ESSLLI Tutorial: Nonmonotonic Logic
Approaches based on Maximal Consistent Subsets
Mathieu Beirlaen1
Christian Straßer1,2
August 23, 2016
1 Institute
2 Center
for Philosophy II, Ruhr-University Bochum in
for Logic and Philosophy of Science, Ghent University
Aims of this session
1. Get familiar with approaches to nonmonotonic logic
based on maximal consistent subsets
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Aims of this session
1. Get familiar with approaches to nonmonotonic logic
based on maximal consistent subsets
2. get familiar with or recall meta-theoretic principles that
play an important role in nonmonotonic logic
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Nonmonotonic Logics based on
Maximal Consistent Subsets
Nonmonotonic Logics based on
Maximal Consistent Subsets
Some background
… as known from:
• Rescher / Manor consequence relations (Rescher and
Manor (1970))
• Brewka’s preferred subtheories (Brewka (1989))
• Benferhat, Dubious, Prade (Benferhat et al. (1997))
• Makinsons’ Default Assumptions (Makinson (2003))
• Batens’ Adaptive Logics and generalizations (Batens
(2007); Straßer (2014); Van De Putte (2013))
• Constrained Input/Output logics (Makinson and Van
Der Torre (2001))
• there are also close connections to argumentation-based
methods (Amgoud and Besnard (2013))
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The setup
(non-defeasible) facts
Σ = ⟨Σ0 , Σd ⟩
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The setup
(non-defeasible) facts
Σ = ⟨Σ0 , Σd ⟩
defeasible premises (assumptions)
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The setup
(non-defeasible) facts
Σ = ⟨Σ0 , Σd ⟩
defeasible premises (assumptions)
• Σd may also be stratified:
Σd = ⟨Σ1 , . . . , Σn , . . .⟩
• 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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The setup
(non-defeasible) facts
Σ = ⟨Σ0 , Σd ⟩
defeasible premises (assumptions)
• Σd may also be stratified:
Σd = ⟨Σ1 , . . . , Σn , . . .⟩
• 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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The Base Logic L with consequence relation Cn
We suppose the L satisfies the following requirements:
• monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ′ )
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The Base Logic L with consequence relation Cn
We suppose the L satisfies the following requirements:
• monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ′ )
• reflexive: Γ ⊆ Cn(Γ)
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The Base Logic L with consequence relation Cn
We suppose the L satisfies the following requirements:
• monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ′ )
• reflexive: Γ ⊆ Cn(Γ)
• compact: if A ∈ Cn(Γ) then A ∈ Cn(Γ′ ) for some finite
Γ′ ⊆ Γ
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The Base Logic L with consequence relation Cn
We suppose the L satisfies the following requirements:
• monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ′ )
• reflexive: Γ ⊆ Cn(Γ)
• compact: if A ∈ Cn(Γ) then A ∈ Cn(Γ′ ) for some finite
Γ′ ⊆ Γ
• cut: where Γ′ ⊆ Cn(Γ) and A ∈ Cn(Γ ∪ Γ′ ), A ∈ Cn(Γ)
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The Base Logic L with consequence relation Cn
We suppose the L satisfies the following requirements:
• monotonic: Cn(Γ) ⊆ Cn(Γ ∪ Γ′ )
• reflexive: Γ ⊆ Cn(Γ)
• compact: if A ∈ Cn(Γ) then A ∈ Cn(Γ′ ) for some finite
Γ′ ⊆ Γ
• cut: where Γ′ ⊆ Cn(Γ) and A ∈ Cn(Γ ∪ Γ′ ), A ∈ Cn(Γ)
• or, transitivity: where Γ′ ⊆ Cn(Γ) and A ∈ Cn(Γ′ ),
A ∈ Cn(Γ).
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Note
Given reflexivity and monotonicity, the following are
equivalent:
• cut: where Γ′ ⊆ Cn(Γ) and A ∈ Cn(Γ ∪ Γ′ ), A ∈ Cn(Γ)
• transitivity: where Γ′ ⊆ Cn(Γ) and A ∈ Cn(Γ′ ), A ∈ Cn(Γ).
Proof.
Suppose Γ′ ⊆ Cn(Γ).
• (⇒) Suppose A ∈ Cn(Γ′ ).
• By Monotonicity, A ∈ Cn(Γ ∪ Γ′ ).
• By Cut, A ∈ Cn(Γ).
• (⇐) Suppose A ∈ Cn(Γ ∪ Γ′ ).
• By Reflexivity, Γ ∪ Γ′ ⊆ Cn(Γ).
• By Transitivity, A ∈ Cn(Γ).
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(In)consistency
There are many notions of inconsistency of a set of formulas Γ,
e.g.,
1. Γ ⊢ ⊥
2. Γ ⊢ p ∧ ¬p for a propositional atom p
3. Γ ⊢ A for all wff A
4. etc.
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(In)consistency
There are many notions of inconsistency of a set of formulas Γ,
e.g.,
1. Γ ⊢ ⊥
2. Γ ⊢ p ∧ ¬p for a propositional atom p
3. Γ ⊢ A for all wff A
4. etc.
In what follows we suppose option 1.
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(In)consistency
There are many notions of inconsistency of a set of formulas Γ,
e.g.,
1. Γ ⊢ ⊥
2. Γ ⊢ p ∧ ¬p for a propositional atom p
3. Γ ⊢ A for all wff A
4. etc.
In what follows we suppose option 1.
We call a set of formulas consistent if it is not inconsistent.
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Maximal Consistent Subsets
Ξ is a maximal consistent subset of Σ = ⟨Σ0 , Σd ⟩ iff
1. Ξ ⊆ Σd is such that Ξ ∪ Σ0 is consistent
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Maximal Consistent Subsets
Ξ is a maximal consistent subset of Σ = ⟨Σ0 , Σd ⟩ 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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Maximal Consistent Subsets
Ξ is a maximal consistent subset of Σ = ⟨Σ0 , Σd ⟩ 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.
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Maximal Consistent Subsets
Ξ is a maximal consistent subset of Σ = ⟨Σ0 , Σd ⟩ 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.
We write CS(Σ) for the set of all Ξ for which item 1 holds, i.e.,
for the set of all consistent subsets of Σ.
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ where
• Σ0 = {s}
• Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r}
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ where
• Σ0 = {s}
• Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r}
What are the MCSs?
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ where
• Σ0 = {s}
• Σd = {s ⊃ (p ∧ q), p ∧ ¬q, r}
What are the MCSs?
• Ξ1 = {s ⊃ (p ∧ q), r}
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ 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 Σ = ⟨Σ0 , Σd ⟩, and ΞCS(Σ), there is a Ξ′ ∈ MCS(Σ) s.t.
Ξ ⊆ Ξ′ .
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Existence of MCSs
Where Σ = ⟨Σ0 , Σd ⟩, and ΞCS(Σ), there is a Ξ′ ∈ MCS(Σ) s.t.
Ξ ⊆ Ξ′ . Here’s a proof:
∪
• Let Σd = {A1 , . . .}. Let Ξ′ = i≥0 Ξi where Ξ0 = Ξ
{
Ξi ∪ {Ai+1 } if Ξi ∪ {Ai+1 } ∪ Σ0 ⊬ ⊥
Ξi+1 =
Ξi
else
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Existence of MCSs
Where Σ = ⟨Σ0 , Σd ⟩, and ΞCS(Σ), there is a Ξ′ ∈ MCS(Σ) s.t.
Ξ ⊆ Ξ′ . Here’s a proof:
∪
• Let Σd = {A1 , . . .}. Let Ξ′ = i≥0 Ξi where Ξ0 = Ξ
{
Ξi ∪ {Ai+1 } if Ξi ∪ {Ai+1 } ∪ Σ0 ⊬ ⊥
Ξi+1 =
Ξi
else
• By induction, for each Ξi , Ξi ∪ Σ0 ⊬ ⊥.
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Existence of MCSs
Where Σ = ⟨Σ0 , Σd ⟩, and ΞCS(Σ), there is a Ξ′ ∈ MCS(Σ) s.t.
Ξ ⊆ Ξ′ . Here’s a proof:
∪
• Let Σd = {A1 , . . .}. Let Ξ′ = i≥0 Ξi where Ξ0 = Ξ
{
Ξi ∪ {Ai+1 } if Ξi ∪ {Ai+1 } ∪ Σ0 ⊬ ⊥
Ξi+1 =
Ξi
else
• By induction, for each Ξi , Ξi ∪ Σ0 ⊬ ⊥.
• Assume Ξ′ ∪ Σ0 ⊢ ⊥.
• By compactness, there is a finite Ξ′f ⊆ Ξ′ such that
Ξ′f ∪ Σ0 ⊢ ⊥.
• Thus there is a Ξi ⊇ Ξ′f and by monotonicity,
Ξi ∪ Ξ0 ⊢ ⊥,—a contradiction.
• Hence, Ξ′ ∪ Σ0 ⊬ ⊥.
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Existence of MCSs
Where Σ = ⟨Σ0 , Σd ⟩, and ΞCS(Σ), there is a Ξ′ ∈ MCS(Σ) s.t.
Ξ ⊆ Ξ′ . Here’s a proof:
∪
• Let Σd = {A1 , . . .}. Let Ξ′ = i≥0 Ξi where Ξ0 = Ξ
{
Ξi ∪ {Ai+1 } if Ξi ∪ {Ai+1 } ∪ Σ0 ⊬ ⊥
Ξi+1 =
Ξi
else
• By induction, for each Ξi , Ξi ∪ Σ0 ⊬ ⊥.
• Assume Ξ′ ∪ Σ0 ⊢ ⊥.
• By compactness, there is a finite Ξ′f ⊆ Ξ′ such that
Ξ′f ∪ Σ0 ⊢ ⊥.
• Thus there is a Ξi ⊇ Ξ′f and by monotonicity,
Ξi ∪ Ξ0 ⊢ ⊥,—a contradiction.
• Hence, Ξ′ ∪ Σ0 ⊬ ⊥.
• Where Ai ∈
/ Ξ′ , Ξ′ ∪ {Ai } ∪ Σ0 ⊢ ⊥ since
Ξi−1 ∪ {Ai } ∪ Σ0 ⊢ ⊥ and by monotonicity.
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Nonmonotonic Logics based on
Maximal Consistent Subsets
Defining Consequence Relations
General Idea
Given Σ = ⟨Σ0 , Σd ⟩, what should be in the consequence set?
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General Idea
Given Σ = ⟨Σ0 , Σd ⟩, what should be in the consequence set?
• if Σ0 ∪ Σd is consistent, clearly Cn(Σ0 ∪ Σd )
• but what if not?
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General Idea
Given Σ = ⟨Σ0 , Σd ⟩, what should be in the consequence set?
• if Σ0 ∪ Σd is consistent, clearly Cn(Σ0 ∪ Σd )
• but what if not?
Here we put at use maximal consistent subsets.
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Three central consequence relations
Given Σ = ⟨Σ0 , Σd ⟩, we define:
• Free consequences:
Σ ⊢free A iff Σ0 ∪ Free(Σ) ⊢ A where Free(Σ) =
∩
MCS(Σ)
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Three central consequence relations
Given Σ = ⟨Σ0 , Σd ⟩, we define:
• 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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Three central consequence relations
Given Σ = ⟨Σ0 , Σd ⟩, we define:
• 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)
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ where
What are the MCSs?
• Σ0 = {s}
• Ξ1 = {s ⊃ (p ∧ q), r}
• Σd = {s ⊃ (p∧q), p∧¬q, r}
• Ξ2 = {p ∧ ¬q, r}
Free consequences
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ where
What are the MCSs?
• Σ0 = {s}
• Ξ1 = {s ⊃ (p ∧ q), r}
• Σd = {s ⊃ (p∧q), p∧¬q, r}
• Ξ2 = {p ∧ ¬q, r}
Free consequences
• Free(Σ) = {r}
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ where
What are the MCSs?
• Σ0 = {s}
• Ξ1 = {s ⊃ (p ∧ q), r}
• Σd = {s ⊃ (p∧q), 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 = ⟨Σ0 , Σd ⟩ where
What are the MCSs?
• Σ0 = {s}
• Ξ1 = {s ⊃ (p ∧ q), r}
• Σd = {s ⊃ (p∧q), 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 = ⟨Σ0 , Σd ⟩ where
What are the MCSs?
• Σ0 = {s}
• Ξ1 = {s ⊃ (p ∧ q), r}
• Σd = {s ⊃ (p∧q), p∧¬q, r}
• Ξ2 = {p ∧ ¬q, r}
Free consequences
• Free(Σ) = {r}
• Γ ⊢free A iff {s, r} ⊢ A
Notice: Syntax-Dependency
• ⟨Σ0 , {s ⊃ (p ∧ q), p, ¬q, r}⟩ ⊢free p
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ 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 = ⟨Σ0 , Σd ⟩ 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 = ⟨Σ0 , Σd ⟩ 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
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Question: Is there also some syntax-dependency for ⊢∀ ?
Take Σ′ = ⟨∅, {p ∧ q, ¬p}⟩
{
}
• MCS(Σ′ ) = {p ∧ q}, {¬p}
• Σ′ ⊬∀ q
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Question: Is there also some syntax-dependency for ⊢∀ ?
Take Σ′ = ⟨∅, {p ∧ q, ¬p}⟩
{
}
• MCS(Σ′ ) = {p ∧ q}, {¬p}
• Σ′ ⊬∀ q
While, where Σ′′ = ⟨∅, {p, q, ¬p}⟩
{
}
• MCS(Σ′′ ) = {p, q}, {¬p, q}
• Σ′′ ⊢∀ q.
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Examples
Let Σ = ⟨Σ0 , Σd ⟩ 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 = ⟨Σ0 , Σd ⟩ 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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Exercises
Try to show:
1. Each free consequence is a universal consequence.
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Exercises
Try to show:
1. Each free consequence is a universal consequence.
2. Each universal consequence is a existential consequence.
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Exercises
Try to show:
1. Each free consequence is a universal consequence.
2. Each universal consequence is a existential consequence.
3. The L-consequences of the free consequences of Σ are
free consequences of Σ: in symbols,
Cn({A | Σ ⊢free A}) ⊆ {A | Σ ⊢free A}.
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Exercises
Try to show:
1. Each free consequence is a universal consequence.
2. Each universal consequence is a existential consequence.
3. The L-consequences of the free consequences of Σ are
free consequences of Σ: in symbols,
Cn({A | Σ ⊢free A}) ⊆ {A | Σ ⊢free A}.
4. The same holds for the universal consequences.
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Exercises
Try to show:
1. Each free consequence is a universal consequence.
2. Each universal consequence is a existential consequence.
3. The L-consequences of the free consequences of Σ are
free consequences of Σ: in symbols,
Cn({A | Σ ⊢free A}) ⊆ {A | Σ ⊢free A}.
4. The same holds for the universal consequences.
5. The free consequences are consistent.
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Exercises
Try to show:
1. Each free consequence is a universal consequence.
2. Each universal consequence is a existential consequence.
3. The L-consequences of the free consequences of Σ are
free consequences of Σ: in symbols,
Cn({A | Σ ⊢free A}) ⊆ {A | Σ ⊢free A}.
4. The same holds for the universal consequences.
5. The free consequences are consistent.
6. So are the universal consequences.
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A hierarchy
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Nonmonotonic Logics based on
Maximal Consistent Subsets
What about stratified premise sets?
Making use of Lexicographic Orders
• Let ⟨Σ0 , Σd ⟩ where Σd = ⟨Σ1 , Σ2 , . . .⟩.
• Idea: Order MCSs relative to the strength of their
constituting premises where strength/reliability/etc. is
indirectly proportional to the index Σi .
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Making use of Lexicographic Orders
• Let ⟨Σ0 , Σd ⟩ where Σd = ⟨Σ1 , Σ2 , . . .⟩.
• Idea: Order MCSs relative to the strength of their
constituting premises where strength/reliability/etc. is
indirectly proportional to the index Σi .
Lexicographic Ordering (e.g., Brewka (1989))
Ξ ≺ Ξ′ iff there is an i ≥ 1 such that
1. Ξ ∩ Σj = Ξ′ ∩ Σj for all 1 ≤ j < i, and
2. Ξ ∩ Σi ⊃ Ξ′ ∩ Σi
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Making use of Lexicographic Orders
Lexicographic Ordering (e.g., Brewka (1989))
Ξ ≺ Ξ′ iff there is an i ≥ 1 such that
1. Ξ ∩ Σj = Ξ′ ∩ Σj for all 1 ≤ j < i, and
2. Ξ ∩ Σi ⊃ Ξ′ ∩ Σi
Consequence relation
… relative to min≺ (MCS(Σ))
∩
min≺ (MCS(Σ)) ⊢ A
• Σ ⊢≺
free A iff
• Σ ⊢≺
∀ A iff for all Ξ ∈ min≺ (MCS(Σ)), Ξ ⊢ A
• Σ ⊢≺
∃ A iff for some Ξ ∈ min≺ (MCS(Σ)), Ξ ⊢ A
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Time for an example
Lexicographic Ordering
Ξ ≺lex Ξ′ iff there is an i ≥ 1 such that
1. Ξ ∩ Σj = Ξ′ ∩ Σj for all 1 ≤ j < i, and
2. Ξ ∩ Σi ⊃ Ξ′ ∩ Σi
Example
MCSs
• Σ0 = {s}
• Σ1 = {s ⊃ p}
• Σ2 = {¬p, ¬q}
• Σ3 = {q, r}
order: Ξ1 ≺ Ξ2 ≺ Ξ3 ≺ Ξ4
• Ξ1 = {s ⊃ p, ¬q, r}
• Ξ2 = {s ⊃ p, q, r}
• Ξ3 = {¬p, ¬q, r}
• Ξ4 = {¬p, q, r}
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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)).
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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 < Ξ′ iff there there is an i ≥ 1 such that
1. Ξ ∩ Σj = Ξ′ ∩ Σj for all j > i, and
2. Ξ ∩ Σi ⊃ Ξ ∩ Σi
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A problematic example
Ordering
Ξ < Ξ′ iff there there is an i ≥ 1 such that
1. Ξ ∩ Σj = Ξ′ ∩ Σj for all j > i, and
2. Ξ ∩ Σi ⊃ Ξ ∩ Σi
Example
• Take Σ = ⟨Σ0 , Σ1 , Σ2 , . . .⟩
where
• Σ0 = {pi ∨ pj | j > i ≥ 1}
• Σi = {¬pi , s} for each i ≥ 1
MCSs
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A problematic example
Ordering
Ξ < Ξ′ iff there there is an i ≥ 1 such that
1. Ξ ∩ Σj = Ξ′ ∩ Σj for all j > i, and
2. Ξ ∩ Σi ⊃ Ξ ∩ Σi
Example
• Take Σ = ⟨Σ0 , Σ1 , Σ2 , . . .⟩
where
• Σ0 = {pi ∨ pj | j > i ≥ 1}
• Σi = {¬pi , s} for each i ≥ 1
MCSs
• Σi for each i ≥ 1
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A problematic example
Ordering
Ξ < Ξ′ iff there there is an i ≥ 1 such that
1. Ξ ∩ Σj = Ξ′ ∩ Σj for all j > i, and
2. Ξ ∩ Σi ⊃ Ξ ∩ Σi
Example
• Take Σ = ⟨Σ0 , Σ1 , Σ2 , . . .⟩
where
• Σ0 = {pi ∨ pj | j > i ≥ 1}
• Σi = {¬pi , s} for each i ≥ 1
MCSs
• Σi for each i ≥ 1
Note
• min< (MCS(Σ)) = ∅
• however, we would at least expect to derive s
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Some Meta-Theory
Some Meta-Theory
Monotonicity & Co
Monotonicity?
We can distinguish between (where |∼ ∈ {⊢∀ , ⊢∃ , ⊢free }):
Monotonicity in the defeasible premises
If ⟨Σ0 , Σd ⟩ |∼ A then ⟨Σ0 , Σd ∪ Γ⟩ |∼ A.
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Monotonicity?
We can distinguish between (where |∼ ∈ {⊢∀ , ⊢∃ , ⊢free }):
Monotonicity in the defeasible premises
If ⟨Σ0 , Σd ⟩ |∼ A then ⟨Σ0 , Σd ∪ Γ⟩ |∼ A.
Monotonicity in the factual premises
If ⟨Σ0 , Σd ⟩ |∼ A then ⟨Σ0 ∪ Γ, Σd ⟩ |∼ A.
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Can you think of a counter-example to
monotonicity for our consequence relations?
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Exercise
Prove that monotonicity holds for existential consequences in
the defeasible premises.
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Some Meta-Theory
Weakening Monotonicity
Recall that there are weakened forms of
monotonicity, most prominently the following
two principles …
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Cautious Monotonicity
If A1 , . . . , An |∼ B and A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
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Cautious Monotonicity
If A1 , . . . , An |∼ B and A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
Conclusions/Output
Premises/Input
Logic
A1 , . . . , An
B, C
“We do not loose conclusions if we plug in conclusions as
additional input.”
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Cautious Monotonicity
If A1 , . . . , An |∼ B and A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
Conclusions/Output
Premises/Input
Logic
A1 , . . . , An
B, C
“We do not loose conclusions if we plug in conclusions as
additional input.”
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Rational Monotonicity
If it is not the case that A1 , . . . , An |∼ ¬B, and moreover
A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
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Rational Monotonicity
If it is not the case that A1 , . . . , An |∼ ¬B, and moreover
A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
Premises/Input
Conclusions/Output
Logic
A1 , . . . , An ,
C
|
¬B
“Our conclusions are robust under that addition of consistent
information to the input.”
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Rational Monotonicity
If it is not the case that A1 , . . . , An |∼ ¬B, and moreover
A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
Premises/Input
Conclusions/Output
Logic
A1 , . . . , An , B
C
|
¬B
“Our conclusions are robust under that addition of consistent
information to the input.”
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Some Meta-Theory
Cumulativity
We will treat
Cautious Monotonicity
If Σ |∼ A and Σ |∼ B then Σ ∪ {A} |∼ B.
and
Cut
If Σ |∼ A and Σ ∪ {A} |∼ B then Σ |∼ B.
at one single blow …
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Cumulativity
If Σ |∼ A : Σ |∼ B iff Σ ∪ {A} |∼ B
27/77
Cumulativity
If Σ |∼ A : Σ |∼ B iff Σ ∪ {A} |∼ B
• Cautious Monotonicity
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Cumulativity
If Σ |∼ A : Σ |∼ B iff Σ ∪ {A} |∼ B
• Cautious Monotonicity
• Cut
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Cumulativity
If Σ |∼ A : Σ |∼ B iff Σ ∪ {A} |∼ B
• Cautious Monotonicity
• Cut
Cumulativity in defeasible premises:
If ⟨Σ0 , Σd ⟩ |∼ A : ⟨Σ0 , Σd ⟩ |∼ B iff ⟨Σ0 , Σd ∪ {A}⟩ |∼ B
Cumulativity in factual premises:
If ⟨Σ0 , Σd ⟩ |∼ A : ⟨Σ0 , Σd ⟩ |∼ B iff ⟨Σ0 ∪ {A}, Σd ⟩ |∼ B
27/77
Some Meta-Theory
Cumulativity in the factual premises
For this, proving one central lemma will be
sufficient …
27/77
A central lemma
Note that
Cumulativity in the factual premises (|∼ ∈ {⊢∀ , ⊢free })
If ⟨Σ0 , Σd ⟩ |∼ A : ⟨Σ0 , Σd ⟩ |∼ B iff ⟨Σ0 ∪ {A}, Σd ⟩ |∼ B
follows immediately with the following lemma:
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A central lemma
Note that
Cumulativity in the factual premises (|∼ ∈ {⊢∀ , ⊢free })
If ⟨Σ0 , Σd ⟩ |∼ A : ⟨Σ0 , Σd ⟩ |∼ B iff ⟨Σ0 ∪ {A}, Σd ⟩ |∼ B
follows immediately with the following lemma:
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
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A central lemma
Note that
Cumulativity in the factual premises (|∼ ∈ {⊢∀ , ⊢free })
If ⟨Σ0 , Σd ⟩ |∼ A : ⟨Σ0 , Σd ⟩ |∼ B iff ⟨Σ0 ∪ {A}, Σd ⟩ |∼ B
follows immediately with the following lemma:
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
(Simple) Exercise
Try to prove that Cumulativity for the different consequence
relations follows from Lemma 1.
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A central lemma (cont.)
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
Proof of ”⊆”.
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A central lemma (cont.)
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
Proof of ”⊆”.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A and let (†) Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
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A central lemma (cont.)
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
Proof of ”⊆”.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A and let (†) Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
• Thus, Ξ ∪ Σ0 ⊢ A and hence, by (†) and cut,
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(1)
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A central lemma (cont.)
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
Proof of ”⊆”.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A and let (†) Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
• Thus, Ξ ∪ Σ0 ⊢ A and hence, by (†) and cut,
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(1)
• Assume for a contradiction that there is a
Ξ′ ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ′ .
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A central lemma (cont.)
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
Proof of ”⊆”.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A and let (†) Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
• Thus, Ξ ∪ Σ0 ⊢ A and hence, by (†) and cut,
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(1)
• Assume for a contradiction that there is a
Ξ′ ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ′ .
• Hence, Σ0 ∪ {A} ∪ Ξ′ ⊬ ⊥ and hence Σ0 ∪ Ξ′ ⊬ ⊥ by
monotonicity.
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A central lemma (cont.)
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
Proof of ”⊆”.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A and let (†) Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
• Thus, Ξ ∪ Σ0 ⊢ A and hence, by (†) and cut,
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(1)
• Assume for a contradiction that there is a
Ξ′ ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ′ .
• Hence, Σ0 ∪ {A} ∪ Ξ′ ⊬ ⊥ and hence Σ0 ∪ Ξ′ ⊬ ⊥ by
monotonicity.
• This contradicts that Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
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A central lemma (cont.)
Lemma 1
If ⟨Σ0 , Σd ⟩ ⊢∀ A, then MCS(⟨Σ0 , Σd ⟩) = MCS(⟨Σ0 ∪ {A}, Σd ⟩).
Proof of ”⊆”.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A and let (†) Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
• Thus, Ξ ∪ Σ0 ⊢ A and hence, by (†) and cut,
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(1)
• Assume for a contradiction that there is a
Ξ′ ∈ MCS(Σ0 ∪ {A}, Σd ) such that Ξ ⊂ Ξ′ .
• Hence, Σ0 ∪ {A} ∪ Ξ′ ⊬ ⊥ and hence Σ0 ∪ Ξ′ ⊬ ⊥ by
monotonicity.
• This contradicts that Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
• Thus, by (1), Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
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Exercise
Exercise
Proof the other direction of Lemma 1.
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Some Meta-Theory
Cumulativity in the defeasible premises
For this, proving one central lemma will be
sufficient …
(sounds familiar?)
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A lemma central to Cumulativity
Lemma 2
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢∀ A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
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A lemma central to Cumulativity
Lemma 2
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢∀ A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
Proof.
We proceed in steps …
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First we exclude the trivial case …
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Excluding the trivial case
Lemma 2
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢∀ A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
Proof (cont.)
If A ∈ Σd the lemma is trivial. Suppose thus that A ∈
/ Σd .
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Next, we should check whether π is
well-defined.
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Is π well-defined?
Proof (cont.)
• Let Ξ ∈ MCS(Σ).
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Is π well-defined?
Proof (cont.)
• Let Ξ ∈ MCS(Σ).
• We show that Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
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Is π well-defined?
Proof (cont.)
• Let Ξ ∈ MCS(Σ).
• We show that Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• Note that Ξ ∪ Σ0 ⊢ A. Thus, by cut and since Ξ ∈ MCS(Σ),
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(2)
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Is π well-defined?
Proof (cont.)
• Let Ξ ∈ MCS(Σ).
• We show that Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• Note that Ξ ∪ Σ0 ⊢ A. Thus, by cut and since Ξ ∈ MCS(Σ),
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(2)
• Assume for a contradiction that there is a Ξ′ ⊃ Ξ ∪ {A}
such that Ξ′ ∈ MCS(Σ0 , Σd ∪ {A}).
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Is π well-defined?
Proof (cont.)
• Let Ξ ∈ MCS(Σ).
• We show that Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• Note that Ξ ∪ Σ0 ⊢ A. Thus, by cut and since Ξ ∈ MCS(Σ),
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(2)
• Assume for a contradiction that there is a Ξ′ ⊃ Ξ ∪ {A}
such that Ξ′ ∈ MCS(Σ0 , Σd ∪ {A}).
• Thus, Σd ⊇ Ξ′ \ {A} ⊃ Ξ.
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Is π well-defined?
Proof (cont.)
• Let Ξ ∈ MCS(Σ).
• We show that Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• Note that Ξ ∪ Σ0 ⊢ A. Thus, by cut and since Ξ ∈ MCS(Σ),
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(2)
• Assume for a contradiction that there is a Ξ′ ⊃ Ξ ∪ {A}
such that Ξ′ ∈ MCS(Σ0 , Σd ∪ {A}).
• Thus, Σd ⊇ Ξ′ \ {A} ⊃ Ξ.
• Since Σ0 ∪ (Ξ′ \ {A}) ⊬ ⊥ (by monotonicity), this is a
contradiction to Ξ being in MCS(Σ).
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Is π well-defined?
Proof (cont.)
• Let Ξ ∈ MCS(Σ).
• We show that Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• Note that Ξ ∪ Σ0 ⊢ A. Thus, by cut and since Ξ ∈ MCS(Σ),
Σ0 ∪ Ξ ∪ {A} ⊬ ⊥
(2)
• Assume for a contradiction that there is a Ξ′ ⊃ Ξ ∪ {A}
such that Ξ′ ∈ MCS(Σ0 , Σd ∪ {A}).
• Thus, Σd ⊇ Ξ′ \ {A} ⊃ Ξ.
• Since Σ0 ∪ (Ξ′ \ {A}) ⊬ ⊥ (by monotonicity), this is a
contradiction to Ξ being in MCS(Σ).
• Thus, the assumption is wrong and in view of (2),
Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
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Next, we show that π is surjective.
(Recall, this means that every entity in the
co-domain of π is in the image of π.)
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π is surjective
Proof (cont.)
We now show that π is surjective. Let for this (‡)
Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
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π is surjective
Proof (cont.)
We now show that π is surjective. Let for this (‡)
Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• By monotonicity and since Σ0 ∪ Ξ ⊬ ⊥, also
Σ0 ∪ (Ξ \ {A}) ⊬ ⊥
(3)
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π is surjective
Proof (cont.)
We now show that π is surjective. Let for this (‡)
Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• By monotonicity and since Σ0 ∪ Ξ ⊬ ⊥, also
Σ0 ∪ (Ξ \ {A}) ⊬ ⊥
(3)
• Assume for a contradiction that there is a
Ξ′ ∈ MCS(⟨Σ0 , Σd ⟩) such that Ξ \ {A} ⊂ Ξ′ ⊆ Σd .
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π is surjective
Proof (cont.)
We now show that π is surjective. Let for this (‡)
Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• By monotonicity and since Σ0 ∪ Ξ ⊬ ⊥, also
Σ0 ∪ (Ξ \ {A}) ⊬ ⊥
(3)
• Assume for a contradiction that there is a
Ξ′ ∈ MCS(⟨Σ0 , Σd ⟩) such that Ξ \ {A} ⊂ Ξ′ ⊆ Σd .
• By the supposition, Σ0 ∪ Ξ′ ⊢ A and hence
Σ0 ∪ Ξ′ ∪ {A} ⊬ ⊥.
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π is surjective
Proof (cont.)
We now show that π is surjective. Let for this (‡)
Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• By monotonicity and since Σ0 ∪ Ξ ⊬ ⊥, also
Σ0 ∪ (Ξ \ {A}) ⊬ ⊥
(3)
• Assume for a contradiction that there is a
Ξ′ ∈ MCS(⟨Σ0 , Σd ⟩) such that Ξ \ {A} ⊂ Ξ′ ⊆ Σd .
• By the supposition, Σ0 ∪ Ξ′ ⊢ A and hence
Σ0 ∪ Ξ′ ∪ {A} ⊬ ⊥.
• This is a contradiction to (‡) since Ξ ⊂ Ξ′ ∪ {A} ⊆ Σd ∪ {A}.
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π is surjective
Proof (cont.)
We now show that π is surjective. Let for this (‡)
Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• By monotonicity and since Σ0 ∪ Ξ ⊬ ⊥, also
Σ0 ∪ (Ξ \ {A}) ⊬ ⊥
(3)
• Assume for a contradiction that there is a
Ξ′ ∈ MCS(⟨Σ0 , Σd ⟩) such that Ξ \ {A} ⊂ Ξ′ ⊆ Σd .
• By the supposition, Σ0 ∪ Ξ′ ⊢ A and hence
Σ0 ∪ Ξ′ ∪ {A} ⊬ ⊥.
• This is a contradiction to (‡) since Ξ ⊂ Ξ′ ∪ {A} ⊆ Σd ∪ {A}.
• Since our assumption is wrong and by (3),
Ξ \ {A} ∈ MCS(Σ).
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What is left, is to show that π is injective.
Fortunately, this is easy …
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π is injective
Proof (cont.)
That π is injective is trivial in view of the supposition that
A∈
/ Σd .
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So, how does this help in proving Cumulativity?
Let’s see and start with ⊢∀ …
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Cumulativity
Theorem 3
Where ⟨Σ0 , Σd ⟩ ⊢∀ A: ⟨Σ0 , Σd ⟩ ⊢∀ B iff ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B.
We again proceed in steps:
1. we exclude a trivial case
2. we proof the (⇒)-direction
3. we proof the (⇐)-direction
This will all be pretty straight-forward, given our lemma.
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Cumulativity (cont.): excluded a trivial case
Proof.
In case A ∈ Σd the proposition is trivial. Suppose thus that
A∈
/ Σd .
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Cumulativity (cont.): the (⇒)-direction
Proof.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A.
• Suppose ⟨Σ0 , Σd ⟩ ⊢∀ B.
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Cumulativity (cont.): the (⇒)-direction
Proof.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A.
• Suppose ⟨Σ0 , Σd ⟩ ⊢∀ B.
• To show: ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B.
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Cumulativity (cont.): the (⇒)-direction
Proof.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A.
• Suppose ⟨Σ0 , Σd ⟩ ⊢∀ B.
• To show: ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B.
• Let Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
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Cumulativity (cont.): the (⇒)-direction
Proof.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A.
• Suppose ⟨Σ0 , Σd ⟩ ⊢∀ B.
• To show: ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B.
• Let Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• Hence, by our Lemma 2, Ξ \ {A} ∈ MCS(⟨Σ0 , Σd ⟩) and thus
Σ0 ∪ (Ξ \ {A}) ⊢ B.
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Cumulativity (cont.): the (⇒)-direction
Proof.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A.
• Suppose ⟨Σ0 , Σd ⟩ ⊢∀ B.
• To show: ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B.
• Let Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• Hence, by our Lemma 2, Ξ \ {A} ∈ MCS(⟨Σ0 , Σd ⟩) and thus
Σ0 ∪ (Ξ \ {A}) ⊢ B.
• By Monotonicity, Σ0 ∪ Ξ ⊢ B.
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Cumulativity (cont.): the (⇐)-direction
Proof.
• Suppose ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B.
• Let Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
• Hence, by our Lemma 2, Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ⟩) and
Σ0 ∪ Ξ ⊢ A.
• Also, Σ0 ∪ Ξ ∪ {A} ⊢ B.
• By cut, Σ0 ∪ Ξ ⊢ B.
• Thus, ⟨Σ0 , Σd ⟩ ⊢∀ B.
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Exercise: Cumulativity for the Free Consequences
Note that with Lemma 2 the following follows immediately:
Lemma 4
Where ⟨Σ0 , Σd ⟩ ⊢∀ A,
∩
∩
MCS(⟨Σ0 , Σd ∪ {A}⟩) = MCS(⟨Σ0 , Σd ⟩) ∪ {A}
Exercise
Try to prove the following two theorems:
Theorem 5
Where ⟨Σ0 , Σd ⟩ ⊢free A: ⟨Σ0 , Σd ⟩ ⊢free B iff
⟨Σ0 , Σd ∪ {A}⟩ ⊢free B.
Theorem 6
Where ⟨Σ0 , Σd ⟩ ⊢∀ A: ⟨Σ0 , Σd ⟩ ⊢free B iff ⟨Σ0 , Σd ∪ {A}⟩ ⊢free B.
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Some Meta-Theory
A stronger form of Cautious Monotonicity
Interestingly, we can strengthen our previous
result by adjusting our lemma a bit …
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A strengthened Lemma
Recall our lemma:
Lemma 2
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢∀ A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
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A strengthened Lemma
Recall our lemma:
Lemma 2
we’re talking about this bit
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢∀ A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
Do you have an idea how to weaken the antecedent of this
lemma?
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A strengthened Lemma
Recall our lemma:
Lemma 2
we’re talking about this bit
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢∀ A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
Do you have an idea how to weaken the antecedent of this
lemma?
Lemma 7
Where Σ ⊬∃ ¬A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
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A strengthened Lemma
Recall our lemma:
Lemma 2
we’re talking about this bit
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢∀ A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
Do you have an idea how to weaken the antecedent of this
lemma?
7
hereLemma
we suppose
to have a classical negation
Where Σ ⊬∃ ¬A,
π : MCS(⟨Σ0 , Σd ⟩) → MCS(⟨Σ0 , Σd ∪ {A}⟩), Λ 7→ Λ ∪ {A} is a
bijection.
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Exercise
Adjust the proof of Lemma 2 to prove Lemma 7.
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A stronger form of Cautious Monotonicity
Our lemma immediately gives rise to the following theorem.
Theorem 8
Where ⟨Σ0 , Σd ⟩ ⊬∃ ¬A: ⟨Σ0 , Σd ⟩ ⊢∀ B implies
⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B.
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A stronger form of Cautious Monotonicity
Our lemma immediately gives rise to the following theorem.
Theorem 8
Where ⟨Σ0 , Σd ⟩ ⊬∃ ¬A: ⟨Σ0 , Σd ⟩ ⊢∀ B implies
⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B.
Proof.
Suppose ⟨Σ0 , Σd ⟩ ⊢∀ A ⟨Σ0 , Σd ⟩ ⊬∃ ¬A.
• Suppose ⟨Σ0 , Σd ⟩ ⊢∀ B.
• Let Ξ ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩).
• Hence, by our Lemma 7, Ξ \ {A} ∈ MCS(⟨Σ0 , Σd ⟩) and thus
Σ0 ∪ (Ξ \ {A}) ⊢ B.
• By Monotonicity, Σ0 ∪ Ξ ⊢ B.
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Wait a moment, doesn’t Σ ⊬∃ ¬A imply Σ ⊢∀ A
and are we not back to where we started?
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Example
Here’s an example to show that Σ ⊬∃ ¬A doesn’t imply Σ ⊢∀ A.
Example
• Take Σ = ⟨∅, ∅⟩.
• We have one MCS: ∅
• Clearly,
• Σ ⊬∃ ¬p
• Σ ⊬∀ p
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Example
Here’s an example to show that Σ ⊬∃ ¬A doesn’t imply Σ ⊢∀ A.
Example
• Take Σ = ⟨∅, ∅⟩.
• We have one MCS: ∅
• Clearly,
• Σ ⊬∃ ¬p
• Σ ⊬∀ p
Simple exercise
Show that Σ ⊢∀ A implies Σ ⊬∃ ¬A.
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Some Meta-Theory
What about Rational Monotonicity?
Our strengthened Cautious Monotonicity is
quite similar to Rational Monotonicity …
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Theorem 8 and Rational Monotonicity
Theorem 8
Where ⟨Σ0 , Σd ⟩ ⊬∃ ¬B: ⟨Σ0 , Σd ⟩ ⊢∀ C implies
⟨Σ0 , Σd ∪ {B}⟩ ⊢∀ C.
Recall our general form of Rational Monotonicity:
If it is not the case that A1 , . . . , An |∼ ¬B, and moreover
A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
Premises/Input
Conclusions/Output
Logic
A1 , . . . , An ,
C
|
¬B
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Theorem 8 and Rational Monotonicity
Theorem 8
Where ⟨Σ0 , Σd ⟩ ⊬∃ ¬B: ⟨Σ0 , Σd ⟩ ⊢∀ C implies
⟨Σ0 , Σd ∪ {B}⟩ ⊢∀ C.
Recall our general form of Rational Monotonicity:
If it is not the case that A1 , . . . , An |∼ ¬B, and moreover
A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
Premises/Input
Conclusions/Output
Logic
A1 , . . . , An ,
C
|
¬B
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Theorem 8 and Rational Monotonicity
Theorem 8
Where ⟨Σ0 , Σd ⟩ ⊬∃ ¬B: ⟨Σ0 , Σd ⟩ ⊢∀ C implies
⟨Σ0 , Σd ∪ {B}⟩ ⊢∀ C.
Recall our general form of Rational Monotonicity:
If it is not the case that A1 , . . . , An |∼ ¬B, and moreover
A1 , . . . , An |∼ C, then A1 , . . . , An , B |∼ C.
Premises/Input
Conclusions/Output
Logic
A1 , . . . , An , B
C
|
¬B
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Should we be optimistic about Rational
Monotonicity in view of this?
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Two forms of Rational Monotonicity (relative to ⊢∀ )
Rational Monotonicity in the defeasible premises
Where ⟨Σ0 , Σd ⟩ ⊬∀ ¬B: ⟨Σ0 , Σd ⟩ ⊢∀ C implies
⟨Σ0 , Σd ∪ {B}⟩ ⊢∀ C.
Rational Monotonicity in the factual premises
Where ⟨Σ0 , Σd ⟩ ⊬∀ ¬B: ⟨Σ0 , Σd ⟩ ⊢∀ C implies
⟨Σ0 ∪ {B}, Σd ⟩ ⊢∀ C.
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Negative result for defeasible premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B?
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Negative result for defeasible premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
• A = p and B = q
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Negative result for defeasible premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
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Negative result for defeasible premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
• but ⟨Σ0 , Σd ∪ {p}⟩ ⊬∀ q.
• Do you see why?
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Negative result for defeasible premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B?
• Nope! Take:
MCS(⟨Σ0 , Σd ⟩) =
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
• but ⟨Σ0 , Σd ∪ {p}⟩ ⊬∀ q.
• Do you see why?
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Negative result for defeasible premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
MCS(⟨Σ0 , Σd ⟩) =
1. {(p ∧ r) ⊃ ¬q, r, ¬p ∧ q}
2. {(p ∧ r) ⊃ ¬q, p ∧ q ∧ ¬r}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
• but ⟨Σ0 , Σd ∪ {p}⟩ ⊬∀ q.
• Do you see why?
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Negative result for defeasible premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
• but ⟨Σ0 , Σd ∪ {p}⟩ ⊬∀ q.
• Do you see why?
MCS(⟨Σ0 , Σd ⟩) =
1. {(p ∧ r) ⊃ ¬q, r, ¬p ∧ q}
2. {(p ∧ r) ⊃ ¬q, p ∧ q ∧ ¬r}
MCS(⟨Σ0 , Σd ∪ {p}⟩) =
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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Also Rational Monotonicity for factual
premises doesn’t work out. We can use the
same example to demonstrate this.
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Negative result for factual premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ B?
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Negative result for factual premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
• A = p and B = q
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Negative result for factual premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
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Negative result for factual premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
• but ⟨Σ0 ∪ {p}, Σd ⟩ ⊬∀ q.
• Do you see why?
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Negative result for factual premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
MCS(⟨Σ0 , Σd ⟩) =
1. {(p ∧ r) ⊃ ¬q, r, ¬p ∧ q}
2. {(p ∧ r) ⊃ ¬q, p ∧ q ∧ ¬r}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
• but ⟨Σ0 ∪ {p}, Σd ⟩ ⊬∀ q.
• Do you see why?
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Negative result for factual premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
MCS(⟨Σ0 , Σd ⟩) =
1. {(p ∧ r) ⊃ ¬q, r, ¬p ∧ q}
2. {(p ∧ r) ⊃ ¬q, p ∧ q ∧ ¬r}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
MCS(⟨Σ0 ∪ {p}, Σd ⟩) =
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
• but ⟨Σ0 ∪ {p}, Σd ⟩ ⊬∀ q.
• Do you see why?
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Negative result for factual premises
If ⟨Σ0 , Σd ⟩ ⊢∀ B and ⟨Σ0 , Σd ⟩ ⊬∀ ¬A then ⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ B?
• Nope! Take:
• Σ0 = ∅
• Σd = {r, p ∧ q ∧ ¬r, (p ∧ r) ⊃
¬q, ¬p ∧ q}
MCS(⟨Σ0 , Σd ⟩) =
1. {(p ∧ r) ⊃ ¬q, r, ¬p ∧ q}
2. {(p ∧ r) ⊃ ¬q, p ∧ q ∧ ¬r}
• A = p and B = q
• ⟨Σ0 , Σd ⟩ ⊢∀ q
MCS(⟨Σ0 ∪ {p}, Σd ⟩) =
• ⟨Σ0 , Σd ⟩ ⊬∀ ¬p
1. {r, (p ∧ r) ⊃ ¬q}
• but ⟨Σ0 ∪ {p}, Σd ⟩ ⊬∀ q.
2. {p ∧ q ∧ ¬r, (p ∧ r) ⊃ ¬q}
• Do you see why?
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Exercise
Find similar counter-examples for Rational Monotonicity and
⊢free .
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Some Meta-Theory
The Resolution Theorem
Two versions:
Where |∼ ∈ {⊢∀ , ⊢free } and we have an implication ⊃ in our
formal language,
The Resolution Theorem in the factual premises
⟨Σ0 ∪ {A}, Σd ⟩ |∼ B if ⟨Σ0 , Σd ⟩ |∼ A ⊃ B.
The Resolution Theorem in the defeasible premises
⟨Σ0 , Σd ∪ {A}⟩ |∼ B if ⟨Σ0 , Σd ⟩ |∼ A ⊃ B.
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• What do you expect?
• Given the centrality of this result in
Classical Logic, do we get it in the
nonmonotonic case (maybe at least where
the base logic is Classical Logic)?
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A negative result for the case of defeasible premises
Where |∼ ∈ {⊢∀ , ⊢free } and we have an implication ⊃ in our
formal language, the following does not hold (in general):
The Resolution Theorem in the defeasible premises
⟨Σ0 , Σd ∪ {A}⟩ |∼ B if ⟨Σ0 , Σd ⟩ |∼ A ⊃ B.
Counter-Example
• Consider Σ = ⟨∅, {¬p}⟩, A = p and B = q.
• Then Σ |∼ p ⊃ q while ⟨∅, {¬p, p}⟩ ̸ |∼ q.
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A negative result for the case of factual premises
Where |∼ ∈ {⊢∀ , ⊢free } and we have an implication ⊃ in our
formal language, the following does not hold:
The Resolution Theorem in the factual premises
⟨Σ0 ∪ {A}, Σd ⟩ |∼ B if ⟨Σ0 , Σd ⟩ |∼ A ⊃ B.
Counter-Example
• Take Σ = ⟨∅, {¬p}⟩ and Σ+ = ⟨{p}, {¬p}⟩.
• Then Σ |∼ p ⊃ q while Σ+ ̸ |∼ q.
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Some Meta-Theory
The Deduction Theorem
Two versions:
Where |∼ ∈ {⊢∀ , ⊢free } and we have an implication ⊃ in our
formal language,
The deduction theorem in the factual premises
⟨Σ0 ∪ {A}, Σd ⟩ |∼ B implies ⟨Σ0 , Σd ⟩ |∼ A ⊃ B.
The deduction theorem in the defeasible premises
⟨Σ0 , Σd ∪ {A}⟩ |∼ B implies ⟨Σ0 , Σd ⟩ |∼ A ⊃ B.
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What do you think? We all know this is a
central property in classical logic. It should
hold here as well, at least if the base logic is
classical, right?
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The free case: a negative result for factual premises
We now give a counter-example to the following principle:
The deduction theorem in the factual premises
⟨Σ0 ∪ {A}, Σd ⟩ ⊢free B implies ⟨Σ0 , Σd ⟩ ⊢free A ⊃ B.
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The free case: a negative result for factual premises
We now give a counter-example to the following principle:
The deduction theorem in the factual premises
⟨Σ0 ∪ {A}, Σd ⟩ ⊢free B implies ⟨Σ0 , Σd ⟩ ⊢free A ⊃ B.
Let
• Σ0 = {p ∨ q, r ∨ q}
• Σd = {¬p, ¬q}
We have
• ⟨Σ0 , Σd ⟩ ⊬free p ⊃ r while
• ⟨Σ0 ∪ {p}, Σd ⟩ ⊢free r
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The free case: a negative result for factual premises
We now give a counter-example to the following principle:
The deduction theorem in the factual premises
⟨Σ0 ∪ {A}, Σd ⟩ ⊢free B implies ⟨Σ0 , Σd ⟩ ⊢free A ⊃ B.
Let
• Σ0 = {p ∨ q, r ∨ q}
• Σd = {¬p, ¬q}
We have
• ⟨Σ0 , Σd ⟩ ⊬free p ⊃ r while
• ⟨Σ0 ∪ {p}, Σd ⟩ ⊢free r
To see this notice that
• MCS(⟨Σ0 , Σd ⟩) =
{{¬p}, {¬q}}
• MCS(⟨Σ0 ∪ {p}, Σd ⟩) =
{{¬q}}
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Hm, so what’s your guess for the universal
consequence: do we get the Deduction
Theorem there?
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A useful and simple lemma
Lemma 9
If Ξ ∈ MCS(⟨Σ0 , Σd ⟩) and Σ0 ∪ Ξ ∪ {A} ⊬ ⊥ then
Ξ ∈ MCS(⟨Σ0 ∪ {A}, Σd ⟩).
Exercise
Try to prove this.
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Deduction Theorem: the universal case in the factual premises
We now suppose the Deduction Theorem holds for the Base
Logic L and prove:
The deduction theorem in the factual premises
⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ B implies ⟨Σ0 , Σd ⟩ ⊢∀ A ⊃ B.
Proof.
• Suppose ⟨Σ0 ∪ {A}, Σd ⟩ ⊢∀ 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} ⊬ ⊥ then by Lemma 9,
Ξ ∈ MCS(⟨Σ0 ∪ {A}, Σd ⟩) and thus Σ0 ∪ Ξ ⊢ B. By
monotonicity, Σ0 ∪ Ξ ∪ {A} ⊢ B. By the deduction theorem,
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Σ0 ∪ Ξ ⊢ A ⊃ B.
This is an interesting asymmetry between the
two consequence relations: while the universal
one validates the Deduction Theorem (in the
factual premises), the free one doesn’t.
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Should we expect the same behavior for the
Deduction Theorem in the defeasible
premises?
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Well, here’s the surprise …
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Deduction Theorem for defeasible premises, Free Consequences
We now suppose the Deduction Theorem holds for the Base
Logic L and prove:
The deduction theorem in the defeasible premises
⟨Σ0 , Σd ∪ {A}⟩ ⊢free B implies ⟨Σ0 , Σd ⟩ ⊢free A ⊃ B.
Proof.
• Suppose ⟨Σ0 , Σd ∪ {A}⟩ ⊢free B and hence
∩
Σ0 ∪ MCS(⟨Σ0 , Σd ∪ {A}⟩) ⊢ B.
• It is easy to see that
∩
∩
MCS(⟨Σ0 , Σd ∪ {A}⟩) ⊆
MCS(⟨Σ0 , Σd ⟩) ∪ {A}
(4)
∩
• By monotonicity, Σ0 ∪ MCS(⟨Σ0 , Σd ⟩) ∪ {A} ⊢ B and by
∩
the deduction theorem, Σ0 ∪ MCS(⟨Σ0 , Σd ⟩) ⊢ A ⊃ B.
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Exercise
Exercise
Prove (4) in the proof above.
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Deduction Thm for defeasible premises, Universal Consequences
We now suppose the Deduction Theorem holds for the Base
Logic L and prove:
The deduction theorem in the defeasible premises
⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B implies ⟨Σ0 , Σd ⟩ ⊢∀ A ⊃ B.
Proof.
• Suppose ⟨Σ0 , Σd ∪ {A}⟩ ⊢∀ B and let Ξ ∈ MCS(Σ0 , Σd ).
• We have two case: (1) Ξ ∪ {A} ∈ MCS(⟨Σ0 , Σd ∪ {A}⟩) or (2)
Ξ ∈ MCS(Σ0 , Σd ∪ {A}) and Ξ ∪ Σ0 ∪ {A} ⊢ ⊥ and thus
Ξ ∪ Σ0 ∪ {A} ⊢ B (by ⊥-explosion (!) and transitivity).
1. Then by the supposition Σ0 ∪ Ξ ∪ {A} ⊢ B and hence
Σ0 ∪ Ξ ⊢ A ⊃ B by the deduction theorem.
2. Σ0 ∪ Ξ ⊢ A ⊃ B by the deduction theorem.
• Thus, Σ0 ∪ Ξ ⊢ A ⊃ B and ⟨Σ0 , Σd ⟩ ⊢∀ A ⊃ B.
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Some Meta-Theory
Summing up
OK, this gets a bit exhausting … maybe time to
sum this up!
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An overview
⊢free
Ded.Thm. (factual)
Ded.Thm. (defeasible)
Res.Thm (factual)
Res.Thm (defeasible)
CM (factual)
CM (defeasible)
Cut (factual)
Cut (defeasible)
RM (factual)
RM (defeasible)
✓
⊢∀
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
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From MCSs to Selection Semantics
From MCSs to Selection Semantics
Some basic definitions and suppositions
… concerning L
We now suppose that L has an adequate semantics with no
inconsistent models.
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… concerning L
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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… concerning L
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({⊥}) = ∅
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The defeasible part of a model
Definition 10
Given Σ = ⟨Σ0 , Σd ⟩,
• where M ∈ M(Σ0 ), its defeasible part d(M) is
{A ∈ Σd | M |= A}
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The defeasible part of a model
Definition 10
Given Σ = ⟨Σ0 , Σd ⟩,
• where M ∈ M(Σ0 ), its defeasible part d(M) is
{A ∈ Σd | M |= A}
• Mm (Σ) = {M ∈ M(Σ0 ) | there are no M′ ∈ M(Σ0 ) for
which d(M) ⊂ d(M′ )}
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Two central lemmas
Lemma 11
Σ0 ∪ Ξ is consistent iff M(Σ0 ∪ Ξ) ̸= ∅
Proof.
• Σ0 ∪ Ξ is inconsistent iff
• Σ0 ∪ Ξ ⊢ ⊥ iff
• Σ0 ∪ Ξ ⊩ ⊥ iff
• M(Σ0 ∪ Ξ) = ∅
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Two central lemmas (cont.)
Lemma 12
MCS(⟨Σ0 , Σd ⟩) = {d(M) | M ∈ Mm (⟨Σ0 , Σd ⟩)}
Proof.
• (⊆) Suppose (†) Ξ ∈ MCS(⟨Σ0 , Σd ⟩).
• 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)) ̸= ∅ and hence Σ0 ∪ d(M) ⊮ ⊥.
• Since then Σ0 ∪ d(M) ⊬ ⊥: contradiction to (†).
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Exercise
Prove the other direction of Lemma 12.
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From MCSs to Selection Semantics
A selection semantics for the universal
consequence
The semantic selection
Definition 13
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊩m A iff for all M ∈ Mm (Σ), M |= A.
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Soundness and Completeness
Theorem 14
Where Σ = ⟨Σ0 , Σd ⟩, 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, (here we use Lemma 12) iff
• Σ ⊩m A.
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From MCSs to Selection Semantics
A selection semantics for the free
consequence
The semantic selection
Definition 15
Where Σ = ⟨Σ0 , Σd ⟩,
• Mf (Σ) = {M ∈ M(Σ0 ) | d(M) ⊇
∩
{d(M′ ) | M′ ∈ Mm (Σ)}}.
• Σ ⊩f A iff for all M ∈ Mf (Σ), M |= A
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Soundness and Completeness
Theorem 16
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢free A iff Σ ⊩f A
Proof.
• Σ ⊢free A, iff
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Soundness and Completeness
Theorem 16
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢free A iff Σ ⊩f A
Proof.
• Σ ⊢free A, iff
• Σ0 ∪ Free(Σ) ⊢ A, iff
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Soundness and Completeness
Theorem 16
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢free A iff Σ ⊩f A
Proof.
• Σ ⊢free A, iff
• Σ0 ∪ Free(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊢ A, iff
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Soundness and Completeness
Theorem 16
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢free A iff Σ ⊩f A
Proof.
• Σ ⊢free A, iff
• Σ0 ∪ Free(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊩ A, iff
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Soundness and Completeness
Theorem 16
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢free A iff Σ ⊩f A
Proof.
• Σ ⊢free A, iff
• Σ0 ∪ Free(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊩ A, iff
∩
• for all M ∈ M(Σ0 ∪ MCS(Σ)), M |= A, iff
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Soundness and Completeness
Theorem 16
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢free A iff Σ ⊩f A
Proof.
• Σ ⊢free A, iff
• Σ0 ∪ Free(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊩ A, iff
∩
• for all M ∈ M(Σ0 ∪ MCS(Σ)), M |= A, iff
∩
• for all M ∈ M(Σ0 ) such that d(M) ⊇ MCS(Σ), M |= A, iff
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Soundness and Completeness
Theorem 16
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢free A iff Σ ⊩f A
Proof.
• Σ ⊢free A, iff
• Σ0 ∪ Free(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊩ A, iff
∩
• for all M ∈ M(Σ0 ∪ MCS(Σ)), M |= A, iff
∩
• for all M ∈ M(Σ0 ) such that d(M) ⊇ MCS(Σ), M |= A, iff
• for all M ∈ M(Σ0 ) such that
∩
d(M) ⊇ {d(M′ ) | M′ ∈ Mm (Σ)}, M |= A, iff
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Soundness and Completeness
Theorem 16
Where Σ = ⟨Σ0 , Σd ⟩, Σ ⊢free A iff Σ ⊩f A
Proof.
• Σ ⊢free A, iff
• Σ0 ∪ Free(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊢ A, iff
∩
• Σ0 ∪ MCS(Σ) ⊩ A, iff
∩
• for all M ∈ M(Σ0 ∪ MCS(Σ)), M |= A, iff
∩
• for all M ∈ M(Σ0 ) such that d(M) ⊇ MCS(Σ), M |= A, iff
• for all M ∈ M(Σ0 ) such that
∩
d(M) ⊇ {d(M′ ) | M′ ∈ Mm (Σ)}, M |= A, iff
• Σ ⊩f A.
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Constrained Input/Output Logic
Constrained Input/Output Logic
Input/Output Logic (Derivational
Perspective)
The setup
• We have a set of input formulas Σ0 in some propositional
language L
• And a set of Input/Output pairs Σio of the form (A, B)
where A, B ∈ L
• Think of them as conditionals or rules to which Modus
Ponens should be applied
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Input/Output Logic
Each Input/Output logic L(R) is based on a base logic L and
comes with a set of rules R on the elements in Σio . Here are
some examples:
•
•
•
•
•
•
•
WO: If A ⊢ C, then from (B, A) derive (B, C).
SI: If C ⊢ A, then from (A, B) derive (C, B).
AND: From (A, B) and (A, C) derive (A, B ∧ C).
OR: From (B, A) and (C, A) derive (B ∨ C, A).
CT: From (A, B) and (A ∧ B, C) derive (A, C).
ID: (A, A)
T: (⊤, ⊤)
Definition 17
Where ΣR
io is the set of all (A, B) derivable from Σ0 via the
rules in R,
outL(R) (⟨Σ0 , Σio ⟩) = {A | ∃(B, A) ∈ ΣR
io , ∃B ∈ Cn(Σ0 )}.
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Constrained Input/Output Logic
Constrained Input/Output Logic
Problem: conflicts
Example
• Suppose Σ0 = {p, q},
• Σio = {(p, ¬q)},
• and we have (at least) the rules ID, AGG, WO, SI in R
• via ID we have (q, q) and via SI (p ∧ q, q)
• via SI we have (p ∧ q, ¬q).
• via AGG, (p ∧ q, q ∧ ¬q) and via WO (p ∧ q, A) for any A
• Then outCL(R) (⟨Σ0 , Σio ⟩) is trivial.
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Solution: “go MCS”
The idea is analogous to what we had before:
• premises in Σ0 are strict
• IO-pairs in Σio are defeasible
• Ξio ⊆ Σio is a maximal consistent subset of ⟨Σ0 , Σio ⟩ iff
1. outL(R) (Σ0 , Ξio ) is consistent
2. there is no Ξ′io ⊆ Σio such that outL(R) (Σ0 , Ξ′io ) is
consistent and Ξio ⊂ Ξ′io .
• We write Ξio ∈ MCSL(R) (⟨Σ0 , Σio ⟩).
Definition 18
• out∀L(R) (⟨Σ0 , Σio ⟩) =
}
∩{
outL(R) (⟨Σ0 , Ξio ⟩) | Ξio ∈ MCSL(R) (⟨Σ0 , Σio ⟩) .
• out∃L(R) (⟨Σ0 , Σio ⟩) =
}
∪{
outL(R) (⟨Σ0 , Ξio ⟩) | Ξio ∈ MCSL(R) (⟨Σ0 , Σio ⟩) .
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Examples!
First our previous example where Σ0 = {p, q} and
Σio = {(p, ¬q)} and we had (at least) the rules ID, AGG, WO, SI
in R.
• As we saw, outCL(R) (⟨Σ0 , Σio ⟩) is trivial.
• Thus, the only member of MCSL(R) (⟨Σ0 , Σio ⟩) is ∅.
∀
• This means that outL(R)
(Σ0 , Σio ) = outL(R) (Σ0 , ∅).
• Which of the following are consequences?
1.
2.
3.
4.
p
q
⊤
p∧q
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Another example
• Let Σ0 = {p, s} and
• Σio = {(p, q), (s, ¬q), (p, r)}
• Let R contain WO, SI, AGG.
• our MCS(⟨Σ0 , Σio ⟩) has two members:
1. {(p, q), (p, r)}
2. {(s, ¬q), (p, r)}
• This means e.g., r ∈ out∀L(R) (⟨Σ0 , Σio ⟩).
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But where are the constraints in “Constrained
Input/Output Logic”?
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The full picture
The idea is analogous to what we had before:
•
•
•
•
premises in Σ0 are strict
IO-pairs in Σio are defeasible
Σc , a set of formulas in L, represent constraints
Ξio ⊆ Σio is a maximal consistent subset of ⟨Σ0 , Σio , Σc ⟩ iff
1. outL(R) (Σ0 , Ξio ) ∪ Σc is consistent
2. there is no Ξ′io ⊆ Σio such that outL(R) (Σ0 , Ξ′io ) ∪ Σc is
consistent and Ξio ⊂ Ξ′io .
• We write Ξio ∈ MCSL(R) (⟨Σ0 , Σio , Σc ⟩).
Definition 19
• out∀L(R) (⟨Σ0 , Σio , Σc ⟩) =
}
∩{
outL(R) (⟨Σ0 , Ξio ⟩) | Ξio ∈ MCSL(R) (⟨Σ0 , Σio , Σc ⟩) .
• out∃L(R) (⟨Σ0 , Σio , Σc ⟩) =
}
∪{
outL(R) (⟨Σ0 , Ξio ⟩) | Ξio ∈ MCSL(R) (⟨Σ0 , Σio , Σc ⟩) .
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Bibliography
Bibliography
References
Amgoud, L. and P. Besnard: 2013, ‘Logical limits of abstract argumentation frameworks’. Journal of Applied
Non-Classical Logics 23(3), 229–267.
Batens, D.: 2007, ‘A Universal Logic Approach to Adaptive Logics’. Logica Universalis 1, 221–242.
Benferhat, S., D. Dubois, and H. Prade: 1997, ‘Some Syntactic Approaches to the Handling of Inconsistent Knowledge
Bases: A Comparative Study. Part 1: The Flat Case’. Studia Logica 58, 17–45.
Brewka, G.: 1989, ‘Preferred Subtheories: An Extended Logical Framework for Default Reasoning.’. In: IJCAI, Vol. 89.
pp. 1043–1048.
Makinson, D.: 2003, ‘Bridges between classical and nonmonotonic logic’. Logic Journal of IGPL 11(1), 69–96.
Makinson, D. and L. Van Der Torre: 2001, ‘Constraints for Input/Output Logics’. Journal of Philosophical Logic 30(2),
155–185.
Rescher, N. and R. Manor: 1970, ‘On inference from inconsistent premises’. Theory and Decision 1, 179–217.
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.: 2013, ‘Default Assumptions and Selection Functions: A Generic Framework for Non-monotonic
Logics’. In: Advances in Artificial Intelligence and Its Applications. Springer, pp. 54–67.
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