Tutorial: Nonmonotonic Logic (Day 2) Christian Straßer Institute for Philosophy II, Ruhr-University Bochum Center for Logic and Philosophy of Science, Ghent University http://homepage.ruhr-uni-bochum.de/defeasible-reasoning/index.html

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Outline

1

Plausible Reasoning

2

Preferential / Selection Semantics (KLM, Shoham)

3

Bibliography

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Topic

1

Plausible Reasoning

2

Preferential / Selection Semantics (KLM, Shoham)

3

Bibliography

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Plausible vs. Defeasible Reasoning (Rescher / Vreeswijk / Prakken) Scheme: Plausible Inferences

deductive standard: CL

A follows nonmonotonically from B iff B ` A ∨ ab and there is no reason to assume ab. A follows nonmonotonically from B iff explicite B ` assdefeasible ⊃ A and assumption there is a reason to assume ass.

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Plausible vs. Defeasible Reasoning (Rescher / Vreeswijk / Prakken) Scheme: Plausible Inferences A follows nonmonotonically from B iff standard: B ` A ∨CL ab and there is no deductive 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

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explicite defeasible assumption

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Plausible vs. Defeasible Reasoning (Rescher / Vreeswijk / Prakken) 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

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explicite defeasible assumption

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Plausible vs. Defeasible Reasoning (Rescher / Vreeswijk / Prakken) 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

explicite defeasible assumption

A ` B ∨ ab iff

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Plausible vs. Defeasible Reasoning (Rescher / Vreeswijk / Prakken) 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

explicite defeasible assumption

A ` B ∨ ab iff ¬B ` ¬A ∨ ab; and

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Plausible vs. Defeasible Reasoning (Rescher / Vreeswijk / Prakken) 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

explicite defeasible assumption

A ` B ∨ ab iff ¬B ` ¬A ∨ ab; and A ` ass ⊃ B iff

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Plausible vs. Defeasible Reasoning (Rescher / Vreeswijk / Prakken) 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

explicite defeasible assumption

A ` B ∨ ab iff ¬B ` ¬A ∨ ab; and A ` ass ⊃ B iff ¬B ` ass ⊃ ¬A

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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))

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General Account: Premise Sets (non-defeasible) facts Σ = hΣ0 , Σd i

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General Account: Premise Sets (non-defeasible) facts Σ = hΣ0 , Σd i Σd may also be stratified: defeasible premises (assumptions) Σd = hΣ1 , . . . , Σn , . . .i

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General Account: Premise Sets (non-defeasible) facts Σ = hΣ0 , Σd i Σd may also be stratified: defeasible premises (assumptions) Σd = hΣ1 , . . . , Σn , . . .i e.g., Σ1 stems from the most reliable (though fallible) source, Σ2 stems from the second most reliable (though fallible) source, etc.

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General Account: Premise Sets (non-defeasible) facts Σ = hΣ0 , Σd i Σd may also be stratified: defeasible premises (assumptions) Σd = hΣ1 , . . . , Σn , . . .i e.g., Σ1 stems from the most reliable (though fallible) source, Σ2 stems from the second most reliable (though fallible) source, etc. in Rescher/Manor consequence relations: Σ0 = ∅

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General Account: Premise Sets (non-defeasible) facts Σ = hΣ0 , Σd i Σd may also be stratified: defeasible premises (assumptions) Σd = hΣ1 , . . . , Σn , . . .i e.g., Σ1 stems from the most reliable (though fallible) source, Σ2 stems from the second most reliable (though fallible) source, etc. in Rescher/Manor consequence relations: Σ0 = ∅ Makinsons’s default assumptions: Σ0 may be non-empty

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Base Logic L with consequence relation Cn monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ0 )

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Base Logic L with consequence relation Cn monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ0 ) reflexive: Γ ⊆ Cn(Γ)

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Base Logic L with consequence relation Cn monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ0 ) reflexive: Γ ⊆ Cn(Γ) compact: if A ∈ Cn(Γ) then A ∈ Cn(Γ0 ) for some finite Γ0 ⊆ Γ

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Base Logic L with consequence relation Cn monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ0 ) reflexive: Γ ⊆ Cn(Γ) compact: if A ∈ Cn(Γ) then A ∈ Cn(Γ0 ) for some finite Γ0 ⊆ Γ cut: where Γ0 ⊆ Cn(Γ) and A ∈ Cn(Γ ∪ Γ0 ), A ∈ Cn(Γ)

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Base Logic L with consequence relation Cn monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ0 ) reflexive: Γ ⊆ Cn(Γ) compact: if A ∈ Cn(Γ) then A ∈ Cn(Γ0 ) for some finite Γ0 ⊆ Γ cut: where Γ0 ⊆ Cn(Γ) and A ∈ Cn(Γ ∪ Γ0 ), A ∈ Cn(Γ) or, Transitivity: where Γ0 ⊆ Cn(Γ) and A ∈ Cn(Γ0 ), A ∈ Cn(Γ).

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Base Logic L with consequence relation Cn monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ0 ) reflexive: Γ ⊆ Cn(Γ) compact: if A ∈ Cn(Γ) then A ∈ Cn(Γ0 ) for some finite Γ0 ⊆ Γ cut: where Γ0 ⊆ Cn(Γ) and A ∈ Cn(Γ ∪ Γ0 ), A ∈ Cn(Γ) or, Transitivity: where Γ0 ⊆ Cn(Γ) and A ∈ Cn(Γ0 ), A ∈ Cn(Γ).

Note: given refl. and mono., CUT iff TRANSITIVITY Suppose Γ0 ⊆ Cn(Γ). (⇒) Suppose A ∈ Cn(Γ0 ). By Monotonicity, A ∈ Cn(Γ ∪ Γ0 ).

(⇐) Suppose A ∈ Cn(Γ ∪ Γ0 ). By Reflexivity, Γ ∪ Γ0 ⊆ Cn(Γ). By Transitivity, A ∈ Cn(Γ).

By Cut, A ∈ Cn(Γ). Notation: Cn(Γ) = {A | Γ ` A} Christian Straßer (RUB, UGENT)

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General Account: Consequence Relations Idea: Use (maximal) consistent subsets of Σd .

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General Account: Consequence Relations Idea: Use (maximal) consistent subsets of Σd .

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 = hΣ0 , Σd i Ξ is a maximal consistent subset of Σ iff 1

Ξ ⊆ Σd is such that Ξ ∪ Σ0 is consistent

2

there is no Γ ⊆ Σd such that Ξ ⊂ Γ and Γ ∪ Σ0 is consistent.

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General Account: Consequence Relations Idea: Use (maximal) consistent subsets of Σd .

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 = hΣ0 , Σd i Ξ is a maximal consistent subset of Σ iff 1

Ξ ⊆ Σd is such that Ξ ∪ Σ0 is consistent

2

there is no Γ ⊆ Σd such that Ξ ⊂ Γ and Γ ∪ Σ0 is consistent.

We write MCS(Σ) for the set of all maximal consistent subsets of Σ. Christian Straßer (RUB, UGENT)

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Examples

Let Σ = hΣ0 , Σd i where Σ0 = {s} Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

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Examples

Let Σ = hΣ0 , Σd i where Σ0 = {s} Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

What are the MCSs?

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Examples

Let Σ = hΣ0 , Σd i where Σ0 = {s} Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

What are the MCSs? Ξ1 = {s ⊃ (p ∧ q), r } Ξ2 = {p ∧ ¬q, r }

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Existence of MCSs Where Ξ ⊆ Σd s.t. Σ0 ∪ Ξ 0 ⊥, there is a Ξ0 ∈ MCS(hΣ0 , Σd i) s.t. Ξ ⊆ Ξ0 .

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Existence of MCSs Where Ξ ⊆ Σd s.t. Σ0 ∪ Ξ 0 ⊥, there is a Ξ0 ∈ MCS(hΣ0 , Σd i) s.t. Ξ ⊆ Ξ0 .

Proof Let Σd = {A1 , . . .}. Let Ξ0 =

S

i≥0 Ξi

where Ξ0 = Ξ

Ξi ∪ {Ai+1 } if Ξi ∪ {Ai+1 } ∪ Σ0 0 ⊥ Ξi else

 Ξi+1 =

By induction, for each Ξi , Ξi ∪ Σ0 0 ⊥. Assume Ξ0 ∪ Σ0 ` ⊥. By compactness, there is a finite Ξ0f ⊆ Ξ0 such that Ξ0f ∪ Σ0 ` ⊥. Thus there is a Ξi ⊇ Ξ0f and by monotonicity, Ξi ∪ Ξ0 ` ⊥,—a contradiction.

Hence, Ξ0 ∪ Σ0 0 ⊥. Where Ai ∈ / Ξ0 , Ξ0 ∪ {Ai } ∪ Σ0 ` ⊥ since Ξi−1 ∪ {Ai } ∪ Σ0 ` ⊥ and by monotonicity. Christian Straßer (RUB, UGENT)

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General Account: Consequence Relations

Question Do you think this also holds if we define "Γ is inconsistent iff Cn(Γ) = L"?

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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

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General Account: Consequence Relations Σ = hΣ0 , Σd i

Free consequences Σ `free A iff Σ0 ∪ Free(Σ) ` A where Free(Σ) =

\

MCS(Σ)

Universal consequences Σ `∀ A iff Σ0 ∪ Ξ ` A for all Ξ ∈ MCS(Σ)

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General Account: Consequence Relations Σ = hΣ0 , Σd i

Free consequences Σ `free A iff Σ0 ∪ Free(Σ) ` A where Free(Σ) =

\

MCS(Σ)

Universal consequences Σ `∀ A iff Σ0 ∪ Ξ ` A for all Ξ ∈ MCS(Σ)

Existential consequences Σ `∃ A iff Σ0 ∪ Ξ ` A for some Ξ ∈ MCS(Σ) Christian Straßer (RUB, UGENT)

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Examples Let Σ = hΣ0 , Σd i where Σ0 = {s} Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

What are the MCSs? Ξ1 = {s ⊃ (p ∧ q), r } Ξ2 = {p ∧ ¬q, r }

Free consequences

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Examples Let Σ = hΣ0 , Σd i where Σ0 = {s} Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

What are the MCSs? Ξ1 = {s ⊃ (p ∧ q), r } Ξ2 = {p ∧ ¬q, r }

Free consequences Free(Σ) = {r }

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Examples Let Σ = hΣ0 , Σd i where Σ0 = {s} Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

What are the MCSs? Ξ1 = {s ⊃ (p ∧ q), r } Ξ2 = {p ∧ ¬q, r }

Free consequences Free(Σ) = {r } Γ `free A iff {s, r } ` A

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Examples Let Σ = hΣ0 , Σd i where Σ0 = {s} Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

What are the MCSs? Ξ1 = {s ⊃ (p ∧ q), r } Ξ2 = {p ∧ ¬q, r }

Free consequences Free(Σ) = {r } Γ `free A iff {s, r } ` A

Notice: Syntax-Dependency

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Examples Let Σ = hΣ0 , Σd i where Σ0 = {s} Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

What are the MCSs? Ξ1 = {s ⊃ (p ∧ q), r } Ξ2 = {p ∧ ¬q, r }

Free consequences Free(Σ) = {r } Γ `free A iff {s, r } ` A

Notice: Syntax-Dependency hΣ0 , {s ⊃ p, s ⊃ q, p, ¬q, r }i `free p Christian Straßer (RUB, UGENT)

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Examples Let Σ = hΣ0 , Σd i where

What are the MCSs?

Σ0 = {s}

Ξ1 = {s ⊃ (p ∧ q), r }

Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

Ξ2 = {p ∧ ¬q, r }

Universal Consequences

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Examples Let Σ = hΣ0 , Σd i where

What are the MCSs?

Σ0 = {s}

Ξ1 = {s ⊃ (p ∧ q), r }

Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

Ξ2 = {p ∧ ¬q, r }

Universal Consequences Γ `∀ A iff {s, r , p} ` A

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Examples Let Σ = hΣ0 , Σd i where

What are the MCSs?

Σ0 = {s}

Ξ1 = {s ⊃ (p ∧ q), r }

Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

Ξ2 = {p ∧ ¬q, r }

Universal Consequences Γ `∀ A iff {s, r , p} ` A floating conclusion: p

Question: Is there also some syntax-dependency for `∀ ?

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Examples Let Σ = hΣ0 , Σd i where

What are the MCSs?

Σ0 = {s}

Ξ1 = {s ⊃ (p ∧ q), r }

Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

Ξ2 = {p ∧ ¬q, r }

Universal Consequences Γ `∀ A iff {s, r , p} ` A floating conclusion: p

Question: Is there also some syntax-dependency for `∀ ? Take Σ0 = h∅, {p ∧ q, ¬p}i  MCS(Σ0 ) = {p ∧ q}, {¬p} Σ0 0∀ q Christian Straßer (RUB, UGENT)

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Examples Let Σ = hΣ0 , Σd i where

What are the MCSs?

Σ0 = {s}

Ξ1 = {s ⊃ (p ∧ q), r }

Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

Ξ2 = {p ∧ ¬q, r }

Universal Consequences Γ `∀ A iff {s, r , p} ` A floating conclusion: p

Question: Is there also some syntax-dependency for `∀ ? Take Σ0 = h∅, {p ∧ q, ¬p}i  MCS(Σ0 ) = {p ∧ q}, {¬p} Σ0 0∀ q Christian Straßer (RUB, UGENT)

While where Σ00 = h∅, {p, q, ¬p}i  MCS(Σ00 ) = {p, q}, {¬p, q} Σ00 `∀ q.

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Examples

Let Σ = hΣ0 , Σd i where

What are the MCSs?

Σ0 = {s}

Ξ1 = {s ⊃ (p ∧ q), r }

Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

Ξ2 = {p ∧ ¬q, r }

Existential Consequences

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Examples

Let Σ = hΣ0 , Σd i where

What are the MCSs?

Σ0 = {s}

Ξ1 = {s ⊃ (p ∧ q), r }

Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r }

Ξ2 = {p ∧ ¬q, r }

Existential Consequences Σ `∃ A iff A ∈ Cn(Ξ1 ∪ {s}) ∪ Cn(Ξ2 ∪ {s})

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General Account: Consequence Relations

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What about stratified premises? Let hΣ0 , Σd i where Σd = hΣ1 , Σ2 , . . .i. Idea: Order MCSs relative to the strength of their constituting premises where strength/reliability/etc. is indirectly proportional to the index Σi .

Lexicographic Ordering (Brewka (1989), Van De Putte and Straßer (2012)) Ξ ≺ Ξ0 iff there is an i ≥ 1 such that 1

Ξ ∩ Σj = Ξ0 ∩ Σj for all 1 ≤ j < i, and

2

Ξ ∩ Σi ⊃ Ξ0 ∩ Σi

Consequence relation . . . relative to min≺ (MCS(Σ)) T Σ `≺ min≺ (MCS(Σ)) ` A free A iff Σ `≺ ∀ A iff for all Ξ ∈ min≺ (MCS(Σ)), Ξ ` A Σ `≺ A iff for some Tutorial: Ξ ∈ min ≺ (MCS(Σ)), Ξ ` A Nonmonotonic Logic (Day 2)

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What about stratified premises? Lexicographic Ordering (Brewka (1989), Van De Putte and Straßer (2012)) Ξ ≺lex Ξ0 iff there is an i ≥ 1 such that 1

Ξ ∩ Σj = Ξ0 ∩ Σj for all 1 ≤ j < i, and

2

Ξ ∩ Σi ⊃ Ξ0 ∩ Σi

Example MCSs

Σ0 = {s}

Ξ1 = {s ⊃ p, ¬q, r }

Σ1 = {s ⊃ p}

Ξ2 = {s ⊃ p, q, r }

Σ2 = {¬p, ¬q}

Ξ3 = {¬p, ¬q, r }

Σ3 = {q, r }

Ξ4 = {¬p, q, r }

order: Ξ1 ≺ Ξ2 ≺ Ξ3 ≺ Ξ4 Christian Straßer (RUB, UGENT)

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Caution: Other orders may not be smooth! Suppose we associate strength in direct proporitionality to the index of Σi (e.g., premises in Σi+1 are more reliably than premises in Σi (i ≥ 1)).

Ordering Ξ @ Ξ0 iff there there is an i ≥ 1 such that 1 2

Ξ ∩ Σj = Ξ0 ∩ Σj for all j > i, and Ξ ∩ Σi ⊃ Ξ ∩ Σi

Example Take Σ = hΣ0 , Σ1 , Σ2 , . . .i where Σ0 = {pi ∨ pj | j > i ≥ 1} Σi = {¬pi , s} for each i ≥ 1

MCSs Ξi = {s, ¬pi } for each i ≥ 1

Note min@ (MCS(Σ)) = ∅ however, we would at least expect to derive s Christian Straßer (RUB, UGENT)

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Some Meta-Properties

Monotonicity?

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Some Meta-Properties

Monotonicity? Clearly fails for free and universal.

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Some Meta-Properties

Monotonicity? Clearly fails for free and universal. What about existential?

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Some Meta-Properties

Monotonicity? Clearly fails for free and universal. What about existential? important to distinguish between:

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Some Meta-Properties

Monotonicity? Clearly fails for free and universal. What about existential? important to distinguish between:

Monotonicity in the defeasible premises If hΣ0 , Σd i |∼ A then hΣ0 , Σd ∪ Γi |∼ A.

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Some Meta-Properties

Monotonicity? Clearly fails for free and universal. What about existential? important to distinguish between:

Monotonicity in the defeasible premises If hΣ0 , Σd i |∼ A then hΣ0 , Σd ∪ Γi |∼ A.

Monotonicity in the factual premises If hΣ0 , Σd i |∼ A then hΣ0 ∪ Γ, Σd i |∼ A.

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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)

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Cumulativity If Σ |∼ A : Σ |∼ B iff Σ ∪ {A} |∼ B

Christian Straßer (RUB, UGENT)

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Cumulativity If Σ |∼ A : Σ |∼ B iff Σ ∪ {A} |∼ B This is: Cautious monotonicity and

Christian Straßer (RUB, UGENT)

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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 hΣ0 , Σd i |∼ A : hΣ0 , Σd i |∼ B iff hΣ0 , Σd ∪ {A}i |∼ B

in factual premises: If hΣ0 , Σd i |∼ A : hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd .

Christian Straßer (RUB, UGENT)

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ).

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i).

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i). Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†).

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i). Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†). Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}).

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i). Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†). Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}). Thus, Σd ⊇ Ξ0 \ {A} ⊃ Ξ.

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i). Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†). Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}). Thus, Σd ⊇ Ξ0 \ {A} ⊃ Ξ. Since Σ0 ∪ (Ξ0 \ {A}) 0 ⊥, this is a contradiction to (†). Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i).

(‡) Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i).

Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†). Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}). Thus, Σd ⊇ Ξ0 \ {A} ⊃ Ξ. Since Σ0 ∪ (Ξ0 \ {A}) 0 ⊥, this is a contradiction to (†). Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i).

(‡) Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i).

Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†).

Clearly, Σ0 ∪ (Ξ \ {A}) 0 ⊥.

Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}). Thus, Σd ⊇ Ξ0 \ {A} ⊃ Ξ. Since Σ0 ∪ (Ξ0 \ {A}) 0 ⊥, this is a contradiction to (†). Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i).

(‡) Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i).

Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†).

Clearly, Σ0 ∪ (Ξ \ {A}) 0 ⊥.

Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}).

Assume that there is a Ξ0 ∈ MCS(hΣ0 , Σd i) such that Ξ \ {A} ⊂ Ξ0 .

Thus, Σd ⊇ Ξ0 \ {A} ⊃ Ξ. Since Σ0 ∪ (Ξ0 \ {A}) 0 ⊥, this is a contradiction to (†). Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i).

(‡) Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i).

Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†).

Clearly, Σ0 ∪ (Ξ \ {A}) 0 ⊥.

Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}).

Assume that there is a Ξ0 ∈ MCS(hΣ0 , Σd i) such that Ξ \ {A} ⊂ Ξ0 . By the supposition, Σ0 ∪ Ξ0 ` A and hence Σ0 ∪ Ξ0 ∪ {A} 0 ⊥.

Thus, Σd ⊇ Ξ0 \ {A} ⊃ Ξ. Since Σ0 ∪ (Ξ0 \ {A}) 0 ⊥, this is a contradiction to (†). Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Where Σ = hΣ0 , Σd i, Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i).

(‡) Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i).

Note that Ξ ∪ Σ0 ` A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥ by cut and (†).

Clearly, Σ0 ∪ (Ξ \ {A}) 0 ⊥.

Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}). Thus, Σd ⊇ Ξ0 \ {A} ⊃ Ξ. Since Σ0 ∪ (Ξ0 \ {A}) 0 ⊥, this is a contradiction to (†). Christian Straßer (RUB, UGENT)

Assume that there is a Ξ0 ∈ MCS(hΣ0 , Σd i) such that Ξ \ {A} ⊂ Ξ0 . By the supposition, Σ0 ∪ Ξ0 ` A and hence Σ0 ∪ Ξ0 ∪ {A} 0 ⊥. This is a contradiction to (‡) since Ξ ⊂ Ξ0 ∪ {A}.

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A.

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B.

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i).

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B.

Christian Straßer (RUB, UGENT)

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B. By Mono., Σ0 ∪ Ξ ` B. Christian Straßer (RUB, UGENT)

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B. By Mono., Σ0 ∪ Ξ ` B. Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B.

Suppose hΣ0 , Σd ∪ {A}i `∀ B.

Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B. By Mono., Σ0 ∪ Ξ ` B. Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i).

Suppose hΣ0 , Σd ∪ {A}i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd i).

Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B. By Mono., Σ0 ∪ Ξ ` B. Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B.

Suppose hΣ0 , Σd ∪ {A}i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd i). Hence, Ξ ∪ {A} ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ` A.

By Mono., Σ0 ∪ Ξ ` B. Christian Straßer (RUB, UGENT)

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B.

Suppose hΣ0 , Σd ∪ {A}i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd i). Hence, Ξ ∪ {A} ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ` A. Also, Σ0 ∪ Ξ ∪ {A} ` B.

By Mono., Σ0 ∪ Ξ ` B. Christian Straßer (RUB, UGENT)

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B. By Mono., Σ0 ∪ Ξ ` B. Christian Straßer (RUB, UGENT)

Suppose hΣ0 , Σd ∪ {A}i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd i). Hence, Ξ ∪ {A} ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ` A. Also, Σ0 ∪ Ξ ∪ {A} ` B. By cut, Σ0 ∪ Ξ ` B.

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The Cumulativity of `∀ : defeasible premises Recall: Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `∀ B iff hΣ0 , Σd ∪ {A}i `∀ B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . Suppose hΣ0 , Σd i `∀ A. Suppose hΣ0 , Σd i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Hence, Ξ \ {A} ∈ MCS(hΣ0 , Σd i) and thus Σ0 ∪ (Ξ \ {A}) ` B. By Mono., Σ0 ∪ Ξ ` B. Christian Straßer (RUB, UGENT)

Suppose hΣ0 , Σd ∪ {A}i `∀ B. Let Ξ ∈ MCS(hΣ0 , Σd i). Hence, Ξ ∪ {A} ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ` A. Also, Σ0 ∪ Ξ ∪ {A} ` B. By cut, Σ0 ∪ Ξ ` B. Thus, hΣ0 , Σd i `∀ B.

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Stronger form of cumulativity? Do you think the antecedent of the previous result can be weakened? Where Σ `∀ A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Suggestions?

Christian Straßer (RUB, UGENT)

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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(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Suggestions?

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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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(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. Suggestions?

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

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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(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ 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 0∃ ¬A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection. proof remains nearly the same (check this!) Christian Straßer (RUB, UGENT)

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Here it is . . . (changes in red) Where L is supraclassical and Σ 0∃ ¬A, π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ {A}i), Λ 7→ Λ ∪ {A} is a bijection.

Sketch of the Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . (†) Let Ξ ∈ MCS(Σ). We show that Ξ∪{A} ∈ MCS(hΣ0 , Σd ∪{A}i). Note that Ξ ∪ Σ0 0 ¬A and thus Σ0 ∪ Ξ ∪ {A} 0 ⊥. Assume that there is a Ξ0 ⊃ Ξ ∪ {A} such that Ξ0 ∈ MCS(Σ0 , Σd ∪ {A}). Thus, Σd ⊇ Ξ0 \ {A} ⊃ Ξ. Since Σ0 ∪ (Ξ0 \ {A}) 0 ⊥, this is a contradiction to (†). Christian Straßer (RUB, UGENT)

(‡) Let Ξ ∈ MCS(hΣ0 , Σd ∪ {A}i). Clearly, Σ0 ∪ (Ξ \ {A}) 0 ⊥. Assume that there is a Ξ0 ∈ MCS(hΣ0 , Σd i) such that Ξ \ {A} ⊂ Ξ0 . By the supposition, Σ0 ∪ Ξ0 0 ¬A and hence Σ0 ∪ Ξ0 ∪ {A} 0 ⊥. This is a contradiction to (‡) since Ξ ⊂ Ξ0 ∪ {A}.

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Christian Straßer (RUB, UGENT)

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd .

Christian Straßer (RUB, UGENT)

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . NoteTthat by our lemma we immediately get: T (†) MCS(hΣ0 , Σd ∪ {A}i) = MCS(Σ) ∪ {A}.

Christian Straßer (RUB, UGENT)

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . NoteTthat by our lemma we immediately get: T (†) MCS(hΣ0 , Σd ∪ {A}i) = MCS(Σ) ∪ {A}. Suppose hΣ0 , Σd i `free B and T hence MCS(hΣ0 , Σd i) ` B.

Christian Straßer (RUB, UGENT)

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . NoteTthat by our lemma we immediately get: T (†) MCS(hΣ0 , Σd ∪ {A}i) = MCS(Σ) ∪ {A}. Suppose hΣ0 , Σd i `free B and T hence MCS(hΣ0 , Σd i) ` B. By (†) and monotonicity, T MCS(hΣ0 , Σd ∪ {A}i) ` B.

Christian Straßer (RUB, UGENT)

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . NoteTthat by our lemma we immediately get: T (†) MCS(hΣ0 , Σd ∪ {A}i) = MCS(Σ) ∪ {A}. Suppose hΣ0 , Σd i `free B and T hence MCS(hΣ0 , Σd i) ` B. By (†) and monotonicity, T MCS(hΣ0 , Σd ∪ {A}i) ` B. Hence, hΣ0 , Σd ∪ {A}i `free B. Christian Straßer (RUB, UGENT)

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . NoteTthat by our lemma we immediately get: T (†) MCS(hΣ0 , Σd ∪ {A}i) = MCS(Σ) ∪ {A}. Suppose hΣ0 , Σd i `free B and T hence MCS(hΣ0 , Σd i) ` B.

Suppose T hΣ0 , Σd ∪ {A}i `free B and hence MCS(hΣ0 , Σd ∪ {A}i) ` B.

By (†) and monotonicity, T MCS(hΣ0 , Σd ∪ {A}i) ` B. Hence, hΣ0 , Σd ∪ {A}i `free B. Christian Straßer (RUB, UGENT)

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . NoteTthat by our lemma we immediately get: T (†) MCS(hΣ0 , Σd ∪ {A}i) = MCS(Σ) ∪ {A}. Suppose hΣ0 , Σd i `free B and T hence MCS(hΣ0 , Σd i) ` B. By (†) and monotonicity, T MCS(hΣ0 , Σd ∪ {A}i) ` B.

Suppose T hΣ0 , Σd ∪ {A}i `free B and hence MCS(hΣ0 , Σd ∪ {A}i) ` B. By T (†), MCS(hΣ0 , Σd i) ∪ {A} ` B.

Hence, hΣ0 , Σd ∪ {A}i `free B. Christian Straßer (RUB, UGENT)

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . NoteTthat by our lemma we immediately get: T (†) MCS(hΣ0 , Σd ∪ {A}i) = MCS(Σ) ∪ {A}. Suppose hΣ0 , Σd i `free B and T hence MCS(hΣ0 , Σd i) ` B. By (†) and monotonicity, T MCS(hΣ0 , Σd ∪ {A}i) ` B. Hence, hΣ0 , Σd ∪ {A}i `free B. Christian Straßer (RUB, UGENT)

Suppose T hΣ0 , Σd ∪ {A}i `free B and hence MCS(hΣ0 , Σd ∪ {A}i) ` B. By T (†), MCS(hΣ0 , Σd i) ∪ {A} ` B. T Since MCS(Σ) ` A, by cut T MCS(hΣ0 , Σd i) ` B.

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The Cumulativity of `free : defeasible premises Where hΣ0 , Σd i `free A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Proof In case A ∈ Σd the proposition is trivial. Suppose thus that A ∈ / Σd . NoteTthat by our lemma we immediately get: T (†) MCS(hΣ0 , Σd ∪ {A}i) = MCS(Σ) ∪ {A}. Suppose hΣ0 , Σd i `free B and T hence MCS(hΣ0 , Σd i) ` B. By (†) and monotonicity, T MCS(hΣ0 , Σd ∪ {A}i) ` B. Hence, hΣ0 , Σd ∪ {A}i `free B. Christian Straßer (RUB, UGENT)

Suppose T hΣ0 , Σd ∪ {A}i `free B and hence MCS(hΣ0 , Σd ∪ {A}i) ` B. By T (†), MCS(hΣ0 , Σd i) ∪ {A} ` B. T Since MCS(Σ) ` A, by cut T MCS(hΣ0 , Σd i) ` B. Thus hΣ0 , Σd i `free B.

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Strengthenings

Strengthening 1 Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Christian Straßer (RUB, UGENT)

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Strengthenings

Strengthening 1 Where hΣ0 , Σd i `∀ A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Strengthening 2, where L is supraclassical Where hΣ0 , Σd i 0∃ ¬A: hΣ0 , Σd i `free B iff hΣ0 , Σd ∪ {A}i `free B.

Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i).

Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i). Thus, Ξ ∪ Σ0 ` A and hence Σ0 ∪ Ξ ∪ {A} 0 ⊥ by (†) and cut.

Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i). Thus, Ξ ∪ Σ0 ` A and hence Σ0 ∪ Ξ ∪ {A} 0 ⊥ by (†) and cut. Assume there is a Ξ0 ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ0 .

Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i). Thus, Ξ ∪ Σ0 ` A and hence Σ0 ∪ Ξ ∪ {A} 0 ⊥ by (†) and cut. Assume there is a Ξ0 ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ0 . Hence, Σ0 ∪ {A} ∪ Ξ0 0 ⊥ and hence Σ0 ∪ Ξ0 0 ⊥ by mono..

Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i). Thus, Ξ ∪ Σ0 ` A and hence Σ0 ∪ Ξ ∪ {A} 0 ⊥ by (†) and cut. Assume there is a Ξ0 ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ0 . Hence, Σ0 ∪ {A} ∪ Ξ0 0 ⊥ and hence Σ0 ∪ Ξ0 0 ⊥ by mono.. Since also Ξ0 ⊆ Σd this contradicts (†). Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i). Thus, Ξ ∪ Σ0 ` A and hence Σ0 ∪ Ξ ∪ {A} 0 ⊥ by (†) and cut.

Let (‡) Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

Assume there is a Ξ0 ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ0 . Hence, Σ0 ∪ {A} ∪ Ξ0 0 ⊥ and hence Σ0 ∪ Ξ0 0 ⊥ by mono.. Since also Ξ0 ⊆ Σd this contradicts (†). Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i). Thus, Ξ ∪ Σ0 ` A and hence Σ0 ∪ Ξ ∪ {A} 0 ⊥ by (†) and cut. Assume there is a Ξ0 ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ0 .

Let (‡) Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i). Assume there is a Ξ0 ∈ MCS(hΣ0 , Σd i) for which Ξ ⊂ Ξ0 .

Hence, Σ0 ∪ {A} ∪ Ξ0 0 ⊥ and hence Σ0 ∪ Ξ0 0 ⊥ by mono.. Since also Ξ0 ⊆ Σd this contradicts (†). Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i). Thus, Ξ ∪ Σ0 ` A and hence Σ0 ∪ Ξ ∪ {A} 0 ⊥ by (†) and cut. Assume there is a Ξ0 ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ0 . Hence, Σ0 ∪ {A} ∪ 0 ⊥ and hence Σ0 ∪ Ξ0 0 ⊥ by mono.. Ξ0

Let (‡) Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i). Assume there is a Ξ0 ∈ MCS(hΣ0 , Σd i) for which Ξ ⊂ Ξ0 . Hence, Σ0 ∪ Ξ0 ` A and Σ0 ∪ Ξ0 0 ⊥.

Since also Ξ0 ⊆ Σd this contradicts (†). Christian Straßer (RUB, UGENT)

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If hΣ0 , Σd i |∼ A: hΣ0 , Σd i |∼ B iff hΣ0 ∪ {A}, Σd i |∼ B (|∼ ∈ {`∀ , `free }).

This follows immediately from the following Lemma If hΣ0 , Σd i `∀ A then MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ {A}, Σd i).

Proof: Suppose hΣ0 , Σd i `∀ A. (†) Let Ξ ∈ MCS(hΣ0 , Σd i). Thus, Ξ ∪ Σ0 ` A and hence Σ0 ∪ Ξ ∪ {A} 0 ⊥ by (†) and cut. Assume there is a Ξ0 ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ0 . Hence, Σ0 ∪ {A} ∪ 0 ⊥ and hence Σ0 ∪ Ξ0 0 ⊥ by mono.. Ξ0

Since also Ξ0 ⊆ Σd this contradicts (†). Christian Straßer (RUB, UGENT)

Let (‡) Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i). Assume there is a Ξ0 ∈ MCS(hΣ0 , Σd i) for which Ξ ⊂ Ξ0 . Hence, Σ0 ∪ Ξ0 ` A and Σ0 ∪ Ξ0 0 ⊥. Thus also Σ0 ∪ {A} ∪ Ξ0 0 ⊥ in contradiction to (‡).

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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(hΣ0 , Σd i) = Cn(hΣ0 , Σd ∪ Cn(hΣ0 , Σd i)i)

2

relative to factual premises: Cn(hΣ0 , Σd i) = Cn(hΣ0 ∪ Cn(hΣ0 , Σd i), Σd i)

Christian Straßer (RUB, UGENT)

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Fixed Point follows from Cumulativity:

(?) All we need is (where |∼ ∈ {`∀ , `free }): 1

Where Γ ⊆ Cn|∼ (hΣ0 , Σd i): Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 , Σd ∪ Γi)

2

Where Γ ⊆ Cn|∼ (hΣ0 , Σd i): Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 ∪ Γ, Σd i)

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Fixed Point follows from Cumulativity:

(?) All we need is (where |∼ ∈ {`∀ , `free }): 1

Where Γ ⊆ Cn|∼ (hΣ0 , Σd i): Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 , Σd ∪ Γi)

2

Where Γ ⊆ Cn|∼ (hΣ0 , Σd i): Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 ∪ Γ, Σd i)

Corollary: If |∼ satisfies (?), Cn|∼ has the fixed point property. Proof: Follows immediately, just let Γ = Cn|∼ (Σ).

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Fixed Point follows from Cumulativity:

(?) All we need is (where |∼ ∈ {`∀ , `free }): 1

Where Γ ⊆ Cn|∼ (hΣ0 , Σd i): Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 , Σd ∪ Γi)

2

Where Γ ⊆ Cn|∼ (hΣ0 , Σd i): Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 ∪ Γ, Σd i)

Corollary: If |∼ satisfies (?), Cn|∼ has the fixed point property. Proof: Follows immediately, just let Γ = Cn|∼ (Σ).

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Proof of (?)

Our previous lemmas can easily be strengthened to: 1

Where Γ ⊆ Cn∀ (hΣ0 , Σd i): π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ Γi), Λ 7→ Λ ∪ Γ is a bijection.

Fixed Point follows from Cumulativity: The proofs for the following strong forms of cumulativity are easily adjusted (where |∼ ∈ {`∀ , `free }): 1

Where Γ ⊆ Cn|∼ : Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 , Σd ∪ Γi)

2

Where Γ ⊆ Cn|∼ : Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 ∪ Γ, Σd i)

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Proof of (?)

Our previous lemmas can easily be strengthened to: 1

2

Where Γ ⊆ Cn∀ (hΣ0 , Σd i): π : MCS(hΣ0 , Σd i) → MCS(hΣ0 , Σd ∪ Γi), Λ 7→ Λ ∪ Γ is a bijection. Where Γ ⊆ Cn∀ (hΣ0 , Σd i): MCS(hΣ0 , Σd i) = MCS(hΣ0 ∪ Γ, Σd i).

Fixed Point follows from Cumulativity: The proofs for the following strong forms of cumulativity are easily adjusted (where |∼ ∈ {`∀ , `free }): 1

Where Γ ⊆ Cn|∼ : Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 , Σd ∪ Γi)

2

Where Γ ⊆ Cn|∼ : Cn|∼ (hΣ0 , Σd i) = Cn|∼ (hΣ0 ∪ Γ, Σd i)

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The Resolution Theorem: in the defeasible premises

Do we have (supposing we have the Res.Thm. for `) hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 , Σd ∪ {A}i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 , Σd ∪ {A}i `free B?

Christian Straßer (RUB, UGENT)

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The Resolution Theorem: in the defeasible premises

Do we have (supposing we have the Res.Thm. for `) hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 , Σd ∪ {A}i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 , Σd ∪ {A}i `free B?

for `∀ : NOPE

Christian Straßer (RUB, UGENT)

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The Resolution Theorem: in the defeasible premises

Do we have (supposing we have the Res.Thm. for `) hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 , Σd ∪ {A}i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 , Σd ∪ {A}i `free B?

for `∀ : NOPE Consider Σ = h∅, {¬p}i, A = p and B = q.

Christian Straßer (RUB, UGENT)

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The Resolution Theorem: in the defeasible premises

Do we have (supposing we have the Res.Thm. for `) hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 , Σd ∪ {A}i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 , Σd ∪ {A}i `free B?

for `∀ : NOPE Consider Σ = h∅, {¬p}i, A = p and B = q. Then Σ `∀ p ⊃ q while h∅, {¬p, p}i 0∀ q.

Christian Straßer (RUB, UGENT)

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The Resolution Theorem: in the defeasible premises

Do we have (supposing we have the Res.Thm. for `) hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 , Σd ∪ {A}i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 , Σd ∪ {A}i `free B?

for `∀ : NOPE Consider Σ = h∅, {¬p}i, A = p and B = q. Then Σ `∀ p ⊃ q while h∅, {¬p, p}i 0∀ q.

for `free : NOPE same counterexample

Christian Straßer (RUB, UGENT)

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The Resolution Theorem for ’facts’ ?

Do we have hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 ∪ {A}, Σd i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 ∪ {A}, Σd i `free B?

Christian Straßer (RUB, UGENT)

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The Resolution Theorem for ’facts’ ?

Do we have hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 ∪ {A}, Σd i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 ∪ {A}, Σd i `free B?

Nope: Take Σ = h∅, {¬p}i and Σ+ = h{p}, {¬p}i.

Christian Straßer (RUB, UGENT)

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The Resolution Theorem for ’facts’ ?

Do we have hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 ∪ {A}, Σd i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 ∪ {A}, Σd i `free B?

Nope: Take Σ = h∅, {¬p}i and Σ+ = h{p}, {¬p}i. Then Σ `∀ p ⊃ q while Σ+ 0∀ q.

Christian Straßer (RUB, UGENT)

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September 3, 2015

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The Resolution Theorem for ’facts’ ?

Do we have hΣ0 , Σd i `∀ A ⊃ B implies hΣ0 ∪ {A}, Σd i `∀ B? hΣ0 , Σd i `free A ⊃ B implies hΣ0 ∪ {A}, Σd i `free B?

Nope: Take Σ = h∅, {¬p}i and Σ+ = h{p}, {¬p}i. Then Σ `∀ p ⊃ q while Σ+ 0∀ q. And Σ `free p ⊃ q while Σ+ 0free q.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises

Question: hΣ0 , Σd ∪ {A}i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B? hΣ0 , Σd ∪ {A}i `free B implies hΣ0 , Σd i `free A ⊃ B?

Christian Straßer (RUB, UGENT)

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September 3, 2015

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The Deduction Theorem for defeasible premises

Question: hΣ0 , Σd ∪ {A}i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B? hΣ0 , Σd ∪ {A}i `free B implies hΣ0 , Σd i `free A ⊃ B?

We suppose in the following discussion 1

2

we have the deduction theorem on the level of L: Γ ∪ {A} ` B implies Γ ` A ⊃ B. explosion: ⊥ ` A for all A ∈ L.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the universal case hΣ0 , Σd ∪ {A}i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B?

Proof Suppose hΣ0 , Σd ∪ {A}i `∀ B and let Ξ ∈ MCS(Σ0 , Σd ).

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the universal case hΣ0 , Σd ∪ {A}i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B?

Proof Suppose hΣ0 , Σd ∪ {A}i `∀ B and let Ξ ∈ MCS(Σ0 , Σd ). We have two case: (1) Ξ ∪ {A} ∈ MCS(hΣ0 , Σd ∪ {A}i) or (2) Ξ ∈ MCS(Σ0 , Σd ∪ {A}) and Ξ ∪ Σ0 ∪ {A} ` ⊥ and thus Ξ ∪ Σ0 ∪ {A} ` B (by explosion and trans).

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the universal case hΣ0 , Σd ∪ {A}i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B?

Proof Suppose hΣ0 , Σd ∪ {A}i `∀ B and let Ξ ∈ MCS(Σ0 , Σd ). We have two case: (1) Ξ ∪ {A} ∈ MCS(hΣ0 , Σd ∪ {A}i) or (2) Ξ ∈ MCS(Σ0 , Σd ∪ {A}) and Ξ ∪ Σ0 ∪ {A} ` ⊥ and thus Ξ ∪ Σ0 ∪ {A} ` B (by explosion and trans). 1

Then by the supposition Σ0 ∪ Ξ ∪ {A} ` B and hence Σ0 ∪ Ξ ` A ⊃ B by the deduction theorem.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the universal case hΣ0 , Σd ∪ {A}i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B?

Proof Suppose hΣ0 , Σd ∪ {A}i `∀ B and let Ξ ∈ MCS(Σ0 , Σd ). We have two case: (1) Ξ ∪ {A} ∈ MCS(hΣ0 , Σd ∪ {A}i) or (2) Ξ ∈ MCS(Σ0 , Σd ∪ {A}) and Ξ ∪ Σ0 ∪ {A} ` ⊥ and thus Ξ ∪ Σ0 ∪ {A} ` B (by explosion and trans). 1

2

Then by the supposition Σ0 ∪ Ξ ∪ {A} ` B and hence Σ0 ∪ Ξ ` A ⊃ B by the deduction theorem. Σ0 ∪ Ξ ` A ⊃ B by the deduction theorem.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the universal case hΣ0 , Σd ∪ {A}i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B?

Proof Suppose hΣ0 , Σd ∪ {A}i `∀ B and let Ξ ∈ MCS(Σ0 , Σd ). We have two case: (1) Ξ ∪ {A} ∈ MCS(hΣ0 , Σd ∪ {A}i) or (2) Ξ ∈ MCS(Σ0 , Σd ∪ {A}) and Ξ ∪ Σ0 ∪ {A} ` ⊥ and thus Ξ ∪ Σ0 ∪ {A} ` B (by explosion and trans). 1

2

Then by the supposition Σ0 ∪ Ξ ∪ {A} ` B and hence Σ0 ∪ Ξ ` A ⊃ B by the deduction theorem. Σ0 ∪ Ξ ` A ⊃ B by the deduction theorem.

Thus, Ξ ` A ⊃ B and hΣ0 , Σd i `∀ A ⊃ B.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the free case

hΣ0 , Σd ∪ {A}i `free B implies hΣ0 , Σd i `free A ⊃ B

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the free case

hΣ0 , Σd ∪ {A}i `free B implies hΣ0 , Σd i `free A ⊃ B

Proof Suppose T hΣ0 , Σd ∪ {A}i `free B and hence Σ0 ∪ MCS(hΣ0 , Σd ∪ {A}i) ` B.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the free case

hΣ0 , Σd ∪ {A}i `free B implies hΣ0 , Σd i `free A ⊃ B

Proof Suppose T hΣ0 , Σd ∪ {A}i `free B and hence Σ0 ∪ MCS(hΣ0 , Σd ∪ {A}i) ` B. It T is easy to see that T MCS(hΣ0 , Σd ∪ {A}i) ⊆ MCS(hΣ0 , Σd i) ∪ {A}.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem for defeasible premises, the free case

hΣ0 , Σd ∪ {A}i `free B implies hΣ0 , Σd i `free A ⊃ B

Proof Suppose T hΣ0 , Σd ∪ {A}i `free B and hence Σ0 ∪ MCS(hΣ0 , Σd ∪ {A}i) ` B. It T is easy to see that T MCS(hΣ0 , Σd ∪ {A}i) ⊆ MCS(hΣ0 , Σd i) ∪ {A}. T By monotonicity, Σ0 ∪ T MCS(hΣ0 , Σd i) ∪ {A} ` B and by the deduction theorem, Σ0 ∪ MCS(hΣ0 , Σd i) ` A ⊃ B.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for Facts’, the universal case

hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

Christian Straßer (RUB, UGENT)

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September 3, 2015

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The Deduction Theorem ’for Facts’, the universal case

hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

First a useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

Christian Straßer (RUB, UGENT)

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September 3, 2015

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The Deduction Theorem ’for Facts’, the universal case

hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

First a useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

(Trivial) Proof Since Σ0 ∪ Ξ ∪ {A} 0 ⊥ there is a Ξ0 ⊇ Ξ such that Ξ0 ∈ MCS(hΣ0 ∪ {A}, Σd i).

Christian Straßer (RUB, UGENT)

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September 3, 2015

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The Deduction Theorem ’for Facts’, the universal case

hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

First a useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

(Trivial) Proof Since Σ0 ∪ Ξ ∪ {A} 0 ⊥ there is a Ξ0 ⊇ Ξ such that Ξ0 ∈ MCS(hΣ0 ∪ {A}, Σd i). Thus, Ξ0 ∪ Σ0 0 ⊥ and by the maximality of Ξ, Ξ = Ξ0 .

Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for Facts’, the universal case hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for Facts’, the universal case hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

Our useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

Christian Straßer (RUB, UGENT)

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September 3, 2015

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The Deduction Theorem ’for Facts’, the universal case hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

Our useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

Proof of the deduction theorem Suppose hΣ0 ∪ {A}, Σd i `∀ B.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for Facts’, the universal case hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

Our useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

Proof of the deduction theorem Suppose hΣ0 ∪ {A}, Σd i `∀ B. Let Ξ ∈ MCS(Σ0 , Σd ).

Christian Straßer (RUB, UGENT)

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September 3, 2015

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The Deduction Theorem ’for Facts’, the universal case hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

Our useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

Proof of the deduction theorem Suppose hΣ0 ∪ {A}, Σd i `∀ B. Let Ξ ∈ MCS(Σ0 , Σd ). If Σ0 ∪ Ξ ∪ {A} ` ⊥ also Σ0 ∪ Ξ ∪ {A} ` B by explosion and transitivity. By the deduction theorem, Σ0 ∪ Ξ ` A ⊃ B.

Christian Straßer (RUB, UGENT)

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September 3, 2015

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The Deduction Theorem ’for Facts’, the universal case hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

Our useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

Proof of the deduction theorem Suppose hΣ0 ∪ {A}, Σd i `∀ B. Let Ξ ∈ MCS(Σ0 , Σd ). If Σ0 ∪ Ξ ∪ {A} ` ⊥ also Σ0 ∪ Ξ ∪ {A} ` B by explosion and transitivity. By the deduction theorem, Σ0 ∪ Ξ ` A ⊃ B. If Σ0 ∪ Ξ ∪ {A} 0 ⊥ then by the Lemma, Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i) and thus Σ0 ∪ Ξ ` B. By monotonicity, Σ0 ∪ Ξ ∪ {A} ` B. By the deduction theorem, Σ0 ∪ Ξ ` A ⊃ B. Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for Facts’, the universal case hΣ0 ∪ {A}, Σd i `∀ B implies hΣ0 , Σd i `∀ A ⊃ B

Our useful Lemma If Ξ ∈ MCS(hΣ0 , Σd i) and Σ0 ∪ Ξ ∪ {A} 0 ⊥ then Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i).

Proof of the deduction theorem Suppose hΣ0 ∪ {A}, Σd i `∀ B. Let Ξ ∈ MCS(Σ0 , Σd ). If Σ0 ∪ Ξ ∪ {A} ` ⊥ also Σ0 ∪ Ξ ∪ {A} ` B by explosion and transitivity. By the deduction theorem, Σ0 ∪ Ξ ` A ⊃ B. If Σ0 ∪ Ξ ∪ {A} 0 ⊥ then by the Lemma, Ξ ∈ MCS(hΣ0 ∪ {A}, Σd i) and thus Σ0 ∪ Ξ ` B. By monotonicity, Σ0 ∪ Ξ ∪ {A} ` B. By the deduction theorem, Σ0 ∪ Ξ ` A ⊃ B. Logic (Day 2) Altogether, hΣ , Σ iTutorial: ` A Nonmonotonic ⊃ B.

Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for facts’, the free case Do we have hΣ0 ∪ {A}, Σd i `free B implies hΣ0 , Σd i `free A ⊃ B?

Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for facts’, the free case Do we have hΣ0 ∪ {A}, Σd i `free B implies hΣ0 , Σd i `free A ⊃ B?

Nope: here’s a counter-example Let Σ0 = {p ∨ q, r ∨ q} Σd = {¬p, ¬q}

Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for facts’, the free case Do we have hΣ0 ∪ {A}, Σd i `free B implies hΣ0 , Σd i `free A ⊃ B?

Nope: here’s a counter-example Let Σ0 = {p ∨ q, r ∨ q} Σd = {¬p, ¬q} We have hΣ0 , Σd i 0free p ⊃ r while hΣ0 ∪ {p}, Σd i `free r

Christian Straßer (RUB, UGENT)

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The Deduction Theorem ’for facts’, the free case Do we have hΣ0 ∪ {A}, Σd i `free B implies hΣ0 , Σd i `free A ⊃ B?

Nope: here’s a counter-example Let Σ0 = {p ∨ q, r ∨ q} Σd = {¬p, ¬q} To see this notice that

We have hΣ0 , Σd i 0free p ⊃ r while hΣ0 ∪ {p}, Σd i `free r

Christian Straßer (RUB, UGENT)

MCS(hΣ0 , Σd i) = {{¬p}, {¬q}} MCS(hΣ0 ∪ {p}, Σd i) = {{¬q}}

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Rational Monotonicity for defeasible premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 , Σd ∪ {A}i `∀ B?

Christian Straßer (RUB, UGENT)

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September 3, 2015

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Rational Monotonicity for defeasible premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 , Σd ∪ {A}i `∀ B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q

Christian Straßer (RUB, UGENT)

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September 3, 2015

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Rational Monotonicity for defeasible premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 , Σd ∪ {A}i `∀ B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q hΣ0 , Σd i `∀ q hΣ0 , Σd i 0∀ ¬p

Christian Straßer (RUB, UGENT)

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September 3, 2015

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Rational Monotonicity for defeasible premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 , Σd ∪ {A}i `∀ B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q hΣ0 , Σd i `∀ q hΣ0 , Σd i 0∀ ¬p but hΣ0 , Σd ∪ {p}i 0∀ q. Do you see why?

Christian Straßer (RUB, UGENT)

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Rational Monotonicity for defeasible premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 , Σd ∪ {A}i `∀ B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q hΣ0 , Σd i `∀ q hΣ0 , Σd i 0∀ ¬p but hΣ0 , Σd ∪ {p}i 0∀ q. Do you see why?

Christian Straßer (RUB, UGENT)

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Rational Monotonicity for defeasible premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 , Σd ∪ {A}i `∀ B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q}

1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

A = p and B = q hΣ0 , Σd i `∀ q hΣ0 , Σd i 0∀ ¬p but hΣ0 , Σd ∪ {p}i 0∀ q. Do you see why?

Christian Straßer (RUB, UGENT)

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Rational Monotonicity for defeasible premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 , Σd ∪ {A}i `∀ B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

MCS(hΣ0 , Σd ∪ {p}i) =

hΣ0 , Σd i `∀ q hΣ0 , Σd i 0∀ ¬p but hΣ0 , Σd ∪ {p}i 0∀ q. Do you see why?

Christian Straßer (RUB, UGENT)

1

1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r , p}

3

{r , p, (p ∧ r ) ⊃ ¬q}

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Rational Monotonicity for factual premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 ∪ {A}, Σd i `∀ B?

Christian Straßer (RUB, UGENT)

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September 3, 2015

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Rational Monotonicity for factual premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 ∪ {A}, Σd i `∀ B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q

Christian Straßer (RUB, UGENT)

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September 3, 2015

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Rational Monotonicity for factual premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 ∪ {A}, Σd i `∀ B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q hΣ0 , Σd i `∀ q hΣ0 , Σd i 0∀ ¬p

Christian Straßer (RUB, UGENT)

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September 3, 2015

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Rational Monotonicity for factual premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 ∪ {A}, Σd i `∀ B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q hΣ0 , Σd i `∀ q hΣ0 , Σd i 0∀ ¬p but hΣ0 ∪ {p}, Σd i 0∀ q. Do you see why?

Christian Straßer (RUB, UGENT)

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Rational Monotonicity for factual premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 ∪ {A}, Σd i `∀ B? Nope! Take: Σ0 = ∅

MCS(hΣ0 , Σd i) =

Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q

1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

hΣ0 , Σd i `∀ q hΣ0 , Σd i 0∀ ¬p but hΣ0 ∪ {p}, Σd i 0∀ q. Do you see why?

Christian Straßer (RUB, UGENT)

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Rational Monotonicity for factual premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 ∪ {A}, Σd i `∀ B? Nope! Take: Σ0 = ∅

MCS(hΣ0 , Σd i) =

Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q hΣ0 , Σd i `∀ q

1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

MCS(hΣ0 ∪ {p}, Σd i) =

hΣ0 , Σd i 0∀ ¬p but hΣ0 ∪ {p}, Σd i 0∀ q. Do you see why?

Christian Straßer (RUB, UGENT)

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Rational Monotonicity for factual premises: the universal case If hΣ0 , Σd i `∀ B and hΣ0 , Σd i 0∀ ¬A then hΣ0 ∪ {A}, Σd i `∀ B? Nope! Take: Σ0 = ∅

MCS(hΣ0 , Σd i) =

Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p and B = q hΣ0 , Σd i `∀ q

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

MCS(hΣ0 ∪ {p}, Σd i) =

hΣ0 , Σd i 0∀ ¬p but hΣ0 ∪ {p}, Σd i 0∀ q. Do you see why?

Christian Straßer (RUB, UGENT)

1

1

{r , (p ∧ r ) ⊃ ¬q}

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Rational Monotonicity relative to defeasible premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 , Σd ∪ {A}i `free B?

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Rational Monotonicity relative to defeasible premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 , Σd ∪ {A}i `free B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q

Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to defeasible premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 , Σd ∪ {A}i `free B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q)

Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to defeasible premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 , Σd ∪ {A}i `free B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q) but hΣ0 , Σd ∪ {p ∧ q}i 0free (p ∧ r ) ⊃ ¬q. Do you see why? Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to defeasible premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 , Σd ∪ {A}i `free B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q}

1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q) but hΣ0 , Σd ∪ {p ∧ q}i 0free (p ∧ r ) ⊃ ¬q. Do you see why? Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to defeasible premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 , Σd ∪ {A}i `free B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q

1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

MCS(hΣ0 , Σd ∪ {p ∧ q}i) =

hΣ0 , Σd i 0free ¬(p ∧ q) but hΣ0 , Σd ∪ {p ∧ q}i 0free (p ∧ r ) ⊃ ¬q. Do you see why? Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to defeasible premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 , Σd ∪ {A}i `free B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q) but hΣ0 , Σd ∪ {p ∧ q}i 0free (p ∧ r ) ⊃ ¬q. Do you see why? Christian Straßer (RUB, UGENT)

1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

MCS(hΣ0 , Σd ∪ {p ∧ q}i) = 1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r , p ∧ q}

3

{r , p ∧ q}

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Rational Monotonicity relative to strict premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 ∪ {A}, Σd i `free B?

Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to strict premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 ∪ {A}, Σd i `free B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q

Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to strict premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 ∪ {A}, Σd i `free B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q)

Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to strict premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 ∪ {A}, Σd i `free B? Nope! Take: Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q) but hΣ0 ∪ {p ∧ q}, Σd i 0free (p ∧ r ) ⊃ ¬q. Do you see why? Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to strict premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 ∪ {A}, Σd i `free B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q

For defeasible premises 1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q) but hΣ0 ∪ {p ∧ q}, Σd i 0free (p ∧ r ) ⊃ ¬q. Do you see why? Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to strict premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 ∪ {A}, Σd i `free B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q)

For defeasible premises 1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

MCS(hΣ0 ∪ {p ∧ q}, Σd i) =

but hΣ0 ∪ {p ∧ q}, Σd i 0free (p ∧ r ) ⊃ ¬q. Do you see why? Christian Straßer (RUB, UGENT)

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Rational Monotonicity relative to strict premises: the free case If hΣ0 , Σd i `free B and hΣ0 , Σd i 0free ¬A then hΣ0 ∪ {A}, Σd i `free B? Nope! Take:

MCS(hΣ0 , Σd i) =

Σ0 = ∅ Σd = {r , p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q, ¬p ∧ q} A = p ∧ q and B = (p ∧ r ) ⊃ ¬q hΣ0 , Σd i `free (p ∧ r ) ⊃ ¬q hΣ0 , Σd i 0free ¬(p ∧ q) but hΣ0 ∪ {p ∧ q}, Σd i 0free (p ∧ r ) ⊃ ¬q. Do you see why? Christian Straßer (RUB, UGENT)

For defeasible premises 1

{(p ∧ r ) ⊃ ¬q, r , ¬p ∧ q}

2

{(p ∧ r ) ⊃ ¬q, p ∧ q ∧ ¬r }

MCS(hΣ0 ∪ {p ∧ q}, Σd i) = 1

{r }

2

{p ∧ q ∧ ¬r , (p ∧ r ) ⊃ ¬q}

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Other forms of ’Rationality’ Negation Rationality Γ ∪ {A} 6 |∼ B Γ ∪ {¬A} 6 |∼ B Γ 6 |∼ B or positively: If Γ |∼ B then Γ ∪ {A} |∼ B or Γ ∪ {¬A} |∼ B.

Disjunctive Rationality Γ ∪ {A} 6 |∼ C Γ ∪ {B} 6 |∼ C Γ ∪ {A ∨ B} 6 |∼ 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)

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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 ↔ A0 , 1

hΣ0 ∪ {A}, Σd i |∼ B iff hΣ0 ∪ {A0 }, Σd i |∼ B

2

hΣ0 , Σd ∪ {A}i |∼ B iff hΣ0 , Σd ∪ {A0 }i |∼ B.

Do we have it? Where |∼ ∈ {`∀ , `∃ , `free }, |∼ satisfies replacement of equivalents.

Christian Straßer (RUB, UGENT)

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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 hΣ0 ∪ {A}, Σd i 6 |∼ C and hΣ0 ∪ {B}, Σd i 6 |∼ C implies hΣ0 ∪ {A ∨ B}, Σd i 6 |∼ C , for all A, B and C .

2

Suppose also hΣ0 ∪ {A ∧ B}, Σd i 6 |∼ C and hΣ0 ∪ {A ∧ ¬B}i 6 |∼ C .

3

By 1, hΣ0 ∪ {(A ∧ B) ∨ (A ∧ ¬B)}, Σd i 6 |∼ C .

4

Since ` A ↔ ((A ∧ B) ∨ (A ∧ ¬B)), hΣ0 ∪ {A}, Σd i 6 |∼ C .

Proof for defeasible premises is analogous.

Christian Straßer (RUB, UGENT)

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Counter-example for Negation Rationality for `∀ Let Σ = hΣ0 , Σd i be such that Σ0 = ∅ and Σd = {¬r ∧ s, ¬r ⊃ (¬q ∧ t), ¬s ∧ r , ¬s ⊃ (q ∧ t)}. Note that

MCS(hΣ0 , Σd ∪ {p}i) 1

2

{¬r ∧ s, ¬r ⊃ (¬q ∧ t), ¬s ⊃ (q ∧ t), p} {¬s ∧ r , ¬r ⊃ (¬q ∧ t), ¬s ⊃ (q ∧ t), p}

MCS(hΣ0 ∪ {p}, Σd i) 1

{¬r ∧ s, ¬r ⊃ (¬q ∧ t), ¬s ⊃ (q ∧ t)}

2

{¬s ∧ r , ¬r ⊃ (¬q ∧ t), ¬s ⊃ (q ∧ t)}

Hence, hΣ0 ∪ {p}i `∀ t.

Hence, hΣ0 ∪ {p}i `∀ t.

MCS(hΣ0 , Σd ∪ {p ∧ q}i) 1

2

{¬s ∧ r , ¬r ⊃ (¬q ∧ t), ¬s ⊃ (q ∧ t)} {¬r ∧ s, ¬r ⊃ (¬q ∧ t), ¬s ⊃ (q ∧ t)}

MCS(hΣ0 ∪ {p ∧ q}, Σd i) 1

{¬s ∧ r , ¬r ⊃ (¬q ∧ t), ¬s ⊃ (q ∧ t)}

{¬r ∧ s, ¬s ⊃ (q ∧ t)} Nonmonotonic Logic (Day 2) September 3, 2015 {¬r ∧ s, ¬s ⊃ (q ∧ t),Tutorial: p ∧ q}

3 Christian Straßer (RUB, UGENT)

2

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Recapture

Let Σ = hΣ0 , Σd i. Where Σ0 ∪ Σd 0 ⊥, Σ |∼ A iff Σ0 ∪ Σd ` A (|∼ ∈ {`∀ , `free , `∃ })

Reason: there is only one MCS, namely Σd .

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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)

X

`∀ X X

X X X X X X

X X X X X X

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From MCSs to Semantic Selections

We now suppose that L has an adequate semantics with no inconsistent models.

Christian Straßer (RUB, UGENT)

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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)

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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

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From MCSs to Semantic Selections

Given Σ = hΣ0 , Σd i, where M ∈ M(Σ0 ), d (M) = {A ∈ Σd | M |= A}

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From MCSs to Semantic Selections

Given Σ = hΣ0 , Σd i, where M ∈ M(Σ0 ), d (M) = {A ∈ Σd | M |= A} Mm (Σ) = {M ∈ M(Σ0 ) | there are no M 0 ∈ M(Σ0 ) for which d (M) ⊂ d (M 0 )}

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From MCSs to Semantic Selections

Given Σ = hΣ0 , Σd i, where M ∈ M(Σ0 ), d (M) = {A ∈ Σd | M |= A} Mm (Σ) = {M ∈ M(Σ0 ) | there are no M 0 ∈ M(Σ0 ) for which d (M) ⊂ d (M 0 )}

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Christian Straßer (RUB, UGENT)

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Proof of 1 Σ0 ∪ Ξ is inconsistent iff

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Proof of 1 Σ0 ∪ Ξ is inconsistent iff Σ0 ∪ Ξ ` ⊥ iff

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Proof of 1 Σ0 ∪ Ξ is inconsistent iff Σ0 ∪ Ξ ` ⊥ iff Σ0 ∪ Ξ ⊥ iff

Christian Straßer (RUB, UGENT)

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Proof of 1 Σ0 ∪ Ξ is inconsistent iff Σ0 ∪ Ξ ` ⊥ iff Σ0 ∪ Ξ ⊥ iff M(Σ0 ∪Ξ) = ∅

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Ad 2: (†) Suppose Ξ ∈ MCS(Σ0 , Σd ).

Proof of 1 Σ0 ∪ Ξ is inconsistent iff Σ0 ∪ Ξ ` ⊥ iff Σ0 ∪ Ξ ⊥ iff M(Σ0 ∪Ξ) = ∅

Christian Straßer (RUB, UGENT)

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Ad 2: (†) Suppose Ξ ∈ MCS(Σ0 , Σd ).

Proof of 1 Σ0 ∪ Ξ is inconsistent iff

Assume there is no M ∈ M(Σ0 ) for which d (M) ⊇ Ξ.

Σ0 ∪ Ξ ` ⊥ iff Σ0 ∪ Ξ ⊥ iff M(Σ0 ∪Ξ) = ∅

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Ad 2: (†) Suppose Ξ ∈ MCS(Σ0 , Σd ).

Proof of 1 Σ0 ∪ Ξ is inconsistent iff

Assume there is no M ∈ M(Σ0 ) for which d (M) ⊇ Ξ. Hence, M(Σ0 ∪ Ξ) = ∅ and thus Σ0 ∪ Ξ ⊥.

Σ0 ∪ Ξ ` ⊥ iff Σ0 ∪ Ξ ⊥ iff M(Σ0 ∪Ξ) = ∅

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Ad 2: (†) Suppose Ξ ∈ MCS(Σ0 , Σd ).

Proof of 1 Σ0 ∪ Ξ is inconsistent iff Σ0 ∪ Ξ ` ⊥ iff

Assume there is no M ∈ M(Σ0 ) for which d (M) ⊇ Ξ. Hence, M(Σ0 ∪ Ξ) = ∅ and thus Σ0 ∪ Ξ ⊥. Since then Σ0 ∪ Ξ ` ⊥: contradiction to (†).

Σ0 ∪ Ξ ⊥ iff M(Σ0 ∪Ξ) = ∅

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Ad 2: (†) Suppose Ξ ∈ MCS(Σ0 , Σd ).

Proof of 1 Σ0 ∪ Ξ is inconsistent iff Σ0 ∪ Ξ ` ⊥ iff Σ0 ∪ Ξ ⊥ iff M(Σ0 ∪Ξ) = ∅

Christian Straßer (RUB, UGENT)

Assume there is no M ∈ M(Σ0 ) for which d (M) ⊇ Ξ. Hence, M(Σ0 ∪ Ξ) = ∅ and thus Σ0 ∪ Ξ ⊥. Since then Σ0 ∪ Ξ ` ⊥: contradiction to (†). Assume there is a M ∈ M(Σ0 ) for which d (M) ⊃ Ξ.

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Ad 2: (†) Suppose Ξ ∈ MCS(Σ0 , Σd ).

Proof of 1 Σ0 ∪ Ξ is inconsistent iff Σ0 ∪ Ξ ` ⊥ iff Σ0 ∪ Ξ ⊥ iff M(Σ0 ∪Ξ) = ∅

Christian Straßer (RUB, UGENT)

Assume there is no M ∈ M(Σ0 ) for which d (M) ⊇ Ξ. Hence, M(Σ0 ∪ Ξ) = ∅ and thus Σ0 ∪ Ξ ⊥. Since then Σ0 ∪ Ξ ` ⊥: contradiction to (†). Assume there is a M ∈ M(Σ0 ) for which d (M) ⊃ Ξ. Then, M(Σ0 ∪ d (M)) 6= ∅ and hence Σ0 ∪ d (M) 1 ⊥.

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Lemma 1

Σ0 ∪ Ξ is L-consistent iff M(Σ0 ∪ Ξ) 6= ∅

2

MCS(Σ0 , Σd ) = {d (M) | M ∈ Mm (hΣ0 , Σd i)}

Proof of 2 Ad 2: (†) Suppose Ξ ∈ MCS(Σ0 , Σd ).

Proof of 1 Σ0 ∪ Ξ is inconsistent iff Σ0 ∪ Ξ ` ⊥ iff Σ0 ∪ Ξ ⊥ iff M(Σ0 ∪Ξ) = ∅

Assume there is no M ∈ M(Σ0 ) for which d (M) ⊇ Ξ. Hence, M(Σ0 ∪ Ξ) = ∅ and thus Σ0 ∪ Ξ ⊥. Since then Σ0 ∪ Ξ ` ⊥: contradiction to (†). Assume there is a M ∈ M(Σ0 ) for which d (M) ⊃ Ξ. Then, M(Σ0 ∪ d (M)) 6= ∅ and hence Σ0 ∪ d (M) 1 ⊥. Since then Σ0 ∪ d (M) 0 ⊥: contradiction to (†).

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From MCSs to Semantic Selections Definition, where Σ = hΣ0 , Σd i Σ m A iff for all M ∈ Mm (Σ), M |= A

Christian Straßer (RUB, UGENT)

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From MCSs to Semantic Selections Definition, where Σ = hΣ0 , Σd i Σ m A iff for all M ∈ Mm (Σ), M |= A

Universal Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `∀ A iff Σ m A

Christian Straßer (RUB, UGENT)

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From MCSs to Semantic Selections Definition, where Σ = hΣ0 , Σd i Σ m A iff for all M ∈ Mm (Σ), M |= A

Universal Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `∀ A iff Σ m A

Proof Σ `∀ A, iff

Christian Straßer (RUB, UGENT)

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From MCSs to Semantic Selections Definition, where Σ = hΣ0 , Σd i Σ m A iff for all M ∈ Mm (Σ), M |= A

Universal Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `∀ A iff Σ m A

Proof Σ `∀ A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ ` A, iff

Christian Straßer (RUB, UGENT)

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From MCSs to Semantic Selections Definition, where Σ = hΣ0 , Σd i Σ m A iff for all M ∈ Mm (Σ), M |= A

Universal Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `∀ A iff Σ m A

Proof Σ `∀ A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ ` A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ A, iff

Christian Straßer (RUB, UGENT)

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From MCSs to Semantic Selections Definition, where Σ = hΣ0 , Σd i Σ m A iff for all M ∈ Mm (Σ), M |= A

Universal Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `∀ A iff Σ m A

Proof Σ `∀ A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ ` A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ A, iff for all Ξ ∈ MCS(Σ) and for all M ∈ M(Σ0 ∪ Ξ), M |= A, iff

Christian Straßer (RUB, UGENT)

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From MCSs to Semantic Selections Definition, where Σ = hΣ0 , Σd i Σ m A iff for all M ∈ Mm (Σ), M |= A

Universal Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `∀ A iff Σ m A

Proof Σ `∀ A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ ` A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ A, iff for all Ξ ∈ MCS(Σ) and for all M ∈ M(Σ0 ∪ Ξ), M |= A, iff for all M ∈ Mm (Σ), M |= A, iff

Christian Straßer (RUB, UGENT)

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From MCSs to Semantic Selections Definition, where Σ = hΣ0 , Σd i Σ m A iff for all M ∈ Mm (Σ), M |= A

Universal Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `∀ A iff Σ m A

Proof Σ `∀ A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ ` A, iff for all Ξ ∈ MCS(Σ), Σ0 ∪ Ξ A, iff for all Ξ ∈ MCS(Σ) and for all M ∈ M(Σ0 ∪ Ξ), M |= A, iff for all M ∈ Mm (Σ), M |= A, iff Σ m A. Christian Straßer (RUB, UGENT)

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Definition, where Σ = hΣ0 , Σd i Σ f A iff for all M ∈ Mf (Σ), M |= A

Christian Straßer (RUB, UGENT)

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Definition, where Σ = hΣ0 , Σd i Σ f A iff for all M ∈ Mf (Σ), M |= A where Mf (Σ) = {M ∈ M(Σ0 ) | A ∈ d (M) iff there is a M 0 ∈ Mm (Σ) such that A ∈ d (M 0 )}.

Christian Straßer (RUB, UGENT)

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Definition, where Σ = hΣ0 , Σd i Σ f A iff for all M ∈ Mf (Σ), M |= A where Mf (Σ) = {M ∈ M(Σ0 ) | A ∈ d (M) iff there is a M 0 ∈ Mm (Σ) such that A ∈ d (M 0 )}.

Christian Straßer (RUB, UGENT)

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Christian Straßer (RUB, UGENT)

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Proof

Σ `free A, iff

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Proof

Σ `free A, iff Σ0 ∪ Free(Σ0 , Σd ) ` A, iff

Christian Straßer (RUB, UGENT)

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Proof

Σ `free A, iff Σ0 ∪ Free(Σ0 , Σd ) ` A, iff T Σ0 ∪ MCS(Σ) ` A, iff

Christian Straßer (RUB, UGENT)

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Proof

Σ `free A, iff Σ0 ∪ Free(Σ0 , Σd ) ` A, iff T Σ0 ∪ MCS(Σ) ` A, iff T Σ0 ∪ MCS(Σ) A, iff

Christian Straßer (RUB, UGENT)

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Proof for all M ∈ M(Σ0 ∪ M |= A, iff

Σ `free A, iff

T

MCS(Σ)),

Σ0 ∪ Free(Σ0 , Σd ) ` A, iff T Σ0 ∪ MCS(Σ) ` A, iff T Σ0 ∪ MCS(Σ) A, iff

Christian Straßer (RUB, UGENT)

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Proof for all M ∈ M(Σ0 ∪ M |= A, iff

Σ `free A, iff Σ0 ∪ Free(Σ0 , Σd ) ` A, iff T Σ0 ∪ MCS(Σ) ` A, iff T Σ0 ∪ MCS(Σ) A, iff

Christian Straßer (RUB, UGENT)

T

MCS(Σ)),

for all M T ∈ M(Σ0 ) such that d (M) ⊇ MCS(Σ), M |= A, iff

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Proof for all M ∈ M(Σ0 ∪ M |= A, iff

Σ `free A, iff Σ0 ∪ Free(Σ0 , Σd ) ` A, iff T Σ0 ∪ MCS(Σ) ` A, iff T Σ0 ∪ MCS(Σ) A, iff

Christian Straßer (RUB, UGENT)

T

MCS(Σ)),

for all M T ∈ M(Σ0 ) such that d (M) ⊇ MCS(Σ), M |= A, iff for all M T ∈ M(Σ0 ) such that d (M) ⊇ {d (M 0 ) | M 0 ∈ Mm (Σ)}, M |= A, iff

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Free Consequence: Semantic Selections, where Σ = hΣ0 , Σd i Σ `free A iff Σ f A

Proof for all M ∈ M(Σ0 ∪ M |= A, iff

Σ `free A, iff Σ0 ∪ Free(Σ0 , Σd ) ` A, iff T Σ0 ∪ MCS(Σ) ` A, iff T Σ0 ∪ MCS(Σ) A, iff

T

MCS(Σ)),

for all M T ∈ M(Σ0 ) such that d (M) ⊇ MCS(Σ), M |= A, iff for all M T ∈ M(Σ0 ) such that d (M) ⊇ {d (M 0 ) | M 0 ∈ Mm (Σ)}, M |= A, iff Σ f A.

Christian Straßer (RUB, UGENT)

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From MCSs to Semantic Selections Definition: where Σ = hΣ0 , Σd i Σ e A iff there is a M ∈ Mm (Σ) such that for all M 0 ∈ Mm (Σ) for which d (M 0 ) = d (M), M 0 |= A.

Existential Consequence Σ `∃ A iff Σ e A

Proof Σ `∃ A, iff, there is a Ξ ∈ MCS(Σ) such that Σ0 ∪ Ξ ` A, iff, there is a M ∈ Mm (Σ) such that Σ0 ∪ d (M) A, iff, there is a M ∈ Mm (Σ) such that for all M 0 ∈ Mm (Σ), M 0 |= A (note that there are no M 00 ∈ M(Σ0 ) for which d (M 00 ) ⊃ d (M)!)

Christian Straßer (RUB, UGENT)

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Adaptive Logics – The basic idea: semantics

1

Take a core logic L (the ’lower limit logic’) that is

Christian Straßer (RUB, UGENT)

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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)

Christian Straßer (RUB, UGENT)

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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 ∨)

Christian Straßer (RUB, UGENT)

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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

Christian Straßer (RUB, UGENT)

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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

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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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’

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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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)

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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.

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Inconsistency-adaptive logics

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Inconsistency-adaptive logics

2. Abnormalities ∼-contradictions: A ∧ ∼A Ω = {A ∧ ∼A | A ∈ L}

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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.

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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’: M |= ¬(p ∧ ∼p) models that validate p ∧ ∼p are sorted out

M |= ∼p

for all remaining models M:

M |= p ∨ q

M 6|= p M |= q

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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

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Inconsistency-adaptive logics

Option 1: minimal abnormality strategy Where Γ ⊆ L, where M is a L-model, Ω(M) = {A ∈ Ω | M |= A} 0 Mm.a. L (Γ) = {M ∈ ML (Γ) | there is no M ∈ ML (Γ) for which 0 Ω(M ) ⊂ Ω(M)}

Γ m.a. A iff for all M ∈ Mm.a. L (Γ), M |= A.

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Inconsistency-adaptive logics

Option 1: minimal abnormality strategy Where Γ ⊆ L, where M is a L-model, Ω(M) = {A ∈ Ω | M |= A} 0 Mm.a. L (Γ) = {M ∈ ML (Γ) | there is no M ∈ ML (Γ) for which 0 Ω(M ) ⊂ Ω(M)}

Γ m.a. A iff for all M ∈ Mm.a. L (Γ), M |= A.

Note Γ m.a. A iff hΓ, Ω¬ i `∀ A

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Example: Minimal abnormality

Γ = {p ∨ s, r ∨ s, ¬p, ¬r , r ∨ p} model M1 M2 M3 M4 M5 M6

p ∧ ∼p 1 0 1 1 0 1

r ∧ ∼r 0 1 1 0 1 1

s ∧∼s 0 0 0 1 1 1

Thus, Γ m.a. s.

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Inconsistency-adaptive logics

Option 2: reliability strategy Where Γ ⊆ L, 0 m.a. Mrel L (Γ) = {M ∈ ML (Γ) | A ∈ Ω(M) iff there is a M ∈ ML (Γ) such that A ∈ Ω(M 0 )} rel Γ rel L A iff for all M ∈ ML , M |= A.

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Inconsistency-adaptive logics

Option 2: reliability strategy Where Γ ⊆ L, 0 m.a. Mrel L (Γ) = {M ∈ ML (Γ) | A ∈ Ω(M) iff there is a M ∈ ML (Γ) such that A ∈ Ω(M 0 )} rel Γ rel L A iff for all M ∈ ML , M |= A.

Note ¬ Γ rel L A iff hΓ, Ω i `free A

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Inconsistency-adaptive logics

Option 2: reliability strategy Where Γ ⊆ L, 0 m.a. Mrel L (Γ) = {M ∈ ML (Γ) | A ∈ Ω(M) iff there is a M ∈ ML (Γ) such that A ∈ Ω(M 0 )} rel Γ rel L A iff for all M ∈ ML , M |= A.

Note ¬ Γ rel L A iff hΓ, Ω i `free A

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Example: Reliability

Γ = {p ∨ s, r ∨ s, ∼p, ∼r , r ∨ p} model M1 M2 M3 M4 M5 M6

p ∧ ∼p 1 0 1 1 0 1

r ∧ ∼r 0 1 1 0 1 1

s ∧∼s 0 0 0 1 1 1

Thus, Γ 6 rel. s.

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Dynamic Proof Theories for Adaptive Logics line number l

Christian Straßer (RUB, UGENT)

A

R; l , l 0

∆

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Dynamic Proof Theories for Adaptive Logics line number l

A

R; l , l 0

∆

formula

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Dynamic Proof Theories for Adaptive Logics line number l

justification A

R; l , l 0

∆

formula

Christian Straßer (RUB, UGENT)

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Dynamic Proof Theories for Adaptive Logics line number l

justification A

formula

Christian Straßer (RUB, UGENT)

R; l , l 0

∆

assumption / condition

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Dynamic Proof Theories for Adaptive Logics line number l

justification A

We have 3 generic rules: formula

Christian Straßer (RUB, UGENT)

R; l , l 0

∆

assumption / condition

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Dynamic Proof Theories for Adaptive Logics line number l

justification A

We have 3 generic rules: formula

R; l , l 0

∆

assumption / condition

Premise introduction (PREM) Where A ∈ Γ, introduce A on the empty condition.

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Dynamic Proof Theories for Adaptive Logics line number l

justification A

We have 3 generic rules: formula

R; l , l 0

∆

assumption / condition

Premise introduction (PREM) Where A ∈ Γ, introduce A on the empty condition.

Unconditional Rule (RU) Where A1 , . . . , An `L B, l1 ... ln l Christian Straßer (RUB, UGENT)

A1 ... An B

J1 ... Jn RU;l1 , . . . , ln

∆1 ... ∆n ∆1 ∪ . . . ∪ ∆n

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Dynamic Proof Theories for Adaptive Logics Conditional Rule (RC) Where A1 , . . . , An `L B ∨ l1 ... ln l

A1 ... An B

W

Θ and Θ ⊆ Ω,

J1 ... Jn RC;l1 , . . . , ln

∆1 ... ∆n ∆1 ∪ . . . ∪ ∆n ∪ Θ

Example: Γ = {p ∨ s, ∼p, p}. 1 2 3 4 5

p∨s ∼p s p p ∧ ∼p

PREM PREM 1,2;RC PREM 2,4;RU

Christian Straßer (RUB, UGENT)

∅ ∅ {p∼p} ∅ ∅

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This calls for retraction. Differs with the strategy!

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Dynamic Proof Theories for Adaptive Logics, Retraction The unreliable abnormalities W Let Us = {∆ ⊆ Ω | ∆ is derivedWon the condition ∅ at some line l and there is no ∆0 ⊂ ∆ such that ∆0 is derived on the condition ∅ at stage s members of Us are the unreliable abnormalities at stage s

Christian Straßer (RUB, UGENT)

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Dynamic Proof Theories for Adaptive Logics, Retraction The unreliable abnormalities W Let Us = {∆ ⊆ Ω | ∆ is derivedWon the condition ∅ at some line l and there is no ∆0 ⊂ ∆ such that ∆0 is derived on the condition ∅ at stage s members of Us are the unreliable abnormalities at stage s

Marking: reliability strategy Line l with condition ∆ is marked at stage s of a proof, iff ∆ ∩ Us 6= ∅.

Example

X

Christian Straßer (RUB, UGENT)

1 2 3 4 5

p∨s ∼p s p p ∧ ∼p

PREM PREM 1,2;RC PREM 2,4;RU

∅ ∅ {p ∧ ∼p} ∅ ∅

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More complicated example Γ = {p ∨ s, r ∨ s, ∼p, ∼r , r ∨ p}.

X

X

1 2 3 4 5 6 7 8

p∨s ∼p s r ∨p ∼r (p ∧ ∼p) ∨ (r ∧ ∼r ) r ∨s s

PREM PREM 1,2;RC PREM PREM 2,4,5;RU PREM 5,7;RC

∅ ∅ {p ∧ ∼p} ∅ ∅ ∅ ∅ {r ∧ ∼r }

U8 = {p ∧ ∼p, r ∧ ∼r }

Minimal abnormality: multiple arguments matter either p ∧ ∼p holds and r ∧ ∼r not

or, vice versa, r ∧ ∼r holds and p ∧ ∼p not

then the assumption of line 8 is then the assumption of line 3 is OK OK Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 2) September 3, 2015 70 / 96

Final derivability

A is finally derived at a line l at stage s of a proof from Γ iff 1 2

l is unmarked at stage s for all extensions of the proof in which l is marked there is a further extension such that l is unmarked.

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Flip/Flop Problem: CLuNs

X X

X

1 2 3 4 5 6 7 8 9 10 11

∼∼q p ∧ ∼p q ∨ (∼q ∧ ∼∼q) q ∼q ∨ r r p∧r ∼p ∨ ∼r ∼(p ∧ r ) (p ∧ r ) ∧ ∼(p ∧ r ) [(p ∧ r ) ∧ ∼(p ∧ r )] ∨ [q ∧ ∼q]

Christian Straßer (RUB, UGENT)

PREM PREM 1;RU 3;RU PREM 4,5;RC 2,6;RU 2;RU 8;RU 7,9;RU 10;RA

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∅ ∅ ∅ ∅ ∅ {q ∧ ∼q} {q ∧ ∼q} ∅ ∅ {q ∧ ∼q} ∅

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Flip/Flop Problem: CLuNs

X X

X

1 2 3 4 5 6 7 8 9 10 11

∼∼q p ∧ ∼p q ∨ (∼q ∧ ∼∼q) q ∼q ∨ r r p∧r ∼p ∨ ∼r ∼(p ∧ r ) (p ∧ r ) ∧ ∼(p ∧ r ) [(p ∧ r ) ∧ ∼(p ∧ r )] ∨ [q ∧ ∼q]

PREM PREM 1;RU 3;RU PREM 4,5;RC 2,6;RU 2;RU 8;RU 7,9;RU 10;RA

∅ ∅ ∅ ∅ ∅ {q ∧ ∼q} {q ∧ ∼q} ∅ ∅ {q ∧ ∼q} ∅

Solution: restrict abnormalities to contradictions in atoms.

Christian Straßer (RUB, UGENT)

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Topic

1

Plausible Reasoning

2

Preferential / Selection Semantics (KLM, Shoham)

3

Bibliography

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Preferential / Selection Semantics

We have seen already an example of these with Adaptive Logics.

Christian Straßer (RUB, UGENT)

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Preferential / Selection Semantics

We have seen already an example of these with Adaptive Logics. they have been systematically investigated

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Preferential / Selection Semantics

We have seen already an example of these with Adaptive Logics. they have been systematically investigated by Shoham (Shoham (1987))

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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))

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The System P Various characterisations can be found, here’s one of them: ` A ↔ B A |∼ C Left Logical Equ. B |∼ C

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The System P Various characterisations can be found, here’s one of them: ` A ↔ B A |∼ C Left Logical Equ. B |∼ C ` A ↔ B C |∼ A Right Weakening C |∼ B

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The System P Various characterisations can be found, here’s one of them: ` A ↔ B A |∼ C Left Logical Equ. B |∼ C ` A ↔ B C |∼ A Right Weakening C |∼ B Reflexivity A |∼ A

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The System P Various characterisations can be found, here’s one of them: ` A ↔ B A |∼ C Left Logical Equ. B |∼ C ` A ↔ B C |∼ A Right Weakening C |∼ B Reflexivity A |∼ A A |∼ B A |∼ C And A |∼ B ∧ C

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The System P Various characterisations can be found, here’s one of them: ` A ↔ B A |∼ C Left Logical Equ. B |∼ C ` A ↔ B C |∼ A Right Weakening C |∼ B Reflexivity A |∼ A A |∼ B A |∼ C And A |∼ B ∧ C A |∼ C B |∼ C Or A ∨ B |∼ C

Christian Straßer (RUB, UGENT)

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The System P Various characterisations can be found, here’s one of them: ` A ↔ B A |∼ C Left Logical Equ. B |∼ C ` A ↔ B C |∼ A Right Weakening C |∼ B Reflexivity A |∼ A A |∼ B A |∼ C And A |∼ B ∧ C A |∼ C B |∼ C Or A ∨ B |∼ C A |∼ B A |∼ C CM A ∧ B |∼ C

Christian Straßer (RUB, UGENT)

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The System P Various characterisations can be found, here’s one of them: ` A ↔ B A |∼ C Left Logical Equ. B |∼ C ` A ↔ B C |∼ A Right Weakening C |∼ B Two perspectives Reflexivity A |∼ A 1 as a consequence relation A |∼ B A |∼ C 2 |∼ as operator in the object And A |∼ B ∧ C language (’conditional logics of A |∼ C B |∼ C normality’) Or A ∨ B |∼ C A |∼ B A |∼ C CM A ∧ B |∼ C

Christian Straßer (RUB, UGENT)

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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

5

A ∧ ¬B |∼ A ∧ ¬B by Reflexivity and thus A ∧ ¬B |∼ B ⊃ C by Right Weakening 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.

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The (Standard) Semantics of P: Preferential models NOPE!

hW , ≺, v i where W is a set of points (worlds)

Christian Straßer (RUB, UGENT)

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The (Standard) Semantics of P: Preferential models NOPE!

hW , ≺, v i where W is a set of points (worlds) v : W → ℘(Atoms) is an assignment

Christian Straßer (RUB, UGENT)

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The (Standard) Semantics of P: Preferential models NOPE!

hW , ≺, v i 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

Christian Straßer (RUB, UGENT)

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The (Standard) Semantics of P: Preferential models NOPE!

hW , ≺, v i 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

Christian Straßer (RUB, UGENT)

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The (Standard) Semantics of P: Preferential models NOPE!

hW , ≺, v i 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

Consequence relation: Preferential Closure let min≺ (A) = {w ∈ W | w |= A and for all w 0 ∈ W such that w 0 |= A, w 0 6≺ w }

Alternative: A `P B iff A |∼ B is derivable from system P. Christian Straßer (RUB, UGENT)

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The (Standard) Semantics of P: Preferential models NOPE!

hW , ≺, v i 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

Consequence relation: Preferential Closure let min≺ (A) = {w ∈ W | w |= A and for all w 0 ∈ W such that w 0 |= A, w 0 6≺ w } M |= A |∼ B iff for all w ∈ min≺ (A), w |= B

Alternative: A `P B iff A |∼ B is derivable from system P. Christian Straßer (RUB, UGENT)

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The (Standard) Semantics of P: Preferential models NOPE!

hW , ≺, v i 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

Consequence relation: Preferential Closure let min≺ (A) = {w ∈ W | w |= A and for all w 0 ∈ W such that w 0 |= A, w 0 6≺ 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)

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Central characterisational result: |∼0 is a preferential consequence relation iff there is a preferential model M that characterises it (ie., A |∼0 B iff M |= A |∼ B.)

Task Build a preferential model hW , ≺, v i that characterises a consequence relation that contains all of {p |∼ b, b |∼ f , p |∼ ¬f } with the four worlds: world w1 w2 w3 w4

p 0 0 1 1

f 0 1 0 1

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)

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The System R, Rational consequence relations and models R is P plus Rational Monotonicity A |∼ C A 6 |∼ ¬B A ∧ B |∼ C

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The System R, Rational consequence relations and models R is P plus Rational Monotonicity A |∼ C A 6 |∼ ¬B A ∧ B |∼ C

Christian Straßer (RUB, UGENT)

Semantics hW , ≺, v i where hW , ≺, v i is a preferential model ≺ is a modular order (i.e., we have a ranking function)

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The System R, Rational consequence relations and models R is P plus Rational Monotonicity A |∼ C A 6 |∼ ¬B A ∧ B |∼ C

Christian Straßer (RUB, UGENT)

Semantics hW , ≺, v i where hW , ≺, v i is a preferential model ≺ is a modular order (i.e., we have a ranking function)

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The System R, Rational consequence relations and models R is P plus Rational Monotonicity A |∼ C A 6 |∼ ¬B A ∧ B |∼ C

Christian Straßer (RUB, UGENT)

Semantics hW , ≺, v i where hW , ≺, v i is a preferential model ≺ is a modular order (i.e., we have a ranking function)

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A Problem with Irrelevance

Christian Straßer (RUB, UGENT)

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A Problem with Irrelevance

{p |∼ b, b |∼ f , p |∼ ¬f } 6 R p ∧ a |∼ ¬f .

Christian Straßer (RUB, UGENT)

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A Problem with Irrelevance

Model M

{p |∼ b, b |∼ f , p |∼ ¬f } 6 R p ∧ a |∼ ¬f . M |= p |∼ b M |= p |∼ ¬f M |= b |∼ f but M 6|= p ∧ a |∼ ¬f . Christian Straßer (RUB, UGENT)

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A Problem with Irrelevance

In fact, despite RM, where K is a conditional knowledge base, K P A |∼ B iff K R A |∼ B.

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Enters: Rational Closure

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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 Christian Straßer (RUB, UGENT)

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Enters: Rational Closure Interpret worlds as normal as possible relative to ≺ and K. i.e., ’drop worlds’ as low as possible Idea: order rational consequence relations according to how normal they interpret worlds and pick out the minimal one.

min≺ (p ∧ a) ⊆ min≺ (p) thus: M |= p ∧ a |∼ ¬f Christian Straßer (RUB, UGENT)

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Enters: Rational Closure Interpret worlds as normal as possible relative to ≺ and K. i.e., ’drop worlds’ as low as possible 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

min≺ (p ∧ a) ⊆ min≺ (p) thus: M |= p ∧ a |∼ ¬f Christian Straßer (RUB, UGENT)

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.

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Enters: Rational Closure Interpret worlds as normal as possible relative to ≺ and K. i.e., ’drop worlds’ as low as possible 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

min≺ (p ∧ a) ⊆ min≺ (p) thus: M |= p ∧ a |∼ ¬f Christian Straßer (RUB, UGENT)

2

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. for all C , D: if (C , D) ∈ |∼1 \ |∼2 , there is a (A, B) ∈ |∼2 \ |∼1 s.t. A <2 C .

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Enters: Rational Closure

Formally: we write A <|∼ B iff A ∨ B |∼ ¬B |∼1 @ |∼2 iff 1

2

Christian Straßer (RUB, UGENT)

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. for all C , D: if (C , D) ∈ |∼1 \ |∼2 , there is a (A, B) ∈ |∼2 |∼1 s.t. A <2 C .

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Enters: Rational Closure

Formally: we write A <|∼ B iff A ∨ B |∼ ¬B |∼1 @ |∼2 iff 1

2

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. 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. |∼3

Christian Straßer (RUB, UGENT)

1

|∼3 @ |∼2 : take (p ∧ a, f ) ∈ |∼2 \ |∼3

2

|∼1 @ |∼3 : take (p, ¬a) ∈ |∼3 \ |∼3

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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)

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RC does not always exist Take K = {pn |∼ pn+2 , pn+2 |∼ ¬pn | n ∈ N− ∪ N+ 0 }. then pn+2 < pn for all n for all odd n: . . . pn+4 < pn+2 < pn < . . . for all even n: . . . pn+4 < pn+2 < pn < . . . Suppose we have the rational closure |∼ of K Four cases: 1 p < p for all odd k and all even n n k 2 p < p for all even k and all odd n n k 3 there are two even n, m and an odd k such that p < p < p n m k 4 there are two odd n, m and an even k such that p < p < p n m k However, Ad 1/2: let f (m) = m + 1. Then f (K) = K, but f (|∼) 6= |∼.  l l is odd Ad 3/4: let f (l ) = Now pm < pk in f (|∼). l − m + n l is even Contradiction. Christian Straßer (RUB, UGENT)

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Drowning Problem

K

Christian Straßer (RUB, UGENT)

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Drowning Problem

K Note that:

Christian Straßer (RUB, UGENT)

1

K 6 P penguin |∼ haswings

2

K 6 R penguin |∼ haswings

3

K 6 RC penguin |∼ haswings

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Drowning Problem Here are our possible worlds: p b w f

w1 0 0 0 0

w2 0 0 0 1

w3 0 0 1 0

w4 0 0 1 1

Christian Straßer (RUB, UGENT)

w5 0 1 0 0

w6 0 1 0 1

w7 0 1 1 0

w8 0 1 1 1

w9 1 0 0 0

w10 1 0 0 1

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w11 1 0 1 0

w12 1 0 1 1

w13 1 1 0 0

w14 1 1 0 1

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w15 1 1 1 0

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w

Drowning Problem Here are our possible worlds: p b w f

w1 0 0 0 0

w2 0 0 0 1

w3 0 0 1 0

w4 0 0 1 1

w5 0 1 0 0

w6 0 1 0 1

w7 0 1 1 0

w8 0 1 1 1

w9 1 0 0 0

w10 1 0 0 1

w11 1 0 1 0

w12 1 0 1 1

w13 1 1 0 0

w14 1 1 0 1

w15 1 1 1 0

The model of the rational closure

Christian Straßer (RUB, UGENT)

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w

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)

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Various Semantic Characterisations

From the abstract point of view S = hhW , v i, hD, ≺i, πi where π : ℘(W ) → D S |= A |∼ B iff π([A ∧ B]) preferable to π([A ∧ ¬B]) in view of ≺.

Christian Straßer (RUB, UGENT)

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π possibility measure π : ℘(W ) → [0, 1] ordinal ranking fkt. κ : ℘(W ) → {0, 1, . . . , ∞} plausibility meas. Pl : ℘(W ) → D

S |= A |∼ B iff π([A]) = 0 or π([A ∧ B]) > π([A ∧ ¬B]) κ([A]) = ∞ or κ([A ∧ B]) < κ([A ∧ ¬B]) Pl([A]) = ⊥ or Pl([A ∧ B]) > Pl([A ∧ ¬B])

. . . structures possibilistic ordinal ranking plausibility

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)

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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

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Probabilities, Naive idea: A |∼ B holds in a structure if P(B | A) > τ where τ is a threshold value (e.g., 21 or 23 , etc.).

Christian Straßer (RUB, UGENT)

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Probabilities, Naive idea: A |∼ B holds in a structure if P(B | A) > τ where τ is a threshold value (e.g., 21 or 23 , etc.).

We loose properties! What about A |∼ C (cut)?! A |∼ B (A = being a Pennsylvanian Dutch, B = being a native speaker of German)

Christian Straßer (RUB, UGENT)

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Probabilities, Naive idea: A |∼ B holds in a structure if P(B | A) > τ where τ is a threshold value (e.g., 21 or 23 , 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)

Christian Straßer (RUB, UGENT)

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Probabilities, Naive idea: A |∼ B holds in a structure if P(B | A) > τ where τ is a threshold value (e.g., 21 or 23 , 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) =

Christian Straßer (RUB, UGENT)

P(A∧B) P(A)

Tutorial: Nonmonotonic Logic (Day 2)

=

0.3 0.4

=

3 4

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Probabilities, Naive idea: A |∼ B holds in a structure if P(B | A) > τ where τ is a threshold value (e.g., 21 or 23 , 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(A∧B) 0.3 3 P(A) = 0.4 = 4 ) 0.2 B) = P(A∧B∧C P(A∧B) = 0.3

P(B | A) = P(C | A ∧

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

=

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2 3

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Probabilities, Naive idea: A |∼ B holds in a structure if P(B | A) > τ where τ is a threshold value (e.g., 21 or 23 , 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(A∧B) 0.3 3 P(A) = 0.4 = 4 ) 0.2 A ∧ B) = P(A∧B∧C P(A∧B) = 0.3 ) 0.2 1 A) = P(A∧C P(A) = 0.4 = 2

P(B | A) = P(C | P(C |

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

=

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Better idea: go for the limes

-semantics, idea: (Adams (1975); Pearl (1989)) K A A |∼ B holds if P(B | A) is converges to 1

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

93 / 96

Better idea: go for the limes

-semantics, idea: (Adams (1975); Pearl (1989)) K A A |∼ B holds if P(B | A) is converges to 1

Formally, for each  ∈]0, 1] there is a δ ∈]0, 1] such that P(A | B) ≥ 1 −  in all probability assignments that satisfy P(C | D) ≥ 1 − δ and P(C ) > 0 for all C |∼ D ∈ K

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

93 / 96

Better idea: go for the limes

-semantics, idea: (Adams (1975); Pearl (1989)) K A A |∼ B holds if P(B | A) is converges to 1

Formally, for each  ∈]0, 1] there is a δ ∈]0, 1] such that P(A | B) ≥ 1 −  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, 2015

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Other probabilistic characterisations Big stepped probabilities A |∼ B iff P(B | A) >

Christian Straßer (RUB, UGENT)

1 2

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Other probabilistic characterisations Big stepped probabilities A |∼ B iff P(B | A) >

1 2

wait, but how?

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

94 / 96

Other probabilistic characterisations Big stepped probabilities A |∼ B iff P(B | A) >

1 2

wait, but how? order W linearly by ≺ and request

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Other probabilistic characterisations Big stepped probabilities A |∼ B iff P(B | A) >

1 2

wait, but how? order W linearly by ≺ and request P P({w }) > {P({w 0 }) | w 0 ≺ w }

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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Other probabilistic characterisations Big stepped probabilities A |∼ B iff P(B | A) >

1 2

wait, but how? order W linearly by ≺ and request P P({w }) > {P({w 0 }) | w 0 ≺ w }

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

94 / 96

Other probabilistic characterisations Big stepped probabilities A |∼ B iff P(B | A) >

1 2

wait, but how? order W linearly by ≺ and request P P({w }) > {P({w 0 }) | w 0 ≺ 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, 2015

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Topic

1

Plausible Reasoning

2

Preferential / Selection Semantics (KLM, Shoham)

3

Bibliography

Christian Straßer (RUB, UGENT)

Tutorial: Nonmonotonic Logic (Day 2)

September 3, 2015

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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, 221–242. Brewka, G.: 1989, ‘Preferred Subtheories: An Extended Logical Framework for Default Reasoning.’. In: IJCAI, Vol. 89. pp. 1043–1048. Dubois, D. and H. Prade: 1990, ‘An introduction to possibilistic and fuzzy logics’. In: Readings in Uncertain Reasoning. pp. 742–761. Friedman, N. and J. Y. Halpern: 1996, ‘Plausibility measures and default reasoning’. Journal of the ACM 48, 1297–1304. Gilio, A.: 2002, ‘Probabilistic reasoning under coherence in System P’. Annals of Mathematics and Artificial Intelligence 34(1-3), 5–34. 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. 661–672. Kraus, S., D. Lehman, and M. Magidor: 1990, ‘Nonmonotonic Reasoning, Preferential Models and Cumulative Logics’. Artifical Intelligence 44, 167–207. Makinson, D.: 2003, ‘Bridges between classical and nonmonotonic logic’. Logic Journal of IGPL 11(1), 69–96. 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. 505–516. Rescher, N. and R. Manor: 1970, ‘On inference from inconsistent premises’. Theory and Decision 1, 179–217. 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. 227–249. 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), 601–641. Christian Straßer (RUB, UGENT)

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