On Indicative Conditionals

Abstract

In this paper we present a new approach to evaluate indicative conditionals with respect to some background information specified by a logic program. Because the weak completion of a logic program admits a least model under the three-valued Lukasiewicz semantics and this semantics has been successfully applied to other human reasoning tasks, conditionals are evaluated under these least L-models. If such a model maps the condition of a conditional to unknown, then abduction and revision are applied in order to satisfy the condition. Different strategies in applying abduction and revision might lead to different evaluations of a given conditional. Based on these findings we outline an experiment to better understand how humans handle those cases.

Conditionals are statements of the form if condition then consequence. In the

literature the condition is also called if part, if clause or protasis, whereas the consequence is called then part, then clause or apodosis. Conditions as well as consequences are assumed to be finite sets (or conjunctions) of ground literals.

Indicative conditionals are conditionals whose condition may or may not be true and, consequently, whose consequence also may or may not be true; however, the consequence is asserted to be true if the condition is true. Examples for indicative conditionals are the following:

If it is raining, then he is inside. (1)

If Kennedy is dead and Oswald did not shoot him, then someone else did. (2)

If rifleman A did not shoot, then the prisoner is alive. (3)

If the prisoner is alive, then the captain did not signal. (4)

If rifleman A shot, then rifleman B shot as well. (5)

If the captain gave no signal and rifleman A decides to shoot, then the prisoner will die and rifleman B will not shoot. (6)

Conditionals may or may not be true in a given scenario. For example, if we are told that a particular person is living in a prison cell, then most people are ? The authors are mentioned in alphabetical order. expected to consider ( 1 ) to be true, whereas if we are told that he is living in the forest, then most people are expected to consider ( 1 ) to be false. Likewise, most people consider ( 2 ) to be true.

The question which we shall be discussing in this paper is how to automate reasoning such that conditionals are evaluated by an automated deduction system like humans do. This will be done in a context of logic programming (cf. [11]), abduction [9], Stenning and van Lambalgen’s representation of conditionals as well as their semantic operator [19] and three-valued Lukasiewicz logic [12], which has been put together in [6,7,5,8,3] and has been applied to the suppression [2] and the selection task [1], as well as to model the belief-bias effect [15] and contextual abductive reasoning with side-effects [16].

The methodology of the approach presented in this paper differs significantly from methods and techniques applied in well-known approaches to evaluate (mostly subjunctive) conditionals like Ramsey’s belief-retention approach [17], Lewis’s maximal world-similarity one [10], Rescher’s systematic reconstruction of the belief system using principles of saliency and prioritization [18], Ginsberg’s possible worlds approach [4] and Pereira and Apar´ıco’s improvements thereof by requiring relevancy [14]. Our approach is inspired by Pearl’s do-calculus [13] in that it allows revisions to satisfy conditions whose truth-value is unknown and which cannot be explained by abduction, but which are amenable to hypothetical intervention instead. 2

Preliminaries We assume the reader to be familiar with logic and logic programming. A (logic) program is a finite set of (program) clauses of the form A ← B1 ∧ . . . ∧ Bn where A is an atom and Bi, 1 ≤ i ≤ n, are literals or of the form > and ⊥, denoting truth- and falsehood, respectively. A is called head and B1 ∧ . . . ∧ Bn is called body of the clause. We restrict terms to be constants and variables only, i.e., we consider so-called data logic programs. Clauses of the form A ← > and A ← ⊥ are called positive and negative facts, respectively.

In the this paper we assume for each program that the alphabet consists precisely of the symbols mentioned in the program. When writing sets of literals we will omit curly brackets if the set has only one element.

Let P be a program. g P denotes the set of all ground instances of clauses occurring in P. A ground atom A is defined in g P iff g P contains a clause whose head is A; otherwise A is said to be undefined. Let S be a set of ground literals. def (S, P) = {A ← body ∈ g P | A ∈ S ∨ ¬A ∈ S} is called definition of S.

Let P be a program and consider the following transformation: 1. For each defined atom A, replace all clauses of the form A ← body 1, . . . ,

A ← body m occurring in g P by A ← body 1 ∨ . . . ∨ body m.

2. If a ground atom A is undefined in g P, then add A ← ⊥ to the program. 3. Replace all occurrences of ← by ↔.

The ground program obtained by this transformation is called completion of P, whereas the ground program obtained by applying only the steps 1. and 3. is called weak completion of P or wcP.

We consider the three-valued Lukasiewicz (or L-) semantics [12] and represent each interpretation I by a pair hI>, I⊥i, where I> contains all atoms which are mapped to true by I, I⊥ contains all atoms which are mapped to false by I, and I> ∩ I⊥ = ∅. Atoms occurring neither in I> not in I⊥ are mapped to unknown. Let hI>, I⊥i and hJ >, J ⊥i be two interpretations. We define

hI>, I⊥i ⊆ hJ >, J ⊥i iff I> ⊆ J > and I⊥ ⊆ J ⊥.

Under L-semantics we find F ∧ > ≡ F ∨ ⊥ ≡ F for each formula F , where ≡ denotes logical equivalence. Hence, occurrences of the symbols > and ⊥ in the bodies of clauses can be restricted to those occurring in facts.

It has been shown in [6] that logic programs as well as their weak completions admit a least model under L-semantics. Moreover, the least L-model of the weak completion of P can be obtained as least fixed point of the following semantic operator, which was introduced in [19]: ΦP (hI>, I⊥i) = hJ >, J ⊥i, where J > = {A | A ← body ∈ g P and body is true under hI>, I⊥i}, J ⊥ = {A | def (A, P) 6= ∅ and

body is false under hI>, I⊥i for all A ← body ∈ def (A, P)}.

=lmwc F iff formula F holds in the least L-model of wcP.

We define P | L

As shown in [2], the L-semantics is related to the well-founded semantics as follows: Let P be a program which does not contain a positive loop and let P+ = P \ {A ← ⊥ | A ← ⊥ ∈ P}. Let u be a new nullary relation symbol not occurring in P and B be a ground atom in

P∗ = P+ ∪ {B ← u | def (B, P) = ∅} ∪ {u ← ¬u}.

Then, the least L-model of wcP and the well-founded model for P∗ coincide.

An abductive framework consists of a logic program P, a set of abducibles AP = {A ← > | A is undefined in g P} ∪ {A ← ⊥ | A is undefined in g P}, a set of integrity constraints IC, i.e., expressions of the form ⊥ ← B1 ∧ . . . ∧ Bn, and the entailment relation |=lLmwc , and is denoted by hP, AP , IC, |=lLmwc i.

One should observe that each finite set of positive and negative ground facts has an L-model. It can be obtained by mapping all heads occurring in this set to true. Thus, in the following definition, explanations are always satisfiable.

An observation O is a set of ground literals; it is explainable in the abductive framework hP, AP , IC, |=lLmwc i iff there exists an E ⊆ AP called explanation such that P ∪ E is satisfiable, the least L-model of the weak completion of P ∪ E satisfies IC, and P ∪ E |=lLmwc L for each L ∈ O. 3

A Reduction System for Indicative Conditionals When parsing conditionals we assume that information concerning the mood of the conditionals has been extracted. In this paper we restrict our attention to indicative mood. In the sequel let cond (T , A) be a conditional with condition T and consequence A, both of which are assumed to be finite sets of literals not containing a complementary pair of literals, i.e., a pair B and ¬B.

Conditionals are evaluated wrt background information specified as a logic program and a set of integrity constraints. More specifically, as the weak completion of each logic program always admits a least L-model, the conditionals are evaluated under these least L-models. In the reminder of this section let P be a program, IC be a finite set of integrity constraints, and MP be the least L-model of wcP such that MP satisfies IC. A state is either an expression of the form ic(P, IC, T , A) or true, false, unknown, or vacuous. 3.1

A Revision Operator

Let S be a finite set of ground literals not containing a complementary pair of literals and let B be a ground atom in

rev (P, S) = (P \ def (S, P)) ∪ {B ← > | B ∈ S} ∪ {B ← ⊥ | ¬B ∈ S}. The revision operator ensures that all literals occurring in S are mapped to true under the least L-model of wc rev (P, S). 3.2

The Abstract Reduction System

Let cond (T , A) be an indicative conditional which is to be evaluated in the context of a logic program P and integrity constraints IC such that the least L-model MP of wcP satisfies IC. The initial state is ic(P, IC, T , A).

If the condition of the conditional is true, then the conditional holds if its consequent is true as well; otherwise it is either false or unknown. ic(P, IC, T , A) −→it true iff ic(P, IC, T , A) −→if false iff ic(P, IC, T , A) −→iu unknown iff

MP (T ) = true and MP (A) = true MP (T ) = true and MP (A) = false

MP (T ) = true and MP (A) = unknown

If the condition of the conditional is false, then the conditional is true under L-semantics. However, we believe that humans might make a difference between a conditional whose condition and consequence is true and a conditional whose condition is false. Hence, for the time being we consider a conditional whose condition is false as vacuous.

ic(P, IC, T , A) −→iv vacuous iff

MP (T ) = false

If the condition of the conditional is unknown, then we could assign a truthvalue to the conditional in accordance with the L-semantics. However, we suggest that in this case abduction and revision shall be applied in order to satisfy the condition. We start with the abduction rule: ic(P, IC, T , A) −→ia ic(P ∪ E , IC, T \ O, A) iff MP (T ) = unknown and E explains O ⊆ T in the abductive framework hP, AP , IC, |=lLmwc i and O 6= ∅. Please note that T may contain literals which are mapped to true by MP . These literals can be removed from T by the rule −→ia because the empty set explains them.

Now we turn to the revision rule:

ic(P, IC, T , A) −→ir ic(rev (P, S), IC, T \ S, A) iff MP (T ) = unknown, S ⊆ T , S 6= ∅, for each L ∈ S we find MP (L) = unknown, and the least L-model of wc rev (P, S) satisfies IC.

Altogether we obtain the reduction system RIC operating on states and consisting of the rules {−→it , −→if , −→iu, −→iv , −→ia , −→ir }. 4 4.1

Examples

Al in the Jailhouse

Rainy Day Suppose we are told that Al is imprisoned in a jailhouse on a rainy day, i.e.., he is living in a cell inside the jailhouse and it is raining:

P1 = {inside(X) ← imprisoned (X), imprisoned (al ) ← >, raining ← >}. The least L-model of wcP1 is h{imprisoned (al ), inside(al ), raining }, ∅i. In order to evaluate conditional ( 1 ) with respect to P1 we observe that this model maps raining and inside to true. Hence,

ic(P1, ∅, raining , inside) −→it true.

Sunny Day

Let us assume that Al is still imprisoned but that it is not raining: P2 = {inside(X) ← imprisoned (X), imprisoned (al ) ← >, raining ← ⊥}. The least L-model of wcP2 is h{imprisoned (al ), inside(al )}, {raining }i. In order to evaluate conditional ( 1 ) wrt P2 we observe that this model maps raining to false. Hence,

ic(P2, ∅, raining , inside) −→iv vacuous.

No Information about the Weather Suppose we are told that Al is imprisoned in a jailhouse but we know nothing about the weather:

P3 = {inside(X) ← imprisoned (X), imprisoned (al ) ← >}.

The least L-model of wcP1 is h{imprisoned (al ), inside(al )}, ∅i. In order to evaluate conditional ( 1 ) wrt P3 we observe that this model maps raining to unknown. Hence, we view raining as an observation which needs to be explained. The only possible explanation wrt hP3, {raining ← >, raining ← ⊥}, ∅, |=lLmwc i is {raining ← >}. Altogether we obtain

ic(P3, ∅, raining , inside) −→ia ic(P1, ∅, ∅, inside) −→it true.

Please note that P3 ∪ {raining ← >} = P1 = rev (P3, raining ). Hence, we could replace −→ia by −→ir in the previous reduction sequence.

The Shooting of Kennedy

President Kennedy was killed. There was a lengthy investigation about who actually shot the president and in the end it was determined that Oswald did it:

P4 = {Kennedy dead ← os shot , Kennedy dead ← se shot , os shot ← >}. The least L-model of wcP4 is h{os shot , Kennedy dead }, ∅i. Evaluating the indicative conditional ( 2 ) under this model we find that its condition T = {Kennedy dead , ¬os shot } is mapped to false. Hence,

ic(P4, ∅, {Kennedy dead , ¬os shot }, se shot ) −→iv vacuous.

Now consider the case that we do not know that Oswald shot the president:

P5 = {Kennedy dead ← os shot , Kennedy dead ← se shot }.

As least L-model of wcP5 we obtain h∅, ∅i and find that it maps T to unknown. We may try to consider T as an observation and explain it wrt the abductive framework hP5, AP5 , ∅, |=lLmwci, where AP5 consists of the positive and negative facts for os shot and se shot . The only possible explanation is E = {os shot ← ⊥, se shot ← >}. As least L-model of wc(P5 ∪ E) we obtain h{Kennedy dead , se shot }, {os shot }i. As this model maps se shot to true we find ic(P5, ∅, {Kennedy dead , ¬os shot }, se shot ) −→ia ic(P5 ∪ E, ∅, ∅, se shot ) −→it true.

In this example we could also apply revision. Let

P6 = rev (P5, T ) = {Kennedy dead ← >, os shot ← ⊥}.

We obtain ic(P5, ∅, {Kennedy dead , ¬os shot }, se shot ) −→ir ic(P6, ∅, ∅, se shot ) −→iu unknown because the least L-model of wcP6 is h{Kennedy dead }, {os shot }i and maps se shot to unknown. However, as conditional ( 2 ) can be evaluated by abduction and without revising the initial program, this derivation is not preferred. 4.3

The Firing Squad

This example is presented in [13]. If the court orders an execution, then the captain will give the signal upon which riflemen A and B will shoot the prisoner. Consequently, the prisoner will be dead. We assume that the court’s decision is unknown, that both riflemen are accurate, alert and law-abiding, and that the prisoner is unlikely to die from any other causes. Let

P7 = { sig ← execution, rmA ← sig, rmB ← sig,

dead ← rmA, dead ← rmB , alive ← ¬dead }.

The least L-model of wcP7 is h∅, ∅i.

( 7 ) Rifleman A did not Shoot To evaluate conditional ( 3 ) wrt this model we first observe that the condition rmA is mapped to unknown by ( 7 ). Considering the abductive framework

hP7, {execution ← >, execution ← ⊥}, ∅, |=lLmwci, ¬rmA can be explained by

{execution ← ⊥}.

Let P8 = P7 ∪ ( 9 ). The least L-model of wcP8 is

h{alive}, {execution, sig, rmA, rmB , dead }i.

Because alive is mapped to true under this model, we obtain

ic(P7, ∅, ¬rmA, alive) −→ia ic(P8, ∅, ∅, alive) −→it true.

The Prisoner is Alive Now consider conditional ( 4 ). Because ( 7 ) maps alive to unknown we treat alive as an observation. Considering again the abductive framework ( 8 ) this observation can be explained by ( 9 ). Hence, we evaluate the consequence of ( 4 ) under ( 10 ) and find that the captain did not signal: ic(P7, ∅, alive, ¬sig) −→ia ic(P8, ∅, ∅, ¬sig) −→it true.

Rifleman A Shot Let us turn the attention to conditional ( 5 ). Because ( 7 ) maps rmA to unknown, we treat rmA as an observation. Considering the abductive framework ( 8 ) this observation can be explained by

{execution ← >}.

Let P9 = P7 ∪ ( 11 ). The least L-model of wcP9 is

h{execution, sig, rmA, rmB , dead }, {alive}i.

Because rmB is mapped to true under this model, we obtain

ic(P7, ∅, rmA, rmB ) −→ia ic(P9, ∅, ∅, rmB ) −→it true.

The Captain Gave no Signal Let us now consider conditional ( 6 ). Its condition T = {¬sig, rmA} is mapped to unknown by ( 7 ). We can only explain ¬sig by ( 9 ) and rmA by ( 11 ), but we cannot explain T because

wc(( 9 ) ∪ ( 11 )) = {execution ↔ > ∨ ⊥} ≡ {execution ↔ >}.

In order to evaluate this conditional we have to consider revisions. ( 8 ) ( 9 ) ( 10 ) ( 11 ) ( 12 ) 1. A brute force method is to revise the program wrt all conditions. Let P10 = rev (P7, {¬sig , rmA})

= (P7 \ def ({¬sig , rmA}, P7)) ∪ {sig ← ⊥, rmA ← >}.

The least L-model of wcP10 is h{rmA, dead }, {sig , rmB , alive}i. ( 13 ) This model maps dead to true and rmB to false and we obtain ic(P7, ∅, {¬sig , rmA}, {dead , ¬rmB }) −→ir ic(P10, ∅, ∅, {dead , ¬rmB }) −→it true. 2. As we prefer minimal revisions let us consider

P11 = rev (P7, rmA) = (P7 \ def (rmA, P7)) ∪ {rmA ← >}.

The least L-model of wcP11 is h{dead , rmA}, {alive}i. Unfortunately, ¬sig is still mapped to unknown by this model, but it can be explained in the abductive framework hP11, {execution ← >, execution ← ⊥}, ∅, |=lLmwc i by ( 9 ). Let P12 = P11 ∪ ( 9 ). Because the least L-model of wcP12 is h{dead , rmA}, {alive, execution, sig , rmB }i ( 14 ) we obtain ic(P7, ∅, {¬sig , rmA}, {dead , ¬rmB }) −→ir ic(P11, ∅, ¬sig , {dead , ¬rmB }) −→ia ic(P12, ∅, ∅, {dead , ¬rmB } −→it true.

The revision leading to P11 is minimal in the sense that only the definition of rmA is revised and without this revision the condition of ( 6 ) cannot be explained. This is the only minimal revision as we will show in the sequel. 3. An alternative minimal revision could be the revision of P7 wrt to ¬sig :

P13 = rev (P7, ¬sig ) = (P7 \ def (¬sig , P7)) ∪ {sig ← ⊥}.

The least L-model of wcP13 is h{alive}, {sig , dead , rmA, rmB }i. ( 15 ) Because this model maps rmA to false we obtain: ic(P7, ∅, {¬sig , rmA}, {dead , ¬rmB }) −→ir ic(P13, ∅, rmA, {dead , ¬rmB }) −→iv vacuous. 4. So far the first step in evaluating the conditional was a revision step. Alternatively, we could start with an abduction step. ¬sig can be explained in the abductive framework ( 8 ) by ( 9 ) leading to the program P8 and the least L-model ( 10 ). Because this model maps rmA to false we obtain: ic(P7, ∅, {¬sig , rmA}, {dead , ¬rmB }) −→ia ic(P8, ∅, rmA, {dead , ¬rmB }) −→iv vacuous. 5. Let us now reverse the order in which the conditions are treated and start by explaining rmA. This has already been done before and we obtain P9 and the least L-model ( 12 ). Because this model maps ¬sig to false we obtain: ic(P7, ∅, {¬sig , rmA}, {dead , ¬rmB }) −→ia ic(P9, ∅, ¬sig , {dead , ¬rmB }) −→iv vacuous.

In the last example we have discussed five different approaches to handle the case that the truth value of the conditions of a conditional is unknown and cannot be explained: maximal (parallel) revision (MaxRev), partial (sequential) revision as well as partial (sequential) explanation, where in the sequential approaches the literals in the condition of the conditionals are treated in different orders: left-to-right and right-to-left, where we consider sets to be ordered (PRevLR,PRevRL,PExLR,PExRL). The results are summarized in Table 1, where the conditional as well as the literals are evaluated wrt the final least L-model computed in the different approaches.

Which approach shall be preferred? Because rifleman A causally depends on the captain’s signal but not vice-versa, plus given that in this example clauses express causes, and effects come after causes, it would make sense to take the cause ordering as the preferred one for abducing the conditions. Hence, PExLR would be preferred. However, because rifleman A is an agent, the causes of his actions can be internal to him, his decisions. Hence, when autonomous agents are involved (or spontaneous phenomena like radioactivity), the ordering of abducing the conditions is independent of causal dependency. 5

Properties In this section, let P be a program, hI>, I⊥i the least L-model of wcP, IC a set of integrity constraints, hP, AP , IC, |=lLmwci an abductive framework, and L a literal.

Proposition 1. If O can be explained by E ⊆ AP and hJ >, J ⊥i is the least L-model of wc(P ∪ E ), then hI>, I⊥i ⊆ hJ >, J ⊥i.

MaxRev PRevRL PRevLR PExLR PExRL final program P10 final least L-model ( 13 )

sig false rmA true dead true rmB false alive false execution unknown conditional ( 6 ) true

P12 ( 14 ) false true true false false false true

P13 P8 P9 ( 15 ) ( 10 ) ( 12 ) false false true false false true false false true false false true true true false unknown false true vacuous vacuous vacuous Proof. The least L-models hI>, I⊥i and hJ >, J ⊥i are the least fixed points of the semantic operators ΦP and ΦP∪E , respectively. Let hIn>, In⊥i and hJn>, Jn⊥i be the interpretations obtained after applying ΦP and ΦP∪E n-times to h∅, ∅i, respectively. We can show by induction on n that hIn>, In⊥i ⊆ hJn>, Jn⊥i. The proposition follows immediately.

Proposition 1 guarantees that whenever −→ia is applied, previously checked conditions of a conditional need not to be re-checked. The following Proposition 3 gives the same guarantee whenever −→ir is applied.

Proposition 2. If the least L-model of wcP maps L to unknown and hJ >, J ⊥i is the least L-model of wc rev (P, L), then hI>, I⊥i ⊂ hJ >, J ⊥i.

Proof. By induction on the number of applications of ΦP and Φrev(P,L).

Proposition 3. RIC is terminating.

Proof. Each application of −→it , −→if , −→iu or −→iv leads to an irreducible expression. Let cond (T , A) be the conditional to which RIC is applied. Whenever −→ir is applied then the definition of at least one literal L occurring in T is revised such that the least L-model of the weak completion of revised program maps L to true. Because T does not contain a complementary pair of literals this revised definition of L is never revised again. Hence, there cannot exist a rewriting sequence with infinitely many occurrences of −→ir . Likewise, there cannot exist a rewriting sequence with infinitely many occurrences of −→ia because each application of −→ia to a state ic(P, IC, T , A) reduces the number of literals occurring in the T .

Proposition 4. RIC is not confluent.

Proof. This follows immediately from the examples presented in Section 4. 6

Open Questions and the Proposal of an Experiment Open Questions The new approach gives rise to a number of questions. Which of the approaches is preferable? This may be a question of pragmatics imputable to the user. The default, because no pragmatic information has been added, is maximals revision for skepticism and minimal revisions for credulity. Do humans evaluate multiple conditions sequentially or in parallel? If multiple conditions are evaluated sequentially, are they evaluated by some preferred order? Shall explanations be computed skeptically or credulously? How can the approach be extended to handle subjunctive conditionals? The Proposal of an Experiment Subjects are given the background information specified in the program P9. They are confronted with the conditionals like ( 6 ) as well as variants with different consequences (e.g., execution instead of {dead , ¬rmB } or conditionals where the order of two conditions are reversed. We then ask the subjects to answer questions like: Does the conditional hold? or Did the court order an execution? Depending on the answers we may learn which approaches are preferred by humans.

Acknowledgements

lier draft of the paper.

We thank Bob Kowalski for valuable comments on an ear

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