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UNCOVERING THE NOTIONS OF INFORMATION IN LOGIC Johan van Benthem, Amsterdam & Stanford, http://staff.science.uva.nl/~johan/ Workshop Philosophy of Information, Lorentz Center, Leiden, 9 February 2010 Abstract Logic contains many notions of 'information', when you read well between the lines. This reflects the vexing more general issue how to define this crucial notion across the sciences. We will discuss three major senses of information in logic, based on semantic 'range', semantic 'correlation' and syntactic 'elucidation', we prove some results on comparing the three, and we discuss how far they can be integrated into one coherent picture of what logic is about. We ask four main questions: * what notions of information are there, * how are they related, * where is the natural boundary with agency, and * how does this relate to philosophy and informatics? References (a) J. van Benthem & M-C Martinez, 2008 ‘The Stories of Logic and Information’, Hand-book of Philosophy of Information, P. Adriaans & J. van Benthem, eds., 2008, Elsevier, Amsterdam. (b) J. van Benthem, 2007, ‘Interview’, in Philosophy of Computing and Information: 5 Questions, L. Floridi, ed., Automatic Press, Copenhagen, 9–19. (c) 2008, 'Merging Observation and Access in Dynamic Epistemic Logic', Studies in Logic 1, 1–16. (d) 2009, 'The Information in Intuitionistic Logic', Synthese 167:2, 251–270. (d) 2010, Logical Dynamics of Information Flow, Cambridge Univ. Press, Cambridge. 1 Introduction: information in the sciences Diversity: channel theory (Shannon), Kolmogov (computation), logic & language.

2 Three sorts of logical information: range, correlation, inference Logical consequence: conclusion ‘adds no information’. But then, why reason at all? The inference AvB, ¬A ⇒ B: successive semantic updates, and information flow. Inferential information: solving a Sudoku puzzle in computational stages. The Restaurant. Semantic information update and inferential information combined. Many entangled informational processes: observation, inference, communication. General ‘Tandem Approach’: representation and process belong together. Ask not what information is without asking what it does (Lewis).

The Mountain Top: Information as correlation between situations. Further issues: ‘aboutness’ of information. But also, information is ‘for’ an agent.

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3 Semantic range, information update, and dynamic epistemic logic Information as range (Carnap): update: the fewer possibilities, the more information. Epistemic logic (Hintikka): Implicit knowledge, models M = (W, {Ri}i∈I, V) with M, s |= Kiϕ iff M, t |= ϕ for all t with Rist. Logic: K(ϕ→ψ) → (Kϕ → Kψ), Kϕ → KKϕ, etc. Information update: events !P of new hard information change current model (M, s): from M, s to M|P, s s s

P ¬P Logical dynamics: action modality M, s |= [!P] φ iff if M, s |=P, then M|P, s |= φ. Complete system of public announcement logic PAL: [!P]q ↔ P → q for atomic facts q [!P]¬φ ↔ P → ¬[!P]φ [!P]φ∧ψ ↔ [!P]φ ∧ [!P]ψ [!P]Kφ ↔ P → K(P → [!P]φ) 4 Information as correlation and situation theory Inspiration from Shannon, and Dretske’s “Knowledge and the Flow of Information”. Constraints: information flows in state spaces with ‘gaps’ in values for components:

If x: p, then y: q. Situations with various relations between them: ‘channel theory’, distributed systems. Simplest version: modal logic of constraint models, operators []xφ: φ is true in all those global states whose values differ only in the situations in the list x. Equivalent: current dependency logics. 5 Merging range and constraint views Easy, levels differ: accessibility between worlds plus internal component structure for worlds. Epistemic plus correlation modalities. Open problem: add in the actions! (‘Situation Theory in the Light of Dynamic Logic’, van Benthem, Israel & Perry.) 6 Inferential information and elucidation Proof theoretic paradigms. Difficulty: crucial role of syntax, different proof formats: what is the right abstraction level? Problem of omniscience K(ϕ→ψ) → (Kϕ → Kψ). ‘Scandal of deduction’ – and also, of computation. 7 Merging semantic and inferential views: dynamic logic of access Access dynamics New operator Iϕ of explicit knowledge. Access models: pairs (w, X) of world w and set of factual formulas X. M, w, X |= Iϕ iff ϕ∈X & M, w, X |= ϕ. Iϕ → ϕ and Iϕ → KIϕ valid. Actions: inference steps, conscious versus unconscious seeing. Over access models, realizing that ϕ (#ϕ) means adding a formula ϕ to all awareness sets in our current range, provided that Kϕ holds in the actual world. Need act of realization: Exp(ϕ→ψ) → (Expϕ → [act] Expψ).

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Complete dynamic epistemic logic of informational access: [#ϕ]q ↔ Kϕ → q for atomic q [#ϕ]¬ψ ↔ Kϕ → ¬[#ϕ]ψ [#ϕ](ψ ∧ χ) ↔ [#ϕ]ψ ∧ [#ϕ]χ [#ϕ]Kψ ↔ Kϕ → K[#ϕ]φ [#ϕ]Iψ ↔ Kϕ → ( Iψ ∨ ψ = ϕ) Further illustration: acts of introspection, need extension beyond factual statemens. 8 Still further notions: the case for ‘procedural information’ Protocols & procedural information Restrict tree of possible assertions/observations. Complete epistemic-temporal logic of protocols: <!P>T no longer equivalent with P. New axiom: [!P]Kiφ ↔ <!P>T → Ki(<!P>T → [!P]φ). No more ‘static reduction’. Decoupling factual and procedural information in temporal logics Branching trees: s

M, h, s |= Gϕ iff for all t ≥ s, M, h, t |= ϕ (‘always in the future’) M, h, s |= []ϕ iff for all k =s h, M, k, s |= ϕ (‘necessarily now’)

Similar phenomena in Beth/Kripke models for intuitionistic logic: p p 9 Grand Unification? More abstract approach across all three notions of logical information? Candidate: categorial/relevant logic. Models of ‘information pieces’, implication based on ‘fusing’ information pieces. Informational and dynamic interpretations of the Lambek Calculus. Alternative view: too high abstraction level; duality of stances: no unification needed. 10 Agency Who is the processor engaged in all these activities? “Zhi: Wen, Shuo, Qin” 知 问 说 亲 (communication, inference, observation)

Large set of attitudes: knowledge, belief, ‘entertain’, doubt. Social stance: information is produced and conveyed socially.

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Meaningful values: hard to separate information from evaluation and purpose. Logical dynamics of intelligent interaction.

11 Philosophy and Informatics Repercussions. What image for the understanding of information?