Classical logic is the default for modelling human reasoning, but has been found to be insu cient to do so. It lacks the exibility so characteristically required of reasoning under uncertainty, incomplete information and new information, as people must. In response, non-classical extensions to propositional logic have been formulated, to provide non-monotonicity. Non-monotonic reasoning refers to making an inference which is not absolute: in light of new information, the inference could change. We focus on three extensions of non-monotonic reasoning, KLM Defeasible Reasoning [6], AGM Belief Revision [2] and KM Belief Update [5]. We have investigated, via surveys, the extent to which each of KLM Defeasible Reasoning, AGM Belief Revision and KM Belief Update correspond with human reasoning. In philosophy, when a conclusion has the potential to be withdrawn, or when a conclusion can be reinforced with additional information, the conclusion is said to be defeasible. Defeasible Reasoning occurs when the evidence available to the reasoner does not guarantee the truth of the conclusion being drawn [6] [9]. For Defeasible Reasoning, we investigated the KLM properties of Left Logical Equivalence, Right Weakening, And, Or and Cautious Monotonicity. We nd evidence for correspondence with the KLM property of Or, which states that any formula that is, separately, a plausible consequence of two di erent formulas, should also be a plausible consequence of their disjunction. We also investigate conformance with human reasoning and two subtypes of Defeasible Reasoning: prototypical [7] and presumptive reasoning [11]. Prototypical reasoning is an approach that suggests each reasoning scenario assumes a prototype with certain typical features, whereas presumptive reasoning suggests that an argument may have multiple possible consequences. We nd that both subtypes of defeasible reasoning conform. In Belief Revision, con icting information indicates awed prior knowledge on the part of the agent, forcing the retraction of conclusions drawn from it [5,8]. Information is then taken into account by selecting the models of the new information closest to the models of the base, where a model of information is a state of the world in which is true [5]. For Belief Revision, we investigated the AGM properties of Clo-
sure, Success, Inclusion, Vacuity, Consistency, Extensionality, Super-expansion
and Sub-expansion. We nd evidence for correspondence with the AGM
property of Success, which expresses that the new information should always be part
of the new belief set. We also nd evidence for correspondence with the AGM
properties of Closure and Vacuity. Closure implies logical omniscience on the
part of the ideal reasoner, including revision of their belief set. Vacuity is
motivated by the principle of minimal change and suggests that if the incoming
sentence is not in the original set, then there is no e ect. Literature suggests a
formal link between Defeasible Reasoning and Belief Revision. We take a step
towards investigating whether this formal link translates to an empirical link.
Thus, in the cases of Defeasible Reasoning and Belief Revision, we discuss the
relationship they have with human reasoning. We nd evidence that suggests,
overall, Defeasible Reasoning has a normative relationship, and Belief Revision a
descriptive relationship. A normative [
SUPPLEMENTARY INFORMATION
We have created a GitHub repository which contains additional resources. In this repository, we include our survey questions, the coding of the survey responses as well as our complete project paper. The GitHub repository can be accessed by clicking here. In addition, a summary of our project work is also showcased on our project website which can be viewed by clicking here. 1.2
KLM Properties Table 1 presents the KLM postulates. We use j to represent that a statement, , defeasibly entails a statement, . Re exivity states that if a formula is satis ed, it follows that the formula can 3. Right Weakening 4. And 5. Or 6. Cautious Monotonicity
K j K j K j K j
K j
j K j
j K j
j K j j j ; K 6j K j ^ j ! ; j j ; K j j ; K j _ j ; K j ^ j ^ j j j j : be a consequence of itself. Left Logical Equivalence states that logically equivalent formulas have the same consequences. Right Weakening expresses the fact that one should accept as plausible consequences all that is logically implied by what one thinks are plausible consequences. And expresses the fact that the conjunction of two plausible consequences is a plausible consequence. Or says that any formula that is, separately, a plausible consequence of two di erent formulas, should also be a plausible consequence of their disjunction. Cautious Monotonicity expresses the fact that learning a new fact, the truth of which could have been plausibly concluded, should not invalidate previous conclusions. Additional Properties Table 2 presents additional defeasible reasoning postulates. 2. Rational Monotonicity K j 3. Transitivity 4. Contraposition
K j j
j : j :
j K j ; j ^ j ; K j
K j j ; K 6j ^ j j
j j : Cut expresses the fact that one may, in his way towards a plausible conclusion, rst add an hypothesis to the facts he knows to be true and prove the plausibility of his conclusion from this enlarged set of facts and then deduce (plausibly) this added hypothesis from the facts. Rational Monotonicity expresses the fact that only additional information, the negation of which was expected, should force us to withdraw plausible conclusions previously drawn. Transitivity expresses that if the second fact is a plausible consequence of the rst and the third fact is a plausible consequence of the second, then the third fact is also a plausible consequence of the rst fact. Contraposition allows the converse of the original proposition to be inferred, by the negation of terms and changing their order. 1.3
Properties Table 3 presents the AGM postulates. K is the sentence representing the knowledge base after revising the knowledge base K with . 1. Closure K = Cn(K ) 2. Success K j= 3. Inclusion K Cn(K _ f g) 4. Vacuity If : 2= K then Cn(K _ f g) K 5. Consistency K = Cn( ^ : ) only if j= : 6. Extensionality If then K = K 7. Super-expansion K ( ^ ) Cn(K _ f g) 8. Sub-expansion If : 2= K then Cn(K _ f g) K ( ^ ) Closure implies logical omniscience on the part of the ideal agent or reasoner, including after revision of their belief set. Success expresses that the new information should always be part of the new belief set. Inclusion and Vacuity is motivated by the principle of minimum change. Together, they express that in the case of information , consistent with belief set or knowledge base K, belief revision involves performing expansion on K by i.e. none of the original beliefs need to be withdrawn. Consistency expresses that the agent should prioritise consistency, where the only acceptable case of not doing so is if the new information, , is inherently inconsistent - in which case, success overrules consistency. Extensionality e ectively expresses that the content i.e. the belief represented, and not the syntax, a ects the revision process, in that logically equivalent sentences or beliefs will cause logically equivalent changes to the belief set. Super-expansion and sub-expansion is motivated by the principle of minimal change. Together, they express that for two propositions and , if in revising belief set K by one obtains belief set K' consistent with , then to obtain the e ect of revising K with ^ , simply perform expansion on K' with . In short, K ( ^ ) = (K ) + . 1.4
Properties Table 4 presents the KM postulates. is the sentence representing the knowledge base after updating the knowledge base represented by with .
U1 states that updating with the new fact must ensure that the new fact is
a consequence of the update. U2 states that updating on a fact that could in
principle be already known has no e ect. U3 states the reasonable requirement
that we cannot lapse into impossibility unless we either start with it, or are
directly confronted by it. U4 requires that syntax is irrelevant to the results of an
update. U5 says that rst updating on then simply adding the new
information is at least as strong (i.e. entails) as updating on the conjunction of and
. U6 states that if updating on 1 entails 2 and if updating on 2 entails 1,
then the e ect of updating on either is equivalent. U7 applies only to complete
, that is which have only one model. If some situation arises from updating a
complete on 1 and it also results from updating that from 2 then it must
also arise from updating that on 1 _ 2 [