I. Weak Completion Semantics 13
4. Contextual Reasoning 53
4.5. Contextual Side-effects and Consequences
In the following, we will formalize the notions of side-effects and consequences within contextual abduction and provide definitions and examples to clarify how contextual abduction enriches the expressiveness of abduction.
4.5.1 Contextual Side-effects
The following definition captures the idea of contextual side-effects:
Definition 4.2. Given a program P and a set of integrity constraints I C, let O1 and O2 be two observations and E1 be a contextual explanation for O1.
O2 is a necessary contextual side-effect of O1 given P and I C iff O2 cannot be contextually explained but O1∪O2 is contextually explained byE1.
O2 is a possible contextual side-effect of O1 given P and I C iff O2 cannot be contextually explained by E1 but O1∪O2 is contextually explained by E1.
The notion behind contextual side-effects is that every explanation E1 for O1 gives us an explanation for O2. Note that a necessary contextual side-effect is also a possible contextual side-effect.
Consider again the Tweety example in Section 4.4.1, where, given the programPf ly3 and the observation Ot = {¬can fly(tweety),featherslikeHair(tweety)}, the only contextual explanation forOt is Et ={featherslikeHair(tweety)← >}. Consider now the observa-tionO0={¬can fly(tweety)} ⊂Ot:
O0 cannot be contextually explained by Et, ascan fly(tweety) does not strongly depend on featherslikeHair(tweety). According to Definition 4.2, O0 is a necessary contextual side-effect of
Ot \O0 ={featherslikeHair(tweety)}.
On the other hand, consider again the Jerry example in Section 4.4.2, where, for the observation Oj = {can fly(jerry),inEurope(jerry)}, the only contextual explanation
4.5. Contextual Side-effects and Consequences is Ej = {inEurope(jerry) ← >}. As O0 = {can fly(jerry)} already follows from the empty explanation, E0 =∅,O0 cannot be considered a contextual side-effect of
Oj \O0 ={inEurope(jerry)}.
O00 = {¬penguin(jerry)} is a possible contextual side-effect of Oj, as O00 cannot be contextually explained by Ej. Note that O00 can also be contextually explained by
E00={blackAndWhite(jerry)← ⊥}.
4.5.2 Contestable Contextual Side-effects
Analogously to Definition 4.2, its counterpart, contestable contextual side-effects, is defined as follows:
Definition 4.3. Given a contextual program P and a set of integrity constraints I C. Let O1 and O2 be two observations and E1 be a contextual explanation for O1. The negation of the observation O2 is ¬O2, where ¬O2={¬L|L∈O2}.
O2 is a necessarily contested contextual side-effect of O1 given P and I C iff
¬O2 cannot be contextually explained by E1 but O1 ∪ ¬O2 can be contextually explained by E1.
O2 is apossibly contested contextual side-effect of O1 given P and I C iff¬O2 cannot be contextually explained by E1 butO1∪ ¬O2 can be contextually explained by E1.
Reconsider the examples of the previous subsection: It is easy to see that given O0¬ = {can fly(tweety)},O¬0 is a necessary contested contextual side-effect ofOt. Analogously, given that O00¬ = {penguin(jerry)}, O00¬ is a possible contested contextual side-effect of Oj.
4.5.3 Contextual Relevant Consequences
We define two notions of contextual relevant consequences as follows:
Definition 4.4. Given a contextual program P and a set of integrity constraints I C. Let O1 and O2 be two observations and E1 be a contextual explanation for O1.
O2 is anecessary contextual relevant consequence of O1 givenP andI C iffO2 cannot be contextually explained by E1 but O1∪O2 can be contextually explained by E2, where E1 ⊂E2.
O2 is a possible contextual relevant consequence of O1 given P and I C iff O2 cannot be contextually explained by E1 but O1∪O2 can be contextually explained by E2, where E1⊂E2.
Furthermore, it might be the case that two observations contain contextual relevant consequences of each other, simultaneously, i.e. they are mutually plausibly explainable together, but not each by itself. This notion is stronger than Definition 4.4:
Definition 4.5. Given a program P and a set of integrity constraints I C. Let O1, O2 be observations.
O1 and O2 are necessarily jointly supported contextual relevant consequences given P and I C iff O1 is a necessary contextual relevant consequence of O2 and O2 is a necessary contextual relevant consequence of O1.
O1 and O2 are possibly jointly supported contextual relevant consequences given P and I C iff O1 is a possible contextual relevant consequence of O2 and O2 is a possible contextual relevant consequence of O1.
Consider the contextual programPfire consisting of the following two clauses:
smoke ← fire∧ctxt(firefighters).
sirens ← ctxt(fire)∧firefighters. Let us observe
Osmoke = {smoke}.
We can abducefire ← > but not firefighters ← >, becausesmoke does not depend on firefighters. On the other hand, by observing
Osirens = {sirens},
we can abducefirefighters but notfire. However, if we observe both, smoke and sirens, fire can be abduced by Osmoke because smoke depends on fire and firefighters can be abduced byOsirens becausesirens depends onfirefighters. Accordingly, the explanation forOsmoke∪Osirens is
E = {firefighters ← >,fire ← >}.
OsmokeandOsir are necessarily jointly supported contextual relevant consequences given Pfire and I Cfire.
4.6. Conclusion 4.6 Conclusion
Motivated by the famous Tweety example, we first show that the Weak Completion Semantics does not yield the desired results. We would like to avoid having to abductively consider all exception cases and to automatically prefer normal explanations to those explanations specifying such exception cases. To do so, we set forth contextual programs, for the purpose of which we introduce ctxt, a new truth-functional operator, which turns out to fit quite well with the interpretation of negation as failure under three-valued semantics. Unfortunately, the ΦP operator is not monotonic with respect to these contextual programs anymore. Even worse, the ΦP operator might not even have a least fixed point for some contextual programs. Nevertheless, we can show that the ΦP operator does always have a least fixed point if we restrict contextual programs to the class of acyclic ones and introduce the concept of contextual abduction, and model the Tweety example as desired. In the last part of this chapter, we specify the relations between observations and explanations under contextual abduction, allowing us to define notions with regard to contextual effects, contestable contextual side-effects and contextual relevant consequences. The main advantage of the contextual reasoning approach here over the approach presented in (Pereira and Pinto 2011) is that the ctxt operator is part of the logic, whereas in (Pereira and Pinto 2011) in order to evaluate the inspection points, a meta-abduction transformation procedure is required.
Some open questions are left to be investigated in the future. For instance, can the requirements for the classes of acyclic contextual programs be relaxed to those that are only acyclic with respect to the truth functional operatorctxt, so that the ΦP operator is still guaranteed to yield a least fixed point? Furthermore, as the Weak Completion Semantics seems to adequately model human reasoning, a natural question to ask is whether the assumptions made for the development of contextual reasoning fit with the findings from Cognitive Science? For this purpose, we are particularly interested in psychological experiments that deal with context sensitive information.
Part II.
Human Reasoning Tasks
5. Byrne’s Suppression Task and Wason’s Selection Task
In this chapter, we first discuss the formalization of Byrne’s (1989) suppression task, which we have briefly explained in the introduction. The first part has originally been presented in (H¨olldobler and Kencana Ramli 2009b; H¨olldobler and Kencana Ramli 2009c) and the second part has originally been presented in (H¨olldobler, Phil-ipp and Wernhard 2011). The second part, Section 5.2, presents a formalization of Wason’s (1968) selection task and Griggs and Cox (1982) isomorphic representation of this task in a social context.1