I lay down a system of structural rules for various notions of ground and establish soundness and completeness.

In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a translation of mHC into (...) K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem). (shrink)

In fact, law is necessarily very vague. So if vagueness is a problem for legal theory, it is a serious problem. The problem has to do with the ideal of the rule of law and with the very idea of law: if vague standards provide no guidance in some cases, how can the life of a community be ruled by law? The problem has long concerned philosophers of law; the papers at this symposium address it afresh by asking what legal (...) theory may have to learn from (or contribute to) work on vagueness in philosophy of language and philosophy of logic. Here I will not try to state the implications of vagueness for philosophy of law; I will try to set the stage by showing that vagueness is both an important and an unavoidable feature of law. (shrink)

There are two paradoxes of satisfaction, and they are of different kinds. The classic satisfaction paradox is a version of Grelling’s: does ‘does not satisfy itself’ satisfy itself? The Unsatisfied paradox finds a predicate, P, such that Px if and only if x does not satisfy that predicate: paradox results for any x. The two are intuitively different as their predicates have different paradoxical extensions. Analysis reduces each paradoxical argument to differing rule sets, wherein their respective pathologies lie. Having different (...) pathologies, they are paradoxes of different kinds. Furthermore, each of these satisfaction paradoxes has an analogue with the same pathology in set theory. Therefore, these analogues are respectively of the same two kinds. This level of abstraction is significant in that it tracks two related but different pathologies. Thus, not all paradoxes of semantics and set theory share the same pathology: there are at least two kinds of paradox cutting across the semantic and set-theoretic distinction. (shrink)

The Pinocchio paradox, devised by Veronique Eldridge-Smith in February 2001, is a counter-example to solutions to the Liar that restrict the use or definition of semantic predicates. Pinocchio’s nose grows if and only if what he is stating is false, and Pinocchio says ‘My nose is growing’. In this statement, ‘is growing’ has its normal meaning and is not a semantic predicate. If Pinocchio’s nose is growing it is because he is saying something false; otherwise, it is not growing. ‘Because’ (...) stands here for a non-semantic relation; it might be supposed to be causal or of some other nature, but it is not semantic. The paradox is discussed in relation to Tarski’s and Kripke’s theories of truth. Although the paradox is not necessarily a counter-example to a theory of a truth predicate, it is a problem for a theory of truth of the kind preserved by validity. (shrink)

Semantic dialetheists astutely dodge Explosion, the logical contagion of everything being true if a single contradiction is true. A dialetheia is contained in their semantics, and sustained by a paraconsistent logic. Graham Priest has shown that this is a solution to the Liar paradox. I use the Pinocchio paradox, devised by Veronique Eldridge-Smith, as a counter-example. The Pinocchio paradox turns on the truth of Pinocchio, whose nose grows if and only if what he is saying is not true, saying ‘My (...) nose is growing’. It is not just a matter of interpretation whether Pinocchio’s nose is and is not growing. (shrink)

The Pinocchio paradox poses one dialetheia too many for semantic dialetheists (Eldridge-Smith 2011). However, Beall (2011) thinks that the Pinocchio scenario is merely an impossible story, like that of the village barber who shaves just those villagers who do not shave themselves. Meanwhile, Beall maintains that Liar paradoxes generate dialetheia. The Barber scenario is self-contradictory, yet the Pinocchio scenario requires a principle of truth for a contradiction. In this and other respects the Pinocchio paradox is a version of the Liar, (...) unlike the Barber. One wonders why some Liars would be impossible if others generate dialetheias. (shrink)

The main thesis of this paper is that we sometimes are disposed to accept false and even jointly inconsistent claims by virtue of our semantic competence, and that this comes to light in the sorites and liar paradoxes. Among the subsidiary theses are that this is an important source of indeterminacy in truth conditions, that we must revise basic assumptions about semantic competence, and that classical logic and bivalence can be upheld in the face of the sorites paradox.

Think of the color spectrum, spread out before you. You can identify the different colors with ease. But if you are asked to indicate the point at which one color ends and the next begins, you are at a loss. "There is no such point", is a natural thought: one color just shades gradually into the next.

The aim of this paper is to propose a formal approach to reasoning about desires, understood as logical propositions which we would be pleased to make true, also acknowledging the fact that desire is a matter of degree. It is first shown that, at the static level, desires should satisfy certain principles that differ from those to which beliefs obey. In this sense, from a static perspective, the logic of desires is different from the logic of beliefs. While the accumulation (...) of beliefs tend to reduce the remaining possible worlds they point at, the accumulation of desires tends to increase the set of states of affairs tentatively considered as satisfactory. Indeed beliefs are expected to be closed under conjunctions, while, in the positive view of desires developed here, one can argue that endorsing \ as a desire means to desire \ and to desire \. However, desiring \ and \ at the same time is not usually regarded as rational, since it does not make much sense to desire one thing and its contrary at the same time. Thus when a new desire is added to the set of desires of an agent, a revision process may be necessary. Just as belief revision relies on an epistemic entrenchment relation, desire revision is based on a hedonic entrenchment relation satisfying other properties, due to the different natures of belief and desire. While epistemic entrenchment relations are known to be qualitative necessity relations, hedonic relations obeying a set of reasonable postulates correspond to another set-function in possibility theory, called guaranteed possibility, that drive well-behaved desire revision operations. Then the general framework of possibilistic logic provides a syntactic setting for encoding desire change. The paper also insists that desires should be carefully distinguished from goals. (shrink)

Definite descriptions, I shall argue, have two possible functions. 1] They are used to refer to what a speaker wishes to talk about, but they are also used quite differently. Moreover, a definite description occurring in one and the same sentence may, on different occasions of its use, function in either way. The failure to deal with this duality of function obscures the genuine referring use of definite descriptions. The best known theories of definite descriptions, those of Russell and Strawson, (...) I shall suggest, are both guilty of this. Before discussing this distinction in use, I will mention some features of these theories to which it is especially relevant. (shrink)

I identify a notion of logical grounding, clarify it, and show how it can be used (i) to characterise various consequence relations, and (ii) to give a precise syntactic account of the notion of “groundedness” at work in the literature on the paradoxes of truth.

We argue that the need for commentary in commonly used linear calculi of natural deduction is connected to the “deletion” of illocutionary expressions that express the role of propositions as reasons, assumptions, or inferred propositions. We first analyze the formalization of an informal proof in some common calculi which do not formalize natural language illocutionary expressions, and show that in these calculi the formalizations of the example proof rely on commentary devices that have no counterpart in the original proof. We (...) then present a linear natural deduction calculus that makes use of formal illocutionary expressions in such a way that unique readability for derivations is guaranteed – thus showing that formalizing illocutionary expressions can eliminate the need for commentary. (shrink)

Critical discussion of Alan Code's paper "Aristotle's Investigation of a Basic Logical Principle: Which Science Investigates the Principle of Non-Contradiction?".

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range (...) of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management. (shrink)

The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence, and extends it to the Logic of Evidence and Truth. The latter is a logic of (...) formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. LETj is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. Adequate semantics and a decision method are presented for both BLE and LETj, as well as some technical results that fit the intended interpretation. (shrink)

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. (shrink)

Taking Carnap’s classic exposition as a starting point, this paper develops a pragmatic account of the method of explication, defends it against a range of challenges and proposes a detailed recipe for the practice of explicating. It is then argued that confusions are involved in characterizing explications as definitions, and in advocating precising definitions as an alternative to explications. Explication is better characterized as conceptual re-engineering for theoretical purposes, in contrast to conceptual re-engineering for other purposes and improving exactness for (...) purely practical reasons. Finally, three limitations which call for further development of the method of explication are discussed. (shrink)

In this paper we introduce a paraconsistent reasoning strategy, Chunk and Permeate. In this, information is broken up into chunks, and a limited amount of information is allowed to flow between chunks. We start by giving an abstract characterisation of the strategy. It is then applied to model the reasoning employed in the original infinitesimal calculus. The paper next establishes some results concerning the legitimacy of reasoning of this kind - specifically concerning the preservation of the consistency of each chunk (...) and concludes with some other possible applications and technical questions. (shrink)

In this paper, the relationship between the e-value of a complex hypothesis, H, and those of its constituent elementary hypotheses, Hj, j = 1… k, is analyzed, in the independent setup. The e-value of a hypothesis H, ev, is a Bayesian epistemic, credibility or truth value defined under the Full Bayesian Significance Testing mathematical apparatus. The questions addressed concern the important issue of how the truth value of H, and the truth function of the corresponding FBST structure M, relate to (...) the truth values of its elementary constituents, Hj, and to the truth functions of their corresponding FBST structures Mj, respectively. (shrink)

Since the ground-breaking contributions of M. Dummett (Dummett 1978), it is widely recognized that anti-realist principles have a critical impact on the choice of logic. Dummett argued that classical logic does not satisfy the requirements of such principles but that intuitionistic logic does. Some philosophers have adopted a more radical stance and argued for a more important departure from classical logic on the basis of similar intuitions. In particular, J. Dubucs and M. Marion (?) and (Dubucs 2002) have recently argued (...) that a proper understanding of anti-realism should lead us to the so-called substructural logics (see (Restall 2000)) and especially linear logic. The aim of this paper is to scrutinize this proposal. We will raise two kinds of issues for the radical anti-realist. First, we will stress the fact that it is hard to live without structural rules. Second, we will argue that, from an anti- realist perspective, there is currently no satisfactory justi cation to the shift to substructural logics. (shrink)

We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.

This paper introduces the special issue on Logic and Religion of the journal Logica Universalis (Springer). The issue contains the following articles: Logic and Religion, by Jean-Yves Beziau and Ricardo Silvestre; Thinking Negation in Early Hinduism and Classical Indian Philosophy, by Purushottama Bilimoria; Karma Theory, Determinism, Fatalism and Freedom of Will, by Ricardo Sousa Silvestre; From Logic in Islam to Islamic Logic, by Musa Akrami; Leibniz’s Ontological Proof of the Existence of God and the Problem of Impossible Objects, by Wolfgang (...) Lenzen; A Logical Analysis of the Anselm’s Unum Argumentum (from Proslogion), by Jean-Pierre Desclés; Monotonic and Non-monotonic Embeddings of Anselm’s Proof, by Jacob Archambault; Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy, by C. Benzmüller, L. Weber and B. Woltzenlogel Paleo. (shrink)

This paper challenges defenders/advocates of the semantic-completeness route towards gluts to explain, in simple and plausible terms, why the ‘trivializer paradox’, framed in terms of closure relatives on theories, fails to undermine their argument.

We investigate effective categoricity for polymodal algebras. We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra.

It is shown that the consequence relations defined from theRescher-Manor Mechanism are all inconsistency-adaptive logics combined with a specific interpretation schema for the premises. Each of the adaptive logics isobtained by applying a suitable adaptive strategy to the paraconsistent logicCLuN.This result provides all those consequence relations with a proof theory and with a static semantics.

A subset A $\subseteq$ M of a totally ordered structure M is said to be convex, if for any a, b $\in A: [a . A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some $\emptyset$ -definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly o-minimal theory (...) T, any expansion M + of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63), that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories. (shrink)