Evidence Dynamics in Neighborhood Models Johan van Benthem University of Amsterdam & Stanford University István Németi 70 Conference, Rényi Institute, Budapest, 8 Sept 2012

Current Sources

Evidence for Beliefs Rational beliefs are grounded in reasons Double aspect: ‘ support ’, and ‘ hooks for refutation ’ Many views of reasons or evidence in epistemology We will pursue only an austere view of evidence, that will still lead us to lush logical pastures.

In Between Syntax and Semantics The barest view: one ’ s current evidence is the range of all epistemically accessible worlds. Ignores how we got here. The richest view: all the details of what we learnt so far, including current syntactic theory. In between: family of subsets of the domain encoding evidence from various sources, at an abstract level. Not always consistent: sources may be unreliable.

Ubiquitous Idea Belief revision Modeling theories in philosophy of science Premise semantics for conditionals Fine-grained views of knowledge Knowledge obtained through sensors

Other Important Approaches

Models for Evidence p

Neighborhood Semantics

Language: Evidence, Belief, Knowledge Language: Evidence, Belief, Knowledge

Truth Conditions

First Stab: Belief is Cautious

Static Base Logic Problem Axiomatize this logic: simple + simple ≠ simple?

One Static Logic of Evidence Models Joint work with David Fernandez: Proof uses quasi-models, representation, NBD bisimulation.

Designing a Better Language Two different sources for choosing logical formalisms: The needs of existing discourse, reasoning scenarios What is suggested by the semantic models themselves Issue in philosophy and cognitive science: What is our repertoire of epistemic attitudes? For instance, belief seems not one notion, but many.

First: Conditional Evidence

Adding Conditional Belief Classic addition needed in many places (though curiously overlooked in doxastic logic, belief revision theory):

Base Logic Extended

Logic of Conditional Belief Is more or less the ‘minimal conditional logic’:

Dynamics of Evidence Change Evidence comes in a stream, with all the events that we know from logical dynamics in general. Main purpose: explore more fine-grained information dynamics living at the level of neighborhood models. Usual tool: recursion axioms as key ‘ dynamic equations ’. But also new angle on DEL: explore optimal design of static base language by dynamic-epistemic techniques.

Evidence Change: Announcement Hard information by public announcement

Recursion Axioms for Announcement

Deconstructing Public Announcement

Evidence Change by Addition

Recursion Axioms as Design Tool Issue: how much of such ‘ culture ’ in a course?

Graphical Pictures Can Help Issue: how much of such ‘ culture ’ in a course?

Conditioning Comes in Varieties Issue: how much of such ‘ culture ’ in a course?

New Logical Distinctions Issue: how much of such ‘ culture ’ in a course?

Enriched Base Language One dynamic task leads to another:

Discarding as an Action Discarding or retraction has seemed a challenge to DEL:

Extending the Language Once More Issue: how much of such ‘ culture ’ in a course?

Recursion Axioms for Removal Issue: how much of such ‘ culture ’ in a course?

The Complete Dynamic Logic of Removal Issue: how much of such ‘ culture ’ in a course?

Internal Operations on Evidence What about new operations without earlier counterparts?

Evidence Combination Issue: how much of such ‘ culture ’ in a course?

Logic of Evidence Combination Issue: how much of such ‘ culture ’ in a course?

The Complete Static Logic?

‘ Second Opinion ’ : Plausibility Models The other main approach to belief: add a plausibility order to epistemic models. How does this relate?

Parallels: Enriched Languages Richer languages: Levesque, Stalnaker, Battigalli & Sinischalci

Semantic Rock Bottom Issue: how much of such ‘ culture ’ in a course?

Not Quite Follow Received Wisdom Issue: how much of such ‘ culture ’ in a course?

Crossing Over: Representation Issue: how much of such ‘ culture ’ in a course?

An Illustration Issue: how much of such ‘ culture ’ in a course?

Translation Between Languages Issue: how much of such ‘ culture ’ in a course?

From Evidence to Plausibility Issue: how much of such ‘ culture ’ in a course?

Illustrations

The Total Picture Issue: how much of such ‘ culture ’ in a course?

Partial Translation Issue: how much of such ‘ culture ’ in a course?

Extends to Dynamic Parallels Through the earlier translations, Evidence addition becomes the relational operation called suggest(  ) changing the current order ≤ to its subrelation where no  -world is preferred to a ¬  -world. But evidence removal has no such relational counterpart. The full extent of harmony between the neighborhood/ evidence and ordering levels remains to be understood. Technical questions remain (cf. Parikh ’ s trick for dynamic game logic): mimick all of the NBD via extra relations?

Interim Conclusion Plausibility models a special case of neighborhood models. They, too, support many static notions of belief, and the usual DEL plausibility update mechanism can analyze their dynamics. But neighborhood models also allow for internal operations, and seem the more fine-grained perspective eventually. Technical questions remain: mimick NBD via extra relations?

Technical Challenges Complete axiomatizations new neighborhood logics. New kinds of bisimulation as expressive power varies. Clarify connections to modal logics of plausibility order. For the algebraists How to deal with dynamic operators that change models? From classical to intuitionistic base, from distributive to just a monotonic base algebra? (Ma, Palmigiano & Sadrzadeh 2011, Palmigiano & Kurz 2012.)

Richer Views of Evidence World dependence and iterated evidence Many agents, with new operations (‘merge’) Resolving conflicts in the current evidence Trust, priority, and authority Manipulating reasons Richer syntactic views than neighborhoods Weiging evidence and numerical approaches.

Conclusion Evidence is a key notion worthy of logical study. From trick, neighborhood models become avant garde topic when given a language of their own. Poses new mathematical issues for modal logics (and perhaps also for algebraic logic). Especially when combined with logical dynamics.