An experiment has revealed that if the antecedent of a conditional sentence is denied, then most participants conclude that the negation of the consequent holds. However, a significant number of participants answered nothing follows if the antecedent of the conditional sentence was non-necessary. The weak completion semantics correctly models the answers of the majority, but cannot explain the number of nothing follows answers. In this paper we extend the weak completion semantics by counter examples. The extension allows to explain the experimental findings.
Conditional sentences are propositions of the form if A then C where A and C are
atomic sentences called antecedent and consequent, respectively. Four kinds of
conditional inference tasks have been a common area of research by psychologists till date:
1. Affirmation of the antecedent (AA): if A then C and A, therefore C.
2. Denial of the antecedent (DA): if A then C and ¬A, therefore ¬C.
3. Affirmation of the consequent (AC): if A then C and C, therefore A.
4. Denial of the consequent (DC): if A then C and ¬C, therefore ¬A.
In classical, two-valued propositional logic, conditional sentences are taken to mean
material implications and biconditionals to mean (material) equivalence. The
conclusion for the DA and the AC are hence considered to be logical fallacies (invalid) for
a conditional sentence whereas they are considered valid for a biconditional. When
replacing the above abstract conditional sentences with everyday ones, however, the
inferences largely depend on the semantics and pragmatics of human communication,
culture, and context. In this paper, we therefore discuss how everyday conditional
sentences can be categorized into four proposed semantic categories. We also share the
results of an experiment reported in [
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The Weak Completion Semantics (WCS) is a three-valued, non-monotonic
cognitive theory, which can not only adequately model the suppression task by [
In order to elaborate on what it really means for a conditional sentence to have a non-necessary antecedent and to propose a solution to the aforementioned problem, we begin by considering the following DA inference tasks: 1. If Maria is drinking alcoholic beverages in a pub, then Maria must be over 19 years of age and Maria is not drinking alcoholic beverages in a pub. 2. If the plants get water, then they will grow and the plants get no water. Both of these examples appeared in the aforementioned experiment, and as was the case for every conditional sentence that was included in the experiment, accompanied with a small background story. A curious reader may find the background stories in the Appendix. In the first example, 28 out of 56 participants answered Maria must not be over 19 years of age, whereas 25 answered nf. In this example the antecedent is nonnecessary; it is not considered necessary for a person to drink alcohol in order for her to be older than 19. There are many people who do not drink alcoholic beverages although they are over 19 years of age. In the second example, 47 out of 56 participants answered the plants will not grow whereas only 8 answered nf. In this case, the antecedent is necessary. Plants do not grow without water. Table 3 gives a complete account of this experimental data.
Based on this observation, we propose an extension which allows WCS to account for the nf answers. In Example 1, the existing framework of the WCS creates a model where given that Maria is not drinking alcoholic beverages, it can be concluded that Maria is not older than 19 years of age. With the proposed extension, however, a counter example can be constructed based on a possible observation that Maria is not drinking alcoholic beverages and yet Maria is older than 19 years of age. This leads to an alternative model, which when compared to the former model and reasoned skeptically, leads to the conclusion that it is unknown whether Maria is older than 19 years of age. In Example 2, WCS creates a model where given that the plants do not get water, it can be concluded that they will not grow. But in this case, a counter example does not readily exist.
The paper is organized as follows: In the next section we formally introduce the
WCS. A classification of conditional sentences is given in Section 3. The experiment is
described in Section 4. We demonstrate how the WCS models the general consensus in
Section 5. The search for counter examples is presented in Section 6. Counter Examples
are modeled in the WCS in Section 7. Finally, in Section 8 we conclude and outline
further possible research.
We assume the reader to be familiar with logic and logic programming as presented in
e.g. [
Throughout this paper, P will denote a program. An atom B is defined in P iff P contains a clause of the form B body . As an example consider the program Pc = {C
A ^ ¬ ab, ab
?}
where A, C, and ab are atoms. C and ab are defined, whereas A is undefined. ab is
an abnormality predicate which is assumed to be false. In the WCS, this program
represents the conditional sentence if A then C. In their everyday lives humans are often
required to reason in situations where the information of all factors affecting the
situation might not be complete. They still reason, unless new information which needs
consideration comes to light. The abnormality predicate in the program serves the
purpose of this (default) assumption, as was suggested in [
Consider the following transformation: (1) For all defined atoms B occurring in P,
replace all clauses of the form B body 1, B body 2, . . . by B body 1 _ body 2 _
. . . . (2) Replace all occurrences of by $ . The resulting set of equivalences is called
the weak completion of P. It differs from the program completion defined in [
As shown in [
J > = {B | there is B J ? = {B | there is B for all B body 2 P body 2 P body 2 P and I body = >}, and we find I body = ?} .
Following [
Given premises, general knowledge, and observations, reasoning in the WCS is
currently modeled in five steps:
1. Reasoning towards a logic program P following [
In Section 5 we will explain how these five steps work in the case of the DA reasoning
tasks considered in this paper. More examples can be found, for example, in [
Following [
Obligation conditionals may have different sources. They may be based on legal
laws like Example 1 and are often called deontic conditionals, in which case words
like must, should or ought may be explicitly used in the conditional sentence. Their
usage however, does not seem mandatory in everyday communication and is skipped
on many occasions. Knowledge or awareness that the consequent is obligatory given
the antecedent suffices in these cases, and yields the same responses as when explicitly
denoting the obligation. Obligation conditionals may also express moral or social
obligations like if somebody’s parents are elderly, then he/she should look after them [
On the other end of the spectrum, if the consequent of a conditional sentence is not obligatory given the antecedent, then it is called a factual conditional. In particular, the truth of the antecedent is inconsequential to that of the consequent; that is (even) if the antecedent is true, the consequent may or may not be true. This has already been exemplified using Example 2. The conditional if the plants get water, then they will grow is a factual one. As another example consider the conditional sentence if Maria is over 19 years, then she may drink alcoholic beverages in a pub. This sentence is a factual one, because given the atomic proposition Maria is over 19 years is true, one can imagine two permitted possibilities, one where Maria drinks alcohol beverages and another where Maria does not drink alcoholic beverages in a pub. 3.2
As discussed in the previous section, the obligation or factual nature of a conditional sentence indicates if the consequent is obligatory or simply possible provided the antecedent is satisfied. The question that may naturally arise at this point is, what happens when the antecedent of a conditional sentence is not satisfied? To that end, the antecedent A of a conditional sentence if A then C is said to be necessary with respect to the consequent C, if and only if C cannot be true unless A is true. This implies that if A does not hold, C cannot either. For example, in Example 2 plants get water is a necessary antecedent for plants will grow. If a plant is not watered at all, it will very likely die.
The above does not imply however, that the antecedent need always be a precondition for the consequent, per se. The antecedent A of a conditional sentence if A then C is said to be non-necessary with respect to the consequent C, if C can be true irrespective of the truth or falsity of A. In particular this implies, if A does not hold, C may or may not hold. In Example 1 the falsity of drinking alcoholic beverages in a pub is inconsequential to the truth of the consequent older than 19 years. There are plenty of adults (over 19 years) who do not drink alcohol. The antecedent of the conditional sentence if Maria is drinking alcoholic beverages in a pub, then Maria must be over 19 years of age, in Example 1 is therefore called non-necessary. 3.3
Generally, humans may recognize conditional sentences as obligation or factual and
antecedents as necessary or non-necessary. This leads to an informal and pragmatic
classification of four kinds: obligation conditional with necessary antecedent (ON) or
nonnecessary antecedent (ONN) and factual conditional with necessary antecedent (FN) or
non-necessary antecedent (FNN). For an abstract conditional if A then C, without an
everyday context, the classification of the conditional into any of the aforementioned
kinds would be as discussed in the above section. The classification of everyday
conditionals, however, often depend on pragmatics: the context, the background knowledge
and experience of a person. For example, the conditional sentence if it is cloudy, then it
is raining discussed in [
The classification of conditional sentences can be taken into account by extending the definition of the set of abducibles:
AeP = AP [ A nPn [ A fP , where AP is as defined above,
nn = {C AP
f AP = {ab > | C is the head of a rule occurring in P representing a
conditional sentence with non-necessary antecedent, > | ab occurs in the body of a rule occurring in P
representing a factual conditional}.
nn contains facts for the consequents of conditional sentences with nonThe set AP necessary antecedents. As mentioned earlier, if an antecedent of a conditional sentence is non-necessary, then the truth of the consequent does not depend on the truth of the antecedent. The abducible C > therefore implies that there may be other unknown reasons for establishing the consequent of the conditional sentence.
f contains facts for the abnormality predicates occurring in the bodies of The set AP the (logic program) representation of factual conditionals. Owing to the factual nature of a conditional sentence, the antecedent of the conditional may be true, however its consequent may not hold, due to various reasons which we might broadly call abnormalities. As mentioned earlier, considerations of other plausible factors at play might override our default assumption that these abnormalities are false. Once we weakly complete our program, the abducible ab > shall cause the abnormality predicate to become true and its negation to become false. Hence, the body of the clause containing its negation will be false, causing the consequent to be false in turn.5 Table 2 illustrates how the set of abducibles can be extended for each classification. 4
In [
In particular, a library not being open prevents a person from studying in it. The participants responded by clicking one of the answer options. They could take as much time as they needed. Participants acted as their own controls.
The participants carried out 48 problems consisting of the 12 conditionals listed in the Appendix and solved all four inference types (AA, DA, AC, DC). They could select one of three responses: nothing follows (nf), the fact that had not been presented in the second premise, and the negation of this fact. E.g., in the case of DA, the first assertion was of the form if A then C, the second assertion was ¬A, and they could answer C, ¬C, or nf . It should also be mentioned that the classification of the conditional sentences into the four aforementioned kinds was done by the authors of the experiment.
As an example consider the following short scenario from the experiment: Peter has a lawn in front of his house. He is keen to make sure that the grass on the lawn does not dry out, so whenever it has been dry for multiple days, he turns on the sprinkler to water the lawn. Along with this context the conditional sentence if it rains, then the lawn is wet and the negated atomic proposition it does not rain were provided. The participants were given three choices of answers: the lawn is wet, the lawn is not wet, and nothing follows.
As mentioned earlier, the WCS could well explain the findings of the experiment
in the cases AA, AC, and DC (see [
Everyday contexts for the DA inference task elicited a high response rate of about 78% (525 out of 672) for ¬C, but in case of nf the rate varied from 8% (14 out of 168) up to 33% (56 out of 168). The number of participants answering C seems irrelevant. Until the present, the WCS could predict the ¬C answered by the majority of the participants, but it could not yet model the significant number of nf responses. We now propose a solution to the latter. Before we elaborate further, one might first observe that as per our data nf was answered much more often in case of conditional sentences with non-necessary antecedents than in the case of conditional sentences with necessary ones (30% vs. 8%, Wilcoxon signed rank, W = 0, p < .001). More importantly, the reader may observe that when the classification of the antecedents changed from necessary to non-necessary the number of ¬C responses decreased to 225 and nf increased to (a significant) 101. The goal of this paper is to extend the WCS in order to model this observed phenomenon. 5
As shown in Table 3 the majority of the participants always answered ¬C when given the premises if A then C and ¬A no matter how the conditional sentence was classified. To illustrate how WCS models the majority consensus, let us consider Example 2 ((8) in Table 3). Assuming it is known that the plants do not get water we obtain the program P1 = {g w ^ ¬ ab1, ab1 ? , w ?} Obligation Conditional (O) Factual Conditional (F) Necessary Antecedent (N) Non-Necessary Antecedent (NN) 10 3% where g and w denote that the plants will grow and the plants get water, respectively, and ab1 is an abnormality predicate. Weakly completing P1 we obtain: {g $ w ^ ¬ ab1, ab1 $ ? , w $ ?} ,
MP1 = h; , {g, ab1, w}i, whose least model is where nothing is true, and g, ab1, and w are all false. As mentioned earlier, because water (the antecedent) is generally considered to be necessary for the growth of a plant (the consequent), the falsity of w allows us to falsify g. Hence, we conclude that the plants will not grow. 6
But the general consensus to answer ¬C when given the premises if A then C and ¬A is sometimes only barely met. Reconsider again Example 1 ((5) in Table 3): 28 out of 56 participants answered ¬C, whereas 25 participants answered nf. In general, the increase in the number of nf responses occurs when the classification of the antecedent of the conditional sentence changes from necessary to non-necessary. This is because unlike a necessary antecedent, a non-necessary one makes room for counter examples where even if the antecedent does not hold, the consequent might still hold. For example, if Maria is not drinking alcoholic beverages in a pub she may nevertheless be over 19 years of age. Maria may simply abstain from alcohol. One should observe that this cannot be modeled within the WCS so far, even if the set of abducibles is extended e , as in the case of DA no abductive reasoning takes place. from AP to AP
In this paper we propose to extend WCS by adding a sixth step to the procedure
presented in Section 2:
1. Reasoning towards a logic program P following [
The sixth step corresponds to the validation step in the mental model theory [
In order to illustrate how WCS along with its extension can explain the significant number of nf answers in case of the non-necessary antecedents, we return to Example 1 and assume that Maria is not drinking alcoholic beverages in a pub. In the WCS this is formalized by
P2 = {o where o and a denote that Maria is over 19 years old and she is drinking alcoholic beverages, respectively, and ab2 is an abnormality predicate which is initially assumed to be false. As the weak completion of P2 we obtain whose least model is {o $ a ^ ¬ ab2, ab2 $ ? , a $ ?} ,
MP2 = h; , {a, ab2, o}i.
Here, a, ab2, and o are all false. An average reasoner following this approach will draw the conclusion Maria is not over 19 years old and stop reasoning at this point. This accounts for the 28 ¬C responses for this conditional in our data.
Classifying an antecedent as non-necessary, however, would allow the consequent to be true or false despite the falsity of the former. In other words, recognizing an antecedent as non-necessary, might allow humans to consider two possibilities: Maria does not drink alcohol in a pub and she is younger than 19 years, and Maria does not drink alcohol in a pub but she is older than 19. Hence, for such a reasoner the extended WCS not only creates the previous model MP2 , where o is mapped to false, but also searches for a counter example by considering o as a possible observation that needs to be explained. As mentioned earlier in Section 3.4, because the conditional sentence is classified as an obligation conditional with non-necessary antecedent, the extended set of abducibles for P2 is
e AP2 = {a > , a ? , o >} .
The abducible {o >} can be used as a minimal explanation for the observation o. Hence, adding this abducible to P2 leads to
P3 = {o whose least model is {o $ (a ^ ¬ ab2) _ > , ab2 $ ? , a $ ?} ,
MP3 = h{o}, {a, ab2}i.
Here o is true, whereas a and ab2 are false. As o is false in the model MP2 but true in MP3 , reasoning skeptically WCS concludes nf . This accounts for the 25 nf responses for this conditional sentence in our data. Similar counter examples can be constructed for the examples (4), (6), and (10)-(12) depicted in Table 3 which explain the nf answers given by a significant number of participants. But similar counter examples cannot be constructed for the remaining examples (1)-(3) and (7)-(9). 8
In this paper, we have presented how the WCS along with its proposed extension can
adequately model the average human reasoner in case of the DA inference task. Whereas
the majority consensus of ¬C suggests reasoners who might not have considered the
necessity or non-necessity of the antecedent, the significant number of nf answers
suggests reasoners who might have. Accordingly, we have discussed how the classification
of conditional sentences and their antecedents help gain an insight into how humans
understand or recognize conditional sentences. This not only allows us to model the
DA reasoning task but also model the average human reasoner in case of the AA, AC,
and DC. In case of the AC (like in the DA) reasoners might recognize the antecedent
as non-necessary which influences their response. In case of the DC, it is possibly the
obligation or factual nature of the conditional sentence which is taken into consideration
(see [
The case for the AA seems to be a ceiling effect however, as an overwhelming majority of our responses were C (640 out of 672). The WCS can well model this majority which indicates that the conditional sentences were taken to be obligatory by most reasoners, meaning, when A was affirmed, they simply concluded C. Nonetheless, we also realize that in case of factual conditionals where although A holds, C may or may not, a reasoner might respond nothing follows. This, however, does not reflect significantly on the current data. Consider a seemingly strange yet everyday conditional sentence uttered by humans, viz. if I take an umbrella then it will not rain. Affirming the antecedent, I take an umbrella the conclusion it will not rain seems arguable. Given the factual nature of the conditional it seems plausible that skeptical reasoners may respond with nothing follows. WCS can also account for these reasoners. Such a discussion motivates further research about the AA and why humans accord with the response C so easily. Coming back to the DA, if we were to deny the antecedent on the other hand, this is, I do not take an umbrella, then although most reasoners might respond ¬C meaning, it will rain, once again, WCS with the extension proposed in this paper can account for the former as well as for reasoners who choose to respond with skepticism that nothing follows.
Acknowledgement
We thank Marcos Cramer for many fruitful discussions.
(1) If it rains, then the roofs must be wet. (2) If water in the cooking pot is heated over 99 C, then the water starts boiling. (3) If the wind is strong enough, then the sand is blowing over the dunes.
(4) If Paul rides a motorbike, then Paul must wear a helmet. (5) If Maria is drinking alcoholic beverages in a pub, then Maria must be over 19 years of age. (6) If it rains, then the lawn must be wet.
(7) If the library is open, then Sabrina is studying late in the library. (8) If the plants get water, then they will grow. (9) If my car’s start button is pushed, then the engine will start running.
(10) If Nancy rides her motorbike, then Nancy goes to the mountains. (11) If Lisa plays on the beach, then Lisa will get sunburned. (12) If Ron scores a goal, then Ron is happy.
The classification was done by the authors of [
Maria and her friends are visiting a local pub to enjoy the evening with drinks and good food. Maria knows the local rules and regulations and obeys them.
The Presleys have moved into their newly built house and have hired a gardener to lay out the garden. They are sitting on their terrace and are looking at the bushes, small trees, and shrubs which were planted by the gardener two months ago.