An A-theory without tense operators

Abstract

A-theorists think there is a fundamental difference between the present and other times. This concern shows up in what kinds of properties they take to be instantiated, what objects they think exist and how they formalize their views. Nearly every contemporary A-theorist assumes that her metaphysics requires a tense logic – a logic with operators like (‘it was the case that ...’) and (‘it will be the case that ...’). In this paper, I show that there is at least one viable A-theory that does not require a logic with tense operators. And I will argue that three common indispensability arguments for tense operators are unsound.

Keywords:

• A-theory
• predicate modifiers
• eternalism
• permanentism
• tense logic

Acknowledgements

Thanks to Andy Egan, Ted Sider, Jeff Speaks, Dean Zimmerman, Tim Williamson and audiences at Rutgers University, University of Toronto, University of Virginia, University of Iowa, University of Arizona, the Metaphysical Virtues Conference at WMU, the Carolina Metaphysics Workshop, the Birmingham Time Workshop, and the Barcelona As and Bs of Time Workshop.

Notes

1 Examples of various kinds of A-theories include Broad (1923), Prior (1967), Adams (1989), Zimmerman (1998), Crisp (2003), Markosian (2004), Forrest (2006), and Merricks (2007).

2 Thomson (1983) and van Inwagen (1990) give early formulations of the relational B-theory, though van Inwagen does not treat times as physical regions of a manifold. For a more contemporary, relativistic version see Gibson and Pooley (2006).

3 Examples of temporal part theorists include Russell (1915), Quine (1950), Lewis (1986), Price (1996), and Sider (2001). Some, like Sider, are stage theorists – instead of using a primitive temporal parthood relation, they use a primitive counterpart relation. These differences in formulation do not matter for our purposes.

4 For the opposing, deflationist view, see (Hirsch , 2009). For a response to various deflationist views about change, see Chapter 11 of Sider (2011).

5 Throughout what follows, I will understand an object x’s lacking a property C just as it’s being false that x is C. x lacks C iff .

6 Prior (1967, 19).

7 Prior and Fine (1977, 32). Emphasis is his.

8 Zimmerman (2005). In particular, Zimmerman identifies Lewis and Mellor as ‘serious tenser B-theorists’. See Lewis (1983) and Mellor (1998).

9 Zimmerman offers something like this argument in response to Lewis’s charges that only a four-dimensionalist theory can consistently accommodate changing, non-time-relational properties. See Zimmerman (1998). In that paper, he is responding on behalf of presentist A-theories to the problem of temporary intrinsics developed in Lewis (1986, 202–204). For Lewis’s reply (which specifically targets presentism) see Lewis (2002, 2).

10 For more on the connection between A-theories and propositional temporalism, see Sullivan (2014).

11 See van Inwagen (1998) and the introduction to Sider (2001).

12 At least they agree on the language of quantification, and so are having a genuine debate about claims like ‘There are dinosaurs’ or ‘There are temporal parts’. The usual A-theorists will have more fundamental ideology than B-theorists, since they postulate primitive tense operators and the B-theorists do not. See Sider (2011) for considerations in favor of the substantivity of this debate over tense operators.

13 Some examples of presentist views are Prior (1998), Zimmerman (1998), Crisp (2003), Markosian (2004), and Merricks (2007).

14 Some examples of growing block views include Broad (1923), Adams (1989), Tooley (1997), and Forrest (2006). Note that many growing blockers are perdurantists about ordinary objects – they may think all ordinary objects only have B-properties. But they must hold that the growing edge of the block – the distinguished, present space–time region – changes in accord with A-property change, since it goes from being present to being past.

15 I treat A-property change as the distinctive commitment of the A-theories of time. But could there be a meaningful dispute between A and B-theorists even if there was no property change in the world? Suppose there is a world where bare particulars come into and out of existence, but no other change occurs. I’m inclined to say that neither the A-theory nor the B-theory is true in such a world, since nothing has the property of being located in space or time. Compare: we typically do not classify worlds with only abstract objects as either A or B-theory worlds. If the particulars do have locations, then one or the other of the property change principles will hold. Still, if you think worlds where only temporary existence holds count as A-theoretic worlds, this should not affect the arguments of this paper, since the indispensability arguments are arguments about characterizing actual change.

16 Compare this to moving spotlight theorists, who think the only form of A-property change is times gaining or losing the properties of being past, present or future. Or eternalist A-theorists who believe objects change with respect to properties of being spatiotemporally located or concrete, but do not gain or lose existence. Both of these versions of the A-theory deny ontological change, but they deny it in different ways. For more on eternalist A-theories, see Sullivan (2012a).

17 Zimmerman (1998, 211).

18 For discussion of this, see Chapter 11 of Sider (2011).

19 For a survey of some of the difficulties using tense operators to capture intuitive truth conditions, see Kamp (1968), Partee (1973), Enc (1986), Lewis (2004), Szabo (2007) and King (2007).

20 See Sullivan (2012b) and Chapter 2 of Burgess (2009) for discussion of the Barcan formulas in tense logic. See Williamson (2013) for a general defense of Barcan formulas in modal logic and tense logic. These problems vexed Prior throughout his career, and ultimately resulted in his awkward System Q tense logic; see Prior (1967).

21 For instance, one could adapt the logic of predicate modifiers outlined in Chapter 4 of Gamut (1991). The main change will be offering a functional application clause in the formal semantics for what I call ‘expanders’ below – functions that map well-formed formulae of arity n to formulae of arity n+1.

22 Much of the initial philosophical work on these kinds of predicate modifiers was done by Parsons and Clark in the 1970s. See Clark (1970), Parsons (1970), Clark (1975), and Clark (1974). I’ll roughly follow their notation here with some of my own adjustments. Parsons later developed his theory into a book: Parsons (1990). For a rigorous treatment of the logic of predicate modifiers using the theory of types, see Chapter 4 of Gamut (1991). I follow Gamut’s system in the formal appendix.

23 Which isn’t to say reducers aren’t philosophically interesting. For example, reducers would help with a program for eliminating constants, if we reinterpret constants as predicate reducers of the form . On this approach, we could define an atomic sentence as any complex predicate with arity 0. The proposal has a certain affinity with Quine’s program for eliminating variables with ‘de-relativizing’ operators in Quine (1960).

24 Reichenbach (1947) and Davidson (1980).

25 The main benefit of Option 1 may come in formalizing reasoning about changing properties of times themselves. Suppose we wanted to express ‘2008 was an election year’. With a simple past modifier, we might express this as . With expanders, we would have to relativize to a time. For example, we might express it as . But this construction looks strange – it expresses that 2008 was an election year at 2008. Some past and future properties of times themselves seem non-relational, and so might be better captured with simple tenses along the lines of Option 1.

26 For examples of A-theorists who offer theories for quantifying over times or past/future states, see Prior and Fine (1977), Chisholm (1978), Davidson (2003), Crisp (2007), Zimmerman (2011) and Meyer (2012). Parsons (1990) and Szabo (2007) give semantic arguments for quantifying over merely past states. It is also somewhat common to think of times as abstracta when giving a semantics for tense logic, akin to the way we might treat worlds as abstracta in modal logic.

27 I am not the first philosopher to focus on the role that tensed copulae might play in a theory of time. Johnston, for example, develops a theory where we understand tensed copulae as denoting an instantiation relation with three argument places. Sentences like ‘Obama was President’ are to be understood as expressing the same proposition as ‘Obama has-at-2008 the (non-relational) property is President’. Haslanger (1989) develops a similar theory. For a critical discussion of these proposals, see Lewis (2002). Lewis objects that their approach to tensed copulae over-emphasizes the instantiation relation, inviting Bradley’s regress.

28 See, for example, Prior (1996), Williamson (2002), Sullivan (2014) and Chapter 11 of Sider (2011).

29 See Cappelen and Lepore (2005).

30 Haslanger calls it the ‘proper subject’ objection and expresses some sympathy for it. See Haslanger (2003, 331–334).

31 Sider (2001, 215).

32 We could, for instance, treat the axioms of our favored ‘old’ tense logic as meaning postulates. For example, suppose you wanted to recover the old System K axiom in the meaning postulates of the new, operator-free theory. A temporalist of the kind discussed above could do this by quantifying over propositions: . In English, for any proposition , if is true then the proposition that there will be a time when will be true has always been true. The main difference is this isn’t an axiom of a proof theory anymore.

33 Prior (1971).

34 We might also choose to modify the identity relation, if we assume that the identity predicate can be used to express the sense of existence of interest to metaphysics.

35 Williamson (2013). Though note, Williamson is a primitivist about modal operators.

36 Note that Wiggins develops a theory of ‘necessarily’ as a predicate modifier and uses it to defend his essentialism from Quinean objections to quantifying into modal contexts. See Wiggins (1976). And in an appendix to the Wiggins paper, Peacocke develops a formal semantics for the necessity modifier. See Peacocke (1976). But Wiggins’ system is supposed to be added to a logic with modal operators; it is not intended as a replacement.