Conditionals play a prominent role in human reasoning and, hence, all cognitive theories try to evaluate conditionals like humans do. In this paper, we are particularly interested in the Weak Completion Semantics, a new cognitive theory based on logic programming, the weak completion of a program, the three-valued Łukasiewicz logic, and abduction. We show tha t the evaluation of conditionals within the Weak Completion Semantics as defined so far leads to counterintuitive results. We propose to distinguish between obligation and factual conditionals with necessary or sufficient conditions, and adapt the set of abducibles accordingly. This does not only remove the previously encountered counterintuitive results, but also leads to a new model for the Wason Selection Task.
∗ The authors are mentioned in alphabetical order. F ¬ F ⊤ ⊥ U ⊥ ⊤ U ∧ ⊤ U ⊥ ⊤ ⊤ U ⊥ U U U ⊥ ⊥ ⊥ ⊥ ⊥ ∨ ⊤ U ⊥ ⊤ ⊤ ⊤ ⊤ U ⊤ U U ⊥ ⊤ U ⊥ ← ⊤ U ⊥ ⊤ U ⊥ ⊤ ⊤ ⊤ U ⊤ ⊤ ⊥ U ⊤ ↔ ⊤ U ⊥ ⊤ U ⊥ ⊤ U ⊥ U ⊤ U ⊥ U ⊤
If the condition is unknown, then a technique called minimal revision followed by abduction (MRFA) is applied, where
the program is revised as little as possible and abduction is applied afterwards in order to explain the condition of the
conditional. MRFA was able to solve all conditionals in the firing squad scenario described in [
However, as we will show in Section 3 of this paper, MRFA may lead to counterintuitive results and does not generate
the expected results in some cases. This happens because MRFA does not consider the semantics of the conditionals,
which is a crucial aspect when it comes to their evaluation. In particular, in Section 4 we discuss obligation and factual
conditionals with necessary or sufficient conditions. Based on the semantics of a given conditional, in Section 4.3 we vary
the set of abducibles used in the abduction process and demonstrate, that by using this technique, the counterintuitive results
obtained before can be avoided. In Section 5 we verify that the new technique does not alter the answers given by WCS
for the suppression task [
In a three-valued logic, the truth values are not only true or false, symbolized by ⊤ and ⊥, respectively. But there exists also a third value, which, in the sequel, we will call unknown and use the symbol U to denote it. More specifically, we will be using the three-valued Łukasiewicz (or Ł-) logic. As shown in Table 1, the expressions U ← U and U ↔ U are evaluated to true under the Ł-Logic. This is the main di fference between this and the three-valued logics introduced by
2.2
Clauses are expressions of the forms A ← L1 ∧...∧Ln (called rules), A ← ⊤ (called facts), and A ← ⊥ (called assumptions), where n ≥ 1, A is an atom, and each Li, 1 ≤ i ≤ n, is a literal. A is called the head and L1 ∧ ... ∧ Ln as well as ⊤ and ⊥ are called bodies of the corresponding clauses. A (propositional logic) program P is a finite set of clauses. A is defined in P if and only if P contains a clause of the form A ← Body. A is undefined in P if and only if A is not defined in P. The definition of A in P is defined as def (A, P) = { A ← Body | A ← Body is a clause in P} .
A set of literals is consistent if it does not contain an atom and its negation. Let S be a finite and consistent set of literals in rev(P, S) = (P \ def (P, S)) ∪ { A ← ⊤ | A ∈ S} ∪ { A ← ⊤ | ¬ A ∈ S} , where A denotes an atom. rev(P, S) is called the revision of P with respect to S. 2.3
The definitions bellow are based on [
guage. We represent interpretations by pairs hI⊤, I⊥i, where the set I⊤ consists of all atoms which are mapped to ⊤, the set I⊥ consists of all atoms which are mapped to ⊥, and I⊤ ∩ I⊥ = ∅. All atoms which occur neither in I⊤ nor I⊥ are mapped to U. The logical value of formulas can be derived from Table 1 as usual.
(i.e., I(F) = ⊤). Let P be a program of a language L and let I be an interpretation of L. We say I is a model of P iff I is a
model of each clause in P. Two formulas F and G are said to be semantically equivalent if and only if both have the same
truth value under all interpretations. The least model of wcP can be obtained as the least fixed point of the semantic ΦP
operator [
J⊤ J⊥ = { A | there exists A ← Body ∈ P with I(Body) = ⊤} , = { A | there exists A ← Body ∈ P and for all A ← Body ∈ P we find I(Body) = ⊥} .
As has been shown in [
respect to the least models of the weak completion of these programs. We write P |=wcs F iff formula F holds in lfp ΦP, which is identical to the least model of wcP. In the sequel, MP denotes the least model of wcP. 2.5
An abductive framework consists of a logic program P, a set of abducibles A ⊆ AP, where
AP = { A ← ⊤ | A is undefined in P} ∪ { A ← ⊥ | A is undefined in P} , and the entailment relation |=wcs. An abductive framework is denoted by hP, A |=wcsi. One should observe that each P and, in particular, each finite set of positive and negative facts has an least model of the weak completion. For the latter, this can be obtained by mapping all heads occurring in this set to true. Thus, explanations as well as the union of a program and an explanation are always satisfiable.
An observation O is a set of literals; O is explainable in the framework hP, A, IC, |=wcsi iff there exists an E ⊆ A called explanation such that MP∪E |=wcs L for all L ∈ O and P ∪ E satisfies IC. We require explanations to be minimal, i.e. they cannot be subsumed by any other explanation.
There are two possible ways of abductive reasoning: credulous and skeptical reasoning. Let hP, A, IC, |=wcsi be an abductive framework, O an observation and F a formula: • F • F follows credulously from P and O iff there exists an explanation E for O such that P ∪ E |=wcs F .
follows skeptically from P and O iff for all explanations E for O we find P ∪ E |=wcs F . 2.6
We consider conditionals of the form if C then D, where C and D are finite and consistent sets of literals viewed as
conjunctions of literals. Conditionals are evaluated with respect to some background information specified as a program.
Let P be a program, MP be the least model of wcP and if C then D be a conditional. [
MP(C) = true. • hP , C, Di −→c hrev(P, S), C \ S, Di iff
MP(C) = false, where S = { L ∈ C | M P(L) = ⊥} . • hP , C, Di −→a hP ∪ E, C, Di iff
MP(C) = unknown, O ⊆ C, O , ∅, for each L ∈ O we find MP(L) = unknown, and E explains O in the abductive framework hP, AP, |=wcsi. • hP , C, Di −→r hrev(P, S), C \ S, Di iff
MP(C) = unknown, S ⊆ C, S , ∅, for all L ∈ S we find MP(L) = unknown.
[
For the following examples in this section, the conditionals will be evaluated by means of MRFA and we will assume that the following conditionals are known: If it rains, then the streets are wet. (1) If it rains, then she takes her umbrella (2)
= { wet streets ← rain ∧ ¬ ab1,
P The set of abducibles is AP = { rain ← ⊤, rain ← ⊥} and the least model of wcP is MP = h∅, { ab1, ab2i} . umbrella ← rain ∧ ¬ ab2, ab1 ← ⊥, ab2 ← ⊥ } . 3.1
Assume that the streets are not wet. Do humans conclude that it did not rain? Or differently said, given the background knowledge P, how should we evaluate the following conditional? (−→r) Example 1 If the streets are not wet, then it did not rain.
hP, ¬ wet streets, ¬ raini −→a hP ∪ { rain ← ⊥} , ¬ wet streets, ¬ raini −→s true, where first, as MP(¬ wet streets) = U, rule 3b (−→a) from the MRFA evaluation of the previous section applies, where E = { rain ← ⊥} is the only explanation for ¬ wet streets. Given that MP∪E = h∅, { rain, wet streets, umbrella, ab1, ab2i} , ¬ wet streets is evaluated to ⊤, thus rule 1 (−→s) applies, i.e. the conditional is evaluated with respect to the truth value of ¬ rain under MP∪E, which is ⊤. Summing up, the conditional if ¬ wet streets then ¬ rain is evaluated to true.
background knowledge P, how should we evaluate the following conditional? Example 2 If she did not take her umbrella, then it did not rain.
Analogously to the previous example, given that if ¬ umbrella then ¬ rain, according to MRFA, we obtain hP, ¬ umbrella, ¬ raini −→a hP ∪ { rain ← ⊥} , ¬ umbrella, ¬ raini −→s true.
This conditional has the same structure with respect to P as the previous one, thus the evaluation strategy is identical. 3.2
Example 3 If the streets are wet, then it rained.
hP, wet streets, raini −→a hP ∪ { rain ← ⊤} , wet streets, raini −→s true, Now the explanation for wet streets is rain ← ⊤ and if wet streets then rain is evaluated to true.
Example 4 If she took her umbrella, then it rained.
hP, umbrella, raini −→a hP ∪ { rain ← ⊤} , umbrella, raini −→s true, and thus the conditional if umbrella then rain is evaluated to true.
Although MRFA evaluates all the conditionals shown in Section 3 to true, it is not clear that given the information the streets are not wet, humans would conclude it did not rain in the same way that it would be concluded if the information that she did not take her umbrella would be given. Similarly, this is the case for the examples in Section 3.2: Given the information she took her umbrella, it is again not clear that one would conclude it rained in the same way that it would be concluded if the information that the streets are wet would be given.
4.1
tory if the condition of the conditional is true, while in the factual conditionals this is not the case. More precisely, given an obligation conditionals it can never be the case that the condition is true and the consequence is not true.
obligation conditional. This happens because for this type of conditionals, people keep in mind a forbidden possibility where condition and not consequence happens together and, in this case, if the consequence is known to be false, then it cannot be the case that condition is true, otherwise the forbidden possibility is violated. Thus, not condition is concluded. But, since in a factual conditional this forbidden possibility does not exist, conditionals with the consequence denied should be evaluated to unknown.
conditional (1): How plausible is it, that it rained (condition is true) and the streets are not wet (consequent is false)?
4.2
If a conditional has a necessary condition, then the consequence cannot be true unless its condition is true. But if a conditional has only a sufficient condition, then the circumstance where this condition is true gives us the adequate grounds to conclude that the consequence is true as well, but there is no necessity involved.
though it did not rain (condition is false)? We may be able to imagine cases like a flooding or a tsunami has occurred, but we would expect that such an extraordinary reason would have been mentioned in the context. Similarly, while considering conditional (2): How plausible is it that she took her umbrella (consequent is true) even though it did not rain (condition is false)? It seems plausible that, for instance, the weather forecast was wrong and she took her umbrella even though it did not rain. In another hand, we cannot easily imagine a situation where the streets are wet and it did not rain.
4.3
In order to allow MRFA to distinguish between the different types of conditionals discussed in Section 4.2 and 4.1, the following two modifications will be introduced to the approach:
AP = { A ← ⊤ | A is undefined in P} ∪ { A ← ⊥ | A is undefined in P} ∪ { A ← ⊤ | A is the head of a conditional with a sufficient condition in P} ∪ { abi ← ⊤ | abi is in the body of a factual conditional in P} . C ¬ wet streets ¬ umbrella wet streets umbrella
D ¬ rain ¬ rain rain rain
S = ∅ and E = { rain ← ⊥} true S = ∅ and E = { rain ← ⊥} true S = ∅ and E = { rain ← ⊤} true S = ∅ and E = { rain ← ⊤} true
S = ∅ and E = { rain ← ⊥} true S = ∅ and E1 = { rain ← ⊥} or E2 = { ab2 ← ⊤} unknown S = ∅ and E = { rain ← ⊤} true S = ∅ and E1 = { rain ← ⊤} or E2 = { umbrella ← ⊤} unknown We reconsider the examples discussed in Section 3.1 and 3.2 with the newly introduced semantic approach, where we take the following assumption: if it rains, then the streets are wet is an obligation conditional with a necessary condition and if it rains, then she takes her umbrella is a factual conditional with a sufficient condition. Thus, the set of abducibles is defined as follows:
AP = { rain ← ⊤, rain ← ⊥} ∪ { umbrella ← ⊤} ∪ { ab2 ← ⊤} .
Example 1 If the streets are not wet, then it did not rain.
The evaluation of this conditional remains the same, since the explanation used for the condition ¬ wet streets is unique and, thus, the consequence ¬ rain follows skeptically from the program P and the observation ¬ wet streets. Because of this, the conditional is still evaluated to true by following the same steps showed in Section 3.1.
Example 2 If she did not take her umbrella, then it did not rain.
We have two possible explanations E1 = { rain ← ⊥} and E2 = { ab2 ← ⊤} for O = ¬{ umbrella} , where P ∪ E1 |=wcs ¬ rain, but P ∪ E2 6|=wcs ¬ rain. According to semantic MRFA, we obtain the following two evaluation strategies: hP, ¬ umbrella, ¬ raini −→a hP ∪ E1, ¬ umbrella, ¬ raini −→s true, hP, ¬ umbrella, ¬ raini −→a hP ∪ E2, ¬ umbrella, ¬ raini −→s unknown.
The consequence does not follow from all explanations, and therefore, the conditional is evaluated to unknown. This complies with the point of view expressed in Section 4.1.
Example 3 If the streets are wet, then it rained.
Example 4 If she took her umbrella, then it rained.
Similar to Example 1, we have two explanations E1 = { rain ← ⊤} and E2 = { umbrella ← ⊤} , where P ∪ E1 |=wcs rain, but P ∪ E2 6|=wcs rain. According to semantic MRFA, we obtain the following two evaluation strategies: hP, umbrella, raini −→a hP ∪ E1, umbrella, raini −→s true, hP, umbrella, raini −→a hP ∪ E2, umbrella, raini −→s unknown.
Because we are reasoning skeptically, the conditional is now evaluated to unknown. This complies with the point of view expressed in Section 4.2. 5
Byrnes’ suppression task [
D 89%
F 16%
3 62%
7 25% beer 95%
S ocial coke
Case 22yrs
In [
programs. Following the conditional classification in the Section 4, conditonals 1 and 2 are classified as obligation
conditionals whereas conditional 3 is classified as a factual conditional. Individually, each of the three conditionals have a
necessary condition. However, in the second group when conditionals 1 and 2 happen together, their conditions are no
longer necessary, but sufficient. Therefore, for Pe, the set of abducibles remains as in [
Yet, for Pe+Alt, the set of abducibles, originally defined as APe+Alt = { e ← ⊤, e ← ⊥, t ← ⊤, t ← ⊥} is now extended by { ℓ ← ⊤} , because in the second group the conditionals have sufficient conditions. Consider Pe+Alt together with the observation Oℓ = { ℓ ← ⊤} : There are three possible explanations for O, E1 = { e ← ⊤} , E2 = { t ← ⊤} and E3 = { ℓ ← ⊤} . As Pe+Alt ∪ E2 6|=wcs e, 2 she has essay to write does not follow skeptically from Pe+Alt and Oℓ. Consider Pe+Alt together with the observation O¬ ℓ = { ℓ ← ⊥} : The only possible explanation for O is E = { e ← ⊥, t ← ⊥} and Pe+Alt ∪ E |=wcs ¬ e. Thus, She does not have an essay to write follows skeptically from Pe+Alt and O¬ ℓ.
Moreover, for Pe+Add, the set of abducibles, originally defined as APe+Add = { e ← ⊤, e ← ⊥, o ← ⊤, o ← ⊥} is now extended by { ab3 ← ⊤} , because conditional 3 is a factual conditional. Consider Pe+Add together with the observation Oℓ = { ℓ ← ⊤} : The only possible explanation for O is E = { e ← ⊤, o ← ⊤} and Pe+Add ∪ E |=wcs e. Thus, She has an essay to write follows skeptically from Pe+Add and Oℓ. Consider Pe+Add together with the observation O¬ ℓ = { ℓ ← ⊥} : O has three possible explanations, E1 = { e ← ⊥} , E2 = { o ← ⊥} and E3 = { ab3 ← ⊤} . As Pe+Add ∪ E2 6|=wcs ¬ e,2 She does not have an essay to write does not follow skeptically from Pe+Add and O¬ ℓ.
In both programs above, the derived result for the observations Oℓ = { ℓ ← ⊤} and O¬ ℓ = { ℓ ← ⊥} considered complies with the majority of the participants’ answers. 6
Wasons’ [
if there is a D on one side of the card, then there is 3 on the other side and had to consider four cards, showing the letters D and F as well as the numbers 3 and 7. Furthermore, they were told that each card had a letter on one side and a number on the other side. Next, the participants where asked which cards must be turned to prove that the conditional holds.
From a classical logic point of view, the conditional can be represented as the implication D → 3. From a classical logical point of view we must turn the cards showing D (modus ponens) and 7 (modus tollens). As repeated experiments have shown consistently (see Table 3), from a classical logical point of view, the majority of the participants correctly chose card D. However, they failed to choose card 7 and incorrectly chose card 3. In other words, the overall correctness of the answers for the abstract selection task if modeled by an implication in classical two-valued logic is pretty bad.
[
in the abstract case as a belief. For instance, the participants perceived the task to examine whether the conditional is either true or false. On the other hand, in the social case, the participants perceived the conditional as a social constraint, a conditional that ought to be true. People intuitively aim at preventing the violation of such a constraint, which is normally done by observing whether the state of the world complies with the rule.
In [
To generalize the problem, lets’ consider cards with type X o n one side and type Y on the other side. Given the conditional if X, then Y we would like to decide which cards must be turned in order to assure that the conditional holds. The program representing the background knowledge is given by P = { Y ← X ∧ ¬ ab, ab ← ⊥} and the set AP of abducibles depends on the type of the given conditional.
1. X and Y follow skeptically from P and O, or 2. For all explanations E for observation O = { X} given P, either E = { Y ← ⊤} or E = { Y ← ⊥} .
The case where the observed side has type Y can be solved likewise. We encode and evaluate both abstract and social cases using this approach. Table 4 and Table 5 contain the evaluation of the observations for each case and shows that in both cases the results of our approach coincide with the experimental results. 6.2
In the abstract case we have cards with type number on one side and letter on the other side. The background knowledge is represented by the program
Pabstract = { 3 ← D ∧ ¬ ab, ab ← ⊥} .
As 3 ← D ∧ ¬ ab is classified as a factual conditional, the set of abducibles is
APabstract = { D ← ⊤, D ← ⊥, ab ← ⊤} .
Table 4 shows that in the cases where D and 3 are observed, the cards are turned. In both cases, E = { D ← ⊤} is the only explanation and, in the two cases, Pabstract ∪ E |=wcs D and Pabstract ∪ E |=wcs 3, satisfying condition 1. On the other hand, the cards are not turned in the cases where F and 7 are observed. For the case where F is observed, ¬ D must to be explained and the only possible explanation is E = { D ← ⊥} . Because Pabstract ∪ E 6|=wcs D and Pabstract ∪ E 6|=wcs 3, condition 1 is not satisfied, and, further as 7 is not the head of a rule in E, condition 2 is also not satisfied. Finally, in the case where 7 is observed, besides the explanation E above, there is another possible explanation E′ = { ab ← ⊤} . But, because we already know that E does not satisfy any of the conditions, it cannot be the case that one of them would be satisfied by an additional explanation. In the first case, because we reason skeptically, and in the second one, because the condition must hold for all explanations. 6.3
In the social case we have cards with type drink on one side and age on the other side. The background knowledge is represented by
Psocial = { over19 ← beer ∧ ¬ ab, ab ← ⊥} .
As over19 ← beer ∧ ¬ ab is classified as an obligation conditional the set of abducibles is defined as APsocial = { beer ← ⊤, beer ← ⊥, over19 ← ⊤} , turn turn
O E MPsocial∪E beer { beer ← ⊤} h{ beer, over19} , { abi} coke (¬ beer) { beer ← ⊥} h∅ , { beer, over19, abi} 22yrs (over19) { beer ← ⊤} h{ beer, over19} , { abi}
{ over19 ← ⊤} h{ over19} , { abi} 16yrs (¬ over19) { beer ← ⊥} h∅ , { beer, over19, abi} turn turn
80% because beer is undefined in the program Psocial and over19 is the head of a sufficient conditional.
Table 5 shows the results with respect to the social case, where the cards are turned when beer and 16yrs are observed, while for the observation coke and 22yrs, the cards are not turned. When beer is observed, E = { beer ← ⊤} is the only possible explanation and, because Psocial ∪ E |=wcs beer ∧ over19, condition 1 is satisfied. If coke is observed, ¬ beer must be explained and E = { beer ← ⊥} is the only possible explanation. Because neither Psocial ∪ E |=wcs beer ∧ over19 holds nor over19 is the head of a rule in E, none of the conditions are satisfied. In the case where 22yrs is observed, over19 must be explained and there are two possible explanations: E1 = { beer ← ⊤} and E2 = { over19 ← ⊤} . Because Psocial ∪ E1 |=wcs beer ∧ over19, but Psocial ∪ E2 6|=wcs beer ∧ over19, beer and over19 dont’ follow skeptically from Psocial and 22yrs. Thus, condition 1 is not satisfied. Besides this, because beer is the head of a rule in E1, but not in E2, condition 2 is also not satisfied. Finally, if 16yrs is observed, then ¬ over19 must be explained, which is possible only by explanation E = { beer ← ⊥} . Since beer is the head of a rule in E, condition 2 is satisfied.
To conclude, the modeling proposed in this section can be seen as an improvement of the approach in [
We started out by analyzing the MRFA approach with respect to how conditionals are evaluated based on the Weak
Completion Semantics and identified some of its limitations. Our hypothesis is that the semantics of these conditionals play a
crucial role when it comes to their evaluation in the context of human reasoning. We have shown examples that support our
hypothesis and saw that MRFA could not evaluate these examples as expected. Therefore, we propose a semantic approach
to the evaluation of conditionals, in a way that we can specify different types of conditionals, viz. obligation or factual
conditionals with necessary or sufficient conditions, based on their meaning. From a technical point of view, this new
approach consists of extending the set of abducibles and varying this set with respect to the semantics of the conditionals.
The suppression task can still be adequately modeled. Furthermore, a way of modeling the abstract and the social case of
the selection task without changing the background knowledge can be done and improves the approach in [
they using? Can an answer I dont’ know be qualified as a truth value assignment or is it a meta-remark ? At least, it appears
that three-valued logics are needed for modeling human reas oning [
than one condition, and one of the conditions is necessary whereas another one is sufficient? How important is the order
in which multiple conditions of a conditional are considered? From human spatial reasoning [