The Hardest Paradox for Closure
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The Hardest Paradox for Closure Martin Smith1 Received: 10 September 2019 / Accepted: 20 June 2020 © The Author(s) 2020
Abstract According to the principle of Conjunction Closure, if one has justification for believing each of a set of propositions, one has justification for believing their conjunction. The lottery and preface paradoxes can both be seen as posing challenges for Closure, but leave open familiar strategies for preserving the principle. While this is all relatively well-trodden ground, a new Closure-challenging paradox has recently emerged, in two somewhat different forms, due to Backes (Synthese 196(9):3773– 3787, 2019a) and Praolini (Australas J Philos 97(4):715–726, 2019). This paradox synthesises elements of the lottery and the preface and is designed to close off the familiar Closure-preserving strategies. By appealing to a normic theory of justification, I will defend Closure in the face of this new paradox. Along the way I will draw more general conclusions about justification, normalcy and defeat, which bear upon what Backes (Philos Stud 176(11):2877–2895, 2019b) has dubbed the ‘easy defeat’ problem for the normic theory.
1 Background: The Lottery and Preface Paradoxes Consider the following principle: If one has justification for believing each of P1 , P2 , … , Pn , then one has justification for believing P1 ∧ P2 ∧ … ∧ Pn. According to this principle, the set of propositions that one has justification for believing is closed under the operation of taking conjunctions—we might call it Conjunction Closure or, simply, Closure. In one way, the principle seems difficult to deny. Suppose I endeavour to believe all and only those propositions for which I have justification. If Closure fails, there will be possible situations in which I ought to believe each of a series of propositions while refraining from believing their conjunction. And yet, it’s unclear how I would actually go about following such a recommendation, even if I accepted it. It’s unclear that there is any * Martin Smith [email protected] 1
University of Edinburgh, Edinburgh, UK
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psychological difference between believing P 1, believing P 2, … believing P n and believing P1 ∧ P2 ∧ … ∧ Pn (see Evnine 1999, section 7; Douven 2002, p. 395). If a person has expressed belief in P 1, and in P 2, … and in P n we would, without a second thought, describe the person as believing P1 ∧ P2 ∧ … ∧ Pn. Suppose I’m filling in a large truth table. If I put a ‘T’ in the P1 column and a ‘T’ in the P2 column … and a ‘T’ in the P n column, you would describe me as believing P1 ∧ P2 ∧ … ∧ Pn . Even if I hesitated when it came to the final P1 ∧ P2 ∧ … ∧ Pn column, this wouldn’t obviously make any difference—it would merely look as though I don’t know how to complete a truth table for ‘ ∧ ’. One might claim that it is, at least, possible to assert each of a series of propositions without asserting their conjunction. But even this is unclear. If I assert, in sequence, P1 , P2 , … , Pn it would be n
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