The Static Pulse of A-Theory: Re-evaluating Temporality in Williamson's Modal Metaphysics — Epoche C1
Two ways of ordering events In a celebrated paper of 1908, J. M. E. McTaggart distinguished two ways of ordering events in time. The B-series orders them by the relations earlier than and later than : the trial of Socrates is earlier than the fall of Rome, and that relation never alters — it held a thousand years ago and will hold a thousand years hence. The A-series instead classifies events as past, present or future, and these classifications do alter: the trial of Socrates was once future, was then present, and is now ever more remotely past. The metaphysics of time has largely been a quarrel over which series is fundamental. A-theories hold that the A-series is real and basic: there is an objective, non-perspectival fact about which moment is present, and that fact changes — time genuinely flows. B-theories hold that only the B-series is fundamental, and that 'now' works like 'here': a word each speaker uses to mark her own position, not a feature that reality itself singles out. To this dispute a second is customarily attached. A-theorists usually combine the flow of time with a flowing ontology — 'ontology' here meaning simply the stock of what exists. Presentism, the most widely held A-theory, says that only present things exist: Socrates does not exist, he merely did; my grandchildren do not exist, though perhaps some will. The growing-block theory says that past and present things exist while future things do not, so that reality accumulates. On either view, existence itself changes from moment to moment. B-theorists, by contrast, are usually eternalists: past, present and future things all exist, tenselessly, in one four-dimensional block. Hence the received dichotomy that this essay's title questions: A-theory dynamic through and through, B-theory static through and through. A close reading of Timothy Williamson's Modal Logic as Metaphysics (2013) suggests the dichotomy is badly drawn. Williamson's machinery separates two things the tradition runs together: dynamism about which time is present , and dynamism about what exists . One can keep the first — the pulse — while the second turns out to be static. Modal logic taken as metaphysics Williamson's book is officially about modal logic, the logic of necessity and possibility. Its two operators are $\Box$, read 'necessarily' ($\Box p$: it could not have failed to be that $p$), and $\Diamond$, read 'possibly' ($\Diamond p$: it could have been that $p$). Saul Kripke (1963) gave this logic its now-standard semantics: a statement is possibly true if it is true at some possible world , necessarily true if it is true at every possible world, where a possible world is a complete way for things to be — complete in every detail and, crucially, across all of time. A world is not a snapshot; it is a total course of history. This detail, easily missed, is what the original version of this essay meant by saying that the actual world 'is not a fleeting moment' but the entire history of what has been and will be: the actual world is the one total history that obtains, and it is the same total history whether one asks at breakfast or at supper. The metaphysical stakes rise when quantifiers are added — $\forall$ ('everything') and $\exists$ ('something') — to form quantified modal logic. Here a design decision must be made: as the semantics moves from world to world, does the domain of quantification , the collection of things the quantifiers range over, vary, or does every world share one constant domain? In 1946 Ruth Barcan (later Barcan Marcus) formulated the schema now called the Barcan formula: $\Diamond \exists x\, Fx \rightarrow \exists x\, \Diamond Fx$ — if there could have been something that was $F$, then there is something that could have been $F$. It looks innocent; its consequences are not. Wittgenstein in fact had no children, but surely could have had. So: possibly, something is a child of Wittgenstein. The Barcan formula then delivers: there is something that could have been a child of Wittgenstein. But which actually existing thing could that be? No actual person, artefact or particle seems a candidate for that role. So one must either weaken the logic to block the inference, or admit into one's ontology something quite unfamiliar. The converse Barcan formula, $\Box \forall x\, Fx \rightarrow \forall x\, \Box Fx$, bites even harder. Substitute for $F$ the predicate 'is something', formally $\exists y\,(x = y)$. That everything is something is a truth of pure logic, hence necessary; the converse formula then yields that everything is necessarily something — nothing that exists could have failed to exist. Whether to accept all this is decided, in Williamson's hands, by the book's real methodological novelty: rival systems of quantified modal logic are to be assessed the way rival scientific theories are, abductively — by strength, simplicity, economy and fit with what we already know. On those criteria he finds the constant-domain logic, the one that validates both Barcan formulas, clearly superior to its 'free' rivals, which weaken the rules of inference clause by clause precisely to dodge the ontological consequences. If the strongest and simplest logic carries metaphysical commitments, Williamson argues, we should accept the commitments rather than mutilate the logic. Necessitism and its temporal twin The commitment in question is necessitism : necessarily, everything is necessarily something — $\Box \forall x\, \Box \exists y\,(x = y)$. Its denial, contingentism , is the common-sense view that some things exist only contingently. How could necessitism possibly be true of Wittgenstein's merely possible child? Williamson's answer is to prise apart two notions that ordinary talk fuses: being something — in the widest sense of being a value of the variables, available to be quantified over and referred to — and being concrete — having location, mass, causal powers, a life. The something that could have been Wittgenstein's child