Soundness Γ ⊢ D implies Γ � D . Proof. ◮ Take an arbitrary proof of D from Γ. Let M be an arbitrary model of Γ. ◮ We proof by induction over the length of the proof that for each formula E derived at a line l , M | = E . ◮ “ l = 1”: E is either (i) an axiom or (ii) E ∈ Γ. (ii) is trivial. Suppose (i). Suppose E = A ⊃ ( B ⊃ A ) (see ( A ⊃ 1)). Let M be a model of Γ. M | = A ⊃ ( B ⊃ A ) iff v M ( A ⊃ ( B ⊃ A )) = 1 iff max( v M (1 − v M ( A ) , max(1 − v M ( B ) , v M ( A )))) = 1. Note that the latter holds. The proof is similar for the other axioms. ◮ “ l ⇒ l +1”: Either (i) E is an axiom, or (ii) E ∈ Γ or (iii) E is derived via MP from F ⊃ E and F where F ⊃ E and F are derived at lines l ′ ≤ l and l ′′ ≤ l resp. Only (iii) is non-trivial. By the induction hypothesis, M | = F and M | = F ⊃ E . Thus, v M ( F ) = v M ( F ⊃ E ) = max(1 − v M ( F ) , v M ( E )) = 1. Thus, M | = E . 31/111
Completeness Γ � A implies Γ ⊢ A . Proof: we need some preparation for that. 32/111
◮ A set Γ is inconsistent iff Γ ⊢ A for all A ∈ W . ◮ Γ is consistent iff Γ is not inconsistent. Proposition 1 If Γ � A then Γ ∪ {¬ A } is consistent. Proof. ◮ Suppose Γ ∪ {¬ A } is inconsistent. ◮ Hence, Γ ∪ {¬ A } ⊢ ¬¬ A . ◮ Hence, by the deduction theorem, Γ ⊢ ¬ A ⊃ ¬¬ A . ◮ By ( A ¬ 1) and MP, Γ ⊢ ¬¬ A . ◮ Since ¬¬ A ⊢ A (see above), Γ ⊢ A . 33/111
Explosion in Hilbert Show { A , ¬ A } ⊢ B . Tip: use ( A ¬ 2) and MP. 34/111
Proposition 2 If Γ is consistent then there is a model of Γ . Proof. ◮ Let W be enumerated by A 1 , A 2 , . . . . ◮ Let Γ 0 = Γ and define � Γ i ∪ { A i } if Γ i ∪ { A i } is consistent Γ i +1 = Γ i ∪ {¬ A i } else. ◮ Let Γ ∗ = � i ≥ 0 Γ i . ◮ Claim: Γ ∗ is consistent. Proof: by induction. IB: Γ 0 is consistent by supposition. “ i → i +1”: by definition of Γ i +1 . Hence, each Γ i is consistent. Assume Γ ∗ is inconsistent. Hence, Γ ∗ ⊢ A 1 and Γ ∗ ⊢ ¬ A 1 . Hence (by compactness), there is a Γ i such that Γ i ⊢ A 1 and there is a Γ j such that Γ j ⊢ ¬ A 1 . Take k = max( i , j ). Then (by monotonicity), Γ k ⊢ A 1 and Γ k ⊢ ¬ A 1 . Hence, Γ k is inconsistent (since A , ¬ A ⊢ B ). 35/111
◮ Claim: Γ ∗ ⊢ A implies A ∈ Γ ∗ ( Deductive Closure ). Suppose Γ ∗ ⊢ A and assume A / ∈ Γ ∗ . Hence, ¬ A ∈ Γ ∗ . But then Γ ∗ ⊢ A and Γ ∗ ⊢ ¬ A and hence Γ ∗ is inconsistent,—a contradiction. ◮ Claim: B , C ∈ Γ ∗ iff B ∧ C ∈ Γ ∗ . Suppose B , C ∈ Γ ∗ . Hence, by ResThm and ( A ∧ 3), Γ ∗ ⊢ B ∧ C . Hence, B ∧ C ∈ Γ ∗ . Suppose B ∧ C ∈ Γ ∗ . Hence, Γ ∗ ⊢ B (by ( A ∧ 1)). Thus B ∈ Γ ∗ . ◮ Claim: B ∨ C ∈ Γ ∗ iff ( B ∈ Γ ∗ or C ∈ Γ ∗ ). Let B ∨ C ∈ Γ ∗ . ∈ Γ ∗ and Since B ∨ C ⊢ ¬ B ⊃ C , ¬ B ⊃ C ∈ Γ ∗ . Suppose B / hence ¬ B ∈ Γ ∗ . Hence, by MP, C ∈ Γ ∗ . Now suppose B ∈ Γ ∗ . Since B ⊃ ( B ∨ C ) also B ∨ C ∈ Γ ∗ . The case for C ∈ Γ ∗ is analogous. ◮ Claim: A ∈ Γ ∗ iff ¬ A / ∈ Γ ∗ . Suppose A ∈ Γ ∗ . Assume ¬ A ∈ Γ ∗ , then Γ ∗ is inconsistent,—a contradiction. The other way is analogous. 36/111
◮ Define M via the assignment ◮ We show by induction over the length of a formula that = A iff A ∈ Γ ∗ . M | ◮ A ∈ A : by definition. ◮ Let B , C be such that M | = B [ C ] iff B [ C ] ∈ Γ ∗ . ◮ Let A = B ∧ C . Let B ∧ C ∈ Γ ∗ . Hence, B , C ∈ Γ ∗ . Hence, by the induction hypothesis, M | = B and M | = C . Hence, M | = B ∧ C . The other way around is analogous. ◮ Let A = B ∨ C . Let B ∨ C ∈ Γ ∗ . Hence, B ∈ Γ ∗ or C ∈ Γ ∗ . By the induction hypothesis, M | = B or M | = C . Hence, M | = B ∨ C . The other way around is analogous. ◮ Let A = ¬ B . Let ¬ B ∈ Γ ∗ . Hence, B / ∈ Γ ∗ . By the induction hypothesis, M �| = B and hence M | = ¬ B . The other way around is analogous. ◮ etc. 37/111
Completeness: Γ � A implies Γ ⊢ A . Proof. 38/111
Abnormalities ◮ They determine the normality assumptions by means of which the AL strengthens the LLL. ◮ In other words, they determine what is means to interpret premises “normal”. (we will come to the “as possible part” later) ◮ characterized by a logical form F ◮ the set of all abnormalities denoted by Ω . . . in our example . . . ◮ recall: the normality assumption was that if ◦ A then A is the case ◮ hence, Ω = {◦ A ∧ ¬ A } 39/111
3. The adaptive strategy effects both, 1. the proof theory, and 2. the semantics Let’s start with the proof theory. 40/111
Adaptive proofs 41/111
Adaptive proofs: the generic rules ◮ PREM ises are introduced on the empty condition (no normality assumption is needed for that) . . . . . . If A ∈ Γ : (PREM) A ∅ ◮ the U nconditional R ule: A 1 ∆ 1 . . . . . . If A 1 , . . . , A n ⊢ LLL B : (RU) A n ∆ n B ∆ 1 ∪ · · · ∪ ∆ n These two rules give us the full power of the LLL: If Γ ⊢ LLL A , then Γ ⊢ AL A . 42/111
Adaptive proofs: the conditional rule A 1 ∆ 1 . . . . . . If A 1 , . . . , A n ⊢ LLL B ˇ ∨ Dab(Θ) : A n ∆ n B ∆ 1 ∪ · · · ∪ ∆ n ∪ Θ (RC) ◮ Dab(Θ) is a notational convention that denotes the disjunction of abnormalities in Θ, ◮ where Θ ⊆ Ω is a finite set of abnormalities ◮ as in RU, the conditions of the used lines (∆ 1 , . . . , ∆ n ) are carried forward The rational of RC: From A 1 , . . . , A n follows by the LLL that either B is true or one of the abnormalities in Θ is true. The AL allows us to conditionally derive B under the assumption that neither of the abnormalities in Θ is true. 43/111
Time for examples . . . ◮ recall: ◦ A ⊢ CL ◦ A ∨ ( ◦ A ∧ ¬ A ) ◮ a conditional derivation by means of RC: 1 ◦ A PREM ∅ � � ◦ A ∧ ¬ A 2 A 1;RC ◮ also: {◦ A , A ⊃ B } ⊢ CL ◦ B ∨ ( ◦ A ∧ ¬ A ) 3 A ⊃ B PREM ∅ � � 4 B 1,3;RC ◦ A ∧ ¬ A Now what if we also have ◦¬ A ? 44/111
Marking of lines in adaptive proofs Example 1: a simple case of marking 1 ◦ A PREM ∅ � � 6 2 A 1;RC ◦ A ∧ ¬ A 3 A ⊃ B PREM ∅ � � 6 4 B 1,3;RC ◦ A ∧ ¬ A ¬ A ∅ 5 PREM 6 ◦ A ∧ ¬ A 1,5;RU ∅ ◮ the condition has been derived (on the empty condition) ◮ so –obviously– it’s not save anymore to assume that ◦ A ∧ ¬ A is not the case (resp. that ◦ A implies A ) ◮ thus, mark all lines with this assumption 45/111
Marking of lines in adaptive proofs Example 2: a more complex case of marking 1 ◦ A PREM ∅ � � 7 2 ◦ A ∧ ¬ A A 1;RC 3 A ⊃ B PREM ∅ � � 7 4 ◦ A ∧ ¬ A B 1,3;RC 5 ¬ A ∨ ¬ B PREM ∅ ◦ B ∅ 6 PREM 7 ( ◦ A ∧ ¬ A ) ∨ ( ◦ B ∧ ¬ B ) 1,5,6;RU ∅ ◮ here ◦ A ∧ ¬ A is part of a disjunction of abnormalities that has been derived on the empty condition (line 7) ◮ note that the formula at line 7 is a Dab-formula ◮ this disjunction is minimal : right now we have no means to decide whether ◦ A ∧ ¬ A or ◦ B ∧ ¬ B is the case (or even both) ◮ hence, we’re cautious and mark lines that intersect with members of the minimal disjunction of abnormalities on line 7 46/111
Adaptive strategies and marking ◮ the specifics of the marking definition of an AL depend on the adaptive strategy that is used ◮ there are two standard strategies 1. reliability strategy 2. minimal abnormality strategy ◮ we write AL r for an AL characterized by a triple � LLL , Ω , r eliability � ◮ we write AL m for an AL characterized by a triple � LLL , Ω , m inimal abnormality � 47/111
The reliability strategy: marking ◮ a stage of a proof is a list of consecutive lines ◮ where Dab(∆ 1 ) , Dab(∆ 2 ) , . . . are the minimal disjunctions of abnormalities that are derived at some stage s on the empty condition from the premise set Γ, let Σ s (Γ) = df { ∆ 1 , ∆ 2 , . . . } ◮ the set of unreliable formulas at stage s is defined by U s (Γ) = df ∆ 1 ∪ ∆ 2 ∪ . . . = � Σ s (Γ) Marking definition for the reliability strategy A line l with condition ∆ is marked at stage s iff ∆ ∩ U s (Γ) � = ∅ . ◮ in words : a line is marked iff its condition contains unreliable formulas. ◮ put differently : a line is marked if its condition contains formulas that are part of minimal disjunctions of abnormalities ◮ lets take a look at our examples . . . 48/111
Example 1: a simple case of marking 1 ◦ A PREM ∅ � � 6 2 A 1;RC ◦ A ∧ ¬ A A ⊃ B ∅ 3 PREM � � 6 4 B 1,3;RC ◦ A ∧ ¬ A ¬ A ∅ 5 PREM 6 ◦ A ∧ ¬ A 1,5;RU ∅ ◮ Σ 6 (Γ) = ◮ U 6 (Γ) = 49/111
Example 2: a more complex case of marking 1 ◦ A PREM ∅ � � 7 2 ◦ A ∧ ¬ A A 1;RC 3 A ⊃ B PREM ∅ � � 7 4 ◦ A ∧ ¬ A B 1,3;RC 5 ¬ A ∨ ¬ B PREM ∅ 6 ◦ B PREM ∅ 7 ( ◦ A ∧ ¬ A ) ∨ ( ◦ B ∧ ¬ B ) 1,5,6;RU ∅ ◮ Σ 7 (Γ) = ◮ U 7 (Γ) = 50/111
1 ◦ A PREM ∅ � � 2 A 1;RC ◦ A ∧ ¬ A 3 A ⊃ B PREM ∅ � � ◦ A ∧ ¬ A 4 B 1,3;RC 5 ¬ A ∨ ¬ B PREM ∅ ◦ B ∅ 6 PREM 7 ( ◦ A ∧ ¬ A ) ∨ ( ◦ B ∧ ¬ B ) 1,5,6;RU ∅ 8 ¬ B PREM ∅ 9 ◦ B ∧ ¬ B 6,8;RU ∅ ◮ Σ 7 (Γ) = {{ ( ◦ A ∧ ¬ A ) , ( ◦ B ∧ ¬ B ) }} ◮ U 7 (Γ) = {◦ A ∧ ¬ A , ◦ B ∧ ¬ B } ◮ note that at line 9 the formula at stage 7 looses its status of being a minimal(!) Dab-formula ◮ Σ 9 (Γ) = ◮ U 9 (Γ) = 51/111
Markings come and go: lines which are unmarked may be marked at a later stage, and be unmarked again at an even later stage. BUT: when does a formula count as a consequence of the AL? Final derivability A formula A is finally derived at line l at a finite stage s iff (i) l is unmarked at stage s and (ii) for every extension of the proof in which l is marked, there is a further extension in which l is unmarked. (Note: the extensions in question may be infinite.) The adaptive derivability relation ⊢ AL and the adaptive consequence set Cn AL Γ ⊢ AL A iff there is an AL -proof from Γ in which A is finally derived. Cn AL (Γ) = { A | Γ ⊢ AL A } 52/111
Some nice properties of the consequence/derivability relation � Σ s (Γ) where Σ s (Γ) = df { ∆ | Dab(∆) is a ◮ recall: U s (Γ) = df minimal Dab-formula at stage s } ◮ let Σ(Γ) be the set of all ∆ for which Γ ⊢ LLL Dab(∆) and for all ∆ ′ ⊂ ∆, Γ �⊢ LLL Dab(∆ ′ ). � Σ(Γ) ◮ let U (Γ) = df Theorem 1 Γ ⊢ AL r A iff there is a ∆ ⊆ Ω for which Γ ⊢ LLL A ˇ ∨ Dab(∆) and ∆ ∩ U (Γ) = ∅ . 53/111
Another example Suppose a reliable although not infallible witness report that ◮ Mr. X wore a long black coat in the bar in which he was seen half an hour before the murder. — ◦ l Another reliable although not infallible source however witnesses that ◮ Mr. X wore a short dark blue jacket and black trousers at the same time. — ◦ j Obviously ¬ ( l ∧ j ), since both cannot be the case. Moreover, we have ◮ If Mr. X was dressed in a long black coat, then he was dressed in a dark way. — l ⊃ m ◮ If Mr. X was dressed in a short dark blue jacket and black trousers, then he was dressed in a dark way. — j ⊃ m 54/111
1 ◦ l PREM ∅ ◦ j ∅ 2 PREM 3 ¬ ( l ∧ j ) PREM ∅ l ⊃ m ∅ 4 PREM 5 j ⊃ m PREM ∅ � � 10 6 l 1; RC ◦ l ∧ ¬ l � � ? 7 m 4, 6; RU ◦ l ∧ ¬ l � � 10 8 j 2; RC ◦ j ∧ ¬ j � � ? 9 m 5, 8; RU ◦ j ∧ ¬ j 10 ( ◦ l ∧ ¬ l ) ∨ ( ◦ j ∧ ¬ j ) 1,2,3; RU ∅ ◮ according to the reliability strategy lines 7 and 9 are marked ◮ the rationale is: since ( ◦ l ∧ ¬ l ) ∨ ( ◦ j ∧ ¬ j ) is a minimal Dab-formula, one of the two abnormalities is the case or even both ◮ in case both are the case, neither of the lines 7 and 9 is safe 55/111
1 ◦ l PREM ∅ 2 ◦ j PREM ∅ 3 ¬ ( l ∧ j ) PREM ∅ 4 l ⊃ m PREM ∅ 5 j ⊃ m PREM ∅ � � 10 6 ◦ l ∧ ¬ l l 1; RC � � ? 7 m 4, 6; RU ◦ l ∧ ¬ l � � 10 8 ◦ j ∧ ¬ j j 2; RC � � ? 9 m 5, 8; RU ◦ j ∧ ¬ j 10 ( ◦ l ∧ ¬ l ) ∨ ( ◦ j ∧ ¬ j ) 1,2,3; RU ∅ ◮ another rationale: interpreting the premises as normally as possible means that we assume that as less abnormalities as possible are the case ◮ for the disjunction at line 10 this means that we assume that only one of the two abnormalities is the case (we don’t know which one though) ◮ however, then at least one of the two assumptions at line 7 and 9 can be considered as safe and thus it is still (defeasibly) warranted to infer m 56/111
The minimal abnormality strategy ◮ Recall: Σ s (Γ) = df { ∆ | Dab(∆) is a minimal Dab-formula at stage s } . ◮ Φ s (Γ) is the set of all minimal choice sets of Σ s (Γ) ◮ a choice set of { ∆ i | i ∈ I } is a set that contains a member of each ∆ i ( i ∈ I ) ◮ ϕ is a minimal choice set of iff there is no choice set ϕ ′ such that ϕ ′ ⊂ ϕ ◮ example: Let S = {{ 1 , 2 } , { 2 , 3 }} . ◮ { 1 } is not a choice set of S since { 1 } ∩ { 2 , 3 } = ∅ ◮ { 1 , 2 } is a choice set of S ◮ { 1 , 3 } and { 2 } are the minimal choice sets of S ◮ each set ϕ ∈ Φ s (Γ) offers a minimally abnormal interpretation of the given premises resp. minimal Dab-formulas according to the current stage of the proof s . By minimally abnormal we mean that as few abnormalities as possible are interpreted as true. 57/111
Marking for minimal abnormality A line l with conditions ∆ and formula A is marked at stage s iff (i) there is no ϕ ∈ Φ s (Γ) such that ϕ ∩ ∆ = ∅ or (ii) for some ϕ ∈ Φ s (Γ) there is no line at which A is derived on a condition Θ for which Θ ∩ ϕ = ∅ . what does this mean, intuitively . . . ◮ condition (i) expresses that the assumption ∆ on which A is derived is not warranted in any minimally abnormal interpretation offered by Φ s (Γ), since in each ϕ ∈ Φ s (Γ) there is an abnormality that is also in ∆ and since the assumption expressed by the condition ∆ is that no abnormality in ∆ is true. ◮ condition (ii) expresses that there is a minimally abnormal interpretation ϕ ∈ Φ s (Γ) such that A is not derived under any condition that is warranted in ϕ . 58/111
So, when is a line unmarked according to minimal abnormality? A line l with formula A and condition ∆ us not marked at stage s iff (i) there is a ϕ ∈ Φ s (Γ) such that ∆ ∩ ϕ = ∅ and (ii) for each ϕ ∈ Φ s (Γ) there is a ∆ ϕ such that ∆ ϕ ∩ ϕ = ∅ and A is derived on the condition ∆ ϕ at stage s . What does this mean, intuitively . . . ◮ condition (i) expresses that there is a minimally abnormal interpretation ϕ ∈ Φ s (Γ) in which the assumption ∆ is warranted ◮ condition (ii) expresses that for each minimally abnormal interpretation ϕ ∈ Φ s (Γ) our A is derived on an assumption ∆ ϕ that is warranted in ϕ 59/111
◦ l ∅ 1 PREM 2 ◦ j PREM ∅ ¬ ( l ∧ j ) ∅ 3 PREM 4 l ⊃ m PREM ∅ 5 j ⊃ m PREM ∅ � � 10 6 l 1; RC ◦ l ∧ ¬ l � � 7 m 4, 6; RU ◦ l ∧ ¬ l � � 10 8 j 2; RC ◦ j ∧ ¬ j � � 9 m 5, 8; RU ◦ j ∧ ¬ j ( ◦ l ∧ ¬ l ) ∨ ( ◦ j ∧ ¬ j ) ∅ 10 1,2,3; RU ◮ Σ 10 (Γ) = {{ ( ◦ l ∧ ¬ l ) , ( ◦ j ∧ ¬ j ) }} ◮ Φ 10 (Γ) = {{◦ l ∧ ¬ l } , {◦ j ∧ ¬ j }} ◮ lines 6 and 8 are marked since they violate condition (ii) ◮ lines 7 and 9 are not marked: ◮ concerning (i): there is a minimal choice set with which the condition has empty intersection ◮ concerning (ii): there is no choice set that intersects with both conditions, {◦ l ∧ ¬ l } and {◦ j ∧ ¬ j } 60/111
Floating conclusions and the adaptive strategies Floating conclusion A is a floating conclusion in case it is reach be various conflicting arguments. ◮ reliability blocks the floating conclusion m from Γ = {◦ l , ◦ j , ¬ ( l ∧ j ) , l ⊃ m , j ⊃ m } : Γ �⊢ CL r ◦ m ◮ minimal abnormalities derives the floating conclusion: Γ ⊢ CL m ◦ m 61/111
Some nice property ◮ recall: Φ s (Γ) is the set of minimal choice sets of Σ s (Γ) ◮ Σ(Γ) is the set of all ∆ for which Γ ⊢ LLL Dab(∆) and for all ∆ ′ ⊂ ∆, Γ �⊢ LLL Dab(∆ ′ ). ◮ Let Φ(Γ) be the set of minimal choice sets of Σ(Γ) Theorem 2 Γ ⊢ AL m A iff for every ϕ ∈ Φ(Γ) there is a ∆ ⊆ Ω for which ∆ ∩ ϕ = ∅ and Γ ⊢ LLL A ˇ ∨ Dab(∆) . 62/111
Final derivability revisted? ◮ recall: A formula A is finally derived at a finite stage s at line l iff (i) l is unmarked at stage s and (ii) for every extension of the proof in which l is marked, there is a further extension in which l is unmarked. ◮ say: A formula A is finally *-derived at a finite stage s at line l iff (i) l is unmarked at stage s and (ii) for every finite extension of the proof in which l is marked there is a finite further extension in which l is not marked. ◮ Let Γ ⊢ ∗ AL A iff there is a proof in which A is finally *-derived. Does this work? Γ ⊢ AL A iff Γ ⊢ ∗ AL A . Nope E.g. Γ = { ( ◦ A i ∧ ¬ A i ) ∨ ( ◦ A j ∧ ¬ A j ) | j > i > 0 } ∪ { B ∨ A i | i > 1 } . Here Γ �⊢ AL B while Γ ⊢ ∗ AL B . 63/111
Diderik Batens. A universal logic approach to adaptive logics. Logica Universalis , 1:221–242, 2007. Diderik Batens. Towards a dialogic interpretation of dynamic proofs. In C´ edric D´ egremont, Laurent Keiff, and Helge R¨ uckert, editors, Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman , pages 27–51. College Publications, London, 2009. Peter Verd´ ee. Adaptive logics using the minimal abnormality strategy are π 1 1 -complex. Synthese , 167:93–104, 2009. 64/111
Semantics for Adaptive Logics: The basic idea ◮ Take the set of LLL -models of a premise set Γ ◮ order them according to their abnormal part , i.e. Ab ( M ) = { A ∈ Ω | M | = A } M 6 M 4 M 5 M 3 M 1 M 2 ◮ in flat adaptive logics in standard format this is done by means of: M 1 ≺ M 2 iff Ab ( M 1 ) ⊂ Ab ( M 2 ) ◮ select models that are beyond a certain threshold M 6 M 4 M 5 M 3 M 1 M 2 65/111
What threshold? the threshold depends on the strategy: Minimal Abnormality ◮ Idea: take the minimally abnormal models Reliability ◮ M ∈ M AL m (Γ) iff ◮ Idea: take models whose M ∈ M LLL (Γ) and for all M ′ ∈ M LLL (Γ), if abnormal part only consists of unreliable abnormalities Ab ( M ′ ) ⊆ Ab ( M ) then ◮ we call this models “reliable” Ab ( M ′ ) = Ab ( M ). ◮ M ∈ M AL r (Γ) iff M 6 M ∈ M LLL (Γ) and M 4 M 5 Ab ( M ) ⊆ U (Γ) M 3 M 1 M 2 66/111
Let’s go back to our example... ◮ Γ = { ♦ l , ♦ j , ¬ ( l ∧ j ) , l ⊃ m , j ⊃ m } ◮ Γ ⊢ T ( ◦ l ∧ ¬ l ) ∨ ( ◦ j ∧ ¬ j ) ◮ Γ �⊢ T ◦ l ∧ ¬ l , ◮ Γ �⊢ T ◦ j ∧ ¬ j ◮ hence, U (Γ) = {◦ l ∧ ¬ l , ◦ j ∧ ¬ j } ◮ we have for instance the following models M 1 , . . . , M 6 where ◮ Ab ( M 1 ) = {◦ l ∧ ¬ l } , ◮ Ab ( M 2 ) = {◦ j ∧ ¬ j } , ◮ Ab ( M 3 ) = {◦ l ∧ ¬ l , ◦ j ∧ ¬ j } , ◮ Ab ( M 4 ) = {◦ l ∧ ¬ l , ◦ k ∧ ¬ k } ◮ Ab ( M 5 ) = {◦ j ∧ ¬ j , ◦ o ∧ ¬ o } ◮ Ab ( M 6 ) = {◦ l ∧ ¬ l , ◦ j ∧ ¬ j , ◦ k ∧ ¬ k , ◦ o ∧ ¬ o } ◮ models M 1 and M 2 are minimally abnormal ◮ models M 1 , M 2 , and M 3 are reliable 67/111
M 6 M 6 M 6 M 4 M 5 M 4 M 5 M 4 M 5 M 3 M 3 M 3 M 1 M 2 M 1 M 2 M 1 M 2 (a) (b) (c) (a) the ordering of the models according to their abnormal part (b) the threshold for the reliable models (c) the threshold for the minimal abnormal models Note that every reliable model is minimal abnormal: M AL m (Γ) ⊆ M AL r (Γ) 68/111
Is the ordering of models smooth? The danger: infinite descending chains without minima M 6 M 4 M 5 M 3 M 1 M 2 ∞ ∞ ◮ w.r.t. the infinite chains without minima there are no minimally abnormal models ◮ e.g. if there are only infinite chains without minima there are no minimally abnormal models: Γ | = AL ⊥ (although there are LLL -models of Γ and hence Γ �| = LLL ⊥ ) 69/111
Smoothness and Reassurance ◮ a partial order � X , ≺� is well-founded iff there are no infinitely descending chains. ◮ a partial order � X , ≺� is smooth (resp. stoppered) iff for each x ∈ X there is a minimal element y ∈ x such that y ≺ x or y = x ◮ what we need is: �{ Ab ( M ) | M ∈ M LLL (Γ) } , ⊂� is smooth. (Note it may be smooth but not well-founded (e.g. invert the order on the natural numbers)) Theorem 3 1. For every LLL -model M of Γ , M is minimally abnormal or there is an LLL -model M ′ of Γ such that Ab ( M ′ ) ⊂ Ab ( M ) and M ′ is minimally abnormal. 2. �{ Ab ( M ) | M ∈ M LLL (Γ) } , ⊂� is smooth. 3. If Γ has LLL -models, then there are minimally abnormal models of Γ . 4. If Γ �| = LLL ⊥ then Γ �| = AL ⊥ . 70/111
Simple facts about choice sets Let in the following Σ = Σ(Γ). Fact 4 Where ϕ is a choice set of Σ and A ∈ ϕ : If A satisfies there is a ∆ ∈ Σ : ϕ ∩ ∆ = { A } ( † ) then ϕ \ { A } is not a choice set of Σ . Fact 5 Where ϕ is a choice set of Σ and A ∈ ϕ : If A doesn’t satisfy ( † ) then ϕ \ { A } is also a choice set of Σ . Fact 6 Where ϕ is a choice set of Σ : each A ∈ ϕ satisfies ( † ) iff ϕ is a minimal choice set of Σ . 71/111
Lemma 7 ϕ = � Where ϕ = { A 1 , A 2 , . . . } is a choice set of Σ let ˆ i ∈ N ϕ i where ϕ 1 = ϕ and � ϕ i if there is a ∆ ∈ Σ s.t. ϕ i ∩ ∆ = { A i } ϕ i +1 = ϕ i \ { A i } else ϕ is a minimal choice set of Σ . ˆ Proof. ◮ note that ϕ i is a choice set of Σ for each i ∈ N ◮ Assume for some ∆ ∈ Σ, ˆ ϕ ∩ ∆ = ∅ . Note that since ∆ is finite ∆ ∩ ϕ 1 = { B 1 , . . . , B n } for some n ∈ N . Assume there no B j s.t. for all i ∈ N , B j ∈ ϕ i ∩ ∆. Hence, for all B j ’s there is a i j such that ϕ i j ∩ ∆ = ∅ . Take k = max( { i j | 1 ≤ j ≤ n } ), then B j / ∈ ϕ k ∩ ∆ since ( ⋆ ) { B 1 , . . . , B n } ⊇ ϕ i ∩ ∆ ⊇ ϕ i +1 ∩ ∆. This is a contradiction since ϕ k is a choice set of Σ and ( ⋆ ). ◮ Suppose some A i ∈ ˆ ϕ does not satisfy ( † ). Hence, for all ∆ ∈ Σ, ϕ ∩ ∆ � = { A i } . Hence, ϕ i ∩ ∆ � = { A i } for all ∆ ∈ Σ. But then ˆ A i / ∈ ˆ ϕ ,—a contradiction. ◮ Hence, by the fact above, ˆ ϕ is a minimal choice set. 72/111
Simple facts about the relation between choice sets and the abnormal parts of models 73/111
Lemma 8 If ϕ ∈ Φ(Γ) then there is a M ∈ M LLL (Γ) for which Ab ( M ) ⊆ ϕ . Proof. ◮ assume � ∃ M ∈ M LLL (Γ) s.t. Ab ( M ) ⊆ ϕ ◮ then Γ ∪ (Ω \ ϕ ) ¬ has no LLL -models (where Θ ¬ = df {¬ A | A ∈ Θ } ) ◮ by the compactness of LLL , there is a finite ∆ ⊆ Ω \ ϕ such that Γ ∪ ∆ ¬ has no LLL -model ◮ hence Γ � LLL Dab(∆) and hence Γ ⊢ LLL Dab(∆) ◮ this is a contradiction to ϕ ∈ Φ(Γ) Lemma 9 Where M ∈ M LLL (Γ) , Ab ( M ) is a choice set of Σ(Γ) . Proof. Let ∆ ∈ Σ(Γ), then Γ ⊢ LLL Dab(∆). Hence, Ab ( M ) ∩ ∆ � = ∅ . 74/111
Corollary 10 Where M ∈ M LLL (Γ) , Ab ( M ) �⊂ ϕ for all ϕ ∈ Φ(Γ) . Corollary 11 For all ϕ ∈ Φ(Γ) there is a M ∈ M LLL (Γ) such that (i) Ab ( M ) = ϕ and (ii) M ∈ M AL m (Γ) . Corollary 12 Where M ∈ M LLL (Γ) , M ∈ M AL m (Γ) iff Ab ( M ) ∈ Φ(Γ) . Corollary 13 (Strong Reassurance) For each M ∈ M LLL (Γ) there is a M ′ ∈ M AL m (Γ) such that Ab ( M ′ ) ⊆ Ab ( M ) . Proof. By Lemma 9 and Lemma 7 there is a ϕ ∈ Φ(Γ) such that ϕ ⊆ Ab ( M ). By Corollary 11 there is a M ′ ∈ M AL m (Γ) for which Ab ( M ′ ) = ϕ . 75/111
Links between the marking and the semantic selection: Reliability Syntax Theorem 14 Γ ⊢ AL r A iff there is a ∆ ⊆ Ω for which Γ ⊢ LLL A ∨ Dab(∆) and ∆ ∩ U (Γ) = ∅ . Semantics Γ | = AL r A iff (for each M ∈ M LLL (Γ), if Ab ( M ) ⊆ U (Γ), then M | = A ). 76/111
Links between the marking and the semantic selection: Minimal Abnormality Syntax Theorem 15 Γ ⊢ AL m A iff for every ϕ ∈ Φ(Γ) there is a ∆ ⊆ Ω for which ∆ ∩ ϕ = ∅ and Γ ⊢ LLL A ∨ Dab(∆) . Semantics Theorem 16 Let M LLL (Γ) be non-empty. 1. M AL m (Γ) = � ϕ ∈ Φ(Γ) { M ∈ M LLL (Γ) | Ab ( M ) = ϕ } 2. ϕ ∈ Φ(Γ) iff there is an M ∈ M AL m (Γ) for which Ab ( M ) = ϕ . 77/111
Conflicts in adaptive proofs A conflict between a defeasible inference and a “hard fact” ◮ “hard facts”: derived on empty condition ◮ Type 1 : hard facts conflict with defeasible assumptions ◮ → marking ◮ Type 2 : hard facts conflict with defeasible conclusions l A . . . ∆ l ′ ¬ A ∅ . . . ◮ in this case Γ ⊢ LLL Dab(∆) ◮ line will be marked ◮ shortcut rule A ∆ ¬ A ∅ (RC0) ∅ Dab(∆) Lemma 17 An AL -proof contains a line at which A is derived on the condition ∆ iff Γ ⊢ LLL A ∨ Dab(∆) . 78/111
A conflict between two defeasible inferences ◮ Type 1 : concerning the defeasible assumption l A . . . ∆ l ′ Dab(∆) . . . Θ ◮ in this case Γ ⊢ LLL Dab(∆ ∪ Θ) ◮ shortcut rule: A ∆ Dab(∆) Θ (RD1) Dab(∆ ∪ Θ) ∅ ◮ Type 2 : concerning defeasible consequences l A . . . ∆ l ′ ¬ A . . . Θ ◮ in this case Γ ⊢ LLL Dab(∆ ∪ Θ) ◮ shortcut rule A ∆ ¬ A Θ (RD2) Dab(∆ ∪ Θ) ∅ 79/111
Some trouble with the classical connectives ◮ we need some classical connectives in order to express Dab-formulas (i.e. the classical disjunction) ◮ but what if the LLL has already a classical disjunction? ◮ suppose ∨ is classical and part of the language of the LLL ◮ Let ! A = df ♦ A ∧ ♦ ¬ A ◮ Let Γ = Γ 1 ∪ Γ 2 where ◮ Γ 1 = { ! A i ∨ ! A j | 1 ≤ i < j } ◮ Γ 2 = { � i ≤ i < j ≤ n (! A i ∨ ! A j ) ⊃ ( A ∨ ! A n − 1 ) | 1 < n } ◮ Note, Φ(Γ) = { ϕ i | i > 0 } where ϕ i = Ω \ { ! A i } . ◮ Moreover, Γ ⊢ LLL A ∨ ! A i ◮ Hence, for all M ∈ M T m (Γ), M | = A and whence Γ | = T m A . 80/111
1 ! A 1 ∨ ! A 2 PREM ∅ (! A 1 ∨ ! A 2 ) ⊃ ( A ∨ ∅ 2 PREM ! A 1 ) 3 A ∨ ! A 1 1, 2; RU ∅ � � 1 4 A 3; RC ! A 1 5 ! A 1 ∨ ! A 3 PREM ∅ 6 ! A 2 ∨ ! A 3 PREM ∅ � 7 1 ≤ i < j ≤ 3 (! A i ∨ PREM ∅ ! A j ) ⊃ ( A ∨ ! A 2 ) 8 A ∨ ! A 2 1, 4, 6, 7; RU ∅ � � 6 9 A 8; RC ! A 2 ◮ Φ 4 (Γ) = {{ ! A 1 } , { ! A 2 }} ◮ Φ 9 (Γ) = {{ ! A 1 , ! A 2 } , { ! A 1 , ! A 3 } , { ! A 2 , ! A 3 }} ◮ Γ �⊢ ∗ AL m A 81/111
How to save the day? ◮ classical “checked” symbols are superimposed on the language of LLL ◮ where W is the set of wffs of the LLL, W + is the � ˇ ∨ , ˇ ∧ , ˇ ¬ , . . . � -closure of W ◮ premise sets are considered to be formulated in W ◮ sometimes authors distinguish btw. LLL and LLL + ◮ Dab-formulas are formulated with ˇ ∨ ◮ Why does this solve our problem? 1 ! A 1 ∨ ! A 2 PREM ∅ 2 (! A 1 ∨ ! A 2 ) ⊃ ( A ∨ PREM ∅ ! A 1 ) A ∨ ! A 1 ∅ 3 1, 2; RU � � 4 A 3; RC ! A 1 ◮ line 4 is not marked anymore since ! A 1 ∨ ! A 2 is not a Dab-formula 82/111
The Upper Limit Logic ◮ Recall: the upper limit logic rigorously interprets the premises normal ◮ hence, ⊢ ULL ˇ ¬ A for all A ∈ Ω ◮ the consequence relation of the upper limit logic is then defined as follows: Γ ⊢ ULL A iff Γ ∪ { ˇ ¬ A | A ∈ Ω } ⊢ LLL A ◮ semantically ULL is characterized by all LLL -models M of Γ that are “normal”, i.e. that have an empty abnormal part, Ab ( M ) = ∅ . ◮ these are precisely the LLL -models of Γ ∪ { ˇ ¬ A | A ∈ Ω } . 83/111
ALs approximate ULL Theorem 18 Cn LLL (Γ) ⊆ Cn AL (Γ) ⊆ Cn ULL (Γ) Definition 19 A premise set Γ is normal iff it has one of the following equivalent properties 1. Γ ∪ { ˇ ¬ A | A ∈ Ω } is LLL -non-trivial 2. there are LLL -models M of Γ that are normal, i.e. for which Ab ( M ) = ∅ Theorem 20 If Γ is normal, then Cn AL (Γ) = Cn ULL (Γ) . If a premise set can rigorously be interpreted as normal, then the adaptive logic does so. 84/111
Properties of the Standard Format Theorem 21 (Soundness and Completeness) Γ ⊢ AL A iff Γ | = AL A. Theorem 22 (Reflexivity) Γ ⊆ Cn AL (Γ) Theorem 23 (Hierarchy of the Consequence Relations) Cn LLL (Γ) ⊆ Cn AL r (Γ) ⊆ Cn AL m (Γ) ⊆ Cn ULL (Γ) Theorem 24 (Redundancy of LLL w.r.t. AL ) Cn LLL ( Cn AL (Γ)) = Cn AL (Γ) Theorem 25 Cn AL ( Cn LLL (Γ)) = Cn AL (Γ) Theorem 26 (Fixed Point) Cn AL (Γ) = Cn AL ( Cn AL (Γ)) 85/111
Properties of the Standard Format Theorem 27 (Cautious Cut / Cumulative Transitivity) If Γ ′ ⊆ Cn AL (Γ) then Cn AL (Γ ∪ Γ ′ ) ⊆ Cn AL (Γ) . Theorem 28 (Cautious Monotonicity) If Γ ′ ⊆ Cn AL (Γ) then Cn AL (Γ) ⊆ Cn AL (Γ ∪ Γ ′ ) . Corollary 29 (Cautious Indifference) If Γ ⊆ Cn AL (Γ) then Cn AL (Γ) = Cn AL (Γ ∪ Γ ′ ) . Theorem 30 (Non-Monotonicity/Non-Transitivity) If Cn LLL (Γ) ⊂ Cn AL (Γ) then AL is non-monotonic and non-transitiv. 86/111
The “rational” properties Theorem 31 In general AL is not rational monotonous, i.e. the following does not hold: If A ∈ Cn AL (Γ) and A / ∈ Cn AL (Γ ∪ { B } ) , then ˇ ¬ B ∈ Cn AL (Γ) Theorem 32 Rational distributivity does not hold for ALs in general, i.e. the following does not hold: ∈ Cn AL (Γ ∪ { B ˇ If A / ∈ Cn AL (Γ ∪ { B } ) and A / ∈ Cn AL (Γ ∪ { C } ) , then A / ∨ 87/111
Some open questions for you What about some well-known weakenings of Rational Monotonicity? ◮ If B ∈ Cn L (Γ) and ˇ ¬ ( B ∧ C ) / ∈ Cn L (Γ), then B ∈ Cn L (Γ ∪ { C } ). (proposed by Lou Goble) ◮ If B ∈ Cn L (Γ) and ˇ ¬ B / ∈ Cn L (Γ ∪ { C } ), then B ∈ Cn L (Γ ∪ { C } ). (proposed by Giordano et al.) Goble L., A Proposal for Dealing with Deontic Dilemmas, in A. Lomuscio, D. Nute (eds), DEON , vol. 3065 of Lecture Notes in Computer Science , Springer, p. 74-113, 2004. Giordano L., Olivetti N., Gliozzi V., Pozzato G. L., ALC + T: a Preferential Extension of Description Logics., Fundamenta Informaticae p. 341-372, 2009. 88/111
Other strategies: the simple strategy ◮ applicable in case all minimal Dab-consequences are abnormalities ◮ then: U (Γ) = Φ(Γ) and hence the reliability strategy and the minimal abnormality strategy result in the same consequence set ◮ then: all adaptive models have the same abnormal part ◮ simplified marking condition ◮ semantic selection ala minimal abnormality or reliability (both select the same models in this case) ◮ Task: understand why. Definition 33 (Marking for the Simple Strategy) A line l with condition ∆ is marked at stage s iff some B ∈ ∆ is derived on the empty condition. Definition 34 (Marking for the Simple Strategy 2) A line l with condition ∆ is marked at stage s iff for some ∆ ′ ⊆ ∆, Dab(∆ ′ ) is derived on the empty condition. 89/111
Other strategies: normal selections ◮ Rescher-Manor consequence relations: ◮ strong: � MCS(Γ) ◮ weak: � MCS(Γ) ◮ Default reasoning ◮ skeptical: in all extensions of the given default theory ◮ credulous: in some extension of the given default theory ◮ Abstract argumentation ◮ skeptical: in all extensions of a given argumentation framework ◮ credulous: in some extension of a given argumentation framework ◮ Adaptive Logics ◮ standard format: � M ∈M AL (Γ) { A | M | = A } ◮ normal selections 90/111
Normal Selections Strategy: going “weak” resp. “credulous” Semantics ◮ equivalence relation on the LLL -models: M ∼ M ′ iff Ab ( M ) = Ab ( M ′ ) ◮ partition of the minimally abnormal models: [ M 1 ] ∼ [ M 2 ] ∼ [ M 3 ] ∼ . . . . . . ◮ Γ � n AL A iff there is a M ∈ M AL m (Γ) such that for all M ′ ∈ [ M ] ∼ , M ′ | = A . 91/111
Normal Selections Note: not what is valid in some adaptive model is a consequence! Γ = { ! A ∨ ! B , X ∨ ! A } . Minimally abnormal models: ◮ models with abnormal part { ! A } : ◮ some validate C (some arbitrary non-abnormal formula) ◮ some validate ¬ C ◮ models with abnormal part { ! B } : these validate X . We have Γ � n AL X but Γ � � n AL C . 92/111
Normal Selections: Marking Definition 35 (Marking for Normal Selections) A line l with condition ∆ is marked at stage s iff Dab(∆) is derived on the empty condition at stage s . Take Γ = { ! A ∨ ! B , X ∨ ! A , Y ∨ ! A ∨ ! B } . 1 ! A ∨ ! B PREM ∅ 2 X ∨ ! A PREM ∅ 3 Y ∨ ! A ∨ ! B PREM ∅ � � 4 X 2; RC ! A � � 6 5 Y 3; RC ! A , ! B ! A ˇ ∨ ! B ∅ 6 1; RC 93/111
Combining ALs References: ◮ Diderik Batens’ forthcoming book ◮ Frederik Van De Putte, Hierarchic Adaptive Logics [Logic Journal of the IGPL, 2011] ◮ Frederik Van De Putte and Christian Straßer, Extending the Standard Format of Adaptive Logics to the Prioritized Case [Logique et Analyse, To appear] ◮ Frederik Van De Putte and Christian Straßer, Three Formats of Prioritized Adaptive Logics: a Comparative Study [Under review,] 94/111
Combining ALs 1. diachronic combinations / sequential combination / vertical combination / superposing ALs à AL 1 AL 2 . . . consequences 2. synchronic combinations / horizontal combination / HAL AL 1 AL 2 AL 3 . . . � �� � LLL consequences 95/111
Sequential Combinations Consequence sets ◮ finite case: � �� � � � Cn SAL (Γ) = Cn AL sn Cn AL . . . Cn AL Cn AL 1 (Γ) . . . sn − 1 s2 s1 n 2 n − 1 ◮ infinite case: � � � � Cn SAL i (Γ) = Cn AL . . . ( Cn AL Cn AL 1 (Γ) ) si s2 s1 i 2 This is generalized to the infinite case as follows: Cn SAL (Γ) = lim inf i →∞ Cn SAL i (Γ) = lim sup Cn SAL i (Γ) i →∞ 96/111
Sequential Combinations Semantics ◮ take all AL 1 -models: M 1 ◮ in case s 2 = m take all minimally abnormal models (w.r.t. Ω 2 ) from M 1 ◮ in case s 2 = r take all reliable models (w.r.t Ω 2 ) from M 1 : select all models M ∈ M 1 for which Ab ( M ) ⊆ � { Ab ( M ′ ) | M ′ ∈ M m 2 } where M m 2 is the set of all minimally abnormal models (w.r.t. Ω 2 ) from M 1 ◮ this way we get M 2 ◮ repeat this procedure until you’re through with all the ALs in the sequence 97/111
Problems with Sequential Combinations: No Fixed Point Suppose we have s 1 = s 2 = r and Γ = { ! A 1 ∨ ! A 2 , ! A 1 ∨ ! B , X ∨ ! A 2 } where ! A 1 , ! A 2 ∈ Ω 1 \ Ω 2 and ! B ∈ Ω 2 \ Ω 1 . Take a look at the following AL 1 -proof: 1 ! A 1 ∨ ! A 2 PREM ∅ 2 ! A 1 ∨ ! B PREM ∅ ! A 1 ˇ ∨ ! A 2 ∅ 3 1; RU 4 X ∨ ! A 2 PREM ∅ � � 3 5 X 4; RC ! A 2 In AL 2 we can proceed as follows (with the premise set Cn AL 1 (Γ)): 1 ! A 1 ∨ ! B PREM ∅ � � 2 ! A 1 1; RC ! B Hence ! A 1 ∈ Cn AL 2 ( Cn AL 1 (Γ)). 98/111
Problems with Sequential Combinations: No Fixed Point 1 ! A 1 ∨ ! A 2 PREM ∅ 2 ! A 1 ∨ ! B PREM ∅ ! A 1 ˇ 3 ∨ ! A 2 1; RU ∅ X ∨ ! A 2 ∅ 4 PREM � � 3 5 X 4; RC ! A 2 In AL 2 we can proceed as follows (with the premise set Cn AL 1 (Γ)): 1 ! A 1 ∨ ! B PREM ∅ � � 2 ! A 1 1; RC ! B Hence ! A 1 ∈ Cn AL 2 ( Cn AL 1 (Γ)). Let’s now apply AL 1 to the premise set Cn AL 2 ( Cn AL 1 (Γ)): 1 ! A 1 PREM ∅ 2 X ∨ ! A 2 PREM ∅ � � 3 X 2; RC ! A 2 Now, X is a consequence of Cn AL 1 ( Cn AL2 ( Cn AL 1 (Γ))) and hence also of Cn AL 2 ( Cn AL 1 ( Cn AL 2 ( Cn AL 1 (Γ)))). 99/111
Problems with Sequential Combinations ◮ Note: lack of deduction theorem is the culprit: Γ ∪ { ! A 1 } ⊢ AL 1 X but Γ �⊢ AL 1 B ˇ ⊃ X . ◮ This also shows that we don’t have Cautious Transitivity: Cn SAL (Γ ∪ { ! A } ) �⊆ Cn SAL (Γ) although ! A ∈ Cn SAL (Γ). 100/111
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