Conditionals under Weak Completion Semantics · Isabelly Louredo Rocha Conditionals under Weak Completion Semantics 2. Evaluating conditionals I MRFA: Minimal Revision Followed by

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


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


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) = >.


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← ⊥}


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 E = {rain← ⊥}

I MP∪E = 〈∅, {rain, wet streets, umbrella, ab1, ab2}〉

I The conditional is true


Examples

I If the streets are wet, then it rained

. cond({wet streets}, {rain})

I MP = 〈∅, {ab1, ab2}〉

I AP = {rain← >, rain← >}


I E = {rain← >}

I MP∪E = 〈{rain, wet streets, umbrella}, {ab1, ab2}〉



Examples



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


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)


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)


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


Examples: revisited


. If it rains, then the streets are wet (Obligation and necessity)

. If it rains, then I take my umbrella (Factual and sufficiency)


I MP = 〈∅, {ab1, ab2}〉

I AP = {rain← >, rain← ⊥, umbrella ← >, ab2 ← >}


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 E = {rain← ⊥}

I MP∪E = 〈∅, {rain, wet streets, umbrella, ab1, ab2}〉



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}〉



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


Examples: revisited (Obligation and necessity)

I If the streets are wet, then it rained

. cond({wet streets}, {rain})

I MP = 〈∅, {ab1, ab2}〉



I E = {rain← >}

I MP∪E = 〈{rain, wet streets, umbrella}, {ab1, ab2}〉



Examples: revisited (Factual and sufficiency)

I If I took my umbrella, then it rained

. cond({umbrella}, {rain})

I MP = 〈∅, {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


Examples: revisited



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


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%


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%


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}


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}〉


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%


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


Documents

Conditionals under Weak Completion Semantics · Isabelly Louredo Rocha Conditionals under Weak Completion Semantics 2. Evaluating conditionals I MRFA: Minimal Revision Followed by