I Introduction
I Evaluating conditionals
I Obligation vs Factual conditionals
I Necessity vs sufficiency
I The Wason Selection Task
Isabelly Louredo RochaConditionals under Weak Completion Semantics 1
Conditionals under Weak Completion Semantics
Isabelly Louredo RochaInternational Center for Computational LogicTechnische Universitat DresdenGermany
Conditionals
I Statements of the form if condition then consequence
I Indicative conditionals
. Condition and consequence can be true, false or unknown
. If the condition is true, the consequence is asserted to be true
I Subjunctive conditionals (counterfactuals)
. Condition is false
. Consequence can be true, false or unknown
. In the counterfactual circumstance of the condition being true, theconsequence is asserted to be true
Isabelly Louredo RochaConditionals under Weak Completion Semantics 2
Evaluating conditionals
I MRFA: Minimal Revision Followed by Abduction
. Model background knowledge as a logic program
. Reason wrt the least model of the weak completion of the program
. Explain the conditions by revision and abduction
. Evaluate the conclusion, once the conditions are explained
. Minimize revision and maximize abduction
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MRFA: Minimal Revision Followed by Abduction
I Given P and cond(C,D)
. IfMP(C) = > then cond(C,D) is assigned toMP(D).
. IfMP(C) = ⊥, then evaluate cond(C,D) with respect toMrev(P,S), whereS = {L ∈ C | MP(L) = ⊥}.
. IfMP(C) = U, then evaluate cond(C,D) with respect toMP′ , where
II P′ = rev(P,S) ∪ E,
II S is a smallest subset of C and E ⊆ Arev(P,S) is a minimal explanationfor C \ S such thatMP′(C) = >.
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Examples
I Background knowledge
. If it rains, then the streets are wet and I take my umbrella
I P = {wet streets ← rain ∧ ¬ab1,umbrella ← rain ∧ ¬ab2,ab1 ← ⊥,ab2 ← ⊥}
I MP = 〈∅, {ab1, ab2}〉
I AP = {rain← >, rain← ⊥}
Isabelly Louredo RochaConditionals under Weak Completion Semantics 5
Examples
I If the streets are not wet, then it did not rain
. cond({¬wet streets}, {¬rain})
I MP = 〈∅, {ab1, ab2}〉
I AP = {rain← >, rain← ⊥}
I P = {wet streets ← rain ∧ ¬ab1,umbrella ← rain ∧ ¬ab2,ab1 ← ⊥,ab2 ← ⊥}
I E = {rain← ⊥}
I MP∪E = 〈∅, {rain, wet streets, umbrella, ab1, ab2}〉
I The conditional is true
Isabelly Louredo RochaConditionals under Weak Completion Semantics 6
Examples
I If the streets are wet, then it rained
. cond({wet streets}, {rain})
I MP = 〈∅, {ab1, ab2}〉
I AP = {rain← >, rain← >}
I P = {wet streets ← rain ∧ ¬ab1,umbrella ← rain ∧ ¬ab2,ab1 ← ⊥,ab2 ← ⊥}
I E = {rain← >}
I MP∪E = 〈{rain, wet streets, umbrella}, {ab1, ab2}〉
I The conditional is true
Isabelly Louredo RochaConditionals under Weak Completion Semantics 7
Examples
I Background knowledge
. If it rains, then the streets are wet and I take my umbrella
I Denying the consequent (Modus tollens)
. If the streets are not wet, then it did not rain
. If I did not take my umbrella, then it did not rain
I Affirming the consequent
. If the streets are wet, then it rained
. If I took my umbrella, then it rained
Isabelly Louredo RochaConditionals under Weak Completion Semantics 8
Obligation vs factual conditional
If condition, then consequence
I Background knowledge. If it rains, then the streets are wet and I take my umbrella
I Obligation conditional: consequence is obligatory. Permitted possibility: condition and consequence. Forbidden possibility: condition and not consequence. Example: If the streets are not wet, then it did not rain (true)
I Factual conditional. Permitted possibility: condition and consequence. Example: If I did not take my umbrella, then it did not rain (not true)
Isabelly Louredo RochaConditionals under Weak Completion Semantics 9
Necessity vs sufficiency
If condition, then consequence
I Background knowledge. If it rains, then the streets are wet and I take my umbrella
I Necessity. The consequence cannot be true unless the condition is true. Example: If the streets are wet, then it rained (true)
I Sufficiency. The condition to be true is adequate grounds to conclude the consequence. Example: If I took my umbrella, then it rained (not true)
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Handling obligation vs factual conditionals and necessity vs sufficiency
I Modify the set of abducibles to model the difference between
. obligation and facts
. necessary and sufficient conditions
I AP = {A← > |A is undefined in P}∪ {A← ⊥ |A is undefined in P}∪ {A← > |A is the head of a sufficient conditional in P}∪ {abi ← > |abi is in the body of a factual conditional in P}
I Reason skeptically wrt the consequences
I F follows skeptically from P andOiff for all explanations E forO we find P ∪ E |=wcs F
Isabelly Louredo RochaConditionals under Weak Completion Semantics 11
Examples: revisited
I Background knowledge
. If it rains, then the streets are wet (Obligation and necessity)
. If it rains, then I take my umbrella (Factual and sufficiency)
I P = {wet streets ← rain ∧ ¬ab1,umbrella ← rain ∧ ¬ab2,ab1 ← ⊥,ab2 ← ⊥}
I MP = 〈∅, {ab1, ab2}〉
I AP = {rain← >, rain← ⊥, umbrella ← >, ab2 ← >}
Isabelly Louredo RochaConditionals under Weak Completion Semantics 12
Examples: revisited (Obligation and necessity)
I If the streets are not wet, then it did not rain
. cond({¬wet streets}, {¬rain})
I MP = 〈∅, {ab1, ab2}〉
I AP = {rain← >, rain← ⊥, umbrella ← >, ab2 ← >}
I P = {wet streets ← rain ∧ ¬ab1,umbrella ← rain ∧ ¬ab2,ab1 ← ⊥,ab2 ← ⊥}
I E = {rain← ⊥}
I MP∪E = 〈∅, {rain, wet streets, umbrella, ab1, ab2}〉
I The conditional is true
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Examples: revisited (Factual and sufficiency)
I If I did not take my umbrella, then it did not rain
. cond({¬umbrella}, {¬rain})
I MP = 〈∅, {ab1, ab2}〉
I AP = {rain← >, rain← ⊥, umbrella ← >, ab2 ← >}
I P = {wet streets ← rain ∧ ¬ab1,umbrella ← rain ∧ ¬ab2,ab1 ← ⊥,ab2 ← ⊥}
E MP∪E{rain← ⊥} 〈∅, {rain, wet streets, umbrella, ab1, ab2}〉 P ∪ E |=wcs ¬rain{ab2 ← >} 〈{ab2}, {umbrella, ab1}〉 P ∪ E 6|=wcs ¬rain
I The conditional is unknown
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Examples: revisited (Obligation and necessity)
I If the streets are wet, then it rained
. cond({wet streets}, {rain})
I MP = 〈∅, {ab1, ab2}〉
I AP = {rain← >, rain← ⊥, umbrella ← >, ab2 ← >}
I P = {wet streets ← rain ∧ ¬ab1,umbrella ← rain ∧ ¬ab2,ab1 ← ⊥,ab2 ← ⊥}
I E = {rain← >}
I MP∪E = 〈{rain, wet streets, umbrella}, {ab1, ab2}〉
I The conditional is true
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Examples: revisited (Factual and sufficiency)
I If I took my umbrella, then it rained
. cond({umbrella}, {rain})
I MP = 〈∅, {ab1, ab2}〉
I AP = {rain← >, rain← ⊥, umbrella ← >, ab2 ← >}
I P = {wet streets ← rain ∧ ¬ab1,umbrella ← rain ∧ ¬ab2,ab1 ← ⊥,ab2 ← ⊥}
E MP∪E{rain← >} 〈{rain, wet streets, umbrella}, {ab1, ab2}〉 P ∪ E |=wcs rain{umbrella ← >} 〈{umbrella}, {ab1, ab2}〉 P ∪ E 6|=wcs rain
I The conditional is unknown
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Examples: revisited
I Background knowledge
. If it rains, then the streets are wet and I take my umbrella
I Obligation conditionals with necessary conditions: Evaluated as true
. If the streets are not wet, then it did not rain
. If the streets are wet, then it rained
I Factual conditionals with sufficient conditions: Evaluated as unknown
. If I did not take my umbrella, then it did not rain
. If I took my umbrella, then it rained
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The Wason Selection Task: Abstract Case (Wason 1968)
I Consider cards with a letter on one side and a number on the other side
I Given the conditional
If there is a D on one side of the card, then there is a 3 on the other side
I Which cards must be turned to show that the conditional holds?
D F 3 7
89% 16% 62% 25%
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The Wason Selection Task: Social Case (Griggs and Cox 1982)
I Consider cards with a person’s age on one side and what the person isdrinking on the other side
I Given the conditional
If a person is drinking beer, then the person must be over 19 years of age
I Which cards must be turned to show that the conditional holds?
beer coke 22yrs 16yrs
95% 0.025% 0.025% 80%
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Modelling the Wason Selection Task
I Consider cards with X on one side and Y on the other side
I Conditional: If X, then Y
I P = {Y ← X ∧ ¬ab,ab ← ⊥}
I SetAP of abducibles will depend on the type of the conditional
I AssumeO = {X} st X can be negative or positive
I Turn the card, when
. X and Y follow skeptically from P andO, or
. Y is the head of a rule in each E ∈ {E |P ∪ E |= X}
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The Wason Selection Task: Abstract Case (Wason 1968)
I If there is a D on one side of the card, then there is a 3 on the other side
. Defined as a belief by Kowalski
. Factual conditional and necessity
I P = {3← D ∧ ¬ab,ab ← ⊥}
I MP = 〈∅, {ab}〉
I AP = {D ← >, D ← ⊥, ab ← >}
O E MP∪E Experimental resultsD {D ← >} 〈{D, 3}, {ab}〉 turn 89%F (¬D) {D ← ⊥} 〈∅, {D, 3, ab}〉 16%3 {D ← >} 〈{D, 3}, {ab}〉 turn 62%7 (¬3) {D ← ⊥} 〈∅, {D, 3, ab}〉 25%
{ab ← >} 〈{ab}, {3}〉
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The Wason Selection Task: Social Case (Griggs and Cox 1982)
I If a person is drinking beer, then the person must be over 19 years of age
. Defined as a goal by Kowalski
. Obligation conditional and sufficiency
I P = {over 19← beer ∧ ¬ab,ab ← ⊥}
I MP = 〈∅, {ab}〉
I AP = {beer ← >, beer ← ⊥, over 19← >}
O E MP∪E Resultsbeer {beer ← >} 〈{beer, over 19}, {ab}〉 turn 95%coke (¬beer ) {beer ← ⊥} 〈∅, {beer, over 19, ab}〉 0.025%22yrs (over 19) {beer ← >} 〈{beer, over 19}, {ab}〉 0.025%
{over 19← >} 〈{over 19}, {ab}〉16yrs (¬over 19) {beer ← ⊥} 〈∅, {beer, over 19, ab}〉 turn 80%
Isabelly Louredo RochaConditionals under Weak Completion Semantics 22
Summary
I MRFA is not sufficient to evaluate all types of conditionals
I Proposed modifications solve the problem
. Change the set of abducibles
. Reason skeptically
I Solve the Wason Selection Task
I Future work
. More categories of conditionals
. Analyze more examples
. Experimental data
Isabelly Louredo RochaConditionals under Weak Completion Semantics 23
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