MODELING COUNTERFACTUALS USING LOGIC PROGRAMMING

Transcription

1 MODELING COUNTERFACTUALS USING LOGIC PROGRAMMING Katrin Schulz ILLC, Amsterdam

2 1. INTRODUCTION 1.1 OUTLINE Logic Programming

3 1. INTRODUCTION 1.1 OUTLINE Logic Programming Counterfactuals Causality Philosophy

4 1. INTRODUCTION 1.1 OUTLINE Logic Programming NOT: Reasoning with conditionals Cognitive Sciences

5 1. INTRODUCTION 1.1 OUTLINE 1.If she has an essay to write she will study late in the library. 2.She has an essay to write. C.She will work late in the library. NOT: Reasoning with conditionals Cognitive Sciences

6 1. INTRODUCTION 1.1 OUTLINE 1.If she has an essay to write she will study late in the library. 2.She has an essay to write. 3.If the library is open she will study late in the library. C.She will work late in the library. NOT: Reasoning with conditionals Cognitive Sciences

7 1. INTRODUCTION 1.1 OUTLINE (1)If I had struck that match, it would have lit. Counterfactuals Causality Philosophy

8 1. INTRODUCTION 1.1 OUTLINE Logic Programming Counterfactuals Causality Philosophy

9 1. INTRODUCTION 1.1 OUTLINE Goals of this talk: 1. Show you that there are interesting applications of Computational Sciences tools outside of the Computational Sciences. 2.Show you that working on these applications can bring about new perspectives on these tools.

10 1. INTRODUCTION 1.1 OUTLINE How will we proceed: 1. introduce counterfactual conditionals and how they are normally treated in the philosophical literature; 2.convince you that in order to account for the meaning of counterfactuals we need to take into account causal dependence; 3.do so using logic programming; 4.link this to the standard philosophical approaches

11 2. PHILOSOPHICAL APPROACHES 2.1 WHAT IS A COUNTERFACTUAL? (1)If I had struck that match, it would have lit. definition by meaning: counterfactuals = conditionals with false antecedents

12 2. PHILOSOPHICAL APPROACHES 2.2 THE MEANING OF COUNTERFACTUALS Define the meaning of a binary sentence connective > (1)If I had struck that match, it would have lit. A > C Antecedent Consequent Truth Conditions M, w 0 = A > C iff...

13 2. PHILOSOPHICAL APPROACHES 2.3 MATERIAL IMPLICATION Truth Conditions: M, w 0 = A > C iff M, w 0 = A or M, w 0 = C / A C w1 0 0 w2 0 1 w3 1 0 w4 1 1 Problem: no relation between antecedent and consequent counterfactuals are trivially true (3) If Amsterdam was a river, I would be wearing a blue hat today. (4) If I had dropped this glass, it would have grown wings and flown away.

14 2. PHILOSOPHICAL APPROACHES 2.4 STRICT CONDITIONALS

15 2. PHILOSOPHICAL APPROACHES 2.4 STRICT CONDITIONALS Truth Conditions: M, w 0 = A > C iff [[A]] M [[C]] M M C A Problem: too strong; not all A worlds should be considered (1)If I had struck that match, it would have lit. > don t consider worlds where the match is wet or broken.

16 2. PHILOSOPHICAL APPROACHES 2.5 SIMILARITY APPROACH

17 2. PHILOSOPHICAL APPROACHES 2.5 SIMILARITY APPROACH Truth conditions: M,w 0 = A > C iff Min( M,w0, [[A]] M ) [[C]] M M,w0 M A C w 0

18 2. PHILOSOPHICAL APPROACHES 2.6 PREMISE SEMANTICS A counterfactual conditional If A then C is true iff: A L C

19 2. PHILOSOPHICAL APPROACHES 2.6 PREMISE SEMANTICS A counterfactual conditional If A then C is true iff: A + Facts of w 0 L C maximal consistent subset premise set P(w0) general laws special case of the similarity approach: w 1 w 2 iff {p P(w 0 ) w 1 = p} {p P(w 0 ) w 2 = p}

20 2. PHILOSOPHICAL APPROACHES 2.7 SUMMARY Two main approaches to the semantics of counterfactuals A sentence of the form A > C is true in a world w 0 given model M iff (a) C is true in all possible worlds in which A is true, and which in other respects differ minimally from w 0. (b) C follows from A, the laws L and all maximal subset of the premisses P(w 0 ) consistent with the antecedent.

21 NIXON, LIGHT SWITCHES AND KING LUDWIG OF BAVARIA How to model similarity Katrin Schulz ILLC/UvA

22 3. HOW TO MODEL SIMILARITY 3.1 FINE S NIXON CASE The counterfactual (1)If Nixon had pressed the button there would have been a nuclear holocaust. is true or can be imagined to be so. Now suppose that there never will be a nuclear holocaust. Then that counterfactual is, on Lewis' analysis, very likely false. For given any world in which the antecedent and consequent are both true it will be easy to imagine a closer world in which the antecedent is true and the consequent false. For we need only imagine a change that prevents the holocaust but that does not require such a great divergence from reality." (Fine 1975: 452) C A w0

23 3. HOW TO MODEL SIMILARITY 3.2 REACTION Stalnaker:...the relevant conception of minimal difference needs to be spelled out with care."

24 3. HOW TO MODEL SIMILARITY 3.1 FINE S NIXON CASE The counterfactual (1)If Nixon had pressed the button there would have been a nuclear holocaust. is true or can be imagined to be so. Now suppose that there never will be a nuclear holocaust. Then that counterfactual is, on Lewis' analysis, very likely false. For given any world in which the antecedent and consequent are both true it will be easy to imagine a closer world in which the antecedent is true and the consequent false. For we need only imagine a change that prevents the holocaust but that does not require such a great divergence from reality." (Fine 1975: 452) C A w0

25 3. HOW TO MODEL SIMILARITY 3.3 OTHER CASES A coin is going to be flipped and you bet on tails. Unfortunately, heads comes up and you lose. Now you can say: (1)If I had bet on heads, I would have won. betting flipping losing/winning antecedent consequent w0

26 3. HOW TO MODEL SIMILARITY 3.3 OTHER CASES Consider a man - call him Jones who is possessed of the following disposition as regards wearing his hat. If the man on the news predicts bad weather, Mr Jones invariably wears his hat the next day. A weather forecast in favor of fine weather, on the other hand, affects him neither way: in this case he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that yesterday bad weather was prognosed, so Jones is wearing his hat. In this case,.... (1)If the weather forecast had been in favor of fine weather, Jones would have been wearing his hat." [Tichy (1976)]

27 3. HOW TO MODEL SIMILARITY 3.3 OTHER CASES Suppose that Jones always flips a coin before he opens the curtains to see what the weather is like. `Heads' means he is going to wear his hat in case the weather is fine, whereas `tails' means he is not going to wear his hat in that case. Like before, bad weather invariably makes him wear his hat. Now suppose that today heads came up when he flipped the coin, and that it is raining. So, again, Jones is wearing his hat. Would you accept the statement: (4)If the weather had been fine, Jones would have been wearing his hat.

28 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS A conditional sentence If A then C is true iff: A + Facts of w 0 L C maximal consistent subset premise set P(w0) general laws

29 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS A conditional sentence If A then C is true iff: A + Facts of w 0 L C maximal consistent subset basis B(w0) general laws Basis(w 0 ) = minimal set of primitive facts of w 0 from which, given L, all other facts of w 0 can be derived. Particular variant of Premise Semantics

30 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Consider a man - call him Jones who is possessed of the following disposition as regards wearing his hat. If the man on the news predicts bad weather, Mr Jones invariably wears his hat the next day. A weather forecast in favor of fine weather, on the other hand, affects him neither way: in this case he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that yesterday bad weather was prognosed, so Jones is wearing his hat. In this case,.... (1)If the weather forecast had been in favor of fine weather, Jones would have been wearing his hat." [Tichy (1976)] Laws: bad hat A + Facts of w 0 L C Basis(w 0 ): bad

31 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Consider a man - call him Jones who is possessed of the following disposition as regards wearing his hat. If the man on the news predicts bad weather, Mr Jones invariably wears his hat the next day. A weather forecast in favor of fine weather, on the other hand, affects him neither way: in this case he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that yesterday bad weather was prognosed, so Jones is wearing his hat. In this case,.... (1)If the weather forecast had been in favor of fine weather, Jones would have been wearing his hat." [Tichy (1976)] Laws: bad hat bad + Facts of w 0 L hat Basis(w 0 ): bad

32 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Consider a man - call him Jones who is possessed of the following disposition as regards wearing his hat. If the man on the news predicts bad weather, Mr Jones invariably wears his hat the next day. A weather forecast in favor of fine weather, on the other hand, affects him neither way: in this case he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that yesterday bad weather was prognosed, so Jones is wearing his hat. In this case,.... (1)If the weather forecast had been in favor of fine weather, Jones would have been wearing his hat." [Tichy (1976)] Laws: bad hat bad + Basis(w 0 ) L hat Basis(w 0 ): bad

33 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Consider a man - call him Jones who is possessed of the following disposition as regards wearing his hat. If the man on the news predicts bad weather, Mr Jones invariably wears his hat the next day. A weather forecast in favor of fine weather, on the other hand, affects him neither way: in this case he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that yesterday bad weather was prognosed, so Jones is wearing his hat. In this case,.... (1)If the weather forecast had been in favor of fine weather, Jones would have been wearing his hat." [Tichy (1976)] Laws: bad hat bad + bad L hat Basis(w 0 ): bad

34 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Consider a man - call him Jones who is possessed of the following disposition as regards wearing his hat. If the man on the news predicts bad weather, Mr Jones invariably wears his hat the next day. A weather forecast in favor of fine weather, on the other hand, affects him neither way: in this case he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that yesterday bad weather was prognosed, so Jones is wearing his hat. In this case,.... (1)If the weather forecast had been in favor of fine weather, Jones would have been wearing his hat." [Tichy (1976)] Laws: bad hat bad + bad bad hat hat Basis(w 0 ): bad

35 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Consider a man - call him Jones who is possessed of the following disposition as regards wearing his hat. If the man on the news predicts bad weather, Mr Jones invariably wears his hat the next day. A weather forecast in favor of fine weather, on the other hand, affects him neither way: in this case he puts his hat on or leaves it on the peg, completely at random. Suppose, moreover, that yesterday bad weather was prognosed, so Jones is wearing his hat. In this case,.... (1)If the weather forecast had been in favor of fine weather, Jones would have been wearing his hat." [Tichy (1976)] Laws: bad hat bad bad hat hat Basis(w 0 ): bad

36 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Suppose that Jones always flips a coin before he opens the curtains to see what the weather is like. `Heads' means he is going to wear his hat in case the weather is fine, whereas `tails' means he is not going to wear his hat in that case. Like before, bad weather invariably makes him wear his hat. Now suppose that today heads came up when he flipped the coin, and that it is raining. So, again, Jones is wearing his hat. Would you accept the statement: (4)If the weather had been fine, Jones would have been wearing his hat. Laws: bad hat, heads hat Basis(w 0 ): bad, heads bad + Facts of w 0 L hat

37 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Suppose that Jones always flips a coin before he opens the curtains to see what the weather is like. `Heads' means he is going to wear his hat in case the weather is fine, whereas `tails' means he is not going to wear his hat in that case. Like before, bad weather invariably makes him wear his hat. Now suppose that today heads came up when he flipped the coin, and that it is raining. So, again, Jones is wearing his hat. Would you accept the statement: (4)If the weather had been fine, Jones would have been wearing his hat. Laws: bad hat, heads hat Basis(w 0 ): bad, heads bad + bad, heads L hat

38 3. HOW TO MODEL SIMILARITY 3.4 VELTMAN S PREMISE SEMANTICS Suppose that Jones always flips a coin before he opens the curtains to see what the weather is like. `Heads' means he is going to wear his hat in case the weather is fine, whereas `tails' means he is not going to wear his hat in that case. Like before, bad weather invariably makes him wear his hat. Now suppose that today heads came up when he flipped the coin, and that it is raining. So, again, Jones is wearing his hat. Would you accept the statement: (4)If the weather had been fine, Jones would have been wearing his hat. Laws: bad hat, heads hat Basis(w 0 ): bad, heads bad, heads L hat

39 3. HOW TO MODEL SIMILARITY 3.6 CAUSALITY MATTERS Lifschitzʼs Circuit Example Suppose there is a circuit such that the light is on (L) exactly when both switches are in the same position (up or not up). At the moment switch 1 is down ( S1), switch two is up (S2) and the lamp is out ( L). (1) If switch 1 had been up, the lamp would have been on. (S1 > L) Laws: (S1 S2) L Basis(w 0 ): { S1, S2}, { S1, L}, {S2, L} A + Facts of w 0 L C

40 3. HOW TO MODEL SIMILARITY 3.6 CAUSALITY MATTERS Lifschitzʼs Circuit Example Suppose there is a circuit such that the light is on (L) exactly when both switches are in the same position (up or not up). At the moment switch 1 is down ( S1), switch two is up (S2) and the lamp is out ( L). (1) If switch 1 had been up, the lamp would have been on. (S1 > L) Laws: (S1 S2) L Basis(w 0 ): { S1, S2}, { S1, L}, {S2, L} S1 + Facts of w 0 L L

41 3. HOW TO MODEL SIMILARITY 3.6 CAUSALITY MATTERS Lifschitzʼs Circuit Example Suppose there is a circuit such that the light is on (L) exactly when both switches are in the same position (up or not up). At the moment switch 1 is down ( S1), switch two is up (S2) and the lamp is out ( L). (1) If switch 1 had been up, the lamp would have been on. (S1 > L) Laws: (S1 S2) L Basis(w 0 ): { S1, S2}, { S1, L}, {S2, L} S1 + Basis(w 0 ) (S1 S2) L L

42 3. HOW TO MODEL SIMILARITY 3.6 CAUSALITY MATTERS The basis B of the evaluation world w0 is the set of primitive facts (literals) of w) from which, given the rules R, all other facts of w0 can be derived. Not: epistemic derivation But: causal derivation A + Basis(w 0 ) L C

43 3. HOW TO MODEL SIMILARITY 3.6 CAUSALITY MATTERS We need a causal notion of consequence: reasoning from cause to effect.

44 MODELING CAUSAL REASONING WITH LOGIC PROGRAMMING Katrin Schulz ILLC/UvA

45 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS We need a causal notion of consequence: reasoning from cause to effect. we want: take A and take causally independent facts of w 0 and calculate the effects the causal notion of consequence should be assymmetric Logic Programming

46 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L ((partial) model)

47 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Standard definition The operator τ P associated to a program P is a function on the set of three valued models. For every propositionletter p the value τ P (M)(p) is defined as follows. 1. τ P (M)(p) = 1 if there is a clause p Φ in P such that M(Φ) = τ P (M)(p) = 0 if for all clauses p Φ in P it holds M (Φ) = τ P (M)(p) = u, otherwise.

48 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L ((partial) model)

49 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L1 ((partial) model)

50 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L2 ((partial) model)

51 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L3 ((partial) model )

52 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L* ((partial) model ) = C iterated fixed point as designated model to define causal inference

53 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L* ((partial) model ) = C iterated fixed point as designated model to define causal inference Laws: A B, C A Conditional: A > C B A C w0

54 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L* ((partial) model ) = C iterated fixed point as designated model to define causal inference Laws: A B, C A Conditional: A > C B w0

55 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L* ((partial) model ) = C iterated fixed point as designated model to define causal inference Laws: A B, C A B A w0 Conditional: A > C

56 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L* ((partial) model ) = C iterated fixed point as designated model to define causal inference Laws: A B, C A Conditional: A > C B A C w0

57 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Logic Programming A + Basis(w 0 ) L C sentence (partial) model program τ L* ((partial) model ) = C iterated fixed point as designated model to define causal inference Laws: A B, C A Conditional: A > C B A C w0

58 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS What s the problem? the program will overwrite the antecedent because it goes against the laws!

59 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS What s causing the problem? program explanation of the mechanics of reality (partial) model (observed) facts your explanation of the world shouldn t be able to wipe out your observations observations always take priority! This goes against all standard work in logic programming

60 4. A CAUSAL NOTION OF CONSEQUENCE 4.1 INGREDIENTS Two ways to deal with this revise the program to incorporate the new observations as laws change the way programs are applied to models Pearl 00 Schulz 11

61 4. A CAUSAL NOTION OF CONSEQUENCE 4.2 SEMANTIC OPERATORS Standard definition The operator τ P associated to a program P is a function on the set of three valued models. For every propositionletter p the value τ P (M)(p) is defined as follows. 1. τ P (M(p) = 1 if there is a clause p Φ in P such that M(Φ) = τ P (M)(p) = 0 if for all clauses p Φ in P it holds M (Φ) = τ P (M)(p) = u, otherwise.

62 4. A CAUSAL NOTION OF CONSEQUENCE 4.2 SEMANTIC OPERATORS Stenning & van Lambalgen The operator τ P associated to a program P is a function on the set of three valued models. For every propositionletter p the value τ P (M)(p) is defined as follows. 1. τ P (M(p) = 1 if there is a clause p Φ in P such that M(Φ) = τ P (M)(p) = 0 if there is a clause p Φ in P and for all such clauses M (Φ) = τ P (M)(p) = u, otherwise.

63 4. A CAUSAL NOTION OF CONSEQUENCE 4.2 SEMANTIC OPERATORS The new definition The operator τ P associated to a program P is a function on the set of three valued models. For every propositionletter p the value τ P (M)(p) is defined as follows. 1. If M(p) u, then τ P (M)(p) = M(p). 2. If M(p) = u, then a. τ P (M)(p) = 1 if there is a clause p Φ in P such that M(Φ) = 1. b. τ P (M)(p) = 0 if there is a clause p Φ in P and for all such clauses M (Φ) = 0.

64 4. A CAUSAL NOTION OF CONSEQUENCE 4.2 SEMANTIC OPERATORS Results: Program: 1.The operation τ P is monotone w.r.t. the order on models 2.τ P has fixed points and the least fixed point can be reached in finitely many steps. p q p q M1 M2, but τ P (M1) τ P (M2) M1 1 u M2 1 0 τ P (M1) 1 1 τ P (M2) 1 0

65 4. A CAUSAL NOTION OF CONSEQUENCE 4.2 SEMANTIC OPERATORS Results: 1.For all M we have M τ P (M). 2.The iterative application of τ P to a model terminates in a fixed point of τ P after finitely many steps. Definition: For a set of literals Σ (the facts), Σ P C iff C is true on the fixed point τ P* (Σ) of τ P reached by iterating τ P starting from Σ. A + Basis(w 0 ) L C

66 4. A CAUSAL NOTION OF CONSEQUENCE 4.2 SEMANTIC OPERATORS Results: 1.For all M we have M τ P (M). 2.The iterative application of τ P to a model terminates in a fixed point of τ P after finitely many steps. Definition: For a set of literals Σ (the facts), Σ P C iff C is true on the fixed point τ P* (Σ) of τ P reached by iterating τ P starting from Σ. A + Basis(w 0 ) P C

67 BACK TO THE SIMILARITY APPROACH Katrin Schulz ILLC/UvA

68 3. BACK TO SIMILARITY 3.1 THE BASIC CASE Question: Is this notion of inference still modeling reasoning with minimal models? all models Answer: Σ P C iff Min (Σ,P) = C YES!... in a way... How?: Iterated order: you minimize stepwise differences from τ P* (Σ). Σ complete models Min (Σ,P) complete Σ models

69 3. BACK TO SIMILARITY 3.1 THE BASIC CASE Question: Is this notion of inference still modeling reasoning with minimal models? Program: Σ P C iff Min (Σ,P) = C Answer: YES!... in a way... ( )p 1 ( )p 2 q 1 p2 ( )p 3 ( )p 4 q 3 q2 ( )q 1 ( )q 2 ( )q 2 r 1 How?: Iterated order: you minimize stepwise differences from τ P* (Σ). p 1 p 2 q 1 p 3 q 3 q 2 r 1 p 4 which one comes closest to τ P* (Σ)

70 3. BACK TO SIMILARITY 3.1 THE BASIC CASE Question: Is this notion of inference still modeling reasoning with minimal models? Program: Σ P C iff Min (Σ,P) = C Answer: YES!... in a way... ( )p 1 ( )p 2 q 1 p2 ( )p 3 ( )p 4 q 3 q2 ( )q 1 ( )q 2 ( )q 2 r 1 How?: Iterated order: you minimize stepwise differences from τ P* (Σ). p 1 p 2 q 1 p 3 q 3 q 2 r 1 p 4 which one comes closest to τ P* (Σ)

71 3. BACK TO SIMILARITY 3.1 THE BASIC CASE Question: Is this notion of inference still modeling reasoning with minimal models? Program: Σ P C iff Min (Σ,P) = C Answer: YES!... in a way... ( )p 1 ( )p 2 q 1 p2 ( )p 3 ( )p 4 q 3 q2 ( )q 1 ( )q 2 ( )q 2 r 1 How?: Iterated order: you minimize stepwise differences from τ P* (Σ). p 1 p 2 q 1 p 3 q 3 q 2 r 1 p 4 which one comes closest to τ P* (Σ)

72 3. BACK TO SIMILARITY 3.1 THE BASIC CASE Question: Is this notion of inference still modeling reasoning with minimal models? Program: Σ P C iff Min (Σ,P) = C Answer: YES!... in a way... ( )p 1 ( )p 2 q 1 p2 ( )p 3 ( )p 4 q 3 q2 ( )q 1 ( )q 2 ( )q 2 r 1 How?: Iterated order: you minimize stepwise differences from τ P* (Σ). p 1 p 2 q 1 p 3 q 3 q 2 r 1 p 4 which one comes closest to τ P* (Σ)

73 3. BACK TO SIMILARITY 3.2 CONDITIONALS A conditional sentence If A then C is true iff: A + Facts of w 0 P C maximal consistent subset causal notion of basis causal notion of consequence general laws Basis(w 0 ) = minimal set of primitive facts of w 0 from which, given L, all other facts of w 0 can be causally derived. Particular variant of Premise Semantics but based on a different logic!

74 3. BACK TO SIMILARITY 3.2 CONDITIONALS Question: Is this version of premise semantics still a special case of minimal world reasoning? Can we recast this approach in terms of the Lewis/ Stalnaker similarity approach? Answer: YES!

75 CONCLUSIONS Katrin Schulz ILLC/UvA

76 4. CONCLUSIONS 4.1 THE RESULTS with simple tools from logic programming we can account for many central examples in the literature on counterfactuals no open technical questions for this static system but what about updating with counterfactuals? and many open philosophical issues relation counterfactuals causality taking a particular stance towards causality (interventionalist approach)

77 4. CONCLUSIONS 4.2 IS THIS REALLY ABOUT CAUSALITY? conditionals exploit certain invariant relationships, certain asymmetric dependencies what unifies these relationships is that the expressed dependency is one of manipulation and control: A stands in this relation to B if manipulating A will change B in a systematic way; by manipulating A one can control B an invariant relationship with these properties I call a causal relation

78 4. CONCLUSIONS 4.2 IS THIS REALLY ABOUT CAUSALITY? King Ludwig of Bavaria likes to spend his weekends at Leoni Castle. Whenever the Royal Bavarian flag is up and the lights are on, the King is in the Castle. At the moment the lights are on, the flag is down, and the King is away. Suppose now counterfactually that the flag were up. Well, then the King would be in the castle and the lights would still be on. But why wouldn't the lights be out and the King still be away? Kratzer, Lumps of Thought, p. 640

79 4. CONCLUSIONS 4.2 IS THIS REALLY ABOUT CAUSALITY? It is a simple fact of basic math that if you add two natural numbers that are both even or uneven, the sum will be even, but if one of the numbers is even and the other uneven the sum is uneven. Suppose you re explaining this fact to some school kids and you have on the board = 7. You say... (11) If the first number had been even, the result would have been even. (12) If the result had been even, the first number would have been even.

80 4. CONCLUSIONS 4.2 IS THIS REALLY ABOUT CAUSALITY? Beth 55: in everyday usage of causality we make no difference between conventional rules and `real causality.

81 4. CONCLUSIONS 4.3 WHAT IS CAUSALITY? Beth 55: in everyday usage of causality we make no difference between conventional rules and `real causality.

82 4. CONCLUSIONS 4.3 WHAT IS CAUSALITY? Beth, Algemeene Beschouwingen over causaliteit: So beschouwd, is het causaliteitsbeginsel dus geen principe a priori; maar het is ook geen natuurwet en zeker geen conventie. We kunnen misschien het beste zeggen, dat het causaliteitsbeginsel op een mede door historische factoren bepaalde wijze uitdrukking geeft aan een algemeenmenselijke neiging, op grond van spontaan geäpperciëerde causale (en andere) verbanden een meer, en zo mogelijk alles, omvattende causale structuur op te bouwen, die ons in staat stelt het universum waarin wij leven als een kosmos te begrijpen

83 4. CONCLUSIONS 4.3 WHAT IS CAUSALITY? Beth, Algemeene Beschouwingen over causaliteit: From this perspective, the principle of causality is not a principle a priori; but it is also not a law of nature and certainly not a convention. Maybe, the best we can say is that the principle of causality expresses, partly determined by historical factors, a human tendency to build, based on spontaneously experienced causal (and other) relations, a more, and, if possible, all, including causal structure, which enables us to understand the universe we live in.

84 MINIMAL MODELS WITH THREE-VALUED LOGIC Katrin Schulz ILLC, Amsterdam