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## Über dieses Buch

This volume presents recent advances in philosophical logic with chapters focusing on non-classical logics, including paraconsistent logics, substructural logics, modal logics of agency and other modal logics. The authors cover themes such as the knowability paradox, tableaux and sequent calculi, natural deduction, definite descriptions, identity, truth, dialetheism and possible worlds semantics.

The developments presented here focus on challenging problems in the specification of fundamental philosophical notions, as well as presenting new techniques and tools, thereby contributing to the development of the field. Each chapter contains a bibliography, to assist the reader in making connections in the specific areas covered. Thus this work provides both a starting point for further investigations into philosophical logic and an update on advances, techniques and applications in a dynamic field.

The chapters originate from papers presented during the Trends in Logic XI conference at the Ruhr University Bochum, June 2012.

## Inhaltsverzeichnis

### Semantic Defectiveness: A Dissolution of Semantic Pathology

The Liar Paradox and its kin appear to show that there is something wrong with—something pathological about—certain firmly held principles or beliefs. It is our view that these appearances are deceiving. In this paper, we provide both a diagnosis and a treatment of apparent semantic pathology, explaining these appearances without semantic or logical compromise.

### Emptiness and Discharge in Sequent Calculus and Natural Deduction

We investigate the correlation between empty antecedent and succedent of the intutionistic (respectively dual-intuitionistic) sequent calculus and discharge of assumptions and the constants absurdity (resp. discharge of conclusions and triviality) in natural deduction. In order to be able to express and manipulate the sequent calculus phenomena, we add two units to sequent calculus. Depending on the sequent calculus considered, the units can serve as discharge markers or as absurdity and triviality.
Michael Arndt, Luca Tranchini

### The Knowability Paradox in the Light of a Logic for Pragmatics

The Knowability Paradox is a logical argument showing that if all truths are knowable in principle, then all truths are, in fact, known. Many strategies have been suggested in order to avoid the paradoxical conclusion. A family of solutions—called logical revision—has been proposed to solve the paradox, revising the logic underneath, with an intuitionistic revision included. In this paper, we focus on so-called revisionary solutions to the paradox—solutions that put the blame on the underlying logic. Specifically, we analyse a possibile translation of the paradox into a modified intuitionistic fragment of a logic for pragmatics (KILP) inspired by Dalla Pozza and Garola [4]. Our aim is to understand if KILP is a candidate for the logical revision of the paradox and to compare it with the standard intuitionistic solution to the paradox.
Massimiliano Carrara, Daniele Chiffi

### A Dialetheic Interpretation of Classical Logic

According to classical logic, the acceptance of a dialetheia, a proposition that is both true and false, entails trivialism the output that every sentence is true. One way to accept dialetheias but avoid trivialism is to reject the general validity of classical logic, which is the view held by dialetheists, supporters of the existence of dialetheias. In The Logic of Paradox (LP), Priest adopts the material conditional, identifying $$A \rightarrow B$$ with $$\lnot A \vee B$$. He argues that this is not a genuine conditional because it invalidates modus ponens (MP), an essential rule governing the use of the conditional. In subsequent works he introduces a genuine conditional and tries to avoid Curry’s paradox by invoking a highly problematic modal semantics. The aim of our paper is to argue that a dialetheist can stick to the material conditional and recover the whole of classical logic without falling into trivialism. Our strategy sets forth a way of understanding the notion of assumption suitable for the dialetheic perspective. We show the inadequacy of formal classical logic to capture the intended exclusivity of negation. Finally, we argue that the material conditional is adequate to provide a dialetheic solution to semantic paradoxes.
Massimiliano Carrara, Enrico Martino

### Strongly Semantic Information as Information About the Truth

Some authors, most notably Luciano Floridi, have recently argued for a notion of “strongly” semantic information, according to which information “encapsulates” truth (the so-called “veridicality thesis”). We propose a simple framework to compare different formal explications of this concept and assess their relative merits. It turns out that the most adequate proposal is that based on the notion of “partial truth”, which measures the amount of “information about the truth” conveyed by a given statement. We conclude with some critical remarks concerning the veridicality thesis in connection with the role played by truth and information as relevant cognitive goals of inquiry.
Gustavo Cevolani

### Priest’s Motorbike and Tolerant Identity

In his chapter ‘Non-transitive identity’ [8], Graham Priest develops a notion of non-transitive identity based on a second-order version of $$LP$$. Though we are sympathetic to Priest’s general approach to identity we think that the account can be refined in different ways. Here we present two such ways and discuss their appropriateness for a metaphysical reading of indefiniteness in connection to Evans’ argument.
Pablo Cobreros, Paul Egré, David Ripley, Robert van Rooij

### How to Unify Russellian and Strawsonian Definite Descriptions

In this paper I will deal with ambiguities in natural language exemplifying the difference between topic and focus articulation within a sentence. I will show that whereas articulating the topic of a sentence activates a presupposition, articulating the focus frequently yields merely an entailment. Based on analysis of topic-focus articulation, I propose a solution to the almost hundred-year old dispute over Strawsonian versus Russellian definite descriptions. The point of departure is that sentences of the form ‘The $$F$$ is a $$G$$’ are ambiguous. Their ambiguity stems from different topic-focus articulations of such sentences. Russell and Strawson took themselves to be at loggerheads, whereas, in fact, they spoke at cross purposes. My novel contribution advances the research into definite descriptions by pointing out how progress has been hampered by a false dilemma and how to move beyond that dilemma. The point is this. If ‘the $$F$$’ is the topic phrase then this description occurs with de re supposition and Strawson’s analysis appears to be what is wanted. On this reading the sentence presupposes the existence of the descriptum of ‘the $$F$$’. The other option is ‘$$G$$’ occurring as topic and ‘the $$F$$’ as focus. This reading corresponds to Donnellan’s attributive use of ‘the $$F$$’ and the description occurs with de dicto supposition. On this reading the Russellian analysis gets the truth-conditions of the sentence right. The existence of a unique $$F$$ is merely entailed. This paper demonstrates how to unify these disparate insights into one coherent theory of definite descriptions.
Marie Duží

### Tableau Metatheorem for Modal Logics

The aim of the paper is to demonstrate and prove a tableau metatheorem for modal logics. While being effective tableau methods are usually presented in a rather intuitive way and our ambition was to expose the method as rigorously as possible. To this end all notions displayed in the sequel are couched in a set theoretical framework, for example: branches are sequences of sets and tableaus are sets of these sequences. Other notions are also defined in a similar, formal way: maximal, open and closed branches, open and closed tableaus. One of the distinctive features of the paper is introduction of what seems to be the novelty in the literature: the notion of tableau consequence relation. Thanks to the precision of tableau metatheory we can prove the following theorem: completeness and soundness of tableau methods are immediate consequences of some conditions put upon a class of models M and a set of tableau rules MRT. These conditions will be described and explained in the sequel. The approach presented in the paper is very general and may be applied to other systems of logic as long as tableau rules are defined in the style proposed by the author. In this paper tableau tools are treated as an entirely syntactical method of checking correctness of arguments [1, 2].
Tomasz Jarmużek

### On the Essential Flatness of Possible Worlds

The objective of this paper is to introduce and motivate a new semantic framework for modalities. The first part of the paper will be devoted to defending the claim that conventional possible worlds are ill-suited for the semantics of certain types of modal statements. We will see that the source of this expressive limitation comes from what will be dubbed “worldly flatness”, the fact that possible worlds don’t determine modal facts. It will be argued that some modalities are best understood as quantifiers over modal facts and that possible worlds semantics cannot achieve this. In the second part of the paper, I will present a new semantic framework that allows for such an understanding of modalities.
Neil Kennedy

### Collective Alternatives

In the logic of agency individual alternatives of agents have been taken as basic, so far, while the alternatives of groups of agents have been derived from the alternatives of the group members. In many cases, however, groups have additional possibilities. In the paper I propose a generalized theory of collective alternatives that takes them as fundamental.
Franz von Kutschera

### da Costa Meets Belnap and Nelson

There are various approaches to develop a system of paraconsistent logic, and those we focus on in this paper are approaches of da Costa, Belnap, and Nelson. Our main focus is da Costa, and we deal with a system that reflects the idea of da Costa. We understand that the main idea of da Costa is to make explicit, within the system, the area in which you can infer classically. The aim of the paper is threefold. First, we introduce and present some results on a classicality operator which generalizes the consistency operator of Logics of Formal Inconsistency. Second, we show that we can introduce the classicality operator to the systems of Belnap. Third, we demonstrate that we can generalize the classicality operator above to the system of Nelson. The paper presents both the proof theory and semantics for the systems to be introduced, and also establishes some completeness theorems.
Hitoshi Omori, Katsuhiko Sano

### Explicating the Notion of Truth Within Transparent Intensional Logic

The approach of  Transparent Intensional Logic to truth differs significantly from rivalling approaches. The notion of truth is explicated by a three-level system of notions whereas the upper-level notions depend on the lower-level ones. Truth of possible world propositions lies in the bottom. Truth of hyperintensional entities—called constructions—which determine propositions is dependent on it. Truth of expressions depends on truth of their meanings; the meanings are explicated as constructions. The approach thus adopts a particular hyperintensional theory of meanings; truth of extralinguistic items is taken as primary. Truth of expressions is also dependent, either explicitly or implicitly, on language (its notion is thus also explicated within the approach). On each level, strong and weak variants of the notions are distinguished because the approach employs the Principle of Bivalence which adopts partiality. Since the formation of functions and constructions is non-circular, the system is framed within a ramified type theory having foundations in simple theory of types. The explication is immune to all forms of the Liar paradox. The definitions of notions of truth provided here are derivation rules of Pavel Tichý’s system of deduction.
Jiří Raclavský

### Leibnizian Intensional Semantics for Syllogistic Reasoning

Venn diagrams are standardly used to give a semantics for Syllogistic reasoning. This interpretation is extensional. Leibniz, however, preferred an intensional interpretation, according to which a singular and universal sentence is true iff the (meaning of) the predicate is contained in the (meaning of) the subject. Although Leibniz’s preferred interpretation played a major role in his philosophy (in Leibniz [16] he justifies his metaphysical ‘Principle of Sufficient Reason’ in terms of it) he was not able to extend his succesfull intensional interpretation (making use of characteristic numbers) without negative terms to one where also negative terms are allowed. The goal of this paper is to show how syllogistic reasoning with complex terms can be given a natural set theoretic ‘intensional’ semantics, where the meaning of a term is not defined in terms of individuals. We will make use of the ideas behind van Fraassen’s [6, 7] hyperintensional semantics to account for this.
Robert van Rooij

### Inter-Model Connectives and Substructural Logics

The paper provides an alternative interpretation of ‘pair points’, discussed in [3]. Pair points are seen as points viewed from two different ‘perspectives’ and the latter are explicated in terms of two independent valuations. The interpretation is developed into a semantics using pairs of Kripke models (‘pair models’). It is demonstrated that, if certain conditions are fulfilled, pair models are validity-preserving copies of positive substructural models. This yields a general soundness and completeness result for a variety of (positive) substructural logics with respect to multimodal Kripke frames with binary accessibility relations. In addition, an epistemic interpretation of pair models is provided.
Igor Sedlár
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