Action, Failure and Free Will Choice in Epistemic stit Logic

Transcription

1 Action, Failure and Free Will Choice in Epistemic stit Logic Jan Broersen Department of Information and Computing Sciences Utrecht University, The Netherlands Symposium "What is really possible?" Utrecht, April 8th, 2011

2 My research interests Logics of programs, action, choice and agency (dynamic logic, stit logic, situation calculus, event calculus, BIAT, etc.) Logics of action and change (frame problem, qualification problem, non-monotonic logic, etc. ) Logics and theories of norms (deontic logic, deontic temporal logic, norm change, etc.) Logics for reasoning in game situations (multi-agent stit, CL, ATL, PDL, etc. ) Theories of action (knowingly doing, non-successful action, intentional action, attempt, responsibility, deliberate action, etc.) Theories of collective action

3 Actions in computer science In computer science a clear ontology of action is much needed: Computer scientists see actions as the basic instructions of a program Computer scientists often confuse actions, action types, effects, events, etc. Problem: if we specify that the action has a different effect, is it still the same action? Problem: if an action fails, did it have a different effect, or did another action take place? Problem: if an action is performed by another agent, is it still the same action?

4 Outline Introduction / XSTIT Action versus Choice Successful action Choice and failing action Free will action Conclusion

5 Facts and views on STIT stit is an acronym for "Seeing To It That" stit logics originate in philosophy (Belnap, Perloff, Horty) stit theory is a theory of agency and action stit logics define actions as relations between agents and effects: [a stit]ϕ means "agent a ensures the actual world is among those satisfying ϕ" stit: acting = performing a winning strategy against non-determinacy Temporal version of stit (as the present one) take temporal histories as worlds: acting = history selection So, stit may be said to add the witnesses of branching (i.e., choices by agents) to the branching time logics CTL / CTL*

6 Particulars for XSTIT We build up the semantics and the syntax ensuring we stay within the Sahlqvist class, thereby ensuring completeness (*). Effects in next states (without which we would not have (*)) Formal comparison with stit formalisms from philosophy is not straightforward: choices in XSTIT models partition next states, choices in Belnap/Horty stit models partition the current state. The XSTIT language is neither a sub- or superset of the standard temporal stit language of, e.g. Horty. We stay within normal multi-modal / two-dimensional modal logic.

7 XSTIT frame Hb2 Hb4 Hb6 Hb8 Hb1 Hb3 Hb5 Hb7 s2-choices Ag2 s5 s7 s9 s11 s3-choices Ag2 s4 s6 s8 s2-choices Ag1 s10 s3-choices Ag1 s1-choice Ag2 s2 s1-choices Ag1 s3 s1 Figure: Visualization of a partial two agent XSTIT frame

8 h-relative effectivity functions (for individual agency) E : S H Ags 2 S is an h-relative effectivity function yielding for an agent ag the set of next static states allowed by the action taken by the agent relative to a history a. if s h then E(s, h, ag) = b. succ(s, h) E(s, h, ag) c. if s E(s, h, ag) then h : s = succ(s, h ) d. if s = succ(s, h) and s h then s E(s, h, ag) e. E(s, h, ag 1 ) E(s, h, ag 2 )

9 Individual agency: XSTIT.I The XSTIT.I syntax: ϕ := p ϕ ϕ ϕ ϕ [{ag} xstit]ϕ Xϕ

10 Semantics of XSTIT.I Relative to a model M = S, H, E, π, truth s, h = ϕ of a formula ϕ in a dynamic state s, h, with s h, is defined as: s, h = p s π(p) s, h = ϕ not s, h = ϕ s, h = ϕ ψ s, h = ϕ and s, h = ψ s, h = ϕ h : if s h then s, h = ϕ s, h = Xϕ if s = succ(s, h) then s, h = ϕ s, h = [{ag} xstit]ϕ s, h : if s E(s, h, {ag}) and s h then s, h = ϕ Satisfiability, validity on a model, validity on a frame, general validity and the logic are defined as usual.

11 Hilbert style axiomatic characterization of XSTIT.I (Lin) (Sett) (XSett) (Indep) S5 for KD for each [A xstit] X ϕ Xϕ X ϕ [{ag} xstit]ϕ [{ag} xstit]ϕ X ϕ [{ag 1 } xstit]ϕ [{ag 2 } xstit]ψ ([{ag 1 } xstit]ϕ [{ag 2 } xstit]ψ) Plus the standard axioms and rules for propositional logic and normal modal operators. Note: so called deliberative stit is definable: [{ag} dxstit]ϕ def [{ag} xstit]ϕ [{ag} xstit]ϕ

12 Collective agency: XSTIT S5 for KD for each [A xstit] (Det) X ϕ Xϕ ( = SettX) [ xstit]ϕ Xϕ (Ags = XSett) [Ags xstit]ϕ X ϕ (C-Mon) [A xstit]ϕ [A B xstit]ϕ (Indep-G) [A xstit]ϕ [B xstit]ψ ([A xstit]ϕ [B xstit]ψ) for A B = Plus the standard axioms and rules for propositional logic and normal modal operators. ϕ [ xstit]ψ represents that "ϕ causes ψ"

13 Outline Introduction / XSTIT Action versus Choice Successful action Choice and failing action Free will action Conclusion

14 How to interpret independence of agency? Problem: If agent ag 1 opens the door and agent ag 2 closes the door are actions, then independence of agency does not apply to actions in general (the described actions cannot occur simultaneously). There are (at least) two closely related interpretations: 1. the actions of game forms are basic actions (ag 1 pushes to open, ag 2 pushes to close) 2. distinguish between choices and actions (exercising the choice to open the door is different from the action of opening the door)

15 Choice versus action Choices of different agents are independent, actions of different agents not necessarily Game theorists and stit theorists often talk about actions where I would prefer choices A choice can never fail to be chosen A choice can fail to produce the corresponding action How to account for choice/action success/failure? We measure success and failure against an agent s epistemic attitude towards his action

16 Outline Introduction / XSTIT Action versus Choice Successful action Choice and failing action Free will action Conclusion

17 Syntax and semantics Syntax: ϕ... := p ϕ ϕ ϕ ϕ [A xstit]ϕ Xϕ K a ϕ Semantics: M, s, h = K a ϕ s, h a s, h implies that M, s, h = ϕ

18 Epistemic frames Hb1 Hb2 Hb3 Hb4 Hb5 Hb6 Hb7 Hb8 s3-choice 1 s3-choice 2 s3-choice 3 s2-choice s4 s5 s6 s7 s8 s9 s10 s11 s1-choice 1 s1-choice 2 s3 s2 s0-choice s1 Figure: Knowingly doing in a K-extended XSTIT frame

19 XSTIT frame (repeating the earlier slide) Hb2 Hb4 Hb6 Hb8 Hb1 Hb3 Hb5 Hb7 s2-choices Ag2 s5 s7 s9 s11 s3-choices Ag2 s4 s6 s8 s2-choices Ag1 s10 s3-choices Ag1 s1-choice Ag2 s2 s1-choices Ag1 s3 s1 Figure: Visualization of a partial two agent XSTIT frame

20 Examples of (un)knowingly doing If you do not know that you carry a contagious disease, it can be that by knowingly taking a seat next to somebody you unknowingly see to it that the person is infected. By knowingly sending an you may unknowingly see to it that a server breaks down. By knowingly signing a contract you may unknowingly see to it that you loose all your money.

21 How do we express this in the logic? a objectively does p unknowingly: [a xstit]p K a [a xstit]p a does p knowingly: K a [a xstit]p objective possibility for a to see to it that p: [a xstit]p ability of a to see to it that p: K a [a xstit]p

22 Possible logical properties The following Sahlqvist axioms reflect possible logical properties of knowingly doing: (IgnCC) (XK) (Rec-Eff) (R-KS-comm) (L-KS-comm) K a [b xstit]ϕ K a [b xstit]ϕ for a b K a Xϕ K a [a xstit]ϕ K a [a xstit]ϕ [a xstit]k a ϕ K a ϕ K a ϕ K a ϕ K a ϕ

23 Derivable properties The following properties of knowingly doing are derivable: (Indep-K) (HK-confl) (Unif-Str) K a [a xstit]ϕ K b [b xstit]ψ (K a [a xstit]ϕ K b [b xstit]ψ) K a ϕ K a ϕ K a [a xstit]ϕ K a [a xstit]ϕ

24 Knowingly doing is always successful K a [a xstit]ϕ [a xstit]ϕ [a xstit]ϕ X ϕ X ϕ Xϕ How to account for failure?

25 Outline Introduction / XSTIT Action versus Choice Successful action Choice and failing action Free will action Conclusion

26 Belief frames Figure: Unsuccessful action in a B-extended XSTIT frame

27 The logic for failing action Instead of an S5 knowledge operator K a, we now use a KD45 belief operator B a Many axioms from successful action (knowingly doing) are not inherited Important: we have independence of agency for possibly failing action

28 Outline Introduction / XSTIT Action versus Choice Successful action Choice and failing action Free will action Conclusion

29 Free will action that is objectively not coerced Free will action willed action Free will action = action without coercion We define free will action as action an agent could have knowingly refrained from: [a freew ]ϕ def K a [a xstit]ϕ K a [a xstit]ϕ However, here, again, choice is necessarily successful.

30 Free will action that is subjectively not coerced [a freew]ϕ def B a [a xstit]ϕ B a [a xstit]ϕ Figure: A subjectively free will action can be objectively coerced.

31 Logical possibilities The set {[a freew]ϕ, [ xstit]ϕ} is consistent in XSTIT p + extra axioms. {[a freew ]ϕ, [ xstit]ϕ} is not consistent. The sets {[a freew]ϕ, [b freew]ϕ} and {[a freew]ϕ, [b freew] ϕ} are consistent in XSTIT p + extra axioms. What is the relation with Harry Frankfurt s argument?

32 The principle of alternative possibilities (PAP) Def: EAP = "the Existence of Alternative Possibilities" PAP: "Moral Responsibility for an action EAP for an action" Harry Frankfurt: "Moral Responsibility for an action & not EAP for an action" Conclusion: PAP is wrong So far, so good. But now from moral responsibility to compatibilsm and free choice.

33 Arguing against compatibilism Def: compatibilism = consistent (determinism, free will action) Many 1: determinism not EAP for an action Many 2: free will action EAP for an action Conclusion: compatibilism is not true

34 Using Frankfurt to argue against the argument Can Frankfurt s argument be applied to attack assertion many 2? No, because: morally responsible action free will action free will action = action the agent could have knowingly refrained from morally responsible action = action originating in an agent s will, relative to a normative (moral) system (which are independent of the existence of real alternatives)

35 So, what do we have? A non-deterministic framework (stit) The logical possibility to have a free choice for a property that is causally determined compatibilism (Yesterday I learned that I am an agnostic compatibilist) The logical possibility is not simply that agents may believe to have alternatives they have not; the theory is more subtle than that: we do not have that [a freew]ϕ B a B a [a xstit]ϕ

36 Outline Introduction / XSTIT Action versus Choice Successful action Choice and failing action Free will action Conclusion

37 Are alternative choices really possible? Maybe they are, maybe they are not, depending, e.g., on the truth of causal determinism. However, the issue may be irrelevant for deciding on the possibility of free will action. Here we have a logic with an operator for free will action, where we allow for the logical possibility that alternative histories are only possible in a specific subjective sense but not objectively (i.e., really ).

38 Thanks Thanks!