## M&M2012- Abstracts for Modality and Modalities, 2012

## Mai Ajspur "Tableaux-based decision methods for linear time temporal-epistemic logics

with interaction between time and knowledge"

In temporal-epistemic logics, modal operators of time and of knowledge can interact in various ways. For instance, it is possible for an agent to never forget, i.e., have perfect recall, which means that if the agent can distinguish two states at a given point in time, she will always be able to distinguish them later on. Similarly, an agent might never learn, i.e. never acquire new knowledge. Further, agents may share knowledge of global time (in synchronous systems), etc.

Depending on the assumptions of interaction between knowledge and time, the resulting temporal-epistemic logics display very different computational behaviours, ranging from decidable in PSPACE, to decidable in non-elementary time, to highly undecidable.

In this talk I will briefly present a tableau method for deciding satisfiability in the multi-agent temporal-epistemic logic with full coalitional common knowledge and linear time, but with no interaction between the time and knowledge operators. I will then describe some difficulties that occur when interactions are assumed and will explain how the presented tableau method can be modified to work for single agent temporal-epistemic logics with interactions between time and knowledge.

Depending on the assumptions of interaction between knowledge and time, the resulting temporal-epistemic logics display very different computational behaviours, ranging from decidable in PSPACE, to decidable in non-elementary time, to highly undecidable.

In this talk I will briefly present a tableau method for deciding satisfiability in the multi-agent temporal-epistemic logic with full coalitional common knowledge and linear time, but with no interaction between the time and knowledge operators. I will then describe some difficulties that occur when interactions are assumed and will explain how the presented tableau method can be modified to work for single agent temporal-epistemic logics with interactions between time and knowledge.

## Martin Mose Bentzen "Action Type Deontic Logic"

Deontic logics aim at representing correct deontic reasoning, the inferences people ought to make about norms, i.e. what they may, should or must do. Most deontic logics apply deontic operators such as must and may to propositions, as in “it must be the case that Andur runs”. However, an old tradition in modern deontic logic going back to von Wright’s first paper on the topic, applies deontic operators to action types, as in the simpler “Andur must run”. On the proposal presented here, it works in the following way. We have a non-empty set of action tokens which instantiate various action types. A subset of the action tokens are singled out as acceptable. Informally, an action type is permissible, if there is an action token of the type, which is acceptable. An action type is required if all acceptable action tokens are of that type. My aim is to show that a good portion of the problems of deontic logic can be solved by these simple semantic devices. In order to operationalize this claim I present a number of bench mark cases for a deontic logic representing deontic inferences speakers usually make in natural language contexts and review a number of deontic logics from the literature which do not fare well with these. I then present a new formal language and semantics, Action Type Deontic Logic, which validates inferences that in most cases coincide with the bench mark cases. However, in many cases it does not coincide with inferences made in normal or even monotonic modal logic. I conclude that it is unlikely that deontic reasoning can be represented satisfactorily by a normal modal logic. My final remarks concern open problems in Action Type Deontic Logic.

## Patrick Blackburn "Indexical Hybrid Tense Logic (Part Two)"

In this talk, we explore the logic of Now, Today, Yesterday and Tomorrow in the setting of hybrid tense logic. More precisely, we combine the semantic approach pioneered by Hans Kamp (an early form of two-dimensional semantics) in his classic ''Formal Properties of 'Now'´´ with contemporary hybrid logic.

We begin with the logic of Now. As we shall show, Kamp's logic of Now is a sublogic of the basic hybrid logic of Now, and we give a simple semantic proof of Kamp's eliminibility result. We then give a complete tableaux system for the basic hybrid logic of Now (with respect to both logical validity and the stronger notion of contextual validity) and discuss its expressive power.

We then add special propositional constants to handle the logic of Yesterday, Today, and Tomorrow. Our hybrid system avoids the well-known scope problems that operator-based approaches to these indexicals face, and correctly treats sentences like ''Niels will die yesterday'' as unsatisfiable. We give a complete tableaux system for the richer language (for both logical and contextual validity) and make further connections with the literature.

This is Part two of a joint presentation with Klaus Frovin Jørgensen, and the material in this abstract will be spread across the two talks.

We begin with the logic of Now. As we shall show, Kamp's logic of Now is a sublogic of the basic hybrid logic of Now, and we give a simple semantic proof of Kamp's eliminibility result. We then give a complete tableaux system for the basic hybrid logic of Now (with respect to both logical validity and the stronger notion of contextual validity) and discuss its expressive power.

We then add special propositional constants to handle the logic of Yesterday, Today, and Tomorrow. Our hybrid system avoids the well-known scope problems that operator-based approaches to these indexicals face, and correctly treats sentences like ''Niels will die yesterday'' as unsatisfiable. We give a complete tableaux system for the richer language (for both logical and contextual validity) and make further connections with the literature.

This is Part two of a joint presentation with Klaus Frovin Jørgensen, and the material in this abstract will be spread across the two talks.

- Thomas Bolander "Epistemic and doxastic planning for single- and multi-agent systems"

This is joint work with Mikkel Birkegaard Andersen and Martin Holm Jensen.

I will present a framework for automated planning based on dynamic epistemic logic (DEL). DEL-based planning generalises classical planning in several significant ways, most notably by giving the planning agent the ability to reason about the knowledge, beliefs and actions of other agents as part of its planning process. This ability is essential to achieve efficient and intelligent communication and collaboration in multi-agent settings. I will first present the purely epistemic planning framework, in which the states of planning problems are multi-pointed epistemic models and the actions are multi-pointed event models (with postconditions). It is shown that in this framework, plan existence in the single-agent case is decidable, but undecidable in general in the multi-agent case. Second, I will sketch our work in progress on generalising the framework from epistemic models to plausibility models. The use of plausibility models allows the planning agent to reason about the most plausible outcomes of its possible actions, and thus to prune the search space significantly by only considering these outcomes, and rely on replanning if the expected outcomes are not the ones being realised at run-time.

I will present a framework for automated planning based on dynamic epistemic logic (DEL). DEL-based planning generalises classical planning in several significant ways, most notably by giving the planning agent the ability to reason about the knowledge, beliefs and actions of other agents as part of its planning process. This ability is essential to achieve efficient and intelligent communication and collaboration in multi-agent settings. I will first present the purely epistemic planning framework, in which the states of planning problems are multi-pointed epistemic models and the actions are multi-pointed event models (with postconditions). It is shown that in this framework, plan existence in the single-agent case is decidable, but undecidable in general in the multi-agent case. Second, I will sketch our work in progress on generalising the framework from epistemic models to plausibility models. The use of plausibility models allows the planning agent to reason about the most plausible outcomes of its possible actions, and thus to prune the search space significantly by only considering these outcomes, and rely on replanning if the expected outcomes are not the ones being realised at run-time.

- Torben Braüner "Hybrid Logic, Truth, and Proof-Systems"

Hybrid logic is an extension of ordinary modal logic which allows explicit reference to individual points in a model. This additional expressive power is useful for many applications, for example when reasoning about time one often wants to formulate a series of statements about what happens at specific times, and standard modal formalisms do not allow this.

What is less obvious is that the basic mechanism of hybrid logic often actually improves the behaviour of the underlying modal formalism. For example, proof-systems can be formulated in a uniform way, which simply is not possible in ordinary modal logic.

In my talk I first give a brief introduction to hybrid logic and its proof-theory. I then discuss how different proof-systems for hybrid logic relate to different views of truth.

What is less obvious is that the basic mechanism of hybrid logic often actually improves the behaviour of the underlying modal formalism. For example, proof-systems can be formulated in a uniform way, which simply is not possible in ordinary modal logic.

In my talk I first give a brief introduction to hybrid logic and its proof-theory. I then discuss how different proof-systems for hybrid logic relate to different views of truth.

- Max Cresswell "Carnap’s Modal Logic -Topics in the Pre-history of Possible-worlds semantics"

view PDF file

- Sebastian Enqvist "Lindström theorems for modal logic over restricted model classes"

My talk will address some problems in the abstract model theory of modal languages, building on work by Maarten de Rijke and Johan van Benthem. They have obtained two Lindström-style characterization results for the basic modal languages, i.e. the normal modal logics without any additional axioms, valid for the class of all relational structures of a given similarity type. However, in most applications of modal logic a restriction is made to some subclass of models satisfying certain constraints. The results of van Benthem and de Rijke do not automatically transfer to cases like this. I will explain this in more detail and list some problems, preliminary results and directions for future research that I fi?nd particularly promising. The central theme that I wish to explore is under what conditions van Benthem's and de Rijke's results, possibly with some suitable modification, can be transferred to a given restricted class of models.

- Emmanuel Genot & Justine Jacot "Propositional Reasoning with (Modal) Models"

Johnson-Laird and Byrne's paper "Propositional reasoning with models" (1992) presents a set of tasks in which empirical subjects must reason from a set of premises to a relevant conclusion, combining logical and pragmatic inferences. While the instructions describe first-order models, they can be translated so that a propositional representation suffices, making the reasoning task appear almost trivial. However, empirical subjects massively fail to report the expected answers, a difficulty which J-L&B explain by the complexity of manipulating mental models. Although J-L&B's explanation accounts for the relative rates of success in variations of a task, it fails to account for the low base rates. We analyze one of the original tasks (the double-disjunction task) within a learning-theoretic model that makes explicit the underlying semantic reasoning. We argue that the information provided in the instructions suggests a representation equivalent to first-order epistemic models, and that the simplification to propositional reasoning is hindered by the formulation of the task. We conclude suggesting some possible reformulations that could improve the base rate of originally expected answer, by eliciting both the adequate modal representation, and the selection of relevant information, through pragmatic principles.

- Mikkel Gerken "Critique of Modal Rationalism"

I will pursue two distinct but interrelated aims concerning the epistemology of modality and, in particular, the necessary a posteriori. The first aim is to reevaluate Kripke’s landmark critique of Kant’s pure rationalism in modal epistemology. Kripke’s arguments for the necessary a posteriori are widely recognized as an important critique of modal rationalism in general. In contrast, it is insufficiently appreciated that Kripke’s critique of Kant is, in fact, both compatible with, and highly congenial to, an ambitious modal rationalism. Thus, my diagnostic aim consists of characterizing such an ambitious modal rationalism by distinguishing it from Kant’s pure modal rationalism on the one hand and from varieties of modal empiricism on the other. The second aim is to make use of this diagnosis as a basis for questioning whether Kripke’s critique of pure modal rationalism goes deep enough. So, I initiate such a second round of critique by outlining some rationales for thinking that the ambitious modal rationalism, of the sort suggested by Kripke’s remarks, is too ambitious. More specifically, I provide three distinct but interrelated rationales for assuming that certain conditional modal judgments are a posteriori. An example is the judgment that if Socrates is human, it is necessary that Socrates is human (if Socrates exists). The pursuit of this twofold aim is organized as follows: In Sect. 2, I outline Kant’s pure modal rationalism and Kripke’s critique of it. In Sect. 3, I revisit Kripke’s argument for the necessary a posteriori and consider its force against pure modal rationalism. In Sect. 4, I argue that Kripke’s critique is compatible with a weaker but nevertheless ambitious modal rationalism according to which an important class of modal judgments is a priori. In Sect. 5, I set forth a number of rationales contra such an ambitious modal rationalism. In Sect. 6, I conclude by noting the scope and limits of my critique.

- Valentin Goranko "Modalities for multi-agent systems"

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- Jens Ulrik Hansen "Pluralistic ignorance - Recovering logic and rationality in an error phenomenon of social psychology"

Pluralistic ignorance is a phenomenon from social psychology that usually refers to a situation where a group of individuals all have a certain attitude towards a proposition or norm, but all believe that the other members of the group have an opposing attitude towards the proposition or the norm. The phenomenon comes in many versions and I will start by explaining some of them. Pluralistic ignorance is often viewed as an error made by individuals (for instance an error in social comparison) and implicitly assumed to illogical or irrational. However, using the framework of epistemic logic I will try to show that pluralistic ignorance can arise even for logical and rational agents. If time permits I will also mention other insights from the modeling of pluralistic ignorance in epistemic logic.

- Klaus Frovin Jørgensen "Indexical Hybrid Tense Logic (Part One)"

- Klaus Frovin Jørgensen "Indexical Hybrid Tense Logic (Part One)"

In this talk, we explore the logic of Now, Today, Yesterday and Tomorrow in the setting of hybrid tense logic. More precisely, we combine the semantic approach pioneered by Hans Kamp (an early form of two-dimensional semantics) in his classic ''Formal Properties of 'Now'´´ with contemporary hybrid logic.

We begin with the logic of Now. As we shall show, Kamp's logic of Now is a sublogic of the basic hybrid logic of Now, and we give a simple semantic proof of Kamp's eliminibility result. We then give a complete tableaux system for the basic hybrid logic of Now (with respect to both logical validity and the stronger notion of contextual validity) and discuss its expressive power.

We then add special propositional constants to handle the logic of Yesterday, Today, and Tomorrow. Our hybrid system avoids the well-known scope problems that operator-based approaches to these indexicals face, and correctly treats sentences like ''Niels will die yesterday'' as unsatisfiable. We give a complete tableaux system for the richer language (for both logical and contextual validity) and make further connections with the literature.

This is Part one of a joint presentation with Patrick Blackburn, and the material in this abstract will be spread across the two talks.

We begin with the logic of Now. As we shall show, Kamp's logic of Now is a sublogic of the basic hybrid logic of Now, and we give a simple semantic proof of Kamp's eliminibility result. We then give a complete tableaux system for the basic hybrid logic of Now (with respect to both logical validity and the stronger notion of contextual validity) and discuss its expressive power.

We then add special propositional constants to handle the logic of Yesterday, Today, and Tomorrow. Our hybrid system avoids the well-known scope problems that operator-based approaches to these indexicals face, and correctly treats sentences like ''Niels will die yesterday'' as unsatisfiable. We give a complete tableaux system for the richer language (for both logical and contextual validity) and make further connections with the literature.

This is Part one of a joint presentation with Patrick Blackburn, and the material in this abstract will be spread across the two talks.

- Peter Øhrstrøm "A.N. Prior’s Early Contributions to the Logic of Time and Modality"

A.N. Prior spent 1956 in Oxford, where he had been invited to give the ‘John Locke lectures’ of that year. These lectures formed the basis of Prior's book ‘Time and Modality’ (published in 1957), the first work in which Prior’s logic of time and modality was presented systematically. In this talk, I shall present some of Prior’s ideas from the book as well as some of his findings during 1955 when he was preparing the Oxford lectures. In addition, I shall discuss the idea of branching time as it was suggested by Saul Kripke in a letter written to Prior as a reaction to some of the claims presented in ‘Time and Modality’. The talk will partly be based on hitherto unpublished material from the Prior collection at the Bodleian Library in Oxford.

## Ahti-Veikko Pietarinen "Peirce's Systems of Modal Gamma Graphs"

The 1903 gamma part of existential graphs was Peirce's boutique of modal (propositional, multi-modal, first-order) and higher-order logics. I define transformation rules for the broken-cut gamma that correspond to various systems of modal logic and suggest that Peirce's preferred system was deontic. I also define semantic tableaux for modal gammas.

- Carlo Proietti "Non-classical Modal Logics and Tableau Testing"

Non-classical modal logics (intuitionistic, minimal or paraconsistent) are a large – and partially unexplored - field of investigation. For many of them is an open issue to know whether - and to what extent - the fusion of their propositional core with modal derivation rules forces a collapse into a stronger system. For example, Negri et al. (2012) show that intuitionistic propositional rules in conjunction with the standard modal rules for K and reflexivity rule for ◊ collapse into classical modal logic - due to the fact that the non modal accessibility relation ≤ becomes an equivalence relation. This issue is highly relevant in the light of some specific applications such as:

a) The

b) The problem of

Tableau methods are an useful tool for the purpose of testing the conditions under which a modal framework collapses, since they provide a straightforward algorithm to check the validity of a given axiom or the derivability of a specific rule. The present work is structured as follows:

a) The

*Fitch-Church paradox of knowability.*Here, the possibility of weakening classical modal logic into a system of intuitionistic modal logic may block the unwanted derivation of*omniscience*p→Kp from the principle of knowability p→◊Kp (see Negri et al. 2012).b) The problem of

*deontic explosion*. In deontic logic one wants to avoid trivialization of a set of formulas containing conflicting obligations (e.g. Op and O¬p) and for that purpose modal principles for O are combined with a weaker paraconsistent basis (see McGinnis 2006).Tableau methods are an useful tool for the purpose of testing the conditions under which a modal framework collapses, since they provide a straightforward algorithm to check the validity of a given axiom or the derivability of a specific rule. The present work is structured as follows:

- I generalize the method of tableaux to intuitionistic and (Johansson's) minimal modal logic.

- I reproduce via tableaux the result of Negri et al. (2012) – originally derived in a labeled sequent calculus – then extend it to the minimal case and finally investigate the general collapse conditions for minimal and intuitionistic modal logics

- I outline an extension of this method to some paraconsistent logics such as RM3, BN4 and LP.

- Rasmus K. Rendsvig "Connections between Lexical and Truth-Conditional Semantic Competence"

One viewpoint of what it takes for a speaker to be semantically competent is that the speaker should know appropriate T-sentences for the language, i.e. have knowledge that "S is true iff P" where S is a sentence of the object-language with which the agent is supposed to be competent and the remained is a meta-language definition of S's truth-conditions. For a speaker to be fully semantically competent with a language, he should therefore know the 'Davidsonian Program' for the given language. This theory may be criticized for not being fine-grained enough, as it has sentences as it's smallest elements with which agents can be competent, which result in problems of learnability.

In this talk, I will propose a lexical theory of semantic competence, focusing on semantic competence with respect to single words. This will be modeled using a two-sorted second-order epistemic logic, incorporating both the aforementioned object-language and meta-language, as to allow the expression of sentences like "Agent a knows that (S is true iff P)" within the logical language. In this setting, it will be investigated how instances of truth-conditional competence may come about as products of lexical competence and how issues of truth-conditional incompetence may be given more fine-grained reasons using lexical competence.

In this talk, I will propose a lexical theory of semantic competence, focusing on semantic competence with respect to single words. This will be modeled using a two-sorted second-order epistemic logic, incorporating both the aforementioned object-language and meta-language, as to allow the expression of sentences like "Agent a knows that (S is true iff P)" within the logical language. In this setting, it will be investigated how instances of truth-conditional competence may come about as products of lexical competence and how issues of truth-conditional incompetence may be given more fine-grained reasons using lexical competence.

- Adrianne Rini "Aristotle on the Necessity of the Consequent"

This paper investigates what Aristotle says in the Prior Analytics that helps to explain his views on

(i) the necessity of a modal proposition, and

(ii) the necessity of the connection between premises and conclusion of a (valid) syllogism.

Aristotle describes both in terms of ‘necessity’. But to what extent does he understand (ii) as a genuinely modal notion?

(i) the necessity of a modal proposition, and

(ii) the necessity of the connection between premises and conclusion of a (valid) syllogism.

Aristotle describes both in terms of ‘necessity’. But to what extent does he understand (ii) as a genuinely modal notion?

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## Krister Segerberg

abstract to be posted