the Interpolation Property in propositional calculi; then we establish the power of the class of the systems with Craig's property. Next we show that there does not exist a minimal R0-system with Craig-Godel-Lindenbaum's property. Finally, we generalize Sobocinski-Tarskis theorem concerning Sobocinski-Tarski's property.

New B-Theorists of language, while conceding the untranslatability of tensed sentences by tenseless sentences, deny that the ineliminability of tense implies the reality of tensed facts. Thus, New BTheorist Nathan Oaklander explains, For a variety of reasons, ... recent defenders of the tenseless view have come to embrace the thesis that tensed sentences cannot be translated by tenseless ones without loss of meaning. Nevertheless, recent detensers have denied that the ineliminability of tensed language and thought entails the reality of temporal (...) properties. ... Tensed discourse is indeed necessary for timely action, but tensed facts are not, since the truth conditions of tensed sentences can be expressed in a tenseless meta-language that describes unchanging temporal relations between and among events. (shrink)

The manifest image is teeming with activity. Objects are booming and buzzing by, changing their locations and properties, vivid perceptions are replaced, and we seem to be inexorably slipping into the future. Time—or at least our experience in time— seems a very turbulent sort of thing. By contrast, time in the scientist image seems very still. The fundamental laws of physics don’t differentiate between past and future, nor do they pick out a present moment that flows. Except for a minus (...) sign in the relativistic metric, there are few differences between the temporal and spatial coordinates in natural science. We seem to have, to echo another debate, an “explanatory gap” between time as we find it in experience and as we find it in science. Reconciling these two images of the world is the principal goal of philosophy of time. (shrink)

In recent years restitution scholars have expended considerable energy in attempting to reveal the logical structure of the law of proprietary remedies. That project advances on the assumption that the strange rhetoric that pervades this area of law can be stripped away to reveal restitutionary principles. However, the doctrines in question have proved resistant to such endeavors. An appreciation of why this is so requires recognition of the very real anxiety generated by the judicial readjustment of property rights that (...) takes place in this area of law. The use of metaphors, fictions and evidential presumptions that characterizes the law tracing, subrogation and the provision of proprietary relief for wrongs serves to assuage this anxiety by manufacturing the appearance that these remedies are perfectly conventional instances of the enforcement of pre-existing property or consensual transfers of property. However, these property rites have a logic of their own - a logic that influences and constrains the development of the law of proprietary remedies and makes it impossible to neatly reduce this field to a set of restitutionary principles. This is illustrated by the work of Professor Peter Birks, which represents the most comprehensive attempt to cut away the abstruse rhetoric found in this area and reveal the restitutionary substructure of the law of proprietary remedies and makes it impossible to neatly reduce this field to a set of restitutionary principles. This is illustrated by the work of Professor Peter Birks, which represents the most comprehensive attempt to cut away the substructure of the law of proprietary remedies. In Birks' work this process has merely resulted in old metaphors being replaced with new ones. Ultimately, we must accept that if re are to rationalize this area we must be prepared to engage directly with the policy issues that arise in this context. (shrink)

Over the last half century, possible worlds have bled into almost every area of philosophy. In the metaphysics of modality, for example, philosophers have used possible worlds almost exclusively to illuminate discourse about metaphysical necessity and possibility. But recently, some have grown dissatisfied with possible worlds. Why are horses necessarily mammals? Because the property of being a horse bears a special relationship to the property of being a mammal, they say. Not because every horse is a mammal in (...) every possible world. Some have recently begun to use properties to develop rivals to possible worlds semantics which may someday compare in formal power and capture the different systems of modal logic. In this paper, I do two things. I first offer a quick primer on possible worlds semantics. Then I discuss three rivals and the work they have left to do. (shrink)

We sometimes say things like this: “being an animal is part of being a dog.” We associate the part with a precondition for exemplifying the whole. A new semantics for modal logic results when we take this way of speaking seriously. We need not treat necessary truths as truths in all possible worlds. Instead, we may treat them as preconditions for the existence of any world at all. I present this semantics for modal propositional logic and argue that it operates (...) on a more basic level of modal reality than possible world semantics. (shrink)

Libertarianism: For and Against offers dueling perspectives on the scope of legitimate government. Tibor R. Machan, a well-known libertarian philosopher, argues for a minimal government devoted solely to protecting individual rights to life, liberty, and property. Against this view, philosopher Craig Duncan defends democratic liberalism, which aims to ensure that all citizens have fair access to a life of dignity. In a dynamic exchange of arguments, the two philosophers cut to the heart of this important debate.

Leibniz's conceptual containment theory says that singular propositions of the form a is F are true when the complete concept of being a contains the concept of being F. In this paper, I provide a new semantics for first-order logic built around this idea. The semantics resolves longstanding problems for Leibniz's theory and can represent, without possible worlds, both hyperintensional distinctions among properties and a certain kind of presumably impossible situation that standard approaches cannot represent. The semantics also captures the (...) strong soundness and completeness of classical first-order logic. (shrink)

In common treatments of deontic logic, the obligatory is what is true in all deontically ideal possible worlds. In this article, I offer a new semantics for Standard Deontic Logic with Leibnizian intensions rather than possible worlds. Even though the new semantics furnishes models that resemble Venn diagrams, the semantics captures the strong soundness and completeness of Standard Deontic Logic. Since, unlike possible worlds, many Leibnizian intensions are not maximally consistent entities, we can amend the semantics to invalidate the inference (...) rule which ensures that all tautologies are obligatory. I sketch this amended semantics to show how it invalidates the rule in a new way. (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)

A central concern of economics is how society allocates its resources. Modern economies rely on two institutions to allocate: markets and governments. But how much of the allocating should be performed by markets and how much by governments? This collection of readings will help students appreciate the power of the market. It supplements theoretical explanations of how markets work with concrete examples, addresses questions about whether markets actually work well and offers evidence that supposed "market failures" are not as serious (...) as claimed. Featuring readings from Hayek, William Baumol, Harold Demsetz, Daniel Fischel and Edward Lazear, Benjamin Klein and Keith B. Leffler, Stanley J. Liebowitz and Stephen E. Margolis, and John R. Lott, Jr., this book covers key topics such as: • Why markets are efficient allocators • How markets foster economic growth • Property rights • How markets choose standards • Asymmetric Information • Whether firms abuse their power • Non-excludable goods • Monopolies The selections should be comprehended by undergraduate students who have had an introductory course in economics. This reader can also be used as a supplement for courses in intermediate microeconomics, industrial organization, business and government, law and economics, and public policy. (shrink)

This article responds to Tibor Machan's criticisms of government provision for needy citizens. It argues that although charity may be morally worthy, private charity is inadequate to the task of providing our fellow citizens with the security they deserve; the tremendous social good of secure access to a life of dignity can only be produced by a public social safety net. Moreover, individual rights to property do not stand in the way of providing a public social safety net. Since (...) there are social inputs in wealth creation, reasonable levels of taxation do not violate people's rights to their wealth. (shrink)

The Craig interpolation property and interpolation property for deducibility are considered for special kind of normal extensions of the Brouwer logic.

Canon is a concept from aesthetics that has become a regular subject of commonplace discussions. The nature of canon, especially as it is used in these commonplace discussions, has not been subject to adequate philosophical scrutiny. We attempt to remedy that by placing canon in its historical and philosophical context, exploring and rejecting several common accounts, and presenting some basics of how canon works. We reject the accounts that place control with the author or the legal property holder, which (...) appear to be the most commonly held accounts. (shrink)

Suppose we thought of free inquiry as a social matter, a public good. We might ask not only whether individual scholars are free from illegitimate, especially external, censorship or attempts to control their work. We might ask also how much the university as an institution contributes to overall freedom of inquiry. To answer the second question would require assessing how well universities educate students to be participants in free inquiry, how well researchers communicate their work to raise the quality of (...) public discourse, and whether the results of scientific inquiry are made freely available to advance further inquiry or controlled as private property. It would require asking whether the specific structures and practices through which we organize academic work - from disciplinary departments to evaluation procedures to publication systems - do more to facilitate or obstruct free inquiry. This article will fall short of answering all these questions, but I hope it will put them on the agenda. I present them in the context of two successive transformations - the late 19th century reorganization of universities by disciplines devoted to the production of new knowledge and the dramatic 20th increase in scale and cost which challenged the internal integration of universities. These shifted the constraints and conditions under which students and faculty could take up the project of free inquiry. Appreciating the impact of these transformations is also basic to projects of renewing the university and free inquiry today. (shrink)

Suppose we thought of free inquiry as a social matter, a public good. We might ask not only whether individual scholars are free from illegitimate, especially external, censorship or attempts to control their work. We might ask also how much the university as an institution contributes to overall freedom of inquiry. To answer the second question would require assessing how well universities educate students to be participants in free inquiry, how well researchers communicate their work to raise the quality of (...) public discourse, and whether the results of scientific inquiry are made freely available to advance further inquiry or controlled as private property. It would require asking whether the specific structures and practices through which we organize academic work - from disciplinary departments to evaluation procedures to publication systems - do more to facilitate or obstruct free inquiry. This article will fall short of answering all these questions, but I hope it will put them on the agenda. I present them in the context of two successive transformations - the late 19th century reorganization of universities by disciplines devoted to the production of new knowledge and the dramatic 20th increase in scale and cost which challenged the internal integration of universities. These shifted the constraints and conditions under which students and faculty could take up the project of free inquiry. Appreciating the impact of these transformations is also basic to projects of renewing the university and free inquiry today. (shrink)

This chapter contains sections titled: Introduction Did the Universe Begin to Exist? Everything That Begins to Exist Has a Cause The Cause of the Universe Properties of the First Cause Objections Conclusion References.

We define a notion of projective meaning which encompasses both classical presuppositions and phenomena which are usually regarded as non-presuppositional but which also display projection behavior—Horn’s assertorically inert entailments, conventional implicatures (both Grice’s and Potts’) and some conversational implicatures. We argue that the central feature of all projective meanings is that they are not-at-issue, defined as a relation to the question under discussion. Other properties differentiate various sub-classes of projective meanings, one of them the class of presuppositions according to Stalnaker. (...) This principled taxonomy predicts differences in behavior unexpected on other models among the various conventional triggers and conversational implicatures, while holding promise for a general, explanatory account of projection which applies to all the types of meanings considered. (shrink)

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.

We prove that certain natural sequent systems for bi-intuitionistic logic have the analytic cut property. In the process we show that the (global) subformula property implies the (local) analytic cut property, thereby demonstrating their equivalence. Applying a version of Maehara technique modified in several ways, we prove that bi-intuitionistic logic enjoys the classical Craig interpolation property and Maximova variable separation property; its Halldén completeness follows.

This paper investigates a structural void that, especially in the context of poor or developing nations, prevents economic growth from being more inclusive and benefiting wider sections of society. The authors initially examine the imperative for inclusive growth, one encompassing a focus on poverty and development. Utilizing social choice theory, and a capability deprivation perspective, we observe that the poor experience deprivations due to a deficiency in their personal autonomy. This in turn is deeply interwoven with the concept of identity. (...) Legally recognizing the poor as individuals, and providing them with proof of their identity, will empower them and facilitate inclusive growth and poverty alleviation. These conceptual arguments are illustrated with the description of a biometric-linked developmental initiative that is providing proof of identity to 1.2 billion residents of India. By establishing a robust identity management system, the project aims to ensure more inclusive growth and efficiently target welfare programs. The authors further investigate how the establishment of identity rights facilitates financial inclusion, property ownership, and necessity-driven entrepreneurial action. Biometric identification on this scale is, however, fraught with dangers to civil liberties and has other serious ethical consequences. In the last section, issues around privacy and security are debated while highlighting the need for external review and independent monitoring to define the project’s boundaries and usages. (shrink)

In a previous paper [ 21 ] all extensions of Johansson’s minimal logic J with the weak interpolation property WIP were described. It was proved that WIP is decidable over J. It turned out that the weak interpolation problem in extensions of J is reducible to the same problem over a logic Gl, which arises from J by adding tertium non datur. In this paper we consider extensions of the logic Gl. We prove that only finitely many (...) logics over Gl have the Craig interpolation property CIP, the restricted interpolation property IPR or the projective Beth property PBP. The full list of Gl-logics with the mentioned properties is found, and their description is given. We note that IPR and PBP are equivalent over Gl. It is proved that CIP, IPR and PBP are decidable over the logic Gl. (shrink)

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)

ABSTRACT The two main directions pursued in the present paper are the following. The first direction was started by Pigozzi in 1969. In [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property. In this paper we extend Maksimova's theorem 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. To extend the characterization beyond multi-modal logics, we look at arbitrary algebraizable logics. We will introduce an algebraic property equivalent with the Craig interpolation property in algebraizable logics, and prove that the superamalgamation property implies the Craig interpolation property. The problem of extending the characterization result to non-normal non-unary modal logics also will be discussed. In the second direction pursued herein: for non-normal modal logic with one unary modality Lemmon [Lem 66] gave a possible worlds semantics. Here we give a more general possible worlds semantics for not necessarily normal multi-modal logics with arbitrarily many not necessarily unary modalities. Strongly related to the above is the theorem, proved, e.g., in Jóns son-Tarski [JT 52] and Henkin-Monk-Tarski [HMT 71], that every normal Boolean algebra with operators can be represented as a subalgebra of the complex algebra of some relational structure. We extend this result to not necessarily normal BAO's as follows. We define partial relational structures and show that every not necessarily normal BAO is embeddable into the complex algebra of a partial relational structure. This gives a possible worlds semantics for not necessarily normal multi-modal logics. (shrink)

We generalise an interval-related interpolation theorem about abstract-time Interval Temporal Logic, which was first obtained in [GUE 01]. The generalisation is based on the abstract-time variant of a projection operator in the Duration Calculus, which was introduced in [DAN 99] and later studied extensively in [GUE 02]. We propose a way to understand interpolation in the context of formal verification. We give an example showing that, unlike abstract-time ITL, DC does not have the Craig interpolation (...) class='Hi'>property in general, and establish a special form of Craig interpolation for abstract-time DC. Explicit definability after Beth is known to be strongly related to Craig interpolation in general. We show a limitation of a different kind to the scope of Beth definability in ITL by a counterexample too. We call the generalisation of interval-related interpolation that we present projection-related interpolation. The DC-specific restrictions apply to it too. We show that both Craig and projection-related interpolation hold about the ⌈P⌉-subset of DC without such restrictions. Our proofs of these theorems for the ⌈P⌉-subset entail algorithms for the construction of the interpolants. (shrink)

In conclusion, then, the notion of temporal necessity is certainly queer and perhaps a misnomer. It really has little to do with temporality per se and everything to do with counterfactual openness or closedness. We have seen that the future is as unalterable as the past, but that this purely logical truth is not antithetical to freedom or contingency. Moreover, we have found certain past facts are counterfactually open in that were future events or actualities to be other than they (...) will be, these past facts would have been different as a consequence. God's beliefs about the future are such past facts. Moreover, the effects of actions which God would have taken had He believed differently are also such past facts. Oddly enough, then, virtually any past fact is potentially counterfactually open, and the only necessity that remains is purely de facto. We, of course, do not in general know which events of the past depend counterfactually on present actions, and those cases we do know about seem rather trivial. Our intuitions of the necessity, unalterability, and unpreventability of the past as opposed to the future stem from the impossibility of backward causation, which is precluded by the dynamic nature of time and becoming. But the counterfactual dependence of God's beliefs on future events or actualities is not a case of backward causation: rather future-tense propositions are true in virtue of what will happen, given a view of truth as correspondence, and God simply has the essential property of knowing all and only true propositions. With regard to the future, virtually all facts are counterfactually open, and therefore future-tense propositions are not temporally necessary. Propositions thus move from being temporally contingent to being temporally necessary when all the opportunities to affect things counterfactually have slipped by. Hence, the mere fact that an event is past is no indication that it is counterfactually closed. This is especially evident in the case of God's foreknowledge. If we say that God foreknows that I shall do x and therefore I cannot refrain from doing x, lest I change God's past foreknowledge, we are being deceived by a modality which has nothing to do with my power or freedom. All that is impossible is the conjunction of God's foreknowledge that p and of ~ p; but this modality in sensu composito has no bearing on my ability to act such that ~ p would be true and God would have foreknown differently. Temporal necessity, then, turns out to be only obliquely temporal and modally weak, certainly no threat to freedom or divine foreknowledge. (shrink)

We give an axiomatization of first‐order logic enriched with the almost‐everywhere quantifier over finitely additive measures. Using an adapted version of the consistency property adequate for dealing with this generalized quantifier, we show that such a logic is both strongly complete and enjoys Craig interpolation, relying on a (countable) model existence theorem. We also discuss possible extensions of these results to the almost‐everywhere quantifier over countably additive measures.

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)