Abstract
We first give a coarse-grained modal-logical analysis of the four best known second-order false-belief tasks. This preliminary analysis shows that the four tasks share a common logical structure in which a crucial role is played by a “principle of inertia” which says that an agent’s belief is preserved over time unless the agent gets information to the contrary. It also reveals informational symmetries (all four possibilities inherent in the two dimensions of deception versus no-deception and change-in-world versus change-in-belief-only are realized) and reveals a rather puzzling feature common to all four tasks. We then take a closer look at how the principle of inertia is used, which leads to a fine-grained analysis in terms of perspective shifting. We formalize this analysis using a natural deduction system for hybrid logic, and show that the proof modelling the solution to the first-order Sally-Anne task is nested inside the proof modelling the second-order solution.
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Notes
- 1.
The three other task we consider are the bake-sale task, the ice-cream task, and the puppy task; see Tables 4, 5 and 6 in the Appendix. The ice-cream task was the very first second-order false-belief task to be used; it was introduced in 1985 by Wimmer and Perner in [14]. The bake-sale task is a variant of the ice-cream task, and, as is explained in [11], pages 323–324: “The stories were modeled after Wimmer and Perner’s (1985) “ice cream truck story”. In contrast to their stories, we made sure that the beliefs of the two main protagonists in the story did not overlap, both at first-order and second-order level: each protagonist had his or her own distinct belief which was different from that of the other protagonist, as well as from the belief of the participants.” The puppy task was introduced in 1994 in [19], again as a simplification of Wimmer and Perner’s ice-cream task.
- 2.
Some stories use more times than this: the bake-sale story, for example, makes use of (at least) four. But the sequence \(t_0\), \(t_1\), and \(t_2\) constitutes the narrated time of the story, and here it is pointless to distinguish the times when Sam and Maria learn that there are no chocolate cookies for sale.
- 3.
In this section we adopt the following convention: belief-states that are part of this common pattern are typeset in bold (and displayed in blue in the online version), other belief-states are typeset in normal font.
- 4.
Note that rows 4,9,14,19 in Table 3 have the same form
, 
and are part of the common reasoning pattern leading to the correct answer, hence they are typeset in bold (and blue in the online version). Similarly, rows 2,7,12,17 have the same form 
, 
, 
and are part of the common reasoning pattern, so they are also bold (and blue). That is, the information in these rows is part of the experimental design, and is intended to ensure that agent x ends up having a false belief about the belief of agent y. - 5.
The distinction between tasks that do and do not involve deception is considered important for first-order false beliefs, as deception in a story may signal the relevance of detecting falsehood. But it has been little discussed for second-order tasks; see [13], especially pages 48–49, for discussion and pointers to the literature.
- 6.
Which is why this information is not typeset in bold (and why in the online version, it is not in blue), and also why in Table 2 (the coarse-grained reasoning analysis) it has been put in parentheses.
- 7.
There are some interesting possibilities here: we could make our formalization more fine-grained by taking some nominals to stand for times, or go two-dimensional by taking nominals to stand for person-time pairs. But here we stick with the simpler setup just defined, as it has the same granularity as Stenning and Van Lambalgen’s work on first-order false-belief tasks, cf. [18], pages 251–259.
- 8.
Natural deduction was originally developed to model mathematical argumentation, but there is now some experimental backing for the claim that it is a mechanism underlying human deductive reasoning more generally; see [16]. One of the reasons we chose hybrid logic for our analysis (rather than, say, a multi-agent doxastic logic) was because of its well-behaved natural deduction systems; see [5].
- 9.
Incidentally, when using the Term rule we make at least one assumption c, but we can make several, and this is often necessary to drive the proof through.
- 10.
The Name rule tells us that if we can prove the information \(\phi \) by adopting some arbitrary perspective c, then \(\phi \) also holds from the original perspective. As we won’t use this rule in our analysis, we refer to [5] for further discussion.
- 11.
Indicated by the premisses \(\phi _{1} \ldots \phi _{n}\) listed just above the horizontal line in the statement of Term given in Fig. 1.
- 12.
- 13.
As we mentioned earlier, “belief formation” (and “belief manipulation”) is terminology we have borrowed from [18], and we discuss them in more detail shortly. As for the belief formations principles themselves, we have already met Principle \((\mathsf{D})\) which says if we believe that something is false, then we don’t believe it. Principle (P1) states that a belief in \(\phi \) may be formed as a result of seeing \(\phi \); this is principle (9.2) in [18], page 251. Principle (P2) is (pretty clearly) a principle of inertia: a belief that the predicate l is true is preserved from a time t to its successor \(t+1\), unless it is believed that the marble moved at t. This is essentially Principle (9.11) from [18], page 253, and axiom \([A_5]\) in [1], page 20. Principle (P3) encodes the information that seeing the marble being moved is the only way a belief that the marble is being moved can be acquired. Obviously this is not a general truth, but the point of the formalization is simply to capture Peter’s reasoning in the Sally-Anne scenario.
- 14.
As we remarked earlier, we do this to ‘compile down’ the simple propositional reasoning involved. Strictly speaking, deducing \(B\phi \) from \(S\phi \) requires us to apply the propositional rule of modus ponens to \(S\phi \rightarrow B\phi \). Using the belief formation principles as additional natural deduction rules enables us to omit such steps and reduce the size of the proof tree.
- 15.
Stenning and Van Lambalgen do not analyse second-order false-belief tasks.
- 16.
So we are adding natural deduction machinery for the minimal modal logic K and thus treating B as a full-fledged modal operator. In this paper we won’t discuss the model-theoretic changes required — but we do believe that the fact that a semantic enrichment is called for at this point adds weight to our argument that the transition from first- to second-order reasoning involves conceptual change.
- 17.
Indeed, our analysis allows us to tentatively indicate the shift in complexity. The LCD fragment is np-complete. By adding BM we have moved to a pspace-hard modal logic. So our analysis of the first-order Sally-Anne task is carried out in computationally simpler logic than the second-order case (assuming p \(\not =\) np).
- 18.
Our formalization does suggest a hypothesis which may be empirically testable. Although we have talked of acquiring second-order competency, to acquire (something like) the BM rule is to acquire a fully recursive competency. That is, once the child has acquired BM, there should be nothing more to learn, for the rule covers the third, fourth, fifth, ..., and all higher-order levels. That is, we suspect that false-belief competency comes in two stages for typically developing children: first-order competency (at around the age of four) and all the rest (at around the age of six). But designing an experiment to test this is likely to be difficult. Apart from anything else, higher levels of reasoning impose heavy cognitive loads very fast, and it is unclear how such performance effects could be disentangled experimentally.
, 
and are part of the common reasoning pattern leading to the correct answer, hence they are typeset in bold (and blue in the online version). Similarly, rows 2,7,12,17 have the same form 
, 
, 
and are part of the common reasoning pattern, so they are also bold (and blue). That is, the information in these rows is part of the experimental design, and is intended to ensure that agent x ends up having a false belief about the belief of agent y.References
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Acknowledgements
We thank the referees for their valuable comments and questions. The authors acknowledge the funding received from the VELUX FOUNDATION for the project Hybrid-Logical Proofs at Work in Cognitive Psychology (VELUX 33305).
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Appendix
Appendix
The Appendix contains the coarse-grained reasoning for the four tasks (Table 2), the table listing their information content (Table 3), the texts of the bake-sale, ice-cream and puppy tasks (Tables 4, 5 and 6 respectively) and the formalization of the first-order and second-order Sally-Anne tasks (Figs. 3 and 4).
Formalization of the child’s correct response in the first-order Sally-Anne task
Formalization of the child’s correct response in the second-order Sally-Anne task
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Braüner, T., Blackburn, P., Polyanskaya, I. (2016). Second-Order False-Belief Tasks: Analysis and Formalization. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_9
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