THE LOGIC OF INTELLIGENT INTERACTION Johan van Benthem, Amsterdam - PDF document

1 THE LOGIC OF INTELLIGENT INTERACTION Johan van Benthem, Amsterdam & Stanford, http://staff.science.uva.nl/~johan/ ICLA Third Indian Logic Conference, Chennai January 2009 Abstract Logic arose historically out of an interest in social argumentative practice, from conversation to legal procedure, though the ‘success formula’ involved taking a scientific abstract viewpoint. Much of modern logic has arisen from a ‘contraction of concerns’ over the centuries to reasoning by a single agent, or even just mathematical proof. We will sketch how the agenda of logic has been broadening again, partly under the influence of computer science, broadly construed – and we will discuss a number of main issues under the heading of Rational Agency that seem to extend the classical grand foundational issues. Starting Question What is the ‘rational’ in rational animals? 1 To be rational is to reason intelligently in splendid isolation Euclidean model, the pure mathematician as the ideal agent. Be the best theorem prover. 2 To be rational is to handle all information flow intelligently Newton “Principia”, “Optics”: look at experiments. Science: observation & deduction. The Restaurant . In a restaurant, your Father has ordered Fish, your Mother ordered Vegetarian, and you ordered Meat. Out of the kitchen comes some new person carrying three plates. What will happen? 2 questions, 1 inference intertwined. Semantic updates ( 6 ⇒ 2 ⇒ 1 ) + inferential step. Social dynamics!

2 3 Multi-agent aspects are crucial to rational behaviour Asking a question: Conveying information about other people(’s information): “Is this the Tsinghua West Gate?” Many sources of information: observation, inference, communication. Lead from ancient logical traditions (China, 500 BC): 知 问 说 亲 “Zhi: Wen, Shuo, Qin” F. Liu & J. Zhang, 2008, A Note on Mohist Logic , “There are three ways to get knowledge: viz. learning from others, reasoning from what one knows already, and consulting one’s own experience”. 4 Logical dynamics of information flow Current dynamic logics, explicit informational processes (infer, observe, communicate). Public hard information Semantic update !P by eliminating all ¬P –worlds. Theorem PAL axiomatized completely by (a) epistemic logic plus (b) recursion axioms : [!P]q P → q for atomic facts q ↔ [!P]¬ φ ↔ P → ¬[!P] φ [!P] φ∧ψ [!P] φ ∧ [!P] ψ ↔ [!P]Ki φ ↔ P → Ki(P → [!P] φ ) P ∧ [!P] ψ [!P] φ Addenda: [!P][!Q] φ ↔ [!(P ∧ [!P]Q)] φ , [!P]CG ψ φ ↔ C Observational powers Product update ( MxE , (s, e)) has Domain {(s, e) | s a world in M , e an event in E , ( M , s)|= PRE e }, Accessibility: (s, e) ~ i (t, f) iff both s ~ i t and e ~ i f, Valuation for atoms p at (s, e) is that at s in M . (can be generalized to world change) Theorem (BMS) LEA is effectively axiomatizable and decidable. Key recursion axiom: [ E , e]K i φ ↔ PRE e → ∧ { K i [ E , f] φ )) | f ~ i e in E }

3 Trickier problem, information flow in inference : actions of ‘access’ for known facts: Theorem Complete logic: (a) logic of epistemic access with K , I , (b) PAL axioms plus [! ϕ ]I ψ ↔ ( ϕ → I ψ ) (c) recursion laws for the dynamic modality of realization, e.g., [# ϕ ]K ψ ↔ (K ϕ → K[# ϕ ] φ ), [# ϕ ]I ψ ↔ (K ϕ → ( I ψ ∨ ψ = ϕ )) Many further informational powers of agents : memory, introspection ! 5 Other epistemic attitudes in agent repertoire: belief change Richer repertoire: Descartes. Belief as truth in the most plausible accessible worlds: M , s |= B i φ iff M , t |= φ for all worlds t minimal in the ordering λ xy. ≤ i, s xy . P [!P] φ Belief change under hard information [!P] B i φ ↔ P → B i Conditional belief helps pre-encode beliefs we would have if we learnt certain things: ψ φ iff M , t |= φ for all worlds t minimal for λ xy. ≤ i, s xy in {u | M, u |= ψ }. M , s |= B i ψ φ , (b) Theorem The logic of belief change under hard information is (a) base logic of B i ψ φ ↔ P → B i P ∧ [!P] ψ [!P] φ PAL reduction axioms, (c) recursion axiom: [!P] B i Allows for interesting scenarios : Misleading true information. New notion: Safe belief . ‘Soft information’ merely changes the plausibility ordering of the existing worlds. Radical upgrade ⇑ P changes the current model M to a model M ⇑ P : P -worlds now better than all ¬P -worlds; within zones, old order remains . Belief change under soft information ( <> is the epistemic existential modality:) Theorem The dynamic logic of lexicographic upgrade is axiomatized completely by logic of conditional belief + compositional analysis of effects of model change: [ ⇑ P] B ψ φ ↔ (<>( P ∧ [ ⇑ P] ψ ) ∧ B P ∧ [ ⇑ P] ψ [ ⇑ P] φ ) ∨ ( ¬ <>( P ∧ [ ⇑ P] ψ ) ∧ B [ ⇑ P] ψ [ ⇑ P] φ 6 To be rational is to act intelligently From who and what, how to why ? Purposeful behaviour means evaluating outcomes. Goals, preferences, decisions, actions. The paradigmatic scenario: a b x ≤ y Can logic say what should happen? ‘Practical Syllogism’: (i) the agent can do both a and b , (ii) the agent prefers the result of a over the result of b , so, (iii) the agent will/should do b .

4 Modal preference language <pref i > φ : i prefers some node with φ to the current one. Status of the conclusion: intention, or belief update , creating plausibility for x over y . Status of the above inference: ‘bridge principles’ relating action, preference, and belief. Rationality , just one particular agent type? Alternative comparison ‘social dynamics’: Newton mechanics axiom F = m•a , postulate forces to account for observed behaviour. 7 To be rational is to interact intelligently Argumentation, communication. G ames a microcosm for logics of rational agency: Just a minimal social scenario: explaining/predicting multi-agent behaviour? A 1, 0 E 0, 100 99, 99 Single actions & strategies (modal logic, dynamic logic, µ -calculus, conditional logic), preference (preference logic), belief & knowledge (doxastic and epistemic logic) – and beyond: intentions, probabilities, iterated games, etc. Logical strategy: ‘deconstruct’. 8 Long-term behaviour Strategies, plans. Protocols & procedural information Restrict tree of all assertions/ observations. Concrete example: epistemic-temporal logic of protocols: <!P>T no longer equivalent with P . New axiom: [!P]Ki φ ↔ <!P>T → Ki(<!P>T → [!P] φ ) . s h 9 From “I” to “we”: social groups Structured groups as additional agents? ‘Merge’: group order out of individual preference/plausibility orders? Reducibility to individual level, or groups sui generis?

5 10 Discussion: some general technical issues Logical analysis ties together major issues in a number of areas in new ways. Need marriage of computational logic (‘hard’) and philosophical logic (‘soft’). Entanglement of separate notions: alternatives, can we parametrize alternatives? Combination incurs computational cost, total architecture not well-understood. What good does logic approach add to already existing disciplines in this area? Concluding Question What is Rational Agency? 11 What is rational agency, and what is our claim about it? One cannot consult some standard text for this purpose, because there are none. Classical foundations Hilbert’s Program, Gödel’s Theorems, Turing Machines. For our Rational Agent, what are its defining skills, beyond pencil and paper sums? Repertoire of powers, idealized or bounded? Repertoire of agents so far complete? Idealized, or is the heart of rationality optimal performance under tight constraints? Which core tasks? Do rational agents have a ‘core business’? Revision and learning Key phenomenon is not correctness, but correction , learning. Interaction A rational agent is someone who interacts rationally with other agents! Further features: Diversity , Switching (Categorical imperative), Organization . Are there refutable claims and goals? An analogue to Hilbert’s Program? Integrating trends Lacking a Hilbert or Turing, we might at least have a Church.