On Specifying Truth-Conditions

Abstract

Consider a committalist-someone who believes that assertions of a sentence like 'the number of the planets is 8' carry commitment to numbers-and a noncommittalist-someone who believes that all it takes for assertions of this sentence to be correct is that there be eight planets. The committalist wants to know more about the noncommittalist's view. She understands what the noncommittalist thinks is required of the world in order for assertions of simple sentences like 'the number of the planets is 8' to be correct, but she wants to know how the proposal is supposed to work in general. Could the noncommittalist respond by supplying a recipe for translating each arithmetical sentence into a sentence that wears its ontological innocence on its sleave? Surprisingly, there is a precise and interesting sense in which the answer is 'no'. We will see that when certain constraints are in place, it is impossible to specify an adequate translation-method.Fortunately, there is a technique for specifying truth-conditions that is not based on translation, and can be used to explain to the committalist what the noncommittalist thinks is required of the world in order for arithmetical assertions to be correct. I call it 'the φ(w)-technique'. A shortcoming of the φ(w)-technique is that it is of limited dialectical effectiveness. From the perspective of a committalist, the φ(w)-technique only delivers suitable noncommittalist truth-conditions on the assumption that numbers exist. So the φ(w)-technique is not particularly promising as a maneuver for convincing a committalist to give up mathematical Platonism. But I hope it will earn its keep by providing the committalist with an illuminating account of the noncommittalist position, and by clarifying the role of noncommittalism in addressing certain philosophical puzzles.The φ(w)-technique turns out to have applications beyond arithmetic. It can be used to shed light on the truth-conditions of second-order and set-theoretic sentences. It can also * For their many helpful comments I am indebted to

Neo-Fregeanism Reconsidered

“…Throughout the definition of dentation and satisfaction I assume that the range of the metalinguistic variables includes merely possible objects. This is a device to simplify the exposition, and could be avoided by appeal to the technique described in Rayo (2008) and Rayo (typescript ).…”

Nominalism, Trivialism, Logicism

“…Without this assumption-or, alternatively, without the assumption of necesitism (Williamson 2013)-[NARROW]'s right-hand-side will be satisfied by worlds which represent reality as containing dinosaurs but don't represent of any actually existing individual that it is a dinosaur, and it therefore won't succeed in associating with '# x (Dinosaur(x)) = 0' the condition that there be no dinosaurs. Happily, there is a technical trick that allows one get the right results without surrendering modal actualism (or contingentism); see (Rayo 2008), (Rayo 2012) and (Rayo 2013, Chapter 6) for details.…”

Fundamental and Derivative Truths

“…13 Agustín Rayo (2008)has recently shown us how to present a theory of requirements in such a fashion. Following Rayo, let us use the notation [p] w to express that the world w is such that p. The idea now is to formulate the requirements for S to be true by way of a biconditional relating S's truth (at a world) to various conditions on w. 14 Given this notational convention, we are in a position to give a compositional Quinean theory of requirements for a syntactically simple language.…”

“…This is a technical device in the metalanguage that for most purposes can be ignored. Full details and a semantics for the metalanguage are presented in the appendices to Rayo (2008).…”

“…See for example Larson & Segal (1995). 21 See Rayo (2008). The reading of the bracketed formula just given corresponds to what he calls the 'secondary' condition expressed by bracketed formulae.…”