two sources of explosion - stanford university
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Two sources of explosion
Eric Jui-Yi Kaowith Carl Hewitt
Department of Computer ScienceStanford University
Stanford, CA 94305 United States
August 18, 2011Inconsistency Robustness 2011
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Inconsistency Robustness
I Automated logical reasoning form a part of many systems.I security policy systemsI semantic webI knowledge bases
I Some logics are explosive
I.E. {α,¬α} ` β, for any sentences α, β.
I Non-explosion is a minimal requirement for inconsistency robustness.
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Classical logic is explosive
1 α Premise
2 ¬α Premise
3 ¬β
4 α Reiteration, 1
5 ¬α Reiteration, 2
6 ⊥ Contradiction, 4, 5
7 ¬¬β Proof by contradiction, 3–6
8 β Double negation elimintation, 7
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Classical logic is explosive
1 α Premise
2 ¬α Premise
3 β ∨ α ∨-Introduction, 1
4 β Disjunctive syllogism, 1, 3
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Outline
I Idea: restrict the proof theory of classical logic in some “reasonable”way
I Avoid explosionI Retain “maximal” deductive power.
I Many “design decisions” involvede.g., cannot retain both ∨-introduction and disjunctive syllogism
I Direct Logic is one proposal [2]
I Can we increase its deductive power?
I We consider two attempts in increasing its deductive power
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Boolean Direct Logic rules of inference
Core rules of bDL
α ∨ β ¬α ∨ ψβ ∨ ψ
[Resolution]
α β
α ∨ β[Restricted ∨-Introduction]
α ∧ βα
[∧-Elimination]
α β
α ∧ β[∧-Introduction]
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Substitution according boolean equivalences
α
s(α)[Substitution],
where s(α) is the result of substituting in α an occurrence of a subformulaby an equivalent subformula according to a boolean equivalence below.
Distributivity ψ ∨ (α ∧ β) ≡ (ψ ∨ α) ∧ (ψ ∨ β)(ψ ∧ α) ∨ (ψ ∧ β) ≡ ψ ∧ (α ∨ β)
De Morgan Laws ¬(α ∧ β) ≡ ¬α ∨ ¬β¬(α ∨ β) ≡ ¬α ∧ ¬β
Double negation ¬¬α ≡ α
Idempotence α ∨ α ≡ αα ∧ α ≡ α
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Some properties of bDL
I bDL is not explosive
I bDL is “reasonable”
I Can we make bDL more powerful?e.g., bDL cannot prove p ∨ ¬p
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Law of excluded middle
I Intuitively: no sentence can be neither true nor false.
I Axiom schemaα ∨ ¬α [Excluded Middle]
I Not obvious whether bDL+[Excluded Middle] is explosive
E.G. bDL plus the axioms {p ∨ ¬p : p ∈ Propositions} is notexplosive [5].
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Excluded Middle is explosive
1 α Premise
2 ¬α Premise
3 (α ∧ ¬β) ∨ ¬(α ∧ ¬β) Excluded Middle
4 (α ∧ ¬β) ∨ ¬α ∨ ¬¬β De Morgan, 3
5 (α ∧ ¬β) ∨ ¬α ∨ β Double negation, 4
6 (α ∨ ¬α ∨ β) ∧ (¬β ∨ ¬α ∨ β) Distributivity, 5
7 α ∨ ¬α ∨ β ∧-Elimination, 6
8 α ∨ β Resolution, 7, 1
9 β Resolution, 8, 2
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Proof by self-refutation
I Intuitively: If a sentence negates itself, it must be false.
I If a sentence α derives the negation of itself, then we can introduce¬α.
I Axiom schema:
¬α, where α proves ¬α [Self-Refutation]
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Proof 2a: ((¬α ∨ ¬β) ∧ (α ∨ β)) proves ¬((¬α ∨ ¬β) ∧ (α ∨ β))
1 (¬α ∧ ¬β) ∧ (α ∨ β) Premise
2 (¬α ∧ ¬β) ∧-Elimination, 1
3 (α ∨ β) ∧-Elimination, 1
4 (α ∨ β) ∨ (¬α ∧ ¬β) Restricted ∨-Introduction, 2, 3
5 (α ∨ β) ∨ ¬(α ∨ β) De Morgan, 4
6 (α ∨ ¬¬β) ∨ ¬(α ∨ β) Double negation, 5
7 (¬¬α ∨ ¬¬β) ∨ ¬(α ∨ β) Double negation, 6
8 ¬(¬α ∨ ¬β) ∨ ¬(α ∨ β) De Morgan, 7
9 ¬((¬α ∨ ¬β) ∧ (α ∨ β)) De Morgan, 8
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1 α Premise
2 ¬α Premise
3 ¬((¬α ∨ ¬β) ∧ (α ∨ β)) Self-Refutation, Proof 2a
4 ¬(¬α ∧ ¬β) ∨ ¬(α ∨ β) De Morgan, 3
5 (¬¬α ∨ ¬¬β) ∨ ¬(α ∨ β) De Morgan, 4
6 (¬¬α ∨ β) ∨ ¬(α ∨ β) Double negation, 5
7 α ∨ β ∨ ¬(α ∨ β) Double negation, 6
8 α ∨ β ∨ (¬α ∧ ¬β) De Morgan, 7
9 (α ∨ β ∨ ¬α) ∧ (α ∨ β ∨ ¬β) Distributivity, 8
10 α ∨ β ∨ ¬α ∧-Elimination, 9
11 α ∨ β Resolution, 10, 1
12 β Resolution, 11, 2E. Kao (Stanford) Two sources of explosion August 2011 13 / 19
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Design decisions
I Let’s take the boolean equivalences and ∧-Elimination for grantedI The explosiveness of bDL+[Excluded Middle] essentially rely on only
I Excluded Middle andI Disjunctive Syllogism (a special case of Resolution)
I Direct Logic chooses Disjunctive Syllogism and leaves out ExcludedMiddle
I The explosiveness of bDL+[Self-Refutation] essentially rely on onlyI Self-Refutation,I Disjunctive Syllogism, andI Restricted ∨-Introduction (α ∨ β from α and β)
I Direct Logic replaces Self-Refutation with a weaker rule.
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Other logics
I The results apply to other paraconsistent logics that support the rulesused.
I For example, Besnard and Hunter’s quasi-classical logic [1, 4, 3] alsobecomes explosive if either Excluded Middle or Self-Refutation isadded.
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Open questions
Consider the set of valid inference rules in classical boolean logic:
R = {Φ1 · · ·Φn
Ψ: φ1 · · ·φn |= ψ
for any intances φ1, . . . , φn, ψ of Φ1, . . . ,Φn,Ψ}I Find a maximal subset S of R such that the the logic induced by S is
not explosive.
I Can the induced logic be axiomatized by a finite number of inferencerules?
I Is the induced logic decidable?
I Characterize the space of all such S ⊆ R
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Besnard, P., Hunter, A.: Quasi-classical logic: Non-trivializableclassical reasoning from incosistent information. In: Proceedings of theEuropean Conference on Symbolic and Quantitative Approaches toReasoning and Uncertainty. pp. 44–51. Springer-Verlag, London, UK(1995),http://portal.acm.org/citation.cfm?id=646561.695561
Hewitt, C.: Common sense for inconsistency robust informationintegration using direct logic reasoning and the actor model. arXivCoRR abs/0812.4852 (2011)
Hunter, A.: Paraconsistent logics. In: Handbook of DefeasibleReasoning and Uncertain Information. pp. 11–36. Kluwer (1996)
Hunter, A.: Reasoning with contradictory information usingquasi-classical logic. Journal of Logic and Computation 10, 677–703(1999)
Kao, E.J.Y., Genesereth, M.: Achieving cut, deduction, and otherproperties with a variation on quasi-classical logic (2011),
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http://dl.dropbox.com/u/5152476/working-papers/
modified-quasiclassical/main.pdf, working paper
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Two sources of explosion
Eric Jui-Yi Kaowith Carl Hewitt
Department of Computer ScienceStanford University
Stanford, CA 94305 United States
August 18, 2011Inconsistency Robustness 2011
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Proof by contradiction
I If by assuming a sentence α we derive a contradiction, then we canconclude ¬α.
I It can be stated as the following meta-rule:If Σ, α ` ψ and Σ, α ` ¬ψ, then conclude Σ ` ¬α.
I Proof by contradiction easily leads to explosiveness. For anysentences α and β,
{α,¬α},¬β ` α and {α,¬α},¬β ` ¬α,
hence {α,¬α} ` ¬¬β using proof by contradiction.
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Self-refutation is explosive
I show that the addition of the proof by self-refutation rule to bDL leadsto explosiveness.For any pair of sentences α and β, I derive β from premises α and ¬α,using bDL inference rules plus the Self-Refutation axiom schema.First, I show that (¬α ∧ ¬β) ∧ (α ∨ β) proves its own negation¬((¬α ∨ ¬β) ∧ (α ∨ β)). Then I use ¬((¬α ∨ ¬β) ∧ (α ∨ β)), α, and ¬αto prove β.
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