=Paper=
{{Paper
|id=Vol-2540/paper51
|storemode=property
|title=None
|pdfUrl=https://ceur-ws.org/Vol-2540/FAIR2019_paper_34.pdf
|volume=Vol-2540
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==None==
https://ceur-ws.org/Vol-2540/FAIR2019_paper_34.pdf
Cognitive defeasible reasoning:
the extent to which forms of defeasible reasoning
correspond with human reasoning
Clayton Baker1[0000−0002−3157−9989] , Claire Denny1[0000−0002−7999−8699] , Paul
Freund1[0000−0002−2826−6631] , and Thomas Meyer2[0000−0003−2204−6969]
1
University of Cape Town, South Africa
bkrcla003@myuct.ac.za
dnncla004@myuct.ac.za
frnpau013@myuct.ac.za
2
University of Cape Town, South Africa and CAIR
tmeyer@cs.uct.ac.za
Classical logic is the default for modelling human reasoning, but has been
found to be insufficient to do so. It lacks the flexibility so characteristically re-
quired of reasoning under uncertainty, incomplete information and new informa-
tion, 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 rea-
soning, 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withdrawn, or when a conclusion can be reinforced with additional information,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 conclu-
sion being drawn [6] [9]. For Defeasible Reasoning, we investigated the KLM
properties of Left Logical Equivalence, Right Weakening, And, Or and Cautious
Monotonicity. We find evidence for correspondence with the KLM property of
Or, which states that any formula that is, separately, a plausible consequence of
two different formulas, should also be a plausible consequence of their disjunc-
tion. We also investigate conformance with human reasoning and two subtypes
of Defeasible Reasoning: prototypical [7] and presumptive reasoning [11]. Proto-
typical reasoning is an approach that suggests each reasoning scenario assumes
a prototype with certain typical features, whereas presumptive reasoning sug-
gests that an argument may have multiple possible consequences. We find that
both subtypes of defeasible reasoning conform. In Belief Revision, conflicting
information indicates flawed 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 µ is a state of the world in which
µ is true [5]. For Belief Revision, we investigated the AGM properties of Clo-
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
2 Baker et al.
sure, Success, Inclusion, Vacuity, Consistency, Extensionality, Super-expansion
and Sub-expansion. We find evidence for correspondence with the AGM prop-
erty of Success, which expresses that the new information should always be part
of the new belief set. We also find 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 mo-
tivated by the principle of minimal change and suggests that if the incoming
sentence is not in the original set, then there is no effect. 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 find evidence that suggests,
overall, Defeasible Reasoning has a normative relationship, and Belief Revision a
descriptive relationship. A normative [3] [10] relationship suggests that humans
reason according to believed norms accepted in general by other human reason-
ers, whereas a descriptive [1] [4] relationship indicates that humans choose to
consider external sources of information as additional grounds on which to make
an inference. In Belief Update, conflicting information is seen as reflecting the
fact that the world has changed, without the agent being wrong about the past
state of the world. For Belief Update, we investigated the KM postulates U1,
U2, U3, U4, U5, U6, U7 and U8, as seen in Table 4 of the Appendix. We find
evidence for correspondence with postulate U1, which states that updating with
the new fact must ensure that the new fact is a consequence of the update. We
find evidence for correspondence with postulate U3, which states the reasonable
requirement that we cannot lapse into impossibility unless we either start with
it, or are directly confronted by it. We also find evidence for correspondence
with postulates U4 and U6. Postulate U4 asserts that syntax is irrelevant to the
results of an update. Postulate U6 states that if updating on α1 entails α2 and
if updating on α2 entails α1 , then the effect of updating on either is equivalent.
In the literature, the KM postulates for Belief Update have seen less acceptance
than the AGM postulates for Belief Revision. To this end, we discuss counter-
examples to the KM postulates, as tested in the experiment pertaining to it.
While the three forms of non-monotonic reasoning examined are meant to be
a better model of human reasoning than propositional logic, the results of this
project indicate that they are not yet a perfect fit, with participants failing to
reason in accordance with many of the properties of the systems. Future work
involving conducting a study with a larger participant pool is necessary to obtain
more accurate results. It may also be interesting to add blocks, in the form of
different control groups, to the study to explore the effects of different circum-
stances on cognitive reasoning and which logic form is most closely resembled
in each such block. Further avenues include a more direct comparison of survey
results.
Cognitive defeasible reasoning 3
References
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ical variations, and system approaches. AI communications 7(1), 39–59 (1994).
https://doi.org/10.3233/AIC-1994-7104
2. Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change:
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510–530 (1985). https://doi.org/10.2307/2274239
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reasoning pp. 132–145 (1994)
4. Besold, T.R., Uckelman, S.L.: Normative and descriptive rationality: from nature
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30(2), 331–344 (2018). https://doi.org/10.1080/0952813X.2018.1430860, https:
//doi.org/10.1080/0952813X.2018.1430860
5. Katsuno, H., Mendelzon, A.O.: On the difference between updating a knowledge
base and revising it. Belief Revision 29, 183 (2003)
6. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential mod-
els and cumulative logics. Artificial Intelligence 44, 167–207 (1990)
7. Lieto, A., Minieri, A., Piana, A., Radicioni, D.: A knowledge-based sys-
tem for prototypical reasoning. Connection Science 27(2), 137–152 (2015).
https://doi.org/10.1080/09540091.2014.956292
8. Martins, J., Shapiro, S.: A model for belief revision. Artif. Intell. 35, 25–79 (01
1988). https://doi.org/10.1016/0004-3702(88)90031-8
9. Pelletier, F., Elio, R.: The case for psychologism in default and inheritance reason-
ing. Synthese 146(2), 7–35 (2005)
10. Torre, L.V.D., Tan, Y.: Diagnosis and decision making in norma-
tive reasoning. Artificial Intelligence and Law 7(1), 51–67 (1999).
https://doi.org/10.1023/A:1008359312576
11. Verheij, B.: Correct grounded reasoning with presumptive arguments. Eu-
ropean Conference on Logics in Artificial Intelligence pp. 481–496 (2016).
https://doi.org/10.1007/978-3-319-48758-8 31
1 SUPPLEMENTARY INFORMATION
1.1 External resources
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 Defeasible Reasoning
KLM Properties Table 1 presents the KLM postulates. We use α |∼ γ to repre-
sent that a statement, α, defeasibly entails a statement, γ.
Reflexivity states that if a formula is satisfied, it follows that the formula can
4 Baker et al.
Table 1. KLM Postulates
1. Reflexivity K |≈ α |∼ α
K |≈ α |∼ γ, K 6|≈ α |∼ ¬β
2. Left Logical Equivalence
K |≈ α ∧ β |∼ γ
K |≈ α → β, γ |∼ α
3. Right Weakening
K |≈ γ |∼ β
K |≈ α |∼ β, K |≈ α |∼ γ
4. And
K |≈ α |∼ β ∧ γ
K |≈ α |∼ γ, K |≈ β |∼ γ
5. Or
K |≈ α ∨ β |∼ γ
K |≈ α |∼ β, K |≈ α |∼ γ
6. Cautious Monotonicity
K |≈ α ∧ β |∼ γ
be a consequence of itself. Left Logical Equivalence states that logically equiva-
lent 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 different
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 postu-
lates.
Table 2. Additional Postulates
K |≈ α ∧ β |∼ γ, K |≈ α |∼ β
1. Cut
K |≈ α |∼ γ
K |≈ α |∼ γ, K 6|≈ α |∼ ¬β
2. Rational Monotonicity
K |≈ α ∧ β |∼ γ
α |∼ β, β |∼ γ
3. Transitivity
α |∼ γ
α |∼ β
4. Contraposition
¬β |∼ ¬α
Cut expresses the fact that one may, in his way towards a plausible conclusion,
first 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 first and the third fact
is a plausible consequence of the second, then the third fact is also a plausible
Cognitive defeasible reasoning 5
consequence of the first fact. Contraposition allows the converse of the original
proposition to be inferred, by the negation of terms and changing their order.
1.3 Belief Revision
Properties Table 3 presents the AGM postulates. K ∗ α is the sentence rep-
resenting the knowledge base after revising the knowledge base K with α.
Table 3. AGM Postulates
1. Closure K ∗ α = Cn (K ∗ α)
2. Success K ∗ α |= α
3. Inclusion K ∗ α ⊆ Cn (K ∨ {α})
4. Vacuity If ¬α ∈
/ K then Cn (K ∨ {α}) ⊆ K ∗ α
5. Consistency K ∗ α = Cn (α ∧ ¬α) only if |= ¬α
6. Extensionality If α ≡ φ then K ∗ α = K ∗ φ
7. Super-expansion K ∗ (α ∧ φ) ⊆ Cn (K ∗ α ∨ {φ})
8. Sub-expansion If ¬φ ∈ / K then Cn (K ∗ α ∨ {φ}) ⊆ K ∗ (α ∧ φ)
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 in-
formation 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 α, consistent with belief set or knowledge base K,
belief revision involves performing expansion on K by α i.e. none of the origi-
nal 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, α, is inherently inconsistent - in which case, success overrules
consistency. Extensionality effectively expresses that the content i.e. the belief
represented, and not the syntax, affects the revision process, in that logically
equivalent sentences or beliefs will cause logically equivalent changes to the be-
lief set. Super-expansion and sub-expansion is motivated by the principle of
minimal change. Together, they express that for two propositions α and φ, if in
revising belief set K by α one obtains belief set K’ consistent with φ, then to
obtain the effect of revising K with α ∧ φ, simply perform expansion on K’ with
φ. In short, K ∗ (α ∧ φ) = (K ∗ α) + φ.
1.4 Belief Update
Properties Table 4 presents the KM postulates. φ α is the sentence represent-
ing the knowledge base after updating the knowledge base represented by φ with
α.
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 effect. U3 states the reasonable requirement
6 Baker et al.
Table 4. KM Postulates
(U1) φ α |= α
(U2) If φ |= α then φ α = φ
(U3) If both φ and α are satisfiable then φ α is satisfiable
(U4) If φ1 ↔ φ2 and α1 ↔ α2 then φ1 α1 ↔ φ2 α2
(U5) (φ α) ∧ γ |= φ (α ∧ γ)
(U6) If φ α1 |= α2 and φ α2 |= α1 then φ α1 ↔ φ α2
(U7) If φ is complete then (φ α1 ) ∧ (φ α2 ) |= φ (α1 ∨ α2 )
(U8) (φ1 ∨ φ2 ) α ↔ (φ1 α) ∨ (φ2 α)
that we cannot lapse into impossibility unless we either start with it, or are di-
rectly confronted by it. U4 requires that syntax is irrelevant to the results of an
update. U5 says that first updating on α then simply adding the new informa-
tion γ is at least as strong (i.e. entails) as updating on the conjunction of α and
γ. U6 states that if updating on α1 entails α2 and if updating on α2 entails α1 ,
then the effect of updating on either is equivalent. U7 applies only to complete
φ, that is φ which have only one model. If some situation arises from updating a
complete φ on α1 and it also results from updating that φ from α2 then it must
also arise from updating that φ on α1 ∨ α2 [5]. U8 is the disjunction rule.
Cognitive defeasible reasoning 7
1.5 Results
In Figure 1, we show the Hit Rate (%) for each Defeasible Reasoning Postu-
late. In Figure 2, we show the Hit Rate (%) for Prototypical Reasoning and
Presumptive Reasoning. In Figure 3, we show the Hit Rate (%) for each Belief
Revision Postulate. In Figure 4, we show the Hit Rate (%) for each Belief Update
Postulate.
Fig. 1. Hit Rate (%) for each Defeasi- Fig. 2. Hit Rate (%) for Prototypical
ble Reasoning Postulate Reasoning and Presumptive Reasoning
Fig. 3. Hit Rate (%) for Belief Revision Fig. 4. Hit Rate (%) for Belief Update
Postulates Postulates