Online research in philosophy

The papers presented in this volume examine topics of central interest in contemporary philosophy of logic. They include reflections on the nature of logic and its relevance for philosophy today, and explore in depth developments in informal logic and the relation of informal to symbolic logic, mathematical metatheory and the limiting metatheorems, modal logic, many-valued logic, relevance and paraconsistent logic, free logics, extensional v. intensional logics, the logic of fiction, epistemic logic, formal logical and semantic paradoxes, the concept of truth, the formal theory of entailment, objectual and substitutional interpretation of the quantifiers, infinity and domain constraints, the Löwenheim-Skolem theorem and Skolem paradox, vagueness, modal realism v. actualism, counterfactuals and the logic of causation, applications of logic and mathematics to the physical sciences, logically possible worlds and counterpart semantics, and the legacy of Hilbert’s program and logicism. The handbook is meant to be both a compendium of new work in symbolic logic and an authoritative resource for students and researchers, a book to be consulted for specific information about recent developments in logic and to be read with pleasure for its technical acumen and philosophical insights. Key Features - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic -Useful bibliographies in every chapter - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic - Useful bibliographies in every chapter.

We describe a method for obtaining classical logic from intuitionistic logic which does not depend on any proof system, and show that by applying it to the most important implicational relevance logics we get relevance logics with nice semantical and proof-theoretical properties. Semantically all these logics are sound and strongly complete relative to classes of structures in which all elements except one are designated. Proof-theoretically they correspond to cut-free hypersequential Gentzen-type calculi. Another major property of all these logic is that the classical implication can faithfully be translated into them.

We solve a major open problem concerning algorithmic properties of products of 'transitive' modal logics by showing that products and commutators of such standard logics as K4. S4. S4.1 K4.3. GL. or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4. K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if C₁ and C₂ are classes of transitive frames such that their depth cannot be bounded by any fixed n < ω, then the logic of the class {F₁ × F₂| F₁ ∈ C₁. F₂ ∈ C₂} is undecidable. (On the contrary, the product of. say. K4 and the logic of all transitive Kripke frames of depth ≤ n, for some fixed n < ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and $\Pi _{1}^{1}$ -complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics.

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the right set-up is chosen. Our logic L4 has a semantics that also underlies Belnap’s [4] and is related to the logic of bilattices. L4 is in focus most of the time, but it is also shown how results obtained for L4 can be transferred to several variants.

Perhaps no concept is more central to maurice merleau-ponty's philosophy than his concept of depth. not only did merleau-ponty recognize the philosophical significance of depth for articulating a phenomenology of perception, but he saw it as essential for pursuing and expressing a novel, radical ontology. depth, merleau-ponty writes, is ``the most existential dimension,'' ``the dimension of dimensions''; it is the ``sine qua non'' of the world and being. let me elucidate merleau-ponty's radical concept of depth by ``addressing'' the salient contexts in which he approached this notion. in the first part of this paper, i shall examine depth in the experience of perception and schizophrenia. in the second, i shall discuss depth as descriptive of an ontology of ``flesh'' and as the guiding theme for the philosophical enterprise.

This paper shows that the Dawson technique of modelling deontic logics into alethic modal logics to gain insight into deontic formulas is not available for modelling a normal (in the spirit of Anderson) relevance deontic modal logic into either of the normal relevance alethic modal logics R S4or R M. The technique is to construct an extension of the well known entailment matrix set M 0and show that the model of the deontic formula P (A v B). PA v PB is excluded.

LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and $\bigwedge, \bigvee$ (both of bounded arity). Then for every d ≥ 0 and n ≥ 2, there is a set Td n of depth d sequents of total size O(n3 + d) which are refutable in LK by depth d + 1 proof of size exp(O(log2 n)) but such that every depth d refutation must have the size at least exp(nΩ(1)). The sets Td n express a weaker form of the pigeonhole principle.

In this paper we extend the anodic systems introduced in Bueno-Soler (J Appl Non Class Logics 19(3):291–310, 2009) by adding certain paraconsistent axioms based on the so called logics of formal inconsistency , introduced in Carnielli et al. (Handbook of philosophical logic, Springer, Amsterdam, 2007), and define the classes of systems that we call cathodic . These classes consist of modal paraconsistent systems, an approach which permits us to treat with certain kinds of conflicting situations. Our interest in this paper is to show that such systems can be semantically characterized in two different ways: by Kripke-style semantics and by modal possible-translations semantics . Such results are inspired in some universal constructions in logic, in the sense that cathodic systems can be seen as a kind of fusion (a particular case of fibring) between modal logics and non-modal logics, as discussed in Carnielli et al. (Analysis and synthesis of logics, Springer, Amsterdam, 2007). The outcome is inherently within the spirit of universal logic, as our systems semantically intermingles modal logics, paraconsistent logics and many-valued logics, defining new blends of logics whose relevance we intend to show.

Let S and T be countable complete theories. We assume that T is superstable without the dimensional order property, and S is interpretable in T in such a way that every model of S is coded in a model of T. We show that S does not have the dimensional order property, and we discuss the question of whether $\operatorname{Depth}(S) \leq \operatorname{Depth}(T)$ . For Mekler's uniform interpretation of arbitrary theories S of finite similarity type into suitable theories T s of groups we show that $\operatorname{Depth}(S) \leq \operatorname{Depth}(T_S) \leq 1 + \operatorname{Depth}(S)$.

“Paraconsistent” means “beyond the consistent” [3, 15]. Paraconsistent logics tolerate inconsistencies in a way that traditional logics do not. In a paraconsistent logic, the inference of explosion A, ∼AB is rejected. This may be for any of a number of reasons [16]. For proponents of relevance [1, 2] the argument has gone awry when we infer an irrelevant B from the inconsistent premises. Those who argue that inconsistent theories may have some logical content but do not commit us to everything, have reason to think that these theories are closed under a relation of paraconsistent logical consequence [12, 18]. Another reason to adopt a paraconsistent logic is more extreme. You may take the world to be inconsistent [14], and a true theory incorporating this inconsistency must be governed by a paraconsistent logic.