Not ignorance, but ignorance of ignorance, is the death of knowledge.
Alfred North Whitehead
But there are also unknown unknowns. These are things we don’t know we don’t know.
I discuss the question of when knowledge of higher order ignorance is possible and show in particular that, under quite plausible assumptions, knowledge of second order ignorance is impossible.
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I am grateful to Eli Alshanetsky, Tom Donaldson, Tim Williamson, the participants in an Academia discussion session and two referees for helpful comments.
The critical observation that second order ignorance implies first order ignorance (Remark (1) below and Lemma 3 of the appendix) goes back to Montgomery and Routley (1966). The earlier literature also deals with some related issues. Montgomery and Routley (1968), Humberstone (1995) and Kuhn (1995) investigate the question of axiomatizing a modal logic with a contingency (or non-contingency) operator as primitive; Cresswell (1988) and Humberstone (1995) consider the question of when necessity can be defined in terms of contingency; and Montgomery and Routley (1966, 1969) and Mortensen (1976) study logics obtained by adding various special non-contingency axioms to a base logic of non-contingency. Hoek and Lomuscio (2004) is a more recent treatment of some of these questions, though apparently written in ignorance of the earlier literature.
I have talked in this section on ‘orders’ in an informal way. See Williamson (1999) for a more formal discussion of the notion.
See Brogaard and Salerno (2013) for a survey of recent work on the topic.
The system S4M, in which all continency is contingent, features prominently in Suzanne Bobzien’s work on vagueness, as in Bobzien (2010) for example.
Thanks to Martin Pleitz for pointing me in the direction of this observation.
Bobzien, S. (2010). Higher-order vagueness, radical unclarity and absolute agnosticism. Philosophers’ Imprint, 10(10), 1–30.
Brogaard, B., & Salerno, J. (2013) Fitch’s paradox of unknowability. In Stanford encyclopedia of philosophy. https://plato.stanford.edu/entries/fitch-paradox/.
Cresswell, M. J. (1988). Necessity and contingency. Studia Logica, 47, 145–149.
Fine, K. (2008). The impossibility of vagueness. Philosophical Perspectives, 22(1), 111–136.
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Humberstone, I. L. (1995). The logic of non-contingency. Notre Dame Journal of Formal Logic, 36(2), 214–229.
Kuhn, T. S. (1995). Minimal non-contingency logics. Notre Dame Journal of Formal Logic, 36, 230–234.
Montgomery, H., & Routley, R. (1966). Contingency and non-contingency bases for normal modal logics. Logique et Analyse, 9, 318–328.
Montgomery, H., & Routley, R. (1968). Non-contingency axioms for S4 and S5. Logique et Analyse, 11, 422–424.
Montgomery, H., & Routley, R. (1969). Modalities in a sequence of normal non-contingency modal systems. Logique et Analyse, 12, 225–227.
Mortensen, C. (1976). A sequence of normal modal systems with non-contingency bases. Logique et Analyse, 19, 341–344.
Van der Hoek, W., & Lomuscio, A. (2004). A logic for ignorance. Electronic Notes in Computer Science, 85(2), 1–17.
Williamson, Timothy. (1999). On the structure of higher-order vagueness. Mind, 108, 127–142.