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Compositional Signaling in a Complex World

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...a singing creature, only associating thoughts with tones.

von Humboldt (1836)

Abstract

Natural languages are compositional in that the meaning of complex expressions depends on those of the parts and how they are put together. Here, I ask the following question: why are languages compositional? I answer this question by extending Lewis–Skyrms signaling games with a rudimentary form of compositional signaling and exploring simple reinforcement learning therein. As it turns out: in complex worlds, having compositional signaling helps simple agents learn to communicate. I am also able to show that learning the meaning of a function word, once meanings of atomic words are known, presents no difficulty.

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Notes

  1. The earliest statement in the Western tradition appears to be in Frege (1923). Pagin and Westerståhl (2010a) find a similar statement in an Indian text dating from the fourth or fifth century.

  2. A related literature also focuses on making the statement of (C) more precise so as to avoid worries that it is trivial and/or vacuous. See, for an overview, §2.1 of Pagin and Westerståhl (2010b). In what follows, I will assume that (C) has been formulated in an appropriately more precise way.

  3. For learnability, see Davidson (1964). For systematicity and productivity, see Fodor and Pylyshyn (1988), Fodor (1987), and Fodor (1998). Some also take the principle to be a methodological one, guiding inquiry in semantics. See Szabó (2013), Pagin and Westerståhl (2010a, b), and Janssen (1997) for overviews of all of these proposals.

  4. See chapter 3, section 5 of Bergstrom and Dugatkin (2012) for a discussion of constraints on natural selection. Futuyma (2010) considers more constraints than just a bottleneck in variation.

  5. This corresponds to one of two types of informational content for a signal identified in chapter 3 of Skyrms (2010) and reflects the idea that a proposition is a set of possible worlds.

  6. In not having the utility function depend on the signal sent, we are assuming that no signal is more costly to send than any other.

  7. A matrix with only 1s and 0s, where each row has a single 1 and distinct rows have 1s in distinct columns.

  8. Here, \(M^T\) denotes the transpose of M, i.e. \(\left( M^T \right) _{ij} = M_{ji}\).

  9. See Roth and Erev (1995). Sutton and Barto (1998) provides a detailed introduction to reinforcement learning.

  10. That seminal paper has spanwed a large body of research. See Schultz (2004) and Glimcher (2011) for overviews.

  11. For instance, the ball-in-urn metaphor essentially assumes that the utility function only has integer values.

  12. See Skyrms (2010), p. 94.

  13. Similar results hold for the closely related replicator dynamics. See Huttegger (2007), Pawlowitsch (2008). Hofbauer and Huttegger (2008, 2015) study the replicator-mutator dynamics for these games.

  14. See Seyfarth et al. (1980).

  15. See Schlenker et al. (2014) for a detailed semantic analysis of this form of signaling. They in fact find that the basic alarm calls have roots with a morphological suffix.

  16. See Hassani (2003) for the definition and applications.

  17. To fully model classical logical negation, we would also need to require that f is an involution, i.e. that \(f\left( f\left( i \right) \right) = i\). But since our syntax is so impoverished that sending a “double negation” signal is not possible, we can omit this requirement. It is also doubtful that natural language negation satisfies double negation elimination. This remark about involutions does, however, explain why we have 2n states and acts: a permutation that is an involution will only have cycles of length \(\le \)2. And being a derangement requires that there are no cycles of length 1. Together, this means that \(\left[ n \right] \) only has derangements which are involutions if n is even.

  18. Or equivalently:

    $$\begin{aligned} \sigma \left( s_j \right) \left( \boxminus m_i \right)&\propto \sigma \left( f^{-1}\left( s_j \right) \right) \left( m_i \right) \\ \rho \left( \boxminus m_i \right) \left( a_j \right)&= \rho \left( m_i \right) \left( f^{-1}\left( a_j \right) \right) \end{aligned}$$

  19. For Barrett, this means \(\pi \left( \sigma , \rho \right) > 0.8\).

  20. Though it’s worth noting that \(\sigma \left( s_{AB} \right) \left( m_{AB} \right) \le 1/2\) so that \(m_{AB}\) never becomes strongly associated with \(s_{AB}\).

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Acknowledgments

I would like to thank the organizers and participants at IDAS’14 and especially Stefano Demichelis and Roland Muehlenbernd for stimulating discussion. An earlier version was presented at a workshop on Knowledge, Argumentation, and Games in Amsterdam. For discussion there, I thank especially Michael Franke and Alexandru Baltag. Jeffrey Barrett, Johan van Benthem, Thomas Icard, Chris Potts, Carlos Santana, Brian Skyrms, and Michael Weisberg have provided helpful comments on earlier versions of this paper as did three anonymous referees for this journal.

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  1. Department of Philosophy, Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA

    Shane Steinert-Threlkeld

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Steinert-Threlkeld, S. Compositional Signaling in a Complex World. J of Log Lang and Inf 25, 379–397 (2016). https://doi.org/10.1007/s10849-016-9236-9

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