Counterpossibles in Science: The Case of Relative Computability

Abstract

I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable, which is true, and If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable, which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What's more, I argue that these counterpossibles don't just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models.

Williamson on Counterpossibles

“…26 We note that there are others. See Jenny [30] and Nolan [17]. (30) is actually false then so is vacuism, so long as it is necessary that 5 + 7 isn't 13.…”

“…Williamson [50] considers the following examples: 18 17 Williamson [48, p. 174] claims that "some objectors" (that is, nonvacuists) have pointed to reductio arguments in support of nonvacuism, but the only reference in the surroundings is to Nolan [30], which is explicit in rejecting any such support. Similarly, the primary target of Williamson [50] is Brogaard and Salerno [10], which does not discuss reductio arguments.…”

“…17 Reductio arguments are, of course, crucial to mathematics as it is practiced. Williamson attempts to show that nonvacuists about counterpossibles must hold current standard mathematical practice to be mistaken.…”

Impossible Worlds

“…Our support for non-vacuism largely rests on our ordinary-language judgements about the truth of a range of counterpossibles, such as the Hobbes-sentences (1.18) and (1.19), and the Anna-{Anna} sentences (12.1) and (12.2). (Jenny (2018) and Nolan (1997) offer further arguments.) Williamson (2007Williamson ( , 2017 worries about this kind of motivation.…”

Knowledge-first believing the unknowable

“…By contrast, Lewis (1973), Emery and Hill (2017), Vetter (2016), and Williamson (2007Williamson ( , 2016, among others, accept the assumption. See Jenny (2018) and Tan (2019) for further arguments in favor of rejecting the assumption. not know that Hobbes secretly squared the circle and so did not care.…”