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Common Knowledge and the Knowledge Account of Assertion

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Structural Analysis of Non-Classical Logics

Part of the book series: Logic in Asia: Studia Logica Library ((LIAA))

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Abstract

In this chapter, I present the assertion account of common knowledge in the framework of a multi-agent system for the epistemic logic of knowledge and assertion: the propositional content of a formula \(\upvarphi \) is common knowledge to a group of agents G iff everyone in G knows that \(\upvarphi \) is true and that \(\upvarphi \) is asserted. Three current accounts of common knowledge, including the iterated account, the fixed-point account, and shared environment approach, will be examined. I argue that common knowledge arises from communication which results from overtly observable interactions among agents in a group. I then propose that assertion plays a substantial role in communication, and a fortiori, in the acquisition of common knowledge, given the knowledge account of assertion—one must assert \(\upvarphi \) only if one knows \(\upvarphi \). I point out some semantic implications of the knowledge account of assertion in multi-agent systems, specifically, the transmission of individual knowledge to others, the transition of individual knowledge to common knowledge, and the luminosity of common knowledge. The assertion account of common knowledge is then proposed and justified by a class of Kripke models (referred to as TWC-models) appropriate for a multi-agent system of epistemic logic of common knowledge and assertion. The construction of TWC-models will be specified, and the related semantic rules will be given.

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Notes

  1. 1.

    Sometimes, the notation ‘’ can be introduced as an abbreviation of ‘’; by convention, ‘’ is just ‘\(\upvarphi \)’.

  2. 2.

    Several logic systems of knowledge and common knowledge based on this equivalence have been proposed, e.g., Halpern and Moses [13], Mertens and Zamir [21], Fagin et al. [9]. In particular, Baltag et al. [2] construct an epistemic logic containing infinitary operators used in the standard modeling of common knowledge. It is worth mentioning that Lismont and Mongin ([20]: 129, footnote 1) briefly note that some logicians prefer to take certain infinitary logic as the required underlying system for a desired logic of common knowledge, such as Kaneko and Nagashima’s works in 1991 and 1993, and a paper of Heifetz in 1994.

  3. 3.

    It is noteworthy that the characterization of common knowledge based on S5-models would validate negative introspection—. However, as far as multi-human-agent systems of epistemic logic is concerned, it seems rather problematic to claim that to a group of agents G, that \(\upvarphi \) is not common knowledge is common knowledge, provided that \(\upvarphi \) is not common knowledge.

  4. 4.

    For the details of the construction of a logic system of knowledge (S5)\(_{C,}\) by taking ‘’ as primitive, and (FP) as an axiom schema, see Fagin et al. [9]; van Ditmarsch et al. [26].

  5. 5.

    In fact, Yang [29] presented a class of TW-models for an epistemic logic of knowledge and belief which satisfy the main theses of Timothy Williamson’s knowledge-first epistemology, proposed in his Knowledge and its Limits, which can be summarized in what follows:

    • Knowing is a state of mind

    • Knowing is factive

    • The broadness of knowing(Externalist approach)

    • The primeness of knowing (Knowledge first!)

    • Take knowledge as central to our understanding of belief.

    • Cognitive-homeless thesis

    • The knowledge account of assertion—Assert p only if one knows that p

    • The knowledge account of evidence—One’s knowledge is just one’s evidence.

    Note that TWA-models are essentially extensions of TW-models and can be used to justify the knowledge account of assertion. A justification of the knowledge account of evidence needs some other kind of models, which will be proposed somewhere else.

  6. 6.

    As Davidson ([8]: 90) rightly remarked, there are no such conventions governing the formation of intentions. So I can only put forth a primitive function here.

  7. 7.

    Davidson ([8]: 91) notes that ‘It is a mistake to suppose that if an agent is doing something intentionally, he must know that he is doing it.’ This indicates that may not hold. But it seems beyond reasonable doubt to claim that the agent must know that she knows what she asserts, otherwise, it would be difficult to show how she could do this intentionally.

  8. 8.

    I am indebted to an anonymous referee for reminding me of making this remark to show explicitly the implication of the normative character of the knowledge rule of assertion, and its impact on the acquisition of common knowledge. Bearing this in mind, misgivings over could be put aside.

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Acknowledgments

An earlier version of this chapter was presented at the Second Taiwan Philosophical Colloquium (TPLC-2014). I thank the participants and audience for valuable criticism and remarks. I am grateful to anonymous referees for their helpful comments and suggestions. This paper is one of a series of papers working on the construction of models to justify Williamson’s main theses in his knowledge-first epistemology. I am deeply indebted to Robert Goldblatt, Kok Yong Lee, Joshua Sack, and Tomoyuki Yamada for their discussion and comments on this paper and previous work. This paper is partly supported by a grant from the Ministry of Science and Technology (MOST), Taiwan, No.: MOST 103-2410-H-002-174-MY3

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  1. National Taiwan University, 1, Section 4, Roosevelt Road, Taipei, 10617, Taiwan

    Syraya Chin-Mu Yang

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  1. Department of Philosophy, National Taiwan University, Taipei, Taiwan

    Syraya Chin-Mu Yang

  2. Department of Philosophy, National Taiwan University, Taipei, Taiwan

    Duen-Min Deng

  3. Department of Philosophy, University of California, Davis, CA, Australia

    Hanti Lin

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Yang, S.CM. (2016). Common Knowledge and the Knowledge Account of Assertion. In: Yang, SM., Deng, DM., Lin, H. (eds) Structural Analysis of Non-Classical Logics. Logic in Asia: Studia Logica Library. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48357-2_12

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