Two sources of explosion

Eric Jui-Yi Kaowith Carl Hewitt

Department of Computer ScienceStanford University

Stanford, CA 94305 United States

erickao@cs.stanford.edu

August 18, 2011Inconsistency Robustness 2011

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

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

erickao@cs.stanford.edu

August 18, 2011Inconsistency Robustness 2011

E. Kao (Stanford) Two sources of explosion August 2011 17 / 19

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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