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 In philosophy, when a conclusion has the potential to be withdrawn, or when a conclusion can be reinforced with additional information, the conclusion is said to be defeasible.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 1. Aamodt, A., Plaza, E.: Case-based reasoning: Foundational issues, methodolog- 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: Partial meet contraction and revision functions. Journal of Symbolic Logic 50, 510–530 (1985). https://doi.org/10.2307/2274239 3. Artosi, A., Cattabriga, P., Governatori, G.: An automated approach to normative reasoning pp. 132–145 (1994) 4. Besold, T.R., Uckelman, S.L.: Normative and descriptive rationality: from nature to artifice and back. Journal of Experimental & Theoretical Artificial Intelligence 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