TLDR: A new research paper by Jie Fan addresses a critical issue in the logic of ignorance: Rumsfeld ignorance was previously trivially definable by first-order ignorance. By introducing distinct accessibility relations for implicit knowledge within these concepts, the paper establishes a “bi-semantics” that resolves this definability, allowing for a more robust axiomatization and analysis of complex forms of ignorance, while retaining most key logical properties from previous work.
In the realm of philosophy and logic, the concept of ignorance has long been a subject of profound inquiry, dating back to ancient thinkers like Socrates. While traditionally viewed as simply the absence of knowledge, contemporary discussions have expanded to include more nuanced perspectives, such as the “new view” (absence of true belief) and the “logical view” (absence of knowledge of a proposition and its negation).
Beyond these foundational definitions, various forms of ignorance have been identified, including pluralistic, circumscriptive, chronological, and group ignorance. A particularly intriguing form, known as Rumsfeld ignorance, gained prominence from Donald Rumsfeld’s famous “unknown unknowns” remark. This concept refers to being ignorant of the fact that one is ignorant whether a proposition holds.
Recent work by Kit Fine explored the logical properties of different types of ignorance, including first-order, second-order, and Rumsfeld ignorance. Fine’s findings revealed interesting relationships, such as second-order ignorance implying first-order and Rumsfeld ignorance, and the surprising result that one cannot know they are Rumsfeld ignorant or second-order ignorant.
However, a critical issue emerged: under the standard logical framework, Rumsfeld ignorance was found to be definable in terms of first-order ignorance. This definability made some of Fine’s results and the problem of axiomatizing Rumsfeld ignorance somewhat trivial. The core reason for this definability was that the implicit “knowledge” operator within the definitions of both first-order and Rumsfeld ignorance shared the same underlying accessibility relation in the logical models.
To address this, a new logical framework, termed “bi-semantics,” has been proposed. This innovative approach introduces two distinct accessibility relations, R and R‚, for the implicit knowledge operators associated with first-order and Rumsfeld ignorance, respectively. By assuming that one relation is a subset of the other (either R Ď R‚ or R‚ Ď R), the definability issue is resolved. This means Rumsfeld ignorance can no longer be trivially defined by first-order ignorance, restoring the complexity and significance of its logical properties.
The paper focuses on the restriction where R is a subset of R‚, which aligns with the intuition that “knowing whether” (related to first-order ignorance) is a weaker notion than “knowing that” (related to Rumsfeld ignorance’s implicit knowledge). Within this new framework, the paper develops a minimal logic for ignorance and Rumsfeld ignorance, proving its soundness and completeness. It further extends this logic to various classes of frames, such as serial and reflexive frames.
Applying this refined framework to Kit Fine’s original results, the research demonstrates that while some specific implications might change, most of Fine’s key validities, such as second-order ignorance implying Rumsfeld ignorance, and the inability to know one is Rumsfeld ignorant or second-order ignorant, are retained. Crucially, the new framework successfully avoids the definability and triviality issues, providing a more robust and nuanced understanding of these complex forms of ignorance.
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This work significantly advances the logical study of ignorance, offering a more precise and non-trivial foundation for exploring its various forms and their interrelationships. For more in-depth details, you can refer to the full research paper: Axiomatizing Rumsfeld Ignorance.
