Nonclassical Logics

We give a short proof of the fundamental theorem of central element theory. The original proof is constructive and very involved and relies strongly on the fact that the class be a variety. Here we give a more direct nonconstructive proof which applies for the more general case of a first-order class which is both closed under the formation of direct products and direct factors.

This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as (...) completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics. (shrink)

The review gives a short description of the content of the book and discusses the treatment of conditionals in it.

Extensions of traditional syllogistics have been increasingly researched in philosophy, linguistics, and areas such as artificial intelligence and computer science in recent decades. This is mainly due to the fact that syllogistics is seen as a logic that comes very close to natural language abilities. Various forms of extended syllogistics have become established. This paper deals with the question to what extent a syllogistic representation in CL diagrams can be seen as a form of extended syllogistics. It will be shown (...) that the ontology of CL enables numerically exact assertions and inferences. (shrink)

Ewald considered tense operators G, H, F and P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. In 2014, Figallo and Pelaitay introduced the variety IKt of IKt-algebras and proved that the IKt system has IKt-algebras as algebraic counterpart. In this paper, we introduce and study the variety of tense Nelson algebras. First, we give some examples and we prove some properties. Next, we associate an IKt-algebra to each tense Nelson algebras. This result allowed us (...) to determine the congruence of the tense Nelson algebras and also to characterize the subdirectly irreducible tense Nelson algebras and particularly the simple tense Nelson algebras. Finally, we prove that there exists an equivalence between the category of IKt-algebras and the category of tense centered Nelson algebras. (shrink)

This paper presents an algebraic framework for investigating proposed translations of classical logic into intuitionistic logic, such as the four negative translations introduced by Kolmogorov, Gödel, Gentzen and Glivenko. We view these as variant semantics and present a semantic formulation of Troelstra’s syntactic criteria for a satisfactory negative translation. We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops. When a translation fails for a particular class of algebras, we (...) demonstrate that failure via specific finite examples. Using these, we prove that the syntactic version of these translations will fail to satisfy Troelstra’s criteria in the corresponding substructural logical setting. (shrink)

This paper proposes a generalization of the Kripke semantics of intuitionistic logic IL appropriate for intuitionistic Łukasiewicz logic IŁL — a logic in the intersection between IL and (classical) Łukasiewicz logic. This generalised Kripke semantics is based on the poset sum construction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143–1158, 2009) to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that w \Vdash \sigma—which for IL is (...) a relation between worlds w and formulas \sigma, and can be seen as a function taking values in the booleans w \Vdash \sigma \in {B}—becomes a function taking values in the unit interval w \Vdash \sigma \in [0,1]. An appropriate monotonicity restriction (which we call sloping functions) needs to be put on such functions in order to ensure soundness and completeness of the semantics. (shrink)

In this article, I present a semantically natural conservative extension of Urquhart’s positive semilattice logic with a sort of constructive negation. A subscripted sequent calculus is given for this logic and proofs of its soundness and completeness are sketched. It is shown that the logic lacks the finite model property. I discuss certain questions Urquhart has raised concerning the decision problem for the positive semilattice logic in the context of this logic and pose some problems for further research.

In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi.

We compare the logic HYPE recently suggested by H. Leitgeb as a basic propositional logic to deal with hyperintensional contexts and Heyting-Ockham logic introduced in the course of studying logical aspects of the well-founded semantics for logic programs with negation. The semantics of Heyting-Ockham logic makes use of the so-called Routley star negation. It is shown how the Routley star negation can be obtained from Dimiter Vakarelov’s theory of negation and that propositional HYPE coincides with the logic characterized by the (...) class of all involutive Routley star information frames. This result provides a much simplified semantics for HYPE and also a simplified axiomatization, which shows that HYPE is identical with the modal symmetric propositional calculus introduced by G. Moisil in 1942. Moreover, it is shown that HYPE can be faithfully embedded into a normal bi-modal logic based on classical logic. Against this background, we discuss the notion of hyperintensionality. (shrink)

I survey Brouwer’s weak counterexamples to classical theorems, with a view to discovering what useful mathematical work is done by weak counterexamples; whether they are rigorous mathematical proofs or just plausibility arguments; the role of Brouwer’s notion of the creative subject in them, and whether the creative subject is really necessary for them; what axioms for the creative subject are needed; what relation there is between these arguments and Brouwer’s theory of choice sequences. I refute one of Brouwer’s claims with (...) a weak counterexample of my own. I also examine Brouwer’s 1927 proof of the negative continuity theorem, which appears to be a weak counterexample reliant on both the creative subject and the concept of choice sequence; I argue that it provides a good justification for the weak continuity principle, but it is not a weak counterexample and it does not depend essentially on the creative subject. (shrink)

In Barrio et al. Barrio Pailos and Szmuc prove that there are systems of logic that agree with classical logic up to any finite meta-inferential level, and disagree with it thereafter. This article presents a generalized sense of meta-inference that extends into the transfinite, and proves analogous results to all transfinite orders.

We develop a general theory of FDE-based modal logics. Our framework takes into account the four-valued nature of FDE by considering four partially defined modal operators corresponding to conditions for verifying and falsifying modal necessity and possibility operators. The theory comes with a uniform characterization for all obtained systems in terms of FDE-style formula-formula sequents. We also develop some correspondence theory and show how Hilbert-style axiom systems can be obtained in appropriate cases. Finally, we outline how different systems from the (...) literature can be expressed in our framework. (shrink)

We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously variable reasons. Some (...) examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism. (shrink)

A fundamental result from Boolean modal logic states that a first-order definable class of Kripke frames defines a logic that is validated by all of its canonical frames. We generalise this to the level of non-distributive logics that have a relational semantics provided by structures based on polarities. Such structures have associated complete lattices of stable subsets, and these have been used to construct canonical extensions of lattice-based algebras. We study classes of structures that are closed under ultraproducts and whose (...) stable set lattices have additional operators that are first-order definable in the underlying structure. We show that such classes generate varieties of algebras that are closed under canonical extensions. The proof makes use of a relationship between canonical extensions and MacNeille completions. (shrink)

I build a canonical model for constant domain basic first-order logic (BQLCD), the constant domain first-order extension of Visser’s basic propositional logic, and use the canonical model to verify that BQLCD satisfies the disjunction and existence properties.

One way in which we might approach the challenge posed by the Sorites Paradox is considering that Sorites-susceptible predicates have several candidate extensions, or several ways in which these expressions can be made precise. For example, a candidate extension for the predicate ‘is a baby’ is the set of humans of less than two years, but also the set of those less than two years and one second, and of those less than two years and two seconds. In this chapter (...) we present and discuss two theories for vague predicates based on this idea: supervaluationism and subvaluationism. The chapter is structured in three parts. The first presents the super- and subvaluationist theories: their similarities and differences in the semantics, the resulting logics with their most characteristic features. The second reviews the super- and subvaluationist solutions to the Sorites Paradox and provides discussion on several controversies surrounding these theories. The third part introduces proof procedures for s’valuationist logics. (shrink)

In this article, I provide Urquhart-style semilattice semantics for three connexive logics in an implication-negation language (I call these “pure theories of connexive implication”). The systems semantically characterized include the implication-negation fragment of a connexive logic of Wansing, a relevant connexive logic recently developed proof-theoretically by Francez, and an intermediate system that is novel to this article. Simple proofs of soundness and completeness are given and the semantics is used to establish various facts about the systems (e.g., that two of (...) the systems have the variable sharing property). I emphasize the intuitive content of the semantics and discuss how natural informational considerations underly each of the examined systems. (shrink)

The formalisation of Natural Language arguments in a formal language close to it in syntax has been a central aim of Moss’s Natural Logic. I examine how the Quantified Argument Calculus (Quarc) can handle the inferences Moss has considered. I show that they can be incorporated in existing versions of Quarc or in straightforward extensions of it, all within sound and complete systems. Moreover, Quarc is closer in some respects to Natural Language than are Moss’s systems – for instance, is (...) does not use negative nouns. The process also sheds light on formal properties and presuppositions of some inferences it formalises. Directions for future work are outlined. (shrink)

We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert (...) axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K and the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic. (shrink)

We develop a paraconsistent logic by introducing new models for conditionals with acceptive and rejective selection functions which are variants of Chellas’ conditional models. The acceptance and rejection conditions are substituted for truth conditions of conditionals. The paraconsistent conditional logic is axiomatized by a sequent system \ which is an extension of the Belnap-Dunn four-valued logic with a conditional operator. Some acceptive extensions of \ are shown to be sound and complete. We also show the finite acceptive model property and (...) decidability of these logics. (shrink)

This paper introduces the inquisitive extension of R, denoted as InqR, which is a relevant logic of questions based on the logic R as the background logic of declaratives. A semantics for InqR is developed, and it is shown that this semantics is, in a precisely defined sense, dual to Routley-Meyer semantics for R. Moreover, InqR is axiomatized and completeness of the axiomatic system is established. The philosophical interpretation of the duality between Routley-Meyer semantics and the semantics for InqR is (...) also discussed. (shrink)

I show that the lattice of the positive integers ordered by division is characteristic for Urquhart’s positive semilattice relevance logic; that is, a formula is valid in positive semilattice relevance logic if and only if it is valid in all models over the positive integers ordered by division. I show that the same frame is characteristic for positive intuitionistic logic, where the class of models over it is restricted to those satisfying a heredity condition. The results of this article highlight (...) deep connections between intuitionistic and semilattice relevance logic. (shrink)

Book Review of Impossible Worlds, by Francesco Berto and Mark Jago. Oxford: Oxford University Press, 2019.

This paper presents a sound and complete five-sided sequent calculus for first-order weak Kleene valuations which permits not only elegant representations of four logics definable on first-order weak Kleene valuations, but also admissibility of five cut rules by proof analysis.

A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, the famous Dunn–Belnap four-valued logic has exactly one self-extensional four-valued extension which has an implication. We also investigate the main properties of this logic, determine the (...) expressive power of its language, and provide a cut-free Gentzen-type proof system for it. (shrink)

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘non-falsity’ versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case (...) of infectious logics, namely to the case of Deutsch's logic Sfde introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are Setl and Snfl. We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between Setl, Snfl and two well-known systems, strong Kleene three-valued logic K3 and Priest's Logic of Paradox LP. This connection allows us to investigate the characterisation of the entailment relations associated with Setl and Snfl as well as to introduce the notion of ‘infectious analogue’ of a certain logic. We also study implicative extensions of Setl and Snfl and prove soundness and completeness theorems for them as well. (shrink)

There is a profound, but frequently ignored, relationship between the classical conception of logical consequence and the material implication. The first repeats the patterns of the latter, but with a wider modal reach. This relationship suggests that there should be also a connection between the notion of logical consequence and the conditional connective of any given logical system. This implies, among other things, that it is incoherent to propose alternatives to the material implication while maintaining the classical conception of logical (...) consequence. The other important implication is that we need to posit different conceptions of logical consequence that are consistent with different theories of conditionals in order to evaluate their relative merits from new and unexplored angles. As a pilot study of this research program, we evaluate two new notions of logical consequence motivated by conditional-assertion theory and possible world theories. Those alternatives are compared unfavourably with the classical conception of logical consequence. (shrink)

Paraconsistent Weak Kleene Logic is the 3-valued propositional logic defined on the weak Kleene tables and with two designated values. Most of the existing proof systems for PWK are characterised by the presence of linguistic restrictions on some of their rules. This feature can be seen as a shortcoming. We provide a cut-free calculus for PWK that is devoid of such provisos. Moreover, we introduce a Priest-style tableaux calculus for PWK.

Building on recent work by Yale Weiss, we study conditional logics in the intuitionistic setting. We consider a number of semantic conditions which give rise, among others, to intuitionistic counterparts of Lewis’s logic VC and Stalnaker’s C2. We show how to obtain a sound and complete axiomatization of each logic arising from a combination of these conditions. On the way, we remark how, in the intuitionistic setting, certain classically equivalent principles of conditional logic come apart, and how certain logical connections (...) between different principles no longer hold. (shrink)

In 1935/1936 Kurt Gödel wrote three notebooks on the foundations of quantum mechanics, which have now been entirely transcribed for the first time. Whereas a lot of the material is rather technical in character, many of Gödel's remarks have a philosophical background and concentrate on Leibnizian monadology as well as on vitalism. Obviously influenced by the vitalistic writings of Hans Driesch and his ‘proofs’ for the existence of an entelechy in every living organism, Gödel briefly develops the idea of a (...) computing machine which closely resembles Turing's groundbreaking conception. After introducing the notebooks on quantum mechanics, this article describes Gödel's vitalistic Weltbild and the ideas leading to the development of his computing machine. It investigates a notion of lawlike sequence which closely resembles Turing's concept of a computable number and which Gödel himself calls ‘problematic’, and compares it to the opposed concept of randomness, drawing upon the notion of program-size complexity. Finally, Gödel's machine is implemented in a dialect of the Lisp programing language. (shrink)

It was shown recently that epimorphisms need not be surjective in a variety K of Heyting algebras, but only one counter-example was exhibited in the literature until now. Here, a continuum of such examples is identified, viz. the variety generated by the Rieger-Nishimura lattice, and all of its (locally finite) subvarieties that contain the original counter-example K . It is known that, whenever a variety of Heyting algebras has finite depth, then it has surjective epimorphisms. In contrast, we show that (...) for every integer $n \geq 2$ , the variety of all Heyting algebras of width at most n has a non-surjective epimorphism. Within the so-called Kuznetsov-Gerčiu variety (i.e., the variety generated by finite linear sums of one-generated Heyting algebras), we describe exactly the subvarieties that have surjective epimorphisms. This yields new positive examples, and an alternative proof of epimorphism surjectivity for all varieties of Gödel algebras. The results settle natural questions about Beth-style definability for a range of intermediate logics. (shrink)

This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is presented bridging Boole’s theory and the use of his arithmetical logical functions with the axioms of Boolean algebra using sets and quantum logic. It is shown that this algebraic polynomial formulation can be naturally extended to operators in finite vector spaces. Logical operators will appear as (...) commuting projection operators and the truth values, which take the binary values \, are the respective eigenvalues. In this view the solution of a logical proposition resulting from the operation on a combination of arguments will appear as a selection where the outcome can only be one of the eigenvalues. In this way propositional logic can be formalized in linear algebra by using elective developments which correspond here to combinations of tensored elementary projection operators. The original and principal motivation of this work is for applications in the new field of quantum information, differences are outlined with more traditional quantum logic approaches. (shrink)

We study the amalgamation property in positive logic, where we shed light on some connections between the amalgamation property, Robinson theories, model-complete theories and the Hausdorff property.

This study aims to introduce a modal extension M4CC of Arieli, Avron, and Zamansky's ideal paraconsistent four-valued logic 4CC as a Gentzen-type sequent calculus and prove the Kripke-completeness and cut-elimination theorems for M4CC. The logic M4CC is also shown to be decidable and embeddable into the normal modal logic S4. Furthermore, a subsystem of M4CC, which has some characteristic properties that do not hold for M4CC, is introduced and the Kripke-completeness and cut-elimination theorems for this subsystem are proved. This subsystem (...) is also shown to be decidable and embeddable into S4. (shrink)

In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its relationship with the notion of positive closedness and the amalgamation property.

We study an alternative embedding of IPC into atomic system F whose translation of proofs is based, not on instantiation overflow, but instead on the admissibility of the elimination rules for disjunction and absurdity. As compared to the embedding based on instantiation overflow, the alternative embedding works equally well at the levels of provability and preservation of proof identity, but it produces shorter derivations and shorter simulations of reduction sequences. Lambda-terms are employed in the technical development so that the algorithmic (...) content is made explicit, both for the alternative and the original embeddings. The investigation of preservation of proof-reduction steps by the alternative embedding enables the analysis of generation of “administrative” redexes. These are the key, on the one hand, to understand the difference between the two embeddings; on the other hand, to understand whether the final word on the embedding of IPC into atomic system F has been said. (shrink)

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 first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.

In this study, we prove the completeness and cut-elimination theorems for a first-order extension F4CC of Arieli, Avron, and Zamansky’s ideal paraconsistent four-valued logic known as 4CC. These theorems are proved using Schütte’s method, which can simultaneously prove completeness and cut-elimination.

In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic \. Secondly, we introduce a new normal modal logic \ which is a proper extension of \, and prove that there exists a Rosser provability predicate whose provability logic includes \.

Pietruszczak :163–171, 2009. https://doi.org/10.12775/LLP.2009.013) proved that the normal logics \, \ ), \ are determined by suitable classes of simplified Kripke frames of the form \, where \. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of simplified frames if and only if it is a normal extension of \. Furthermore, a modal logic is a normal extension of \ ; \; \) if and only if it is (...) determined by a set consisting of finite simplified frames ; such frames with \ or \; such frames with \). Secondly, for all normal extensions of \, \, \ and \, in particular for extensions obtained by adding the so-called “verum” axiom, Segerberg’s formulas and/or their T-versions, we prove certain versions of Nagle’s Fact :319–328, 1981. https://doi.org/10.2307/2273624) ). Thirdly, we show that these extensions are determined by certain classes of finite simplified frames generated by finite subsets of the set \ of natural numbers. In the case of extensions with Segerberg’s formulas and/or their T-versions these classes are generated by certain finite subsets of \. (shrink)

This article begins with a general and abstract definition of logic and, particularly, of paraconsistent logics, to establish a common ground for the discussion. Briefly stating, these kinds of logics have the property of being non-explosive, that is, it is not possible to infer any conclusion from contradictory premises. Using these definitions, it is possible to analyze some of the philosophical aspects of paraconsistent logics, in particular, the relation between the notion of explosion and the law of non-contradiction, as well (...) as the syntactic/semantic possibility and, above all, the metaphysical possibility of paraconsistent logics. I further analyze a stronger position towards paraconsistency, namely: the claim that there are true contradictions. This articles concludes with some possible critiques to paraconsistent logics – and their refutations as well –, and pose some open questions for further work. -/- . (shrink)

A very natural and philosophically important subclassical logic is FDE. This account of logical consequence can be seen as going beyond the standard two-valued account to a four-valued account. A natural question arises: What account of logical consequence arises from considering further combinations of such values? A partial answer was given by Priest in 2014; Shramko and Wansing had also given a partial result some years earlier, although in a different context. In this note we generalize Priest’s result to show (...) that even if one considers ordinal-many combinations of the previous values, for any ordinal, the resulting consequence relation remains FDE. (shrink)

Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. (...) It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and model-theoretic consequence. Every model-theoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely, the homophonic interpretation that does not substitute anything. Applications to free logic, and theories and languages other than set theory are sketched. (shrink)

The Mesopotamian peoples were never really dominated by the reason the way we conceptualize it. It's to the revelation as direct emanation of the divine that they ascribed the appearance of knowledge.

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more (...) precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics. (shrink)

We will present a three-valued consequence relation for metainferences, called CM, defined through ST and TS, two well known substructural consequence relations for inferences. While ST recovers every classically valid inference, it invalidates some classically valid metainferences. While CM works as ST at the inferential level, it also recovers every classically valid metainference. Moreover, CM can be safely expanded with a transparent truth predicate. Nevertheless, CM cannot recapture every classically valid meta-metainference. We will afterwards develop a hierarchy of consequence relations (...) CMn for metainferences of level n. Each CMn recovers every metainference of level n or less, and can be nontrivially expanded with a transparent truth predicate, but cannot recapture every classically valid metainferences of higher levels. Finally, we will present a logic CMω, based on the hierarchy of logics CMn, that is fully classical, in the sense that every classically valid metainference of any level is valid in it. Moreover, CMω can be nontrivially expanded with a transparent truth predicate. (shrink)

The provability logic of a theory T captures the structural behavior of formalized provability in T as provable in T itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations.The logic IL is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this article (...) we raise the previously known lower bound of IL by exhibiting two series of principles which are shown to be provable in any such theory. Moreover, we compute the collection of frame conditions for both series. (shrink)

We prove that the continuous function${\rm{\hat \Omega }}:2^\omega \to $ that is defined via$X \mapsto \mathop \sum \limits_n 2^{ - K\left} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left\,{\rm{d}}X$ is a left-c.e. $wtt$-complete real having effective Hausdorff dimension ${1 / 2}$.We further investigate the algorithmic properties of ${\rm{\hat{\Omega }}}$. For example, we show that the maximal (...) value of ${\rm{\hat{\Omega }}}$ must be random, the minimal value must be Turing complete, and that ${\rm{\hat{\Omega }}}\left \oplus X{ \ge _T}\emptyset \prime$ for every X. We also obtain some machine-dependent results, including that for every $\varepsilon > 0$, there is a universal machine V such that ${{\rm{\hat{\Omega }}}_V}$ maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that $X{ \le _{tt}}{{\rm{\hat{\Omega }}}_V}\left$; and that there is a real X and a universal machine V such that ${{\rm{\Omega }}_V}\left$ is rational. (shrink)