Counterparts, Essences and Quantified Modal Logic

It is commonplace to formalize propositions involving essential properties of objects in a language containing modal operators and quantifiers. Assuming David Lewis’s counterpart theory as a semantic framework for quantified modal logic, I will show that certain statements discussed in the metaphysics of modality de re, such as the sufficiency condition for essential properties, cannot be faithfully formalized. A natural modification of Lewis’s translation scheme seems to be an obvious solution but is not acceptable for various reasons. Consequently, the only safe way to express some intuitions regarding essential properties is to use directly the language of counterpart theory without modal operators.

Relating Logic and Relating Semantics. History, Philosophical Applications and Some of Technical Problems

Here, we discuss historical, philosophical and technical problems associated with relating logic and relating semantics. To do so, we proceed in three steps. First, Section 1 is devoted to providing an introduction to both relating logic and relating semantics. Second, we address the history of relating semantics and some of the main research directions and their philosophical applications. Third, we discuss some technical problems related to relating semantics, particularly whether the direct incorporation of the relation into the language of relating logic is needed. The starting point for our considerations presented here is the 1st Workshop On Relating Logic and the selected papers for this issue.

Relating Semantics for Epistemic Logic

The aim of this paper is to explore the advantages deriving from the application of relating semantics in epistemic logic. As a first step, I will discuss two versions of relating semantics and how they can be differently exploited for studying modal and epistemic operators. Next, I consider several standard frameworks which are suitable for modelling knowledge and related notions, in both their implicit and their explicit form and present a simple strategy by virtue of which they can be associated with intuitive systems of relating logic. As a final step, I will focus on the logic of knowledge based on justification logic and show how relating semantics helps us to provide an elegant solution to some problems related to the standard interpretation of the explicit epistemic operators.

Alternative Semantics for Normative Reasoning with an Application to Regret and Responsibility

We provide a fine-grained analysis of notions of regret and responsibility (such as agent-regret and individual responsibility) in terms of a language of multimodal logic. This language undergoes a detailed semantic analysis via two sorts of models: (i) relating models, which are equipped with a relation of propositional pertinence, and (ii) synonymy models, which are equipped with a relation of propositional synonymy. We specify a class of strictly relating models and show that each synonymy model can be transformed into an equivalent strictly relating model. Moreover, we define an axiomatic system that captures the notion of validity in the class of all strictly relating models.

Dynamic Probabilistic Entailment. Improving on Adams' Dynamic Entailment Relation

The inferences of contraposition (A ⇒ C ∴ ¬C ⇒ ¬A), the hypothetical syllogism (A ⇒ B, B ⇒ C ∴ A ⇒ C), and others are widely seen as unacceptable for counterfactual conditionals. Adams convincingly argued, however, that these inferences are unacceptable for indicative conditionals as well. He argued that an indicative conditional of form A ⇒ C has assertability conditions instead of truth conditions, and that their assertability ‘goes with’ the conditional probability p(C|A). To account for inferences, Adams developed the notion of probabilistic entailment as an extension of classical entailment. This combined approach (correctly) predicts that contraposition and the hypothetical syllogism are invalid inferences. Perhaps less well-known, however, is that the approach also predicts that the unconditional counterparts of these inferences, e.g., modus tollens (A ⇒ C, ¬C ∴ ¬A), and iterated modus ponens (A ⇒ B, B ⇒ C, A ∴ C) are predicted to be valid. We will argue both by example and by calling to the results from a behavioral experiment (N = 159) that these latter predictions are incorrect if the unconditional premises in these inferences are seen as new information. Then we will discuss Adams’ (1998) dynamic probabilistic entailment relation, and argue that it is problematic. Finally, it will be shown how his dynamic entailment relation can be improved such that the incongruence predicted by Adams’ original system concerning conditionals and their unconditional counterparts are overcome. Finally, it will be argued that the idea behind this new notion of entailment is of more general relevance.

History of Relating Logic. The Origin and Research Directions

In this paper, we present the history of and the research directions in relating logic. For this purpose we will describe Epstein's Programme, which postulates accounting for the content of sentences in logical research. We will focus on analysing the content relationship and Epstein's logics that are based on it, which are special cases of relating logic. Moreover, the set-assignment semantics will be discussed. Next, the Torunian Programme of Relating Semantics will be presented; this programme explores the various non-logical relationships in logical research, including those which are content-related. We will present a general description of relating logic and semantics as well as the most prominent issues regarding the Torunian Programme, including some of its special cases and the results achieved to date.

S5-Style Non-Standard Modalities in a Hypersequent Framework

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.

Analysis of Penrose’s Second Argument Formalised in DTK System

This article aims to examine Koellner’s reconstruction of Penrose’s second argument – a reconstruction that uses the DTK system to deal with Gödel’s disjunction issues. Koellner states that Penrose’s argument is unsound, because it contains two illegitimate steps. He contends that the formulas to which the T-intro and K-intro rules apply are both indeterminate. However, we intend to show that we can correctly interpret the formulas on the set of arithmetic formulas, and that, as a consequence, the two steps become legitimate. Nevertheless, the argument remains partially inconclusive. More precisely, the argument does not reach a result that shows there is no formalism capable of deriving all the true arithmetic propositions known to man. Instead, it shows that, if such formalism exists, there is at least one true non-arithmetic proposition known to the human mind that we cannot derive from the formalism in question. Finally, we reflect on the idealised character of the DTK system. These reflections highlight the limits of human knowledge, and, at the same time, its irreducibility to computation.

Logics for Knowability

In this paper, we propose three knowability logics LK, LK−, and LK=. In the single-agent case, LK is equally expressive as arbitrary public announcement logic APAL and public announcement logic PAL, whereas in the multi-agent case, LK is more expressive than PAL. In contrast, both LK− and LK= are equally expressive as classical propositional logic PL. We present the axiomatizations of the three knowability logics and show their soundness and completeness. We show that all three knowability logics possess the properties of Church-Rosser and McKinsey. Although LK is undecidable when at least three agents are involved, LK− and LK= are both decidable.

Topology of Modal Propositions Depicted by Peirce’s Gamma Graphs: Line, Square, Cube, and Four-Dimensional Polyhedron

This paper presents the topological arrangements in four geometrical figures of modal propositions and their derivative relations by means of Peirce's gamma graphs and their rules of transformation. The idea of arraying the gamma graphs in a geometric and symmetrical order comes from Peirce himself who in a manuscript drew two cubes in which he presented the derivative relations of some (but no all) gamma graphs. Therefore, Peirce's insights of a topological order of gamma graphs are extended here backwards from the cube to the line and the square; and then forwards from the cube to the four-dimensional polyhedron.