Tutorial: Nonmonotonic Logic (Day 2)

Transcription

1 Tutorial: Nonmonotonic Logic (Day 2) Christian Straßer Institute for Philosophy II, Ruhr-University Bochum Center for Logic and Philosophy of Science, Ghent University September 3, 2015 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

2 Outline 1 Plausible Reasoning 2 Preferential / Selection Semantics (KLM, Shoham) 3 Bibliography Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

3 Topic 1 Plausible Reasoning 2 Preferential / Selection Semantics (KLM, Shoham) 3 Bibliography Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

4 Plausible vs. Defeasible Reasoning (Rescher / Vreeswijk / Prakken) deductive standard: CL Scheme: Plausible Inferences A follows nonmonotonically from B iff B A ab and there is no reason to assume ab. A follows nonmonotonically from B iff B ass A and there is a reason to assume ass. classical inference in disguise thus, e.g., contrapositable since A B ab iff B A ab; and A ass B iff B ass A explicite defeasible assumption Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

5 Plausible Reasoning can be found in, for instance,... Rescher / Manor consequence relations (Rescher and Manor (1970)) Brewka s preferred subtheories (Brewka (1989)) Makinsons Default Assumptions (Makinson (2003)) Batens Adaptive Logics and generalisations (Batens (2007); Straßer (2014)) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

6 General Account: Premise Sets (non-defeasible) facts Σ = Σ 0, Σ d Σ d may also be stratified: defeasible premises (assumptions) Σ d = Σ 1,..., Σ n,... e.g., Σ 1 stems from the most reliable (though fallible) source, Σ 2 stems from the second most reliable (though fallible) source, etc. in Rescher/Manor consequence relations: Σ 0 = Makinsons s default assumptions: Σ 0 may be non-empty Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

7 Base Logic L with consequence relation Cn monotonic: Cn(Γ) Cn(Γ Γ ) reflexive: Γ Cn(Γ) compact: if A Cn(Γ) then A Cn(Γ ) for some finite Γ Γ cut: where Γ Cn(Γ) and A Cn(Γ Γ ), A Cn(Γ) or, Transitivity: where Γ Cn(Γ) and A Cn(Γ ), A Cn(Γ). Note: given refl. and mono., CUT iff TRANSITIVITY Suppose Γ Cn(Γ). ( ) Suppose A Cn(Γ ). By Monotonicity, A Cn(Γ Γ ). By Cut, A Cn(Γ). ( ) Suppose A Cn(Γ Γ ). By Reflexivity, Γ Γ Cn(Γ). By Transitivity, A Cn(Γ). Notation: Cn(Γ) = {A Γ A} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

8 General Account: Consequence Relations Idea: Use (maximal) consistent subsets of Σ d. F-Inconsistent set we suppose a specific logical form F (e.g., A A) in the language L of L Γ is inconsistent if Γ B where B is of the form F otherwise Γ is consistent in the following we will write as a placeholder for formulas of the form F Maximal Consistent Subsets of Σ = Σ 0, Σ d Ξ is a maximal consistent subset of Σ iff 1 Ξ Σ d is such that Ξ Σ 0 is consistent 2 there is no Γ Σ d such that Ξ Γ and Γ Σ 0 is consistent. We write MCS(Σ) for the set of all maximal consistent subsets of Σ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

9 Examples Let Σ = Σ 0, Σ d where Σ 0 = {s} Σ d = {s (p q), p q, r} What are the MCSs? Ξ 1 = {s (p q), r} Ξ 2 = {p q, r} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

10 Existence of MCSs Where Ξ Σ d s.t. Σ 0 Ξ, there is a Ξ MCS( Σ 0, Σ d ) s.t. Ξ Ξ. Proof Let Σ d = {A 1,...}. Let Ξ = i 0 Ξ i where Ξ 0 = Ξ { Ξi {A Ξ i+1 = i+1 } if Ξ i {A i+1 } Σ 0 else Ξ i By induction, for each Ξ i, Ξ i Σ 0. Assume Ξ Σ 0. By compactness, there is a finite Ξ f Ξ such that Ξ f Σ 0. Thus there is a Ξ i Ξ f and by monotonicity, Ξ i Ξ 0, a contradiction. Hence, Ξ Σ 0. Where A i / Ξ, Ξ {A i } Σ 0 since Ξ i 1 {A i } Σ 0 and by monotonicity. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

11 General Account: Consequence Relations Question Do you think this also holds if we define "Γ is inconsistent iff Cn(Γ) = L"? What about... L where Cn(Γ) = Γ and L = ω? monotonic reflexive compact transitive Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

12 General Account: Consequence Relations Σ = Σ 0, Σ d Free consequences Σ free A iff Σ 0 Free(Σ) A where Free(Σ) = MCS(Σ) Universal consequences Σ A iff Σ 0 Ξ A for all Ξ MCS(Σ) Existential consequences Σ A iff Σ 0 Ξ A for some Ξ MCS(Σ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

13 Examples Let Σ = Σ 0, Σ d where Σ 0 = {s} Σ d = {s (p q), p q, r} What are the MCSs? Ξ 1 = {s (p q), r} Ξ 2 = {p q, r} Free consequences Free(Σ) = {r} Γ free A iff {s, r} A Notice: Syntax-Dependency Σ 0, {s p, s q, p, q, r} free p Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

14 Examples Let Σ = Σ 0, Σ d where Σ 0 = {s} Σ d = {s (p q), p q, r} What are the MCSs? Ξ 1 = {s (p q), r} Ξ 2 = {p q, r} Universal Consequences Γ A iff {s, r, p} A floating conclusion: p Question: Is there also some syntax-dependency for? Take Σ =, {p q, p} MCS(Σ ) = { {p q}, { p} } Σ q While where Σ =, {p, q, p} MCS(Σ ) = { {p, q}, { p, q} } Σ q. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

15 Examples Let Σ = Σ 0, Σ d where Σ 0 = {s} Σ d = {s (p q), p q, r} What are the MCSs? Ξ 1 = {s (p q), r} Ξ 2 = {p q, r} Existential Consequences Σ A iff A Cn(Ξ 1 {s}) Cn(Ξ 2 {s}) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

16 General Account: Consequence Relations Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

17 What about stratified premises? Let Σ 0, Σ d where Σ d = Σ 1, Σ 2,.... Idea: Order MCSs relative to the strength of their constituting premises where strength/reliability/etc. is indirectly proportional to the index Σ i. Lexicographic Ordering (Brewka (1989), Van De Putte and Straßer (2012)) Ξ Ξ iff there is an i 1 such that 1 Ξ Σ j = Ξ Σ j for all 1 j < i, and 2 Ξ Σ i Ξ Σ i Consequence relation... relative to min (MCS(Σ)) Σ free A iff min (MCS(Σ)) A Σ A iff for all Ξ min (MCS(Σ)), Ξ A Σ A iff for some Ξ min (MCS(Σ)), Ξ A Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

18 What about stratified premises? Lexicographic Ordering (Brewka (1989), Van De Putte and Straßer (2012)) Ξ lex Ξ iff there is an i 1 such that 1 Ξ Σ j = Ξ Σ j for all 1 j < i, and 2 Ξ Σ i Ξ Σ i Example Σ 0 = {s} Σ 1 = {s p} Σ 2 = { p, q} Σ 3 = {q, r} MCSs Ξ 1 = {s p, q, r} Ξ 2 = {s p, q, r} Ξ 3 = { p, q, r} Ξ 4 = { p, q, r} order: Ξ 1 Ξ 2 Ξ 3 Ξ 4 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

19 Caution: Other orders may not be smooth! Suppose we associate strength in direct proporitionality to the index of Σ i (e.g., premises in Σ i+1 are more reliably than premises in Σ i (i 1)). Ordering Ξ Ξ iff there there is an i 1 such that 1 Ξ Σ j = Ξ Σ j for all j > i, and 2 Ξ Σ i Ξ Σ i Example Take Σ = Σ 0, Σ 1, Σ 2,... where Σ 0 = {p i p j j > i 1} Σ i = { p i, s} for each i 1 MCSs Ξ i = {s, p i } for each i 1 Note min (MCS(Σ)) = however, we would at least expect to derive s Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

20 Some Meta-Properties Monotonicity? Clearly fails for free and universal. What about existential? important to distinguish between: Monotonicity in the defeasible premises If Σ 0, Σ d A then Σ 0, Σ d Γ A. Monotonicity in the factual premises If Σ 0, Σ d A then Σ 0 Γ, Σ d A. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

21 Weakening of Monotonicity? What about Cautious Monotonicity in the free and universal case? Recall: Cautious Monotonicity If Σ A and Σ B then Σ {A} B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

22 Cumulativity If Σ A : Σ B iff Σ {A} B This is: Cautious monotonicity and Cut In our case two options: in defeasible premises: If Σ 0, Σ d A : Σ 0, Σ d B iff Σ 0, Σ d {A} B in factual premises: If Σ 0, Σ d A : Σ 0, Σ d B iff Σ 0 {A}, Σ d B Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

23 Where Σ = Σ 0, Σ d, Σ A, π : MCS( Σ 0, Σ d ) MCS( Σ 0, Σ d {A} ), Λ Λ {A} is a bijection. Sketch of the Proof In case A Σ d the proposition is trivial. Suppose thus that A / Σ d. ( ) Let Ξ MCS(Σ). We show that Ξ {A} MCS( Σ 0, Σ d {A} ). Note that Ξ Σ 0 A and thus Σ 0 Ξ {A} by cut and ( ). Assume that there is a Ξ Ξ {A} such that Ξ MCS(Σ 0, Σ d {A}). Thus, Σ d Ξ \ {A} Ξ. Since Σ 0 (Ξ \ {A}), this is a contradiction to ( ). ( ) Let Ξ MCS( Σ 0, Σ d {A} ). Clearly, Σ 0 (Ξ \ {A}). Assume that there is a Ξ MCS( Σ 0, Σ d ) such that Ξ \ {A} Ξ. By the supposition, Σ 0 Ξ A and hence Σ 0 Ξ {A}. This is a contradiction to ( ) since Ξ Ξ {A}. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

24 The Cumulativity of : defeasible premises Recall: Where Σ A, π : MCS( Σ 0, Σ d ) MCS( Σ 0, Σ d {A} ), Λ Λ {A} is a bijection. Where Σ 0, Σ d A: Σ 0, Σ d B iff Σ 0, Σ d {A} B. Proof In case A Σ d the proposition is trivial. Suppose thus that A / Σ d. Suppose Σ 0, Σ d A. Suppose Σ 0, Σ d B. Let Ξ MCS( Σ 0, Σ d {A} ). Hence, Ξ \ {A} MCS( Σ 0, Σ d ) and thus Σ 0 (Ξ \ {A}) B. By Mono., Σ 0 Ξ B. Suppose Σ 0, Σ d {A} B. Let Ξ MCS( Σ 0, Σ d ). Hence, Ξ {A} MCS( Σ 0, Σ d ) and Σ 0 Ξ A. Also, Σ 0 Ξ {A} B. By cut, Σ 0 Ξ B. Thus, Σ 0, Σ d B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

25 Stronger form of cumulativity? Do you think the antecedent of the previous result can be weakened? We d need to weaken the antecedent of our lemma: Where Σ A, π : MCS( Σ 0, Σ d ) MCS( Σ 0, Σ d {A} ), Λ Λ {A} is a bijection. Suggestions? Here s one way Suppose we have a supraclassical core logic (with cl.neg. and = p p), then Where Σ A, π : MCS( Σ 0, Σ d ) MCS( Σ 0, Σ d {A} ), Λ Λ {A} is a bijection. proof remains nearly the same (check this!) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

26 Here it is... (changes in red) Where L is supraclassical and Σ A, π : MCS( Σ 0, Σ d ) MCS( Σ 0, Σ d {A} ), Λ Λ {A} is a bijection. Sketch of the Proof In case A Σ d the proposition is trivial. Suppose thus that A / Σ d. ( ) Let Ξ MCS(Σ). We show that Ξ {A} MCS( Σ 0, Σ d {A} ). Note that Ξ Σ 0 A and thus Σ 0 Ξ {A}. Assume that there is a Ξ Ξ {A} such that Ξ MCS(Σ 0, Σ d {A}). Thus, Σ d Ξ \ {A} Ξ. Since Σ 0 (Ξ \ {A}), this is a contradiction to ( ). ( ) Let Ξ MCS( Σ 0, Σ d {A} ). Clearly, Σ 0 (Ξ \ {A}). Assume that there is a Ξ MCS( Σ 0, Σ d ) such that Ξ \ {A} Ξ. By the supposition, Σ 0 Ξ A and hence Σ 0 Ξ {A}. This is a contradiction to ( ) since Ξ Ξ {A}. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

27 The Cumulativity of free : defeasible premises Where Σ 0, Σ d free A: Σ 0, Σ d free B iff Σ 0, Σ d {A} free B. Proof In case A Σ d the proposition is trivial. Suppose thus that A / Σ d. Note that by our lemma we immediately get: ( ) MCS( Σ 0, Σ d {A} ) = MCS(Σ) {A}. Suppose Σ 0, Σ d free B and hence MCS( Σ0, Σ d ) B. By ( ) and monotonicity, MCS( Σ0, Σ d {A} ) B. Hence, Σ 0, Σ d {A} free B. Suppose Σ 0, Σ d {A} free B and hence MCS( Σ 0, Σ d {A} ) B. By ( ), MCS( Σ0, Σ d ) {A} B. Since MCS(Σ) A, by cut MCS( Σ0, Σ d ) B. Thus Σ 0, Σ d free B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

28 Strengthenings Strengthening 1 Where Σ 0, Σ d A: Σ 0, Σ d free B iff Σ 0, Σ d {A} free B. Strengthening 2, where L is supraclassical Where Σ 0, Σ d A: Σ 0, Σ d free B iff Σ 0, Σ d {A} free B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

29 If Σ 0, Σ d A: Σ 0, Σ d B iff Σ 0 {A}, Σ d B ( {, free }). This follows immediately from the following Lemma If Σ 0, Σ d A then MCS( Σ 0, Σ d ) = MCS( Σ 0 {A}, Σ d ). Proof: Suppose Σ 0, Σ d A. ( ) Let Ξ MCS( Σ 0, Σ d ). Thus, Ξ Σ 0 A and hence Σ 0 Ξ {A} by ( ) and cut. Assume there is a Ξ MCS(Σ 0 {A}, Σ d ) such that Ξ Ξ. Hence, Σ 0 {A} Ξ and hence Σ 0 Ξ by mono.. Since also Ξ Σ d this contradicts ( ). Let ( ) Ξ MCS( Σ 0 {A}, Σ d ). Assume there is a Ξ MCS( Σ 0, Σ d ) for which Ξ Ξ. Hence, Σ 0 Ξ A and Σ 0 Ξ. Thus also Σ 0 {A} Ξ in contradiction to ( ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

30 Fixed Point Let Cn(Σ) = df {A Σ A}. Fixed Point Property Cn(Σ) = Cn(Cn(Σ)). Now: two versions (where {, free }) 1 relative to defeasible premises: Cn( Σ 0, Σ d ) = Cn( Σ 0, Σ d Cn( Σ 0, Σ d ) ) 2 relative to factual premises: Cn( Σ 0, Σ d ) = Cn( Σ 0 Cn( Σ 0, Σ d ), Σ d ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

31 Fixed Point follows from Cumulativity: ( ) All we need is (where {, free }): 1 Where Γ Cn ( Σ 0, Σ d ): Cn ( Σ 0, Σ d ) = Cn ( Σ 0, Σ d Γ ) 2 Where Γ Cn ( Σ 0, Σ d ): Cn ( Σ 0, Σ d ) = Cn ( Σ 0 Γ, Σ d ) Corollary: If satisfies ( ), Cn has the fixed point property. Proof: Follows immediately, just let Γ = Cn (Σ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

32 Proof of ( ) Our previous lemmas can easily be strengthened to: 1 Where Γ Cn ( Σ 0, Σ d ): π : MCS( Σ 0, Σ d ) MCS( Σ 0, Σ d Γ ), Λ Λ Γ is a bijection. 2 Where Γ Cn ( Σ 0, Σ d ): MCS( Σ 0, Σ d ) = MCS( Σ 0 Γ, Σ d ). Fixed Point follows from Cumulativity: The proofs for the following strong forms of cumulativity are easily adjusted (where {, free }): 1 Where Γ Cn : Cn ( Σ 0, Σ d ) = Cn ( Σ 0, Σ d Γ ) 2 Where Γ Cn : Cn ( Σ 0, Σ d ) = Cn ( Σ 0 Γ, Σ d ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

33 The Resolution Theorem: in the defeasible premises Do we have (supposing we have the Res.Thm. for ) Σ 0, Σ d A B implies Σ 0, Σ d {A} B? Σ 0, Σ d free A B implies Σ 0, Σ d {A} free B? for : NOPE Consider Σ =, { p}, A = p and B = q. Then Σ p q while, { p, p} q. for free : NOPE same counterexample Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

34 The Resolution Theorem for facts? Do we have Σ 0, Σ d A B implies Σ 0 {A}, Σ d B? Σ 0, Σ d free A B implies Σ 0 {A}, Σ d free B? Nope: Take Σ =, { p} and Σ + = {p}, { p}. Then Σ p q while Σ + q. And Σ free p q while Σ + free q. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

35 The Deduction Theorem for defeasible premises Question: Σ 0, Σ d {A} B implies Σ 0, Σ d A B? Σ 0, Σ d {A} free B implies Σ 0, Σ d free A B? We suppose in the following discussion 1 we have the deduction theorem on the level of L: Γ {A} B implies Γ A B. 2 explosion: A for all A L. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

36 The Deduction Theorem for defeasible premises, the universal case Σ 0, Σ d {A} B implies Σ 0, Σ d A B? Proof Suppose Σ 0, Σ d {A} B and let Ξ MCS(Σ 0, Σ d ). We have two case: (1) Ξ {A} MCS( Σ 0, Σ d {A} ) or (2) Ξ MCS(Σ 0, Σ d {A}) and Ξ Σ 0 {A} and thus Ξ Σ 0 {A} B (by explosion and trans). 1 Then by the supposition Σ 0 Ξ {A} B and hence Σ 0 Ξ A B by the deduction theorem. 2 Σ 0 Ξ A B by the deduction theorem. Thus, Ξ A B and Σ 0, Σ d A B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

37 The Deduction Theorem for defeasible premises, the free case Σ 0, Σ d {A} free B implies Σ 0, Σ d free A B Proof Suppose Σ 0, Σ d {A} free B and hence Σ 0 MCS( Σ 0, Σ d {A} ) B. It is easy to see that MCS( Σ0, Σ d {A} ) MCS( Σ 0, Σ d ) {A}. By monotonicity, Σ 0 MCS( Σ 0, Σ d ) {A} B and by the deduction theorem, Σ 0 MCS( Σ 0, Σ d ) A B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

38 The Deduction Theorem for Facts, the universal case Σ 0 {A}, Σ d B implies Σ 0, Σ d A B First a useful Lemma If Ξ MCS( Σ 0, Σ d ) and Σ 0 Ξ {A} then Ξ MCS( Σ 0 {A}, Σ d ). (Trivial) Proof Since Σ 0 Ξ {A} there is a Ξ Ξ such that Ξ MCS( Σ 0 {A}, Σ d ). Thus, Ξ Σ 0 and by the maximality of Ξ, Ξ = Ξ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

39 The Deduction Theorem for Facts, the universal case Σ 0 {A}, Σ d B implies Σ 0, Σ d A B Our useful Lemma If Ξ MCS( Σ 0, Σ d ) and Σ 0 Ξ {A} then Ξ MCS( Σ 0 {A}, Σ d ). Proof of the deduction theorem Suppose Σ 0 {A}, Σ d B. Let Ξ MCS(Σ 0, Σ d ). If Σ 0 Ξ {A} also Σ 0 Ξ {A} B by explosion and transitivity. By the deduction theorem, Σ 0 Ξ A B. If Σ 0 Ξ {A} then by the Lemma, Ξ MCS( Σ 0 {A}, Σ d ) and thus Σ 0 Ξ B. By monotonicity, Σ 0 Ξ {A} B. By the deduction theorem, Σ 0 Ξ A B. ChristianAltogether, Straßer (RUB, UGENT) Σ, Σ Tutorial: A Nonmonotonic B. Logic (Day 2) September 3, / 96

40 The Deduction Theorem for facts, the free case Do we have Σ 0 {A}, Σ d free B implies Σ 0, Σ d free A B? Nope: here s a counter-example Let Σ 0 = {p q, r q} Σ d = { p, q} We have Σ 0, Σ d free p r while Σ 0 {p}, Σ d free r To see this notice that MCS( Σ 0, Σ d ) = {{ p}, { q}} MCS( Σ 0 {p}, Σ d ) = {{ q}} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

41 Rational Monotonicity for defeasible premises: the universal case If Σ 0, Σ d B and Σ 0, Σ d A then Σ 0, Σ d {A} B? Nope! Take: Σ 0 = Σ d = {r, p q r, (p r) q, p q} A = p and B = q Σ 0, Σ d q Σ 0, Σ d p but Σ 0, Σ d {p} q. Do you see why? MCS( Σ 0, Σ d ) = 1 {(p r) q, r, p q} 2 {(p r) q, p q r} MCS( Σ 0, Σ d {p} ) = 1 {(p r) q, r, p q} 2 {(p r) q, p q r, p} 3 {r, p, (p r) q} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

42 Rational Monotonicity for factual premises: the universal case If Σ 0, Σ d B and Σ 0, Σ d A then Σ 0 {A}, Σ d B? Nope! Take: Σ 0 = Σ d = {r, p q r, (p r) q, p q} A = p and B = q Σ 0, Σ d q Σ 0, Σ d p but Σ 0 {p}, Σ d q. Do you see why? MCS( Σ 0, Σ d ) = 1 {(p r) q, r, p q} 2 {(p r) q, p q r} MCS( Σ 0 {p}, Σ d ) = 1 {r, (p r) q} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

43 Rational Monotonicity relative to defeasible premises: the free case If Σ 0, Σ d free B and Σ 0, Σ d free A then Σ 0, Σ d {A} free B? Nope! Take: Σ 0 = Σ d = {r, p q r, (p r) q, p q} A = p q and B = (p r) q Σ 0, Σ d free (p r) q Σ 0, Σ d free (p q) but Σ 0, Σ d {p q} free (p r) q. Do you see why? MCS( Σ 0, Σ d ) = 1 {(p r) q, r, p q} 2 {(p r) q, p q r} MCS( Σ 0, Σ d {p q} ) = 1 {(p r) q, r, p q} 2 {(p r) q, p q r, p q} 3 {r, p q} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

44 Rational Monotonicity relative to strict premises: the free case If Σ 0, Σ d free B and Σ 0, Σ d free A then Σ 0 {A}, Σ d free B? Nope! Take: Σ 0 = Σ d = {r, p q r, (p r) q, p q} A = p q and B = (p r) q Σ 0, Σ d free (p r) q Σ 0, Σ d free (p q) but Σ 0 {p q}, Σ d free (p r) q. Do you see why? MCS( Σ 0, Σ d ) = For defeasible premises 1 {(p r) q, r, p q} 2 {(p r) q, p q r} MCS( Σ 0 {p q}, Σ d ) = 1 {r} 2 {p q r, (p r) q} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

45 Other forms of Rationality Negation Rationality Γ {A} B Γ { A} B Γ B or positively: If Γ B then Γ {A} B or Γ { A} B. Disjunctive Rationality Γ {A} C Γ {B} C Γ {A B} C or positively If Γ {A B} C then Γ {A} C or Γ {B} C. We have again two versions: relative to factual and to defeasible premises. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

46 Disjunctive Rationality implies Negation Rationality Replacement of equivalents Let be a consequence relation between (L) (L) and L. satisfies replacement of equivalents iff, where A A, 1 Σ 0 {A}, Σ d B iff Σ 0 {A }, Σ d B 2 Σ 0, Σ d {A} B iff Σ 0, Σ d {A } B. Do we have it? Where {,, free }, satisfies replacement of equivalents. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

47 Disjunctive Rationality implies Negation Rationality Where satisfies replacement of equivalents, Disjunctive Rationality in the factual [defeasible] premises implies negation rationality in the factual [defeasible] premises. Proof 1 Suppose Σ 0 {A}, Σ d C and Σ 0 {B}, Σ d C implies Σ 0 {A B}, Σ d C, for all A, B and C. 2 Suppose also Σ 0 {A B}, Σ d C and Σ 0 {A B} C. 3 By 1, Σ 0 {(A B) (A B)}, Σ d C. 4 Since A ((A B) (A B)), Σ 0 {A}, Σ d C. Proof for defeasible premises is analogous. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

48 Counter-example for Negation Rationality for Let Σ = Σ 0, Σ d be such that Σ 0 = and Σ d = { r s, r ( q t), s r, s (q t)}. Note that MCS( Σ 0, Σ d {p} ) 1 { r s, r ( q t), s (q t), p} 2 { s r, r ( q t), s (q t), p} Hence, Σ 0 {p} t. MCS( Σ 0, Σ d {p q} ) 1 { s r, r ( q t), s (q t)} 2 { r s, r ( q t), s (q t)} 3 { r s, s (q t), p q} MCS( Σ 0 {p}, Σ d ) 1 { r s, r ( q t), s (q t)} 2 { s r, r ( q t), s (q t)} Hence, Σ 0 {p} t. MCS( Σ 0 {p q}, Σ d ) 1 { s r, r ( q t), s (q t)} 2 { r s, s (q t)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

49 Recapture Let Σ = Σ 0, Σ d. Where Σ 0 Σ d, Σ A iff Σ 0 Σ d A ( {, free, }) Reason: there is only one MCS, namely Σ d. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

50 An overview free Ded.Thm.(s) Ded.Thm.(d) Res.Thm(s) Res.Thm(d) CM(s) CM(d) Cut(s) Cut(d) fixed-point(s) fixed-point(d) RM(s) RM(d) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

51 From MCSs to Semantic Selections We now suppose that L has an adequate semantics with no inconsistent models. Σ A iff Σ A (where Σ A iff for all M M(Σ), M = A) M M({ }) = for all of the form F Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

52 From MCSs to Semantic Selections Given Σ = Σ 0, Σ d, where M M(Σ 0 ), d(m) = {A Σ d M = A} M m (Σ) = {M M(Σ 0 ) there are no M M(Σ 0 ) for which d(m) d(m )} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

53 Lemma 1 Σ 0 Ξ is L-consistent iff M(Σ 0 Ξ) 2 MCS(Σ 0, Σ d ) = {d(m) M M m ( Σ 0, Σ d )} Proof of 1 Σ 0 Ξ is inconsistent iff Σ 0 Ξ iff Σ 0 Ξ iff M(Σ 0 Ξ) = Proof of 2 Ad 2: ( ) Suppose Ξ MCS(Σ 0, Σ d ). Assume there is no M M(Σ 0 ) for which d(m) Ξ. Hence, M(Σ 0 Ξ) = and thus Σ 0 Ξ. Since then Σ 0 Ξ : contradiction to ( ). Assume there is a M M(Σ 0 ) for which d(m) Ξ. Then, M(Σ 0 d(m)) and hence Σ 0 d(m). Since then Σ 0 d(m) : contradiction to ( ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

54 From MCSs to Semantic Selections Definition, where Σ = Σ 0, Σ d Σ m A iff for all M M m (Σ), M = A Universal Consequence: Semantic Selections, where Σ = Σ 0, Σ d Proof Σ A, iff Σ A iff Σ m A for all Ξ MCS(Σ), Σ 0 Ξ A, iff for all Ξ MCS(Σ), Σ 0 Ξ A, iff for all Ξ MCS(Σ) and for all M M(Σ 0 Ξ), M = A, iff for all M M m (Σ), M = A, iff Σ m A. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

55 Definition, where Σ = Σ 0, Σ d Σ f A iff for all M M f (Σ), M = A where M f (Σ) = {M M(Σ 0 ) A d(m) iff there is a M M m (Σ) such that A d(m )}. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

56 Free Consequence: Semantic Selections, where Σ = Σ 0, Σ d Proof Σ free A iff Σ f A Σ free A, iff Σ 0 Free(Σ 0, Σ d ) A, iff Σ 0 MCS(Σ) A, iff Σ 0 MCS(Σ) A, iff for all M M(Σ 0 MCS(Σ)), M = A, iff for all M M(Σ 0 ) such that d(m) MCS(Σ), M = A, iff for all M M(Σ 0 ) such that d(m) {d(m ) M M m (Σ)}, M = A, iff Σ f A. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

57 From MCSs to Semantic Selections Definition: where Σ = Σ 0, Σ d Σ e A iff there is a M M m (Σ) such that for all M M m (Σ) for which d(m ) = d(m), M = A. Existential Consequence Σ A iff Σ e A Proof Σ A, iff, there is a Ξ MCS(Σ) such that Σ 0 Ξ A, iff, there is a M M m (Σ) such that Σ 0 d(m) A, iff, there is a M M m (Σ) such that for all M M m (Σ), M = A (note that there are no M M(Σ 0 ) for which d(m ) d(m)!) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

58 Adaptive Logics The basic idea: semantics 1 Take a core logic L (the lower limit logic ) that is a Tarski Logic (its consequence relation is reflexive, transitive and monotonic) supraclassical (it has a classical and a classical ) 2 declare some logic form as being abnormal : formulas of that form are abnormalities let Ω be the set of all formulas that have the abnormal form 3 define a consequence relation by means of selecting L-models that are sufficiently normal i.e., that validate not too many abnormalities Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

59 Inconsistency-adaptive logics 1. Lower limit logic: paraconsistent logic CLuN positive fragment of classical logic and A A incl. classical negation (A 1) A (B A) (A 2) (A (B C)) ((A B) (A C)) (A 3) ((A B) A) A (A 1) (A B) A (A 2) (A B) B (A 3) A (B (A B)) (A 1) A (A B) (A 2) B (A B) (A 3) (A C) ((B C) ((A B) C)) (A 1) (A B) (A B) (A 2) (A B) (B A) (A 3) (A B) ((B A) (A B)) (A 1) (A A) A (A 2) A ( A B) (A 3) A A (A 3) A A If we add de Morgan and double-negation intro/elim then we get CLuNs. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

60 Inconsistency-adaptive logics 2. Abnormalities -contradictions: A A Ω = {A A A L} 3. Interpret premises as normal as possible i.e., interpret (A A) as true as much as possible this allows for defeasible disjunctive syllogism: If A B and A and we assume (A A), then B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

61 Example: Let Γ = {p q, r s, p, r, r}. We cannot assume (r r) since r r is derivable. So, s cannot be derived via disjunctive syllogism. However, we can safely assume (p p). Thus, we can defeasibly apply disjunctive syllogism and derive q. Semantically, we choose models which minimise contradictions : models that validate p p are sorted out for all remaining models M: M = (p p) M = p M = p M = p q M = q Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

62 More complicated example Let Γ = {p s, r s, p, r, r p}. Now we can derive the minimal disjunction of contradictions: (r r) (p p). Should s be derivable? remember: floating conclusion different reasoning strategies Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

63 Inconsistency-adaptive logics Option 1: minimal abnormality strategy Where Γ L, where M is a L-model, Ω(M) = {A Ω M = A} M m.a. L (Γ) = {M M L (Γ) there is no M M L (Γ) for which Ω(M ) Ω(M)} Γ m.a. A iff for all M M m.a. L (Γ), M = A. Note Γ m.a. A iff Γ, Ω A Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

64 Example: Minimal abnormality Γ = {p s, r s, p, r, r p} model p p r r s s M M M M M M Thus, Γ m.a. s. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

65 Inconsistency-adaptive logics Option 2: reliability strategy Where Γ L, M rel L (Γ) = {M M L(Γ) A Ω(M) iff there is a M M m.a. L (Γ) such that A Ω(M )} Γ rel L A iff for all M Mrel, M = A. L Note Γ rel L A iff Γ, Ω free A Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

66 Example: Reliability Γ = {p s, r s, p, r, r p} model p p r r s s M M M M M M Thus, Γ rel. s. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

67 Dynamic Proof Theories for Adaptive Logics line number justification l A R; l, l We have 3 generic rules: formula assumption / condition Premise introduction (PREM) Where A Γ, introduce A on the empty condition. Unconditional Rule (RU) Where A 1,..., A n L B, l 1 A 1 J l n A n J n n l B RU;l 1,..., l n 1... n Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

68 Dynamic Proof Theories for Adaptive Logics Conditional Rule (RC) Where A 1,..., A n L B Θ and Θ Ω, l 1 A 1 J l n A n J n n l B RC;l 1,..., l n 1... n Θ Example: Γ = {p s, p, p}. 1 p s PREM 2 p PREM 3 s 1,2;RC {p p} 4 p PREM 5 p p 2,4;RU This calls for retraction. Differs with the strategy! Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

69 Dynamic Proof Theories for Adaptive Logics, Retraction The unreliable abnormalities Let U s = { Ω is derived on the condition at some line l and there is no such that is derived on the condition at stage s members of U s are the unreliable abnormalities at stage s Marking: reliability strategy Line l with condition is marked at stage s of a proof, iff U s. Example 1 p s PREM 2 p PREM 3 s 1,2;RC {p p} 4 p PREM 5 p p 2,4;RU Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

70 More complicated example Γ = {p s, r s, p, r, r p}. 1 p s PREM 2 p PREM 3 s 1,2;RC {p p} 4 r p PREM 5 r PREM 6 (p p) (r r) 2,4,5;RU 7 r s PREM 8 s 5,7;RC {r r} U 8 = {p p, r r} Minimal abnormality: multiple arguments matter either p p holds and r r not then the assumption of line 8 is OK or, vice versa, r r holds and p p not then the assumption of line 3 is OK Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

71 Final derivability A is finally derived at a line l at stage s of a proof from Γ iff 1 l is unmarked at stage s 2 for all extensions of the proof in which l is marked there is a further extension such that l is unmarked. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

72 Flip/Flop Problem: CLuNs 1 q PREM 2 p p PREM 3 q ( q q) 1;RU 4 q 3;RU 5 q r PREM 6 r 4,5;RC {q q} 7 p r 2,6;RU {q q} 8 p r 2;RU 9 (p r) 8;RU 10 (p r) (p r) 7,9;RU {q q} 11 [(p r) (p r)] [q q] 10;RA Solution: restrict abnormalities to contradictions in atoms. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

73 Topic 1 Plausible Reasoning 2 Preferential / Selection Semantics (KLM, Shoham) 3 Bibliography Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

74 Preferential / Selection Semantics We have seen already an example of these with Adaptive Logics. they have been systematically investigated by Shoham (Shoham (1987)) by Kraus, Lehmann and Magidor (Kraus et al. (1990)) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

75 The System P Various characterisations can be found, here s one of them: A B A C Left Logical Equ. B C Right Weakening A B C A C B Two perspectives Reflexivity A A 1 as a consequence relation And A B A C 2 as operator in the object A B C language ( conditional logics of Or A C B C normality ) A B C CM A B A C A B C Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

76 Some derived properties (S) [Ded.Thm.]: A B C implies A B C 1 Suppose A B C. 2 By Right Weakening, A B B C. 3 A (A B) (A B) 4 A B A B by Reflexivity and thus A B B C by Right Weakening 5 A B C by OR (applied to 2 and 4) and LLE (in view of 3). (Cut): A B and A B C implies A C. 1 Suppose A B and A B C. 2 Thus, A B C by S. 3 A B (B C) by AND (applied to 1 and 2) 4 A C by Right Weakening. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

77 The (Standard) Semantics of P: Preferential models W,, v where W is a set of points (worlds) v : W (Atoms) is an assignment is a strict partial order such that for each A, {w w = A} is smooth NOPE! Consequence relation: Preferential Closure let min (A) = {w W w = A and for all w W such that w = A, w w} M = A B iff for all w min (A), w = B A P B iff for all preferential models M, M = A B. Alternative: A P B iff A B is derivable from system P. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

78 Central characterisational result: is a preferential consequence relation iff there is a preferential model M that characterises it (ie., A B iff M = A B.) Task Build a preferential model W,, v that characterises a consequence relation that contains all of {p b, b f, p f } with the four worlds: world p f w w w w Since W and v are already determined, all you need to do is to specify, ie., to order the worlds in a good way. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

79 The System R, Rational consequence relations and models R is P plus Rational Monotonicity A C A B A B C Semantics W,, v where W,, v is a preferential model is a modular order (i.e., we have a ranking function) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

80 A Problem with Irrelevance Model M {p b, b f, p f } R p a f. M = p b M = p f M = b f but M = p a f. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

81 A Problem with Irrelevance In fact, despite RM, where K is a conditional knowledge base, K P A B iff K R A B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

82 Enters: Rational Closure Interpret worlds as normal as possible relative to and K. i.e., drop worlds as low as possible min (p a) min (p) thus: M = p a f Idea: order rational consequence relations according to how normal they interpret worlds and pick out the minimal one. Formally, where A < B iff A B B 1 2 iff 1 there is a (A, B) 2 \ 1 s.t. for all C for which C < 1 A and for all D s.t. C 1 D, also C 2 D. 2 for all C, D: if (C, D) 1 \ 2, there is a (A, B) 2 \ 1 s.t. A < 2 C. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

83 Enters: Rational Closure Formally: we write A < B iff A B B 1 2 iff 1 there is a (A, B) 2 \ 1 s.t. for all C for which C < 1 A and for all D s.t. C 1 D, also C 2 D. 2 for all C, D: if (C, D) 1 \ 2, there is a (A, B) 2 1 s.t. A < 2 C. 1 vs. 2 vs : take (p a, f ) 2 \ : take (p, a) 3 \ 3 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

84 Existence Lemma: Invariance under renaming Let f : Atoms Atoms be a bijection. 1 where 1 and 2 are rational consequence relations, 1 2 implies f ( 1 ) f ( 2 ). 2 f (RC(K)) = RC(f (K)) 3 if K = f (K) then RC(K) = f (RC(K)) (if RC(K) exists) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

85 RC does not always exist Take K = {p n p n+2, p n+2 p n n N N + 0 }. then p n+2 < p n for all n for all odd n:... p n+4 < p n+2 < p n <... for all even n:... p n+4 < p n+2 < p n <... Suppose we have the rational closure of K Four cases: 1 p k < p n for all odd k and all even n 2 p k < p n for all even k and all odd n 3 there are two even n, m and an odd k such that p n < p k < p m 4 there are two odd n, m and an even k such that p n < p k < p m However, Ad 1/2: let f (m) = m + 1. Then f (K) = K, but f ( ). { l l is odd Ad 3/4: let f (l) = Now p l m + n l is even m < p k in f ( ). Contradiction. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

86 Drowning Problem K Note that: 1 K P penguin haswings 2 K R penguin haswings 3 K RC penguin haswings Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

87 Drowning Problem Here are our possible worlds: w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 8 w 9 w 10 w 11 w 12 w 13 w 14 w 15 w p b w f The model of the rational closure Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

88 Swedes swedes blond swedes tall Should we conclude for the short swede Peter that he is blond? this is blocked in RC note that RM cannot be used to conclude this since: swedes short (where: short implies tall) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

89 Various Semantic Characterisations From the abstract point of view S = W, v, D,, π where π : (W ) D S = A B iff π([a B]) preferable to π([a B]) in view of. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

90 π S = A B iff... structures possibility measure π([a]) = 0 or possibilistic π : (W ) [0, 1] π([a B]) > π([a B]) ordinal ranking fkt. κ([a]) = or ordinal ranking κ : (W ) {0, 1,..., } κ([a B]) < κ([a B]) plausibility meas. Pl([A]) = or plausibility Pl : (W ) D Pl([A B]) > Pl([A B]) possibility measures: Dubois and Prade (1990) 0: impossible states, 1: necessary states ordinal ranking functions Goldszmidt and Pearl (1992) κ([a]): level of surprise if A were to hold maximal surprise plausibility measures Friedman and Halpern (1996) D partially ordered domain with and some simple constraints, e.g., Pl(X ) = Pl(Y ) implies Pl(X Y ) = result in qualitative plausibility structures. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

91 Various Semantic Characterisations Where K is a conditional knowledge base: 1 K P A B iff 2 S = A B for all preferential structures S which validate K iff 3 S = A B for all possibilistic structures S which validate K iff 4 S = A B for all ordinal ranking structures S which validate K iff 5 S = A B for all qualitative plausibility structures S which validate K Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

92 Probabilities, Naive idea: A B holds in a structure if P(B A) > τ where τ is a threshold value (e.g., 1 2 or 2 3, etc.). We loose properties! What about A C (cut)?! A B (A = being a Pennsylvanian Dutch, B = being a native speaker of German) A B C (C = being born in Germany) Take the distribution P(B A) = P(A B) P(A) = = 3 4 P(C A B) = P(A B C) P(A B) = = 2 3 P(C A) = P(A C) P(A) = = 1 2 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

93 Better idea: go for the limes ɛ-semantics, idea: (Adams (1975); Pearl (1989)) K A A B holds if P(B A) is converges to 1 Formally, for each ɛ ]0, 1] there is a δ ]0, 1] such that P(A B) 1 ɛ in all probability assignments that satisfy P(C D) 1 δ and P(C) > 0 for all C D K This results in yet another adequate semantics for P. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

94 Other probabilistic characterisations Big stepped probabilities A B iff P(B A) > 1 2 wait, but how? order W linearly by and request P({w}) > {P({w }) w w} De Finetti (coherence-based probabilities) take conditional probabilities as primitive Gilio (2002) similar: Popper-functions Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

95 Topic 1 Plausible Reasoning 2 Preferential / Selection Semantics (KLM, Shoham) 3 Bibliography Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96

96 Bibliography Adams, E. W.: 1975, The Logic of Conditionals. D. Reidel Publishing Co. Batens, D.: 2007, A Universal Logic Approach to Adaptive Logics. Logica Universalis 1, Brewka, G.: 1989, Preferred Subtheories: An Extended Logical Framework for Default Reasoning.. In: IJCAI, Vol. 89. pp Dubois, D. and H. Prade: 1990, An introduction to possibilistic and fuzzy logics. In: Readings in Uncertain Reasoning. pp Friedman, N. and J. Y. Halpern: 1996, Plausibility measures and default reasoning. Journal of the ACM 48, Gilio, A.: 2002, Probabilistic reasoning under coherence in System P. Annals of Mathematics and Artificial Intelligence 34(1-3), Goldszmidt, M. and J. Pearl: 1992, Rank-based Systems: A Simple Approach to Belief Revision, Belief Update, and Reasoning about Evidence and Actions. In: Proceedings of the Third International Conference on Knowledge Representation and Reasoning. pp Kraus, S., D. Lehman, and M. Magidor: 1990, Nonmonotonic Reasoning, Preferential Models and Cumulative Logics. Artifical Intelligence 44, Makinson, D.: 2003, Bridges between classical and nonmonotonic logic. Logic Journal of IGPL 11(1), Pearl, J.: 1989, Probabilistic semantics for nonmonotonic reasoning: a survey. In: Proceedings of the first international conference on Principles of knowledge representation and reasoning. San Francisco, CA, USA, pp Rescher, N. and R. Manor: 1970, On inference from inconsistent premises. Theory and Decision 1, Shoham, Y.: 1987, A Semantical Approach to Nonmonotonic Logics. In: M. L. Ginsberg (ed.): Readings in Non-Monotonic Reasoning. Los Altos, CA: Morgan Kaufmann, pp Straßer, C.: 2014, Adaptive Logic and Defeasible Reasoning. Applications in Argumentation, Normative Reasoning and Default Reasoning., Vol. 38 of Trends in Logic. Springer. Van De Putte, F. and C. Straßer: 2012, Extending the Standard Format of Adaptive Logics to the Prioritized Case. Logique at Analyse 55(220), Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, / 96