Logic, Rationality, and Interaction

Evidence is the underpinning of beliefs and knowledge. Modeling evidence for an agent requires a more fine-grained semantics than possible worlds models. We do this in the form of “neighbourhood models”, originally proposed for weak modal logics. We show how these models support natural actions of “evidence management”, ranging from update with external new information to internal rearrangement. This perspective leads to richer languages for neighborhood semantics, including modalities for new kinds of conditional evidence and conditional belief. Using these, we indicate how one can obtain relative completeness theorems for the dynamic logic of evidence-changing actions.

We study a game model to highlight the mutual recursiveness of individual rationality and societal rationality. These are games that change intrinsically based on the actions / strategies played by the players. There is an implicit player - the society, who makes actions available to players and incurs certain costs in doing so. If and when it feels that an action

a

is being played by a small number of players and/or it becomes too expensive for it to maintain the action

a

, it removes

a

from the set of available actions. This results in a change in the game and the players strategise afresh taking this change into account.

We study the question: which actions of the players should the society restrict and how should it restrict them so that the social cost is minimised in the eventuality? We address two variations of the question: when the players are maximisers, can society choose an order of their moves so that social cost is minimised, and which actions may be restricted when players play according to given strategy specifications.

The possibility for agents to agree to disagree is considered in an extended epistemic-topological framework. In such an enriched context, Aumann’s impossibility theorem is shown to no longer hold. More precisely, agents with a common prior belief satisfying limit knowledge instead of common knowledge of their posterior beliefs may actually entertain distinct posterior beliefs. Hence, agents can actually agree to disagree. In particular, agreeing to disagree with limit knowledge is illustrated within a representative epistemic-topological situation.

This paper introduces a semantic model for vague quantifiers (VQs) combining Fuzzy Theory (FT) and Supervaluation Theory (ST), which are the two main theories on vagueness, a common source of uncertainty in natural language. After comparing FT and ST, I will develop the desired model and a numerical method for evaluating truth values of vague quantified statements, called the Modified Glöckner’s Method, that combines the merits and overcomes the demerits of the two theories. I will also show how the model can be applied to evaluate truth values of complex quantified statements with iterated VQs.

The paper presents a logical study of abstract argumentation theory. It introduces a second-order modal logic, within which all main known semantics for abstract argumentation can be formalized, and studies the model checking game of this logic. The application of the game to the formalized semantics yields adequate game-theoretic proof procedures for all known extension-based semantics, in both their skeptical and credulous versions.

Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. The classic example is Public Announcement Logic (

PAL

), an extension of epistemic logic based on the idea of information acquisition as elimination of possibilities. In this paper, we address the open question of whether the set of

schematic validities

of

PAL

, the set of formulas all of whose substitution instances are valid, is decidable. We obtain positive answers for multi-agent

PAL

, as well as its extension with relativized common knowledge,

PAL-RC

. The conceptual significance of substitution failure is also discussed.

We introduce and study the notion of a

Public Environment

: a system in which a publicly known program is executed in an environment that is partially observable to agents in the system. Although agents do not directly have access to all variables in the system, they may come to know the values of unobserved variables because they know how the program is manipulating these variables. We develop a logic for reasoning about Public Environments, and an axiomatization of the logic.

This paper investigates strategies for responding rationally to opponents who make mistakes. We identify two distinct interpretations of mistakes in the game theory literature: trembling hand and risk averse mistakes. We introduce the concept of an EFG Scenario, a game plus strategy profile, in order to probe the properties of these different types of mistake. An analysis of equivalence preserving transformations over EFG Scenarios reveals that risk averse mistakes are a form of rational play, while trembling hand mistakes are equivalent to moves by nature.

Imperatives occur ubiquitously in our social communications. In real life we often get conflicting orders issued by different speakers whose authorities are ranked. We propose a new update semantics to interpret the meaning of imperatives with priorities and illustrate what changes they bring about in the addressee’s cognitive state. The general properties of the semantics, as well as its core philosophical ideas are discussed extensively in this paper.

This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculus. The definition of the measure takes two steps: First, we measure the width of a given proof. Then the measure of inference assigns, to a given sequent, the minimum value of the widths of its possible proofs. It counts the indispensable cases for possible proofs of a sequent. This measure expresses the degree of difficulty in proving a given sequent. Although our problem is highly proof-theoretic, we are motivated by some general and specific problems in game theory/economics. In this paper, we will define a certain lower bound function, with which we may often obtain the exact value of the measure for a given sequent. We apply our theory a few game theoretical problems and calculate the exact values of the measure.

In various argumentation systems, under most of situations, only the justification status of some arguments of the systems should be evaluated, while that of other arguments is not necessary to be figured out. Based on this observation, we introduce an efficient method to evaluate the status of a part of arguments in an argumentation framework. This method is based on the notion of unattacked sets of an argumentation framework and the directionality criterion of argumentation semantics. Given an argumentation framework and a subset of arguments within it, we firstly identify the minimal set of arguments that are relevant to the arguments in this subset (called the

minimal unattacked set

). Then, under an argumentation semantics satisfying the directionality criterion, the set of extensions of the sub-framework induced by the minimal unattacked set (called a partial semantics of the original argumentation framework) can be evaluated independently. Then, we analyze two basic properties of the partial semantics of argumentation: monotonicity and combinability.

The paper introduces a logic which allows to represent different kinds of mental states of an agent such as knowledge, graded belief, and graded goal, and the notion of epistemic action (as the action of learning that a certain fact

φ

is true.) The logic is applied to the formalization of expectation-based emotions such as hope, fear, disappointment and relief, and of their intensity.

We describe the planning problem within the framework of dynamic epistemic logic (DEL), considering the tree of sequences of events as the underlying structure. In general, the DEL planning problem is computationally difficult to solve. On the other hand, a great deal of fruitful technical advances have led to deep insights into the way DEL works, and these can be exploited in special cases. We present a few properties that will lead to considerable simplifications of the DEL planning problem and apply them in a toy example.

We study some mathematical aspects of public announcement logic (PAL) and its several variants. By a model-theoretic approach, we explore van Benthem’s result that uses recursion axioms to characterize the submodel operation, and show some model-theoretic results on the respecting phenomena. The second approach to understand public announcements is algebraic. Based on a joint work with A. Palmigiano and M. Sadrzadeh, we treat public announcements as devices for getting a new quotient algebra updated by an element in the original one. Then we show the algebraic soundness and completeness result for PAL and generalize this approach to PAL extension of epistemic intuitionistic modal logic. Finally, we give some observations on the PAL extensions of first-order logic as well as epistemic predicate modal logic.

We explore logics of belief over weighted structures under the supposition that everything believed by an agent has a weight in the range of agent’s belief. We first define static graded belief logics which are complete with respect to the class of all weighted frames. Furthermore, we discuss their public announcement and dynamic epistemic extensions. We may also define notions of plausible belief by comparing weights of formulas at the current state in a weighted model. This approach is not a new one but we provide new logics and their dynamic extensions which can capture some intuitive notions of belief and their dynamics.

The paper analyzes interdependencies between strategies of players in a Nash equilibrium using independence relation between two sets of players. A sound and complete axiomatization of this relation is given. It has been shown previously that the same axiomatic system describes independence in probability theory, information flow, and concurrency theory.

Traditionally, meaning is identified with informative content. The central aim of inquisitive semantics [1,2,4,5, a.o.] is to develop a notion of semantic meaning that embodies both informative and inquisitive content. To achieve this, the proposition expressed by a sentence

ϕ

, [

ϕ

], is not taken to be a set of possible worlds, but rather a set of

possibilities

, where each possibility in turn is a set of possible worlds. In uttering a sentence

ϕ

, a speaker provides the information that the actual world is contained in at least one possibility in [

ϕ

], and at the same time she requests enough information from other participants to establish for at least one possibility

α

∈ [

ϕ

] that the actual world is contained in

α

.

Epistemic game theory has shown the importance of informational contexts in understanding strategic interaction. We propose a general framework to analyze how such contexts may arise. The idea is to view informational contexts as the fixed-points of iterated, “rational responses” to incoming information about the agents’ possible choices. We show general conditions for the stabilization of such sequences of rational responses, in terms of structural properties of both the decision rule and the information update policy.

We provide a multi-agent spatially grounded epistemic logical framework to reason about the knowledge of perception (agent

a

sees agent

b

) whose potential applications are video games and robotics. Contrary to the classical epistemic modal logic, we prove that in some configurations the logic with the common knowledge operator is as expressive as the logic without the common knowledge operator. We give some complexity results about the model-checking.

Vagueness is a ubiquitous feature that we know from many expressions in natural languages. It can invite a serious problem: the Sorites Paradox. The aim of this paper is to propose a new version of complete logic for vague predicates - JND-based vague predicate logic (

JVL

) which can avoid the Sorites Paradox and give answers to all of the Semantic Question, the Epistemological Question and the Psychological Question given by Graff. To accomplish this aim, we provide

JVL

with a probabilistic model by means of measurement theory.

We introduce a variant of the standard epistemic logic S5 for reasoning about knowledge under hypotheses or background assumptions. The modal operator of necessity expressing what is known is parameterised with a hypothesis. The operator can be described as relative necessity, a notion already used by Chellas to describe conditionality. In fact, the parameterised box resembles a conditional operator and it turns out that our logic is a variant of Chellas’ Conditional Logic. We present an axiomatisation of the logic and show that it bears the same expressivity and computational complexity as S5. Then we consider the extension of our logic with operators for distributed knowledge and show how it can be used to represent knowledge of agents whose epistemic capacity corresponds to any system between S4 and S5.

Dynamic modal logics are modal logics that have statements of the form [

π

]

ψ

. The truth value of such statements, when evaluated in a pointed model 〈

$\cal F$

,

V

,

ω

〉, is determined by the truth value that

ψ

takes in the pointed models 〈

$\cal F\prime$

,

V

′,

ω

′〉 that stand in a relation

$\xrightarrow{\pi}$

to 〈

$\cal F$

,

V

,

ω

〉.

This paper introduces new dynamic operators that minimally revise finite classical Kripke models to make almost any satisfiable modal formula

φ

true. To this end, we define the minimal revision relations

$\xrightarrow{\dagger \phi}$

and

$\xrightarrow{\ddagger \phi}$

, where

$\xrightarrow{\dagger \phi}$

revises only the valuation function and

$\xrightarrow{\ddagger \phi}$

also changes the frame.

We show that our language enables us to count the number of accessible worlds and to characterize irreflexive frames. We also demonstrate that any consistent formula can be made true, conditional only on modest seriality and finiteness presumptions.

In the literature, different axiomatizations of Public Announcement Logic (PAL) were proposed. Most of these axiomatizations share a ‘core set’ of the so-called reduction axioms. In particular, there is a composition axiom which stipulates how two consecutive announcements are composed into one. In this paper, by designing non-standard Kripke semantics for the language of PAL, we show that without the composition axiom the core set does not completely axiomatize PAL. In fact, most of the intuitive ‘axioms’ and rules we took for granted could not be derived from the core set. The non-standard semantics we proposed is of its own interest in modelling realistic agents. We show that with the help of different composition axioms we may axiomatize PAL w.r.t. such non-standard semantics.

While dynamic epistemic logics with common knowledge have been extensively studied, dynamic epistemic logics with

distributed

knowledge have so far received far less attention. In this paper we study extensions of public announcement logic (

${\cal PAL}$

) with distributed knowledge, in particular their expressivity and axiomatisations.

$\cal PAL$

extended only with distributed knowledge is not more expressive than standard epistemic logic with distributed knowledge. Our focus is therefore on

$\cal PACD$

, the result of adding both common and distributed knowledge to

$\cal PAL$

, which is more expressive than each of its component logics. Our main result is a completeness result for

$\cal PACD$

. The axiomatisation is not surprising: it is the combination of well-known axioms. The completeness proof, however, is not trivial, and requires novel combinations and extensions of techniques for dealing with

S

5 knowledge, distributed knowledge, common knowledge and public announcements at the same time.

We give an alternative logic for knowability to arbitrary public announcement logic. In the new semantics, ‘knowable’ means ‘known after an information update’ rather than the more specific interpretation of ‘known after an announcement’. An update is modelled by the refining of accessibility relations. We compare our logic to arbitrary announcement logic and subset space logic and thus establish a link between the latter two.

This paper presents a new consequentialist deontic logic in which the relation of preference over sets of possible worlds and the relation of conditional dominance are both transitive. This logic validate the principle that absolute ought can be derived from conditional ought whenever the conditional statement is the agent’s absolute ought. Ought about conditionals is not implied by conditional ought in this logic.

This paper develops LMP — a categorial type logic that is meant to formulate vacuous components in natural language. The vacuous components are those expressions which have no semantic referents, but syntactically function as grammatical links between expressions on its left and right hand side. In order to legitimatize the derivations of sentences containing these components in categorial type logic, the present paper proposes system LMP, which is based on a ternary composite category [ A {B} C]. LMP’s axiomatic presentation and Gentzen presentation are thereafter presented, and LMP’s soundness, completeness and decidability are established.

While people play games, they observe, learn, recollect and update their strategies

during

the game as well as adopting deontological strategies and goals before the game. In this paper, we focus on what we call

move

updates

where some moves become unavailable during the game, and present a formal framework for extending

strategy

logic

which was introduced by Ramanujam and Simon [2].

Real-world problems often call for efficient methods to discovery actionable knowledge on which business can directly act [3]. Some works for discovering actionable knowledge [3,5] view actions as behaviors which render a state of an instance into a preferred state, where a state is represented by feature values of the instance and whether a state is preferred is determined by a classifier. Actions usually have many-to-many relations with features of an instance. That is, an action may affect multiple features of an instance, and vise versa, a feature may be influenced by multiple actions. This type of many-to-many relationships between actions and features is prevalent in real-world applications. However, most existing works [3,5] only deal with one-to-one relationship and ignore manyto- many relationship between actions and features. In these works, an action is treated as a behavior with a fixed execution cost. Restricting to a one-to-one relationship between actions and features may not yield an

action

set

(i.e. a set of actions) with the minimal total execution cost. Moreover, one-to-one relationship is simply a special case of many-to-many relationship, and hence the latter will be applicable to more real-world problems. Therefore we aim to extract action sets from a classifier for which the total execution cost is minimal based on many-to-many relationship between actions and features.

The notion of non-manipulability (or: strategy-proofness) used in the famous Gibbard-Satterthwaite theorem is too strong to make useful distinctions between voting rules.We explore alternative definitions and suggest how these can be used to classify voting rules.

This paper reports the development of a simple tool, NEGEXT, written in the platform-independent Java language. NEGEXT has been constructed to aid real people doing actual negotiations, when the ways to negotiate are simply too many to be computed by a normal human mind. This toolkit will also help in planning one’s strategic moves in negotiation situations when the opponents’ possible moves can be approximated. Even though some visualization tools for extensive form game trees already exist, we believe we are the first to make a tree-based negotiation toolkit that incorporates the possibility of representing learning from game to game, by sequential and parallel composition (cf. [1]). Moreover, the toolkit has a model-checking component which computes whether and how an individual or a specific coalition can achieve a given objective.

Following the general programme of [2], we investigate the effect of social relationships on the dynamics of preference change within a community. Specifically, we are interested in the phenomenon of ‘peer pressure’, according to which a person’s preferences are changed in response to the preferences of a ‘peer group’. This involves both aggregation of preferences, to determine the group’s preferences and preference change. We propose a simple model of peer pressure which is still sufficiently non-trivial to display some interesting dynamics, and show how the stable configurations can be expressed logically.

Epistemic disagreement and agreement between two agents have been considered by game theorists and logicians[1][2]. But they are discussed only from the view of knowledge.

In this paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements. We give the dual characterization of the corresponding submodel-injection map, as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). As an application, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, and prove soundness and completeness of IPAL w.r.t. both algebraic and relational models.

The logic of scientific justification is a central problem in the philosophy of science. Bayesianism is usually taken as the leading theory in this area. After a brief review of Bayesian account of scientific justification and learning theorists’ objection against Bayesianism, this paper proposes an argument defending Bayesianism. It is shown that Bayesian conditionalization has the necessary equipment to capture the idea of elimination induction, which functions as an indispensible component in a satisfactory account of scientific justification.

When making a choice between competing alternatives, we are primarily guided by our preferences. But the process is typically aided by asking questions. The questions serve to expand the set of possibilities we consider. Nonetheless a reasonable condition we might impose on this process is that the order in which questions are asked is ultimately irrelevant. Someone for whom this is not the case can be manipulated into making unfortunate choices by a careful choice of questions. We develop a logic for reasoning about such processes, use this to provide an independent justification for the rationality of having transitive preferences, and explain what goes wrong in situations where preferences are not transitive, such as Condorcet’s voting paradox.

David Lewis developed one of the most prominent versions of epistemic contextualism (EC) in

Elusive Knowledge

[2], which is also known as a version of relevant alternatives theory (RAT). In this work, I propose a simple formalization of Lewis account, which results in our basic Lewisian epistemic logic (LEL). Trivial as it may seem, my formalization focuses directly on Lewis’s theory, and thus provides a firm basis to discuss his theses on anti-skepticism, infallible knowledge, rules of relevance, and especially “knowledge without belief”. Moreover, my formalization not only avoids problems of two former existed formalizations [1],[3], but also partly answers Holliday’s question how a relevant alternatives theorist should handle higher-order knowledge [1].