Branching time (BT) is a multipurpose label, which is mainlyused to denote (i) a family of structures (BT representations or BTframes), possibly along with the axiomatic theories defining them,(ii) a family of semantics for temporal and modal logics (BTsemantics); and (iii) a metaphysical conception concerning ouruniverse and its temporal and modal features (branching conception oftime or BT conception).

In very general terms, aBT representation is a complex ofhistories (orchronicles, orpossibleworlds) andmoments (ornodes), which purportsto represent all possible temporal developments of a given system.Intuitively, histories stand for temporally complete (possible)developments of the system, and moments, for their minimal temporal“slices”. Unlike non-branching modal structures, BTrepresentations allow different histories tooverlap orcoincide over a certain stretch of time and thendivide orbranch.^[1]BT frames are mathematical structures that purport toprovide BT representations, i.e., they are formally defined BTrepresentations. In this entry, we shall primarily focus onstandard BT frames ortrees (seeFig. 1).

Figure 1: A tree, where\(h_0\)–\(h_3\) are histories and \(m_0\)–\(m_6\) aremoments. [Anextended description of figure 1 is in the supplement.]

Standard BT frames are formal tools that can be applied to a varietyof different purposes, but two of their applications play a centralrole in philosophy and deserve to be calledcanonical. Theyare also the applications in view of which the frames were initiallyintroduced and developed, by Saul Kripke and Arthur Prior.

The first canonical application is in defining a family of semanticsfor tense-modal logics, now known asbranching timesemantics, which have been independently developed inphilosophical logic and computer science. BT semantics haveconsiderable philosophical interest, for instance in the analysis ofhistorical modality and the treatment of future contingentstatements.

The second canonical application is in modeling a very special system,namely our (supposedly) indeterministic, agent-populated, universe.TheBT conception is the view that our universe and itstemporal and modal features are better represented by standard BTframes than by other, non-tree-like structures. Given that there aredifferent ways to conceive trees and how they represent reality, theBT conception is best regarded as a family of metaphysical views,which need not share a substantive metaphysical core. Nonetheless, theBT conception is neither empty nor trivial, as whether trees arebetter suited to representing our universe than alternative structuresis a matter of legitimate philosophical disagreement.

Section1 introduces the main ideas behind branching time and BTrepresentations. Section2 covers trees and related representations of indeterministic systems.Sections3 and4 are devoted, respectively, to the canonical applications of BT framesin the semantics of tense-modal logics and in metaphysics.

• 1 A primer to branching time
• 2 Representations of branching time
  • 2.1 Trees. Bundled trees.
  • 2.2 Non-branching representations
  • 2.3 Representations of Branching Space-Times
• 3 Branching time semantics for tense-modal logics
  • 3.1 Recursive semantics
  • 3.2 Postsemantic proposals
  • 3.3 Branching time semantics in computer science
• 4 Philosophical views on branching time
  • 4.1 Varieties of the branching time conception
    • 4.1.1 Branching time realism and Branching Space-Times realism
    • 4.1.2 Actual futurism
    • 4.1.3 Asymmetry views
    • 4.1.4 Semantics and metaphysics
  • 4.2 General challenges to the branching time conception
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. A primer to branching time

Branching time (BT) frames are formally defined BT representations,and are endowed with both structural and representational features.From a structural viewpoint, BT frames arebranchingstructures, that is, intuitively, networks of interconnectedpaths, where each path can be thought of as a chain of nodes.Branching structures are ubiquitous in our experience, appearing forinstance in trees, in rivers, in circulatory systems, and in roadsystems. What makes BT frames special, with respect to other branchingstructures, is their representational function: BT frames aim torepresent all the possible temporal developments of a given system,which in philosophy is generally identified with the whole universe.More specifically, in BT frames, paths are calledhistoriesand represent complete courses of events, while nodes are calledmoments and represent instantaneous or temporally minimalevents. Histories are usually defined as maximal chains of momentsordered by a relation oftemporal (orcausal, ortemporal-modal)precedence \(\prec\). Histories areall connected: by moving backward and/or forward along histories, wecan reach any moment in the frame from any other. Histories can branchout in both temporal direction, that is, we can have bothfuture-directed orforward branching and past-directed orbackward branching (seeFig. 2). We shall speak ofstandard BT frames or (BT)trees^[2] to indicate BT frames in which only forward branching is admitted andbackward branching is not. In this entry, we shall focus on trees anddevote only a few passing comments to backward-branching BTframes.

Figure 2: Forward branching and backwardbranching. [Anextended description of figure 2 is in the supplement.]

In short, standard BT frames or trees areabstractrepresentations that are understood to represent our universe (orsome other system) along with all its possible developments; trees arecomplexes ofmoments andhistories, where historiesare maximal chains of temporally ordered moments; histories are allconnected anddo not branch towards the past. Let usexpand on this characterization by delving a little deeper into eachof these italicized notions.

• Abstract representations. BT frames are generallythought of as abstract representations, which need not be fullyprojected onto reality. The problem of making clear what preciselythey represent is a major philosophical concern, which we shalldiscuss in§4.
• Moments. Moments represent events that arespatially maximal and only have an instantaneous or minimal duration.Trees are commonly defined as pairs \((M,\prec\)), where \(M\) is anon-empty set of moments and the temporal precedence relation\(\prec\) is a strict partial order that complies with the principlesofConnection andNo backward branching (seebelow).

As Belnap, Perloff, and Xu observe,

the concept of a moment … is a Newtonian idea. It is distantfrom our everyday conceptions, and it is non-relativistic. It inheritsfrom Laplace’s demon the implausible presupposition that thefundamental terms of the causal order shall be entire instantaneousworld-slices, instead of smallish local events or point events. It is,however, a worthwhile approximation to the truth. (2001: 139)

The idealization is removed in branching-spacetimes frames (BSTframes, see Belnap 1992; Belnap, Müller, & Placek 2022),where the role of moments as basic building blocks is taken byspatially smaller events (see below,§2.3,§4.1.1).

• Histories. Each history represents aspatially and temporally complete, possible course of events.Histories are usually defined as maximal sets of moments, linearly (ortotally) ordered by \(\prec\).
• Connection. For any two distinct moments in theframe, either one precedes the other or there is a moment that is\(\prec\)-related with both. Intuitively, this means that the wholeframe forms a single, connected structure of histories.

Connection reflects the idea that BT frames model the possibleevolutions of a unified system, and not a plenitude of disjointpossibilities.

• No backward branching. If two different moments\(m,m'\) temporally precede another moment \(w\), then \(m\) and\(m'\) are \(\prec\)-related, which implies that they belong to everyhistory containing \(w\). In other words, the past of every moment isa linear order.

Historically, No backward branching reflects the philosophicalmotivations behind the introduction of BT representations. In 1958, avery young Saul Kripke wrote a letter to Arthur Prior where heprovided the first semi-formal characterization of BT frames(Kripke’s letter, dated 3 September 1958, is in the PriorCollection at Bodleian Library, Oxford, Box 4; see the entry onfuture contingents for more historical background). Prior took Kripke’s suggestionvery seriously, and formally developed it during the 1960s (1960,1962, 1966a,b, 1967a,b, 1968a,b). He saw BT frames as a useful tool ina variety of logico-philosophical areas: the semantics of tense-modallogics, the problem of identity across time and possible worlds, theproblems of agency, free will, and future contingents. In all theseareas, Prior felt that too little attention was paid, amongphilosophers and logicians, to the fact that possibilities changeacross time.

It is always a useful exercise (and one insufficiently practiced byphilosophers), when told that something was possible, i.e., could havehappened, to ask “When was it possible?”“When could it have happened?” (Prior 1960:688)

In contemporary philosophical jargon, the kind of time-sensitivemodalities (possibilities, necessities, contingencies) that Prior hadin mind are calledhistorical. Historical modality is timeasymmetrical: all past and present events are historically necessary,and only future events are historically contingent. For this reason,historically contingent events are also calledfuturecontingents. (With a traditional ambiguity, we use “futurecontingents” also to indicate assertions that predict futurecontingent events and the corresponding sentences.) Along withhistorical possibilities, which are anchored to a time, one can alsoconceive possibilities that are anchored to a space-time region or tocertain agents. These “local” possibilities are sometimescalledreal (see, e.g., Müller 2012; Placek 2012;Rumberg 2016a; Belnap, Müller, Placek 2022). BST frames are toolsfor formally studying real possibilities.^[3]

Intuitively, historical and real contingencies are the kinds ofpossibilities that matter to us as agents. In an abstract sense, it isnow possible that you never started reading this entry. However, giventhat youdid start reading this entry, that is no longer alive possibility. By contrast, it is still a genuine possibility foryou to stop reading this entry. It is difficult to overestimate therole of the distinction between abstract (metaphysical, logical)modality and historical (real, “local”) modality in ourself-representation as agents and in our inferential, epistemic andlinguistic practices. In Prior’s view, the distinction is notjust a human construction and is also rooted in mind-independentreality. Prior subscribed to indeterminism, and thought that at leastsome particular events wereobjectively “open”,not already present in their causes (see, e.g., Prior 1967a: Ch. 7;see also the entry oncausal determinism).

Many philosophers in the contemporary debate about branching timeagree with Prior in regarding historical modality as purely objective(mind-independent) and grounded in concrete reality. They think thereis a deep divide between objective andepistemic modality,and between particular, concrete events andqualitative,repeatable states.

[N]o backward branching makes sense only for objective, concreteevents. […] A given concrete situation could obviously havebeen preceded by any of various inconsistent predecessors, “forall one knows”. It is precisely to preclude this epistemic ordoxastic use of “possible” that we so tiresomely repeatthat our present concern is with “objective”possibilities. Second, No backward branching fails to apply to“states” or other repeatable carriers of partialinformation. There is no doubt whatsoever that a present“state” may be accessible from either of two earlierincompatible states. (Belnap, Perloff, & Xu 2001: 184)

In short, the No backward branching condition reflects the idea thatwhether an event \(e\) is historically possible at a moment \(m\)entirely depends on whether \(e\) is causally-historically accessiblefrom \(m\) (i.e., lies either in the past or in a future of \(m\)).Thus, the historical possibility of \(e\) at \(m\) does not depend onwhether \(e\) is compatible with the knowledge available at \(m\), orwhether other events similar to \(e\) are accessible from \(m\).

When joined with Connection, No backward branching entailsHistorical connection: for any two distinct moments in theframe, either one precedes the other or there is a moment thatprecedes both. The adoption of Historical connection hasphilosophically interesting consequences. Let us make one of theseconsequences explicit.

Historical possibility is a very strict notion of possibility, andwithin standard BT frames, we can also define more inclusive notions.For one, we can define a notion ofpast historicalpossibility: an event \(e\) is a past historical possibility at\(m\) iff \(e\) is historically possible at some moment \(m'\) thateither precedes or is identical to \(m\). And we can define also anabsolute, tree-relative notion of possibility: an event \(e\) ispossible in a tree iff it is historically possible at some moment inthe tree. Historical connection ensures us that these two notions ofpossibility coincide. By Historical connection, past historicalpossibility is the widest, most inclusive notion of possibilitydefinable within BT frames.

It is natural to suppose that the notion of past historicalpossibility coincides with the notion ofphysical possibilitygiven initial conditions, when initial conditions are setarbitrarily far in the past. Many philosophers think that physicalpossibility, so understood, is still restrictive as compared tometaphysical possibility, which is the widest notion ofpossibility traditionally recognized in philosophy. If that iscorrect, by adopting Historical connection, we make BT framesunsuitable to model metaphysical modality (on the relations between BTand metaphysical modality, see, e.g., Prior 1960; J. L. Mackie 1974;Lewis 1986: 4; P. Mackie 2006; MacFarlane 2008).

2. Representations of branching time

2.1 Trees. Bundled trees.

As observed above, trees are the most common representations ofbranching time. These structures are formally defined as pairs\({\mathcal{T}}= (T, \prec )\) in which \(T\) is a set (ofmoments) and \(\prec\) is thetemporal precedencerelation with the following properties (the expression \(m \prec m'\)reads \(m\)temporally precedes \(m'\); \(\preceq\),\(\succ\), and \(\succeq\) are defined in the obvious way; two moments\(m,m'\) are said to becomparable when either \(m\preceqm'\) or \(m'\prec m\)):

• Irreflexivitiy. For no \(m\), \(m \prec m\);
• Transitivity. If \(m \prec m'\) and \(m' \precm''\), then \(m \prec m''\);
• Connection. For any two moments, there exists amoment that is comparable with both;
• No backward branching (or Linearity towards thepast). If \(m' \prec m\) and \(m'' \prec m\), then either\(m' \preceq m''\) or \(m'' \prec m'\).

No backward branching and Connection jointly entail the following:

• Historical connection (or left-connectedness). Forall \(m\) and \(m'\), there exists an \(m''\) such that \(m'' \preceqm\) and \(m''\preceq m'\).

Ahistory in the tree \({\mathcal{T}}= (T, \prec )\) is a set\(h \subseteq T\), which is linearly ordered by \(\prec\) and maximalfor inclusion, that is, if \(h'\) is a linearly ordered subset of\(T\) and \(h \subseteq h'\), then \(h' = h\). It can be proved (withthe Axiom of Choice) that every linearly ordered subset of \(T\) canbe extended to a history. TheFuture of \(m\) is defined as\(\{m' \in T : m \prec m'\}\). Afuture of \(m\) (or an\(m\)-branch) is the intersection of the Future of \(m\) anda history containing \(m\). The symmetrical notion of thepast of \(m\) (or \(m\)-trunk) is defined as \(\{m'\in T : m' \prec m\}\).

Connection entails that we can get from any moment to any other atmost in two (backward or forward) steps. If Connection is given up, weget structures consisting of disjoint trees, which are sometimescalledforests (see, e.g., Goranko, Kellerman, & Zanardo2023).

If No backward branching is omitted, then the only condition imposedon the relation \(\prec\) is that it is an irreflexive connectedpartial order. Thus, at any moment, time may branch out also in thepast and a BT structure can be viewed as a set of crossroads (seeFig. 3). So conceived, crossroads are temporally directed, as we can dividethem into a forward-branching fork and a backward branching one. Theforward-branching fork departing from \(m\) is naturally understood torepresent the historical possibilities open at \(m\), and thebackward-branching fork, theepistemic possibilities open at\(m\), i.e., the possibilities consistent with the knowledge availableat \(m\). However, other interpretations are also possible (e.g., wecan understand both forks as representing epistemic possibilities).Backward branching structures have potential applications that arelargely unexplored to date, for instance, concerning the treatment ofknowledge and other propositional attitudes in a branching reality.^[4] We expect that in future iterations of this entry they will receive awidest share of attention.

Figure 3: A temporally directedcrossroad centered at moment \(m\). [Anextended description of figure 3 is in the supplement.]

In many applications of BT frames, it is of crucial importance thatsome kind of synchronism is imposed between moments belonging todifferent histories. For instance, counterfactual assertions seem toinvolve a trans-history identification of moments: “Were I inLondonnow, I would go straight to the British Museum”(see, e.g., Thomason & Gupta 1980; Di Maio & Zanardo 1994).But it must be noticed that not every tree can be endowed withsynchronisms. The definition of tree does not impose any topologicalstructure on histories. We can have histories that are dense in someparts and discrete in others. In particular, two different historiesmay have different topological structures in the parts where they donot overlap.

When trees are viewed as representations of an indeterministicuniverse, it is natural to assume that all histories are isomorphic tothe set \(\mathbb R\) of real numbers. In this case, given twodistinct histories \(h\) and \(h'\), also their set-theoreticaldifferences \(h \setminus h'\) and \(h' \setminus h\) are isomorphicand any order-preserving bijection \(\sigma\) between these two setscan be thought of as asynchronism between them. So we cansay that the tree is synchronizable. In other words, for any \(m \inh\setminus h'\), \(\sigma(m)\) can be viewed as “the same momentas \(m\)” in \(h' \setminus h\). It is important to observe that\(h \setminus h'\) and \(h' \setminus h\) can be synchronized ininfinitely many ways. So, in general, in order to have a synchronizedtree, we must also explicitly provide a set of bijections betweenhistories.

Formally, we say that the tree \({\mathcal{T}}\) issynchronizable if there exists a set \(\Sigma\) of functions\(\sigma_{h,h'}\), where \(h\) and \(h'\) range over the set ofhistories in \({\mathcal{T}}\), such that, for all \(h, h', h''\): (1)\(\sigma_{h,h'}\) is an order-preserving bijection from \(h\) onto\(h'\), (2) \(\sigma_{h,h'}\) is the identity on \(h \cap h'\), (3)\(\sigma_{h',h}\) is identical to \(\sigma^{-1}_{h,h'}\) (i.e., to theinverse of \(\sigma_{h,h'}\)), and (4) the composition\(\sigma_{h,h'}\circ\sigma_{h',h''}\) is identical to\(\sigma_{h,h''}\). Pairs \(({\mathcal{T}}, \Sigma)\) are calledsynchronized trees.^[5]

In a synchronized tree, the set \(\Sigma\) of bijections determines arelation on \(T\): we can write \(m \leftrightarrow_\Sigma m'\) iff\(m' = \sigma_{h,h'}(m)\) for some \(h, h'\). It can be readilyverified that, by the properties of the functions in \(\Sigma\), therelation \(\leftrightarrow_\Sigma\) is well defined and is anequivalence relation (see Di Maio & Zanardo 1994), which isnaturally understood as a kind of simultaneity. The trees consideredin Belnap’s Logic of Agency (in particular for achievementstit)are synchronized trees, but thesynchronism is determined by means of the primitive notion ofinstant (Belnap, Perloff, & Xu 2001: 7A.5). The set ofmoments in a tree is “horizontally” partitioned intoinstants. Every instant is also assumed to have a unique intersectionwith each history, and instants preserve the temporal relationsbetween moments (seeFig. 4). It turns out that the equivalence relation determined by thepartition into instants has the same properties as\(\leftrightarrow_\Sigma\). So, considering instants is the same asconsidering a synchronism.

Figure 4: Instants. [Anextended description of figure 4 is in the supplement.]

Another way for defining synchronisms is to consider metric trees(Burgess 1979: 577). That is, to assume that there exists a function\(d\) from \(T \times T\) into \(\mathbb{R}\) such that \(d(m,m')\)can be viewed as thedistance between \(m\) and \(m'\). If areference moment \(m_0\) is given, the function \(d\) can be seen as aclock: \(d(m_0,m)\) is the (possibly negative) time of \(m\) withrespect to \(m_0\). Of course, some further technicalities are neededin order to have a satisfactory definition of “the time of themoment \(m\)”. For instance, we have to define this notion formoments that are not comparable with \(m_0\). But eventually we canreach a reasonable definition of synchronism.

The existence of instants, as well as the existence of a synchronism,in a tree-like representation of the universe is a strong ontologicalassumption. This is actually recognized in Belnap, Perloff, andXu:

the doctrine of instants harkens back to the Newtonian doctrine ofabsolute time—and therefore is suspect. We use it, but wedon’t trust it. (2001: 194)

Improving on that is indeed one key motivation for Belnap’stheory of Branching Space-Times (see below,§2.3).

The notion of a tree is often generalized by consideringbundledtrees, that is, pairs \(({\mathcal{T}}, {\mathcal{B}})\) in whichabundle \({\mathcal{B}}\) is set of histories such that, foreach moment \(m\), some history in \({\mathcal{B}}\) includes \(m\).Thus, trees can also be seen astrivial bundled trees\(({\mathcal{T}}, {\mathcal{B}})\), in which \({\mathcal{B}}\) is theset of all histories. Using non-trivial bundled trees as BTrepresentations amounts to holding that not every history represents apossible course of events. In this sense, the set \({\mathcal{B}}\)can be thought of as the set ofadmissible histories.

There are two different attitudes one can have towards bundles and therestriction to admissible histories. One can think, following vanBenthem (1996: 206–208), that since sets (histories) aregenerally considered in a particular situation and with specificpurposes in mind, there is no reason to takeall possiblesets (histories) into account; it is preferable to circumscribe therange of the quantification over sets that is relevant to the purposesat stake. An example of this way of proceeding can be found inStirling (1992), where interesting (first-order and second-order)closure properties are identified for the set of branches (or paths,or runs, or histories) in temporal logic for computer science.

The opposite attitude, based on descriptive adequacy, is insteadadvocated by Thomason (1984: 151–152) and Belnap, Perloff, andXu (2001: 200–203). These authors argue that bundled trees leadto contradictory assertions when certain specific indeterministicscenarios are considered. For a reply to their arguments, seeØhrstrøm and Hasle (1995: 268–269), and the entryonfuture contingents, §5.2, which discuss structures (“Leibniziansystems”) very similar to Kamp frames or Ockhamist frames (seebelow,§2.2).

A further kind of BT representations is the one that van Benthem(1999) callsgeometrical. In the geometrical approach, bothmoments and histories are regarded as primitive entities, with noset-theoretical and ontological dependence of the latter on theformer, in the same way as points and lines are regarded in manypresentations of geometry. In this approach, histories arereified. Then, unlike in standard, moment-based perspectives,geometrical representations treat moments and histories on a par.^[6] Ageometrical structure is a 4-tuple \(\mathcal{G} = (T,H,\prec, E)\) in which \(T\) and \(H\) are sets, \(\prec\) is a treerelation on \(T\), and \(E\) is a binary relation between \(T\) and\(H\). The elements of \(H\) are meant to represent histories and\(E(m,h)\) can be read as “the moment \(m\) lies on\(h\)”. The (first-order) properties of these structures givenin Zanardo (2006b: Def. 6.1) have the effect that the elements of\(H\) actually represent histories: it turns out that, for every \(h\in H\), the set \(\bar h = \{m \in T : E(m,h)\}\) is a history in\((T, \prec)\). Moreover, the set of all \(\bar h\), for \(h\) rangingover \(H\), is a bundle in that tree.

2.2 Non-branching representations

Other modal structures for representing indeterministic systems can befound in the literature. These can be termednon-branchingrepresentations, as they are naturally thought to representdisjoint worlds rather than overlapping courses of events. As we shallsee below, there is a strict connection between these structures andbundled trees. This allows, for example, to use either the former orthe latter interchangeably in the semantics for branching time logics(see§3). However, even if equivalent to bundled trees for many logical andmathematical purposes, non-branching structures seem to reflectdifferent philosophical conceptions of historical modality and futureindeterminacy (see, e.g., Belnap, Müller, & Placek 2022;6–8, 13–22; Restall 2011: 228–231).

AKamp frame (Thomason 1984) is a triple\(\mathcal{K} = ({\mathsf{K}}, W, \approx)\) in which \(W\) is a setofworlds and \({\mathsf{K}}\) is a function assigning alinear order \({\mathsf{K}}_w = (T_w, \lt_w)\) to each element of\(W\). Thus, every \({\mathsf{K}}_w\) can be regarded as a history.This implies that the elements of each \(T_w\) can be viewed asmoments in a temporal structure. The branching aspect of time isrepresented by \(\approx\), which is a function that assigns to eachmoment \(t \in \bigcup_{w \in W}T_w\) an equivalence relation\(\approx_t\) on the set of all worlds \(w\) such that \(t \in T_w\).The intended meaning of \(w \approx_t w'\) is that the histories\({\mathsf{K}}_w\) and \({\mathsf{K}}_{w'}\) coincide up to \(t\).Then, it is also required that: (1) if \(w \approx_t w'\), then \(\{t'\in T_w : t' \lt_w t \} = \{t' \in T_{w'} : t' \lt_{w'} t \}\) and (2)if \(w \approx_t w'\) and \(t' \lt_w t\), then \(w \approx_{t'}w'\).

The definition of Kamp frame makes evident the relationship betweenthese structures and trees. Given a Kamp frame \(\mathcal{K} =({\mathsf{K}}, W, \approx)\), the set of equivalence classes \([(w,t)]= \{(w',t) : w \approx_t w'\}\) can be endowed with a binary relation\(\prec_{\mathcal{K}}\) defined by

(cf. Zanardo 2006b: 4). It can be easily verified that\(\prec_{\mathcal{K}}\) is well-defined and is a tree relation. Then,the classes \([(w,t)]\) can be viewed as moments in a tree\({\mathcal{T}}_{\mathcal{K}}\). Moreover, for any \(w\) in \(W\), theset \(\{ [(w,t)] : t \in T_w \}\) turns out to be a history \(h_w\) in\({\mathcal{T}}_{\mathcal{K}}\).

Trees generate Kamp frames in a natural way. Given a tree\({\mathcal{T}}\), we let \(W_{\mathcal{T}}\) be the set of allhistories in \({\mathcal{T}}\) and let \(\approx_{\mathcal{T}}\) bethe identity function (Fig. 5). For all histories \(h, h'\in W_{\mathcal{T}}\) and all moments \(m\in T\), we set \(h \approx_{{\mathcal{T}}m} h'\) iff \(m \in h \caph'\). The proof that \(\mathcal{K}_{\mathcal{T}}=({\mathsf{K}}_{\mathcal{T}}, W_{\mathcal{T}}, \approx_{\mathcal{T}})\)is a Kamp frame is straightforward.

Figure 5: A tree with the correspondingKamp frame. [Anextended description of figure 5 is in the supplement.]

Despite the close relationship between trees and Kamp frames, thecorrespondences \(\mathcal{K} \to {\mathcal{T}}_{\mathcal{K}}\) and\({\mathcal{T}}\to \mathcal{K}_{\mathcal{T}}\) are not the inverse(modulo isomorphisms) of each other. In general, the structure\(\mathcal{K}_{({\mathcal{T}}_{\mathcal{K}})}\) is not isomorphic to\(\mathcal{K}\) because \({\mathcal{T}}_{\mathcal{K}}\) might havehistories that do not correspond to any world in \(\mathcal{K}\) (cf.Zanardo 2006b). The set of histories of the form \(h_w\) in\({\mathcal{T}}_{\mathcal{K}}\) constitute a possibly proper bundle inthis tree. It is true, instead, that\({\mathcal{T}}_{(\mathcal{K}_{\mathcal{T}})}\) is isomorphic to\({\mathcal{T}}\).

\(\mathbf{T\times W}\) frames can be defined as Kampframes in which \({\mathsf{K}}_w = {\mathsf{K}}_{w'}\) for all \(w, w'\in W\). Thus, what we said above about the correspondence betweenKamp frames and trees also holds for \(\mathrm{T\times W}\) frames. Itmust be noted that the trees generated by these frames aresynchronized trees. Given a \(\mathrm{T\times W}\) frame\(\mathcal{K}\), for any two histories \(h_w\) and \(h_{w'}\) in\({\mathcal{T}}_{\mathcal{K}}\), we define the function\(\sigma_{w,w'}\) by \(\sigma_{w,w'}([(w,t)]) = [(w',t)]\). It isreadily verified that the set of all \(\sigma_{w,w'}\) is asynchronism.

TheOckhamist frames have been introduced in Zanardo(1985) and further investigated in Zanardo (1996). Intuitively,Ockhamist frames can be viewed as deriving from a tree by consideringthe histories in it as disjoint sets, each one linearly ordered by arelation \(\lt\), and by representing the sameness of moments by anequivalence relation \(\sim\). Formally, an Ockhamist frame is atriple \(\mathcal{O} =(W, \lt, \sim)\) in which \(W\) is a non-emptyset and \(\lt\) and \(\sim\) are binary relations on \(W\) with thefollowing properties: (1) \(\lt\) and \(\sim\) are disjoint (\(w \ltw' \Rightarrow w \not\sim w'\)); (2) \(\lt\) is a union of disjointlinear orders; (3) \(\sim\) is an equivalence relation; (4) if \(w\sim w'\), then the restriction of \(\sim\) to \(\{ w'' : w'' \lt w\}\times \{ w'' : w'' \lt w'\}\) is an order-preserving bijection.

The set of pairs \((m,h)\) with \(m \in h\) in a tree\({\mathcal{T}}\) can be given an Ockhamist frame structure by setting\((m,h) \lt (m', h')\) if \(h=h'\) and \(m \prec m'\), and \((m,h)\sim (m', h')\) if \(m=m'\). The frame defined in this way will bedenoted by \(\mathcal{O}_{\mathcal{T}}\) (Fig. 6).

Figure 6: A tree with the correspondingOckhamist frame. [Anextended description of figure 6 is in the supplement.]

Conversely, given an Ockhamist frame \(\mathcal{O} =(W, \lt, \sim)\)we can define a tree \({\mathcal{T}}_{\mathcal{O}}\) in a natural way.The moments in this tree are equivalence classes \([w]_\sim\) and thetree relation \(\prec\) is defined by: \([w]_\sim\prec [w']_\sim\) if\(w'' \lt w'\) for some \(w'' \in [w] _\sim\). Property (4) aboveguarantees that \(\prec\) is (well defined and) a tree relation. Every\(\lt\)-connected component \(C\) of \(W\), that is, every maximalsubset \(C\) of \(W\) such that, for all \(w, w' \in C\), \(w \leqw'\) or \(w' \leq w\), determines a history \(h_C = \{ [w]_\sim : w\in C\}\). Like in the case of Kamp frames, the set of all \(h_C\) isa bundle \({\mathcal{B}}\) in \({\mathcal{T}}_{\mathcal{O}}\), sothat, in general, \(\mathcal{O}_{{\mathcal{T}}_{\mathcal{O}}}\) and\(\mathcal{O}\) are not isomorphic structures (while\({\mathcal{T}}_{\mathcal{O}_{\mathcal{T}}}\) and \({\mathcal{T}}\)are).

2.3 Representations of Branching Space-Times

In Belnap (1992), Nuel Belnap set up the ambitious project ofcombining relativity and indeterminism in a rigorous framework.Various publications on the subject followed and developed that work,including Belnap, Müller, and Placek (2022), which summarizesthose developments in the form of a monograph. In this section, weoutline the main aspects of this innovative approach, also inconnection with the representations of branching time consideredabove.

The main goal of Branching Space-Times (BST) theory is to representpossibilities in the context of relativity. Then, the basic notions inthe theory are those of (possible)point event and ofcausal order relation (\(\lt\)), which are the BSTcounterparts of moments and of the temporal order in BT. As clearlypointed out in Belnap (2012), BST theory aims to move away from aNewtonian framework in two different directions: from determinism toindeterminism and from classical physics to relativity. This includespaying attention to the relativistic notion of causal independence asspace-like separation (see below).

A BST structure (Our World, in Belnap’s terminology) isthen a pair \(\mathcal{W} = (W, \lt)\) in which \(W\) is a non-emptyset (of point events) and \(\lt\) is a strict partial order on \(W\)(\(\leqslant\), \(\gt\), and \(\geqslant\) are defined in the obviousway). If \(e_1 \leqslant e_2\) or \(e_2 \leqslant e_1\), we say that\(e_1\) and \(e_2\) arecomparable. The axioms for BST do notinclude the linearity towards the past (while they imply density andcontinuity of the causal order relation). Thus, unlike in BT, threepoint events may constitute acausal confluence diagram:\(e_1 \lt e_2 \gt e_3\) with \(e_1 \not\leqslant e_3\) and \(e_3\not\leqslant e_1\). A consequence of this is that we can consider twodifferent kinds ofup-forks \(e'_1 \gt e'_2 \lt e'_3\),according to whether \(e'_1\) and \(e'_3\) are followed by causalconfluence or not.

Figure 7: Causal dispersion andbranching dispersion. [Anextended description of figure 7 is in the supplement.]

The branching of possibilities represented via BT does not allow forbackward branching, while relativistic space-time does. Then, in thediagrams of the first kind (that Belnap callscausaldispersion) the point events \(e'_1 , e'_2\) and \(e'_3\) mustbelong to the same space-time, while the forks of the second kind(branching dispersion) are given a strictly modalmeaning.

These considerations directly lead to the notion ofhistory,which has a crucial role also in BST. Like in BT, a structureconsisting of just one history can be viewed as a representation of adeterministic universe. Then, taking relativity into account, any twopoint events \(e_1\) and \(e_2\) in a history \(h\) must have alater witness in \(h\), that is, \(e_1 \leqslant e_3\geqslant e_2\) must hold for some \(e_3 \in h\). Subsets of \(W\)with this property are calleddirected subsets. Thus,histories in BST are directed subsets of \(W\). They are also requiredto be maximal, which implies that every history is downward closed and(upward) closed for causal confluence. \(d\)-dimensional Minkowskispace-times with \(d \geq 2\) are examples of histories. Two pointevents are said to becompatible if there is a history thatcontains them.

In general, histories in BST are not linearly ordered sets, but theyshare many of the properties of histories in BT. In particular, everydirected subset of \(W\) can be extended to a history. Moreover, sinceHistorical connection is assumed in BST, any two histories havenon-empty intersection. The original axioms for BST in Belnap (1992)imply also that the intersection of any two histories has a maximalelement, which is calledchoice point. As shown in Belnap,Müller, and Placek (2022), though, other alternatives can beconsidered.

Given a structure \((W,\lt)\), BST theory allows for the definition ofspace-like separation as incomparability plus compatibility.The point events \(e'_1\) and \(e'_3\) on the left of Figure7 are space-like separated, while, on the right picture, \(e'_1\) and\(e'_3\) are with respect to each other neither causally future norcausally past nor causally contemporaneous. Within a given history,then, space-like separation is just incomparability, which agrees withthe corresponding definition in relativistic space-time.

The relatively simple framework considered in this section is widelydeveloped in Belnap, Müller, and Placek (2022) and in other worksto obtain a rigorous theory that combines relativity and indeterminism(Müller 2010, 2013; Placek 2010; Belnap, Müller, &Placek 2022). We only mention here that BST theory’s analysis ofcausation and probability allows to represent probabilisticcorrelation (see Belnap 2005), and that an application to quantumcorrelation is given in Belnap, Müller, and Placek (2022: 8).

The representations \((W,\lt)\) for BST are susceptible to variations(or alternative presentations) similar to those that we haveconsidered for BT, which would similarly involve discussions about theadequacy of the new structures. As an example, the passage from treesto Ockhamist frames can be adapted to BST. We can consider all thehistories in a structure \((W,\lt)\) as disjoint sets and representequality of point events in \((W,\lt)\) by an equivalence relation. Weobtain structures \((W^*,\lt^*, \sim)\) in which \(W^*\) is the union\(\bigcup_{i \in I} W_i\) of disjoint sets, and \(\lt^* = \bigcup_{i\in I} \lt_i\), where each \(\lt_i\) is a partial order relation on\(W_i\). Since every \((W_i, \lt_i)\) is meant to represent a history,we have also to assume that these structures are directed. Like in thecase of Ockhamist frames, the relation \(\sim\) mimics the equality in\((W,\lt)\). Let us note that, in this context, up-forks \(e_1 \gt^*e_2 \lt^* e_3\) can only be interpreted as instances of causaldispersion. Branching dispersion is represented by diagrams of thisform: \(e_1 \gt^* e_2 \sim e_2' \lt^* e_3\), with \(e_2 \neqe_2'\).

3. Branching time semantics for tense-modal logics

Trees, or the related set-theoretical structures considered above,constitute the fundamental framework for BT semantics for tense-modal logics.^[7] The bulk of this section (§3.1–§3.2) is devoted to BT semantics that are especially relevant to thephilosophical debate on branching time. We discuss most of theseapproaches with reference to a simple Priorean propositional language\(\mathcal{L}\) endowed with a future operator \(\mathsf{F}\)(“it will sometimes be the case that”) with dual\(\mathsf{G}\) (“it will always be the case that”), a pastoperator \(\mathsf{P}\) (“it was sometimes the case that”)with dual \(\mathsf{H}\) (“it was always the case that”),and a historical possibility operator \(\lozenge\) (“it is stillhistorically possible that”) with dual \(\Box\) (“it issettled that”). In presenting these semantics, we presuppose adistinction betweenrecursive semantics (orsemanticsproper) andpostsemantics, in MacFarlane’s (2003,2008, 2014) terminology. Arecursive semantics \(S\) for\(\mathcal{L}\) is a standard, Tarski-style recursive definition oftruth for sentences of \(\mathcal{L}\) relative to certain points ofevaluation. A postsemantics for \(\mathcal{L}\) based on \(S\) is adirect definition oftruth-in-a-context (in a sense akin tothat of Kaplan 1989) that is based on the notion of truth recursivelydefined by \(S\). Intuitively, semantics proper provides each sentencein \(\mathcal{L}\) with truth conditions relative to each point ofevaluation, while postsemantics tells us what sentences of\(\mathcal{L}\) are true relative to what contexts.

In the final subsection (§3.3), we outline a few BT semantics specifically developed for applicationsin computer science.

3.1 Recursive semantics

This section focuses on recursive semantics for BT logics. There arebasically three main strands in this area, all envisaged by Prior,which are known as recursivethin red line (TRL) semantics,Peircean semantics andOckhamist semantics (see theentry ontemporal logic, §5 for an exhaustive definition of the language and semanticsfor Ockhamist and Peircean temporal logics and a discussion oftechnical results). The key difference between these semantics lies inthe interpretation of the future operator \(\mathsf{F}\). Let us startby introducing recursive TRL semantics.

The past operator \(\mathsf{P}\) has the same semantic behavior in allrecursive TRL semantics:

\((\mathrm{T}_{\alpha})\)

\(\mathsf{P} \alpha\) is true at \(m\) iff \(\alpha\) is true atsome \(m' \prec m\).

As for the future operator \(\mathsf{F}\), the general idea is thatformulae of form \(\mathsf{F} \alpha\) are true iff \(\alpha\) is truein the actual future (see Prior 1966: 157; Øhrstrøm1981), that is, in the future on thethin redline—i.e., “the course along which history willgo” (Belnap & Green 1994: 366). This idea can be madeprecise in two different ways. In presenting them, we loosely followthe discussion in Belnap and Green (1994).

1. Inabsolute recursive TRL semantics, models specify a history\(\TRL\), which represents the (unique) actual history in thetree. The future operator \(\mathsf{F}\) is understood in terms oftruth at future moments on the thin red line:

   \((\mathrm{T}_{\mathsf{F}})\)

   \(\mathsf{F}\alpha\) is true at \(m\) iff \(\alpha\) is true at some\(m' \in \TRL\) with \(m \prec m'\).

   This approach gives plausible results when future-tensed statementsare evaluated at actual moments but runs into problems when momentsnot belonging to the \(\TRL\) are at stake (this may happen, forinstance, when the future-tense operator is embedded under modaloperators, see Belnap & Green 1994: 379); see also Thomason 1970:270–271). The standard diagnosis is that a unique \(\TRL\) isnot sufficient, and one must provide each moment in the tree with itsown \(\TRL\). This requirement is satisfied in relative variants of TRLsemantics.

2. Inrelative (orMolinist)^[8] recursive TRL semantics, models define a function \(\TRL()\) that mapseach moment \(m\) to the corresponding thin red line (see McKim &Davis 1976; Thomason & Gupta 1980). The simplest way to specifythe truth conditions of future-tensed statements in this frameworkis:

   \((\mathrm{T}_{\mathsf{F}})\)

   \(\mathsf{F}\alpha\) is true at \(m\) iff \(\alpha\) is true at some\(m'\succ m\) such that \(m'\in \TRL(m)\).

   As Belnap and Green (1994) point out, however, relative recursive TRLsemantics either entails that time is linear or fails to validatenatural principles such as the following:

   • Retrogradation. \(\alpha\to\mathsf{H}\mathsf{F}\alpha\)

   Moreover, from a philosophical viewpoint, it is unclear what the\(\TRL\) should represent in this framework: why suppose thatcounterfactual moments have an actual future, and what should this“quasi-actual” future represent? Even though thesecontentions have been addressed in the literature (see Braüner,Hasle, & Øhrstrøm 1998; Braüner 2023),recursive TRL views have fallen out of favor nowadays, and theattention has shifted to their postsemantic counterparts (see§3.2).^[9]

Let us turn to Peircean semantics. Peircean language is different fromour standard language \(\mathcal{L}\) in that it does not containmodal operators and includes a weak future operator \(\mathsf{f}\),which expresses the notion of future possibility:

\((\mathrm{P}_{\mathsf{f}})\)

\(\mathsf{f}\alpha\) is (Peirce) true at \(m\) iff \(\alpha\) is trueat some \(m'\succ m\).

The Peircean reading of the past operator \(\mathsf{P}\) is identicalto that of recursive TRL semantics. The future operator \(\mathsf{F}\)receives a very strong reading:

\((\mathrm{P}_{\mathsf{F}})\)

\(\mathsf{F}\alpha\) is true at \(m\) iff all histories passingthrough \(m\) contain a moment \(m'\succ m\) where \(\alpha\) istrue.

This clause is meant to capture the idea that “nothing can besaid to be truly ‘going-to-happen’ (futurum)until it is so ‘present in its causes’ as to be beyondstopping” (Prior 1962: 124). In other words, an event is trulyfuture only if it is inevitable. Peircean semantics is formallypleasant and allows us to combine bivalence and excluded middle withindeterminism. However, it has several drawbacks. For instance, itobliterates important semantic differences, such as the one between“will be the case” and “will be inevitably thecase”, and it fails to validate a variety of plausibleprinciples, including Retrogradation and the following:

• Future excluded middle. \(\mathsf{F}\alpha\vee\mathsf{F}\neg \alpha\)

Moreover, let us not forget that Peirceanism predicts that all futurecontingents are false, against strong semantic intuitions. As aresult, it runs up against two very general and strictly relatedproblems, which also affect other approaches to future contingents.First, the so-calledassertion problem: if future contingentsare all false (or untrue) at their moment of use, why are we entitledto assert them? (see, e.g., Belnap & Green 1994; MacFarlane 2014:9; Besson & Hattiangadi 2014; Cariani 2021b: 11). Second, theproblem offuture skepticism: if no future contingent istrue, then we do not have any knowledge at all about future contingentmatters, for knowledge entails truth. But this skeptical conclusion isproblematic both with respect to our semantic intuitions and ongeneral philosophical grounds (see, e.g., Cariani 2021b, 2021a; Iacona2022).

For these and other reasons, nowadays most logicians and semanticistsregard Peircean semantics as inadequate (see, e.g., Belnap, Perloff,& Xu 2001: 160; Cariani 2021b: 4.1; MacFarlane forthcoming). It isfair to add, however, that the key motivations behind Peirceanism aremetaphysical rather than logical or semantical. Prior favored it asespecially well suited tono futurist approaches in theontology of time (see below,§4.1), and variants of Peirceanism have been recently defended on similargrounds (see Todd 2016a, 2021). Furthermore, there are attempts in therecent literature to defend Peirceanism, or related semantics, againstlogical and semantical objections (see, e.g., Iacona 2013; Todd 2021:3–4; De Florio & Frigerio 2024; Iacona & Iaquinto 2023).For instance, Todd (2021) has argued that principles likeRetrogradation are philosophically contentious, and their invalidityis actually a benefit of Peirceanism (see also Andreoletti &Spolaore 2021).

Ockhamist semantics is different from both the recursive TRL semanticsand Peirceanism, in that formulae are evaluated not just relative to agiven moment \(m\) but also to a history passing through \(m\). If, asis standard, we write “\(m/h\)” to indicate amoment-history pair such that \(m\in h\), the truth conditions offormulae of form \(\mathsf{F}\alpha\) and \(\mathsf{P}\alpha\) and canbe written as:

\((\mathrm{O}_{\mathsf{F}})\)

\(\mathsf{F}\alpha\) is (Ockham) true at \(m/h\) iff \(\alpha\) istrue at some \(m'/h\) with \(m \prec m'\).

\((\mathrm{O}_{\mathsf{P}})\)

\(\mathsf{P} \alpha\) is true at \(m/h\) iff \(\alpha\) is true atsome \(m'/h\) with \(m' \prec m\).

Then, in Ockhamist logic, the future and past operators have the samebehavior as in linear temporal logic. The historical possibilityoperator \(\Diamond\) expresses an existential quantification over theset of histories passing through the moment of evaluation:

\((\mathrm{O}_{\Diamond})\)

\(\Diamond \alpha\) is true at \(m/h\) iff \(\alpha\) is true at\(m/h'\) for some history \(h'\) passing through \(m\).^[10]

Like Peirceanism, Ockhamism is an indeterminism-friendly semantics inwhich both bivalence and the excluded middle hold. In addition, likelinear time logics, it validates many plausible principles, includingFuture excluded middle and Retrogradation. Its main downside is thatit is unclear how to apply it to future-tensed utterances (see Belnap,Perloff, & Xu 2001: 231–233; MacFarlane 2014: 9.4). To seewhere the problem lies, suppose you assert the following futurecontingent:

(1)

Tomorrow there will be a sea battle.

In order to apply Ockhamist semantics to your utterance, one mustselect both a moment of evaluation \(m\) and a history of evaluation\(h\). It is natural to identify \(m\) with the moment of (the contextof) use. But what about \(h\)? In general, \(m\) may belong todifferent histories, and Ockhamist semantics, as such, is silent onwhat the history parameter represents and how it has to be selected.But when assessing future contingents, choosing one history overanother can make a very big difference—the same difference asthere is between truth and falsity.

There are two general approaches to this problem in theliterature:

1. Stick to Ockhamist semantics and make explicit what thehistory parameter stands for (Belnap, Perloff, & Xu 2001; Iacona2014);
2. Use Ockham truth postsemantically, to define other,non-history-relative notions of truth.

Approach(b) is the most common today and we shall discuss it in the nextsubsection.

We conclude this subsection by observing that all the recursivesemantics for trees introduced here can be extended tobundledtrees in a natural way, by replacing the quantification overhistories with a quantification over the histories belonging to thebundle \({\mathcal{B}}\). Thus, the second-order quantification overthe set of all histories in the tree is replaced by a first-orderquantification overelements of \({\mathcal{B}}\). Given thecorrespondence between bundled trees and other set-theoreticalstructures considered above, the TRL, Peircean and Ockhamist truthconditions can easily be turned into truth conditions relative, e.g.,to Kamp frames or Ockhamist frames. In this context, it is worthobserving that Ockhamist frames are genuine Kripke frames for a modallogic with three modal operators: \(\mathsf{F}, \mathsf{P}\), and\(\Diamond\).

3.2 Postsemantic proposals

As seen in the previous subsection, it is unclear how to applyOckhamism to ordinary statements, for their moment of use may belongto more than one history. Postsemantic proposals tackle this problemby defining non-history-relative notions of truth in terms of Ockhamtruth.

Let us start with what was historically the first proposal to go inthis direction, namely, Thomason’s (branching time)supervaluationism (1970, 1984), which is based on vanFraassen (1966). At the core of Thomason’s approach lies asupervaluationist, moment-relative notion of truth (super-truth). Forour purposes, we can define super-truth relative to a moment of thecontext \(m_c\), which represents the moment of use (Lewis 1970;Kaplan 1989):

(Sup)

\(\alpha\) is super-true (super-false) at \(m_c\) iff \(\alpha\) isOckham true (false) at \(m_c/h\) for all histories \(h\) passingthrough \(m_c\).

Like the other postsemantic proposals we shall discuss,supervaluationism inherits all Ockhamist validities, such as theexcluded middle, the Future excluded middle and Retrogradation.However, unlike Ockhamism, supervaluationism is not bivalent: thereare sentences that are neither super-true nor super-false at theirmoment of use, namely, future contingents. Thus, on the one hand,supervaluationism agrees with the common philosophical view thatfuture contingents are neither true nor false at their moment of use,but on the other hand, it faces both the assertion problem and theproblem of future skepticism (see§3.1). Unlike Peircean semantics, supervaluationism does not go as far as tocollapse “will be the case” into “will be inevitablythe case” but, still, it collapses (super-) truth intoinevitable truth. Partly because of such “modal nature” ofsuper-truth, supervaluationism requires us to accept importantdepartures from standard logical practice (see, e.g., Fine 1975;Williamson 1994; Asher, Dever, & Pappas 2009). For instance, onecan doubt that super-truth is disquotational (i.e., that“\(\alpha\) is super-true” is equivalent to \(\alpha\))and so, that it is a legitimate notion of truth at all. Moreover, ifsupervaluationist consequence is defined in the most straightforwardway, based on (super-) truth-at-a-moment, we must give up pleasantmetavalidities such as the deduction theorem.

Relativism (MacFarlane 2003, 2008, 2014; see also Belnap,Perloff, & Xu 2001: 175) can be thought of as a generalization ofsupervaluationism. For this reason, virtually all we have said aboutsupervaluationism also applies to relativism. The key motivationbehind relativism is to reconcile two contrasting intuitionsconcerning future contingents (MacFarlane 2003: 323–325).Consider again(1) (“Tomorrow there will be a sea battle”) and suppose youassert it today. On the one hand, MacFarlane says, we have the(‘indeterminacy’) intuition that the truth-value of yourutterance is undetermined in its context of use, which MacFarlanecallscontext of utterance. On the other hand, if tomorrow wefind ourself in the middle of a sea battle, we will have the(‘determinacy’) intuition that your claim was definitelytrue in the context of utterance. If we stick with the view that truthis only sensitive to the context of utterance, MacFarlane argues, thedeterminacy intuition and the indeterminacy intuition are inconsistentwith one another and at best, we can only save one of them. MacFarlaneconcludes that, since we want to save both intuitions, we must give upthe view that truth is only relative to the context of utterance, andrecognize that truth is also sensitive to the context from which anutterance is assessed (context of assessment). If we acceptthis new form of context-sensitivity(assessment-sensitivity), we can reconcile our prima facieconflicting intuitions by saying that(1) is neither true nor false in its content of utterance relative to acontext of assessment located at the present moment, but it is truerelative to a context of assessment located tomorrow, in the middle ofa sea battle.^[11] For our purposes, we can identify the context of utterance and thecontext of assessment, respectively, with two moments \(m_u\) and\(m_a\) such that \(m_u\preceq m_a\). Now relativistic truth can bedefined as follows:

(Rel)

\(\alpha\) is relativistically true (false) at a context of utterance\(m_u\) and a context of assessment \(m_a\) iff \(\alpha\) is Ockhamtrue (false) at \(m_u/h\) for all \(h\) passing through \(m_a\).

Intuitively, a sentence is relativistically true at \(m_u,m_a\) if itscontent in context \(m_u\) is historically necessary at moment\(m_a\). Clearly, relativistic truth collapses into super-truth when\(m_u=m_a\). Relativism can be regarded as an improvement oversupervaluationism: among other things, it is more flexible, and it canbe smoothly combined with a wider variety of philosophical views (seebelow,§4.1). Relativism has also been regarded as a generalization of Ockhamism;in this view, Ockhamism is an extreme version of relativism, whereevery sentence is evaluated as if it were being assessed from theperspective of the end of time (Wawer 2020). Relativism has beenapplied, more or less successfully, to a number of semantical andphilosophical problems, and has given rise to a vast debate, which wecannot address here (see the entry onrelativism, §5 for an overview).

Above, we have introduced two variants of recursive TRL views, onerelative (or Molinist) and the other absolute. Exactly the samedistinction applies to TRL postsemantic views. InabsoluteTRL postsemantics, models specify a history TRL representing theunique actual history in the tree (Iacona 2014; Wawer 2016; Wawer& Malpass 2020). The relevant notion of truth is defined asfollows.

(AbsTRL)

\(\alpha\) is absolutely TRL true at the moment of the context \(m_c\)iff \(\alpha\) is Ockham truth at \(m_c/\TRL\).

This semantics preserves bivalence, but is silent on how to assessstatements made at moments that lie outside the thin red line. Thisfeature has appeared intolerable to some philosophers; e.g., accordingto Belnap, Perloff, and Xu (2001: 162), it is a “logical”defect that makes the theory useless. Absolute TRL theorists havereacted to this contention in two different ways: (i) arguing that itis ill-conceived; (ii) providing an absolute TRL semantics thataccounts for utterances made outside of the \(\TRL\). According to (i), thecontention is based on a controversial piece of metaphysics, i.e., theview that actual and merely possible moments all exist (see below,§4.1). Philosophers who reject this view, and subscribe to actualism (theview that only actual things exist, see again§4.1), have a principled reason to restrict their semantic account to speechacts performed at actual moments. After all, these are all the speechacts that there are (Wawer 2014, 2016; Wawer & Malpass 2020). Anaccount that goes in direction (ii) is thesupervaluationalTRL postsemantics proposed in Malpass and Wawer 2012 and Malpass 2013.This approach is based on the idea that, from an actualistperspective, the utterance of a sentence \(\alpha\) made at a moment\(m\) outside of the \(\TRL\) can be equated with a counterfactual of form“\(\alpha\) would be true if uttered at \(m\)”. If weassume that counterfactuals of this form are true (false) when\(\alpha\) is true (false) on all histories passing through \(m\), weget the following, disjunctive semantic definition:

(SupTRL)

\(\alpha\) is super TRL true (false) at \(m_c\) iff either \(\alpha\)is Ockham true (false) at \(m_c/\TRL\), or \(\alpha\) is Ockham true(false) at \(m_c/h\) for all histories passing through \(m_c\).

This definition allows one to overcome the limitations of (unamended)absolute TRL postsemantics without giving up the idea that thereexists a unique actual history. It does so, however, at the expense ofgiving up bivalence for assertions made outside of the \(\TRL\) and ofinheriting, at least in part, the non-standard logical behavior ofsupervaluationism.

Inrelative (orMolinist) TRL postsemantic views,like in relative recursive TRL semantics (see above,§3.1), models define a function \(\TRL()\) that maps each moment \(m\) to thecorresponding thin red line. Truth at the moment of context \(m_c\) isdefined in the obvious way:

(RelTRL)

\(\alpha\) is relatively TRL true at the moment of the context \(m_c\)iff \(\alpha\) is Ockham truth at \(m_c/\TRL(m_c)\).

This semantics does not share the limitations of its absolutecounterpart, preserves bivalence and is formally very well-behaved,but is not beyond criticism. For instance, according to MacFarlane(2014), it makes it difficult to accommodate counterfactualassessments. Let us to suppose that today, in context \(m_{c}\), Jackasserts:

(2)

Berkeley will be sunny tomorrow.

MacFarlane invites us to consider two alternative moments \(m\in\TRL(m_c)\) and \(m'\notin \TRL(m_c)\), both located tomorrow. At\(m\), Berkeley is sunny while at the counterfactual moment \(m'\),Berkeley is rainy. Statement(2) counts as true at \(m_c\) according to relative TRL postsemantics,for “Berkeley is sunny” is true on the day following\(m_c\) on \(\TRL(m_c)\). But now let us consider an assessor that islocated at the counterfactual moment \(m'\). Based on the definitionof relative TRL truth, the assessor should regard(2) as true at \(m_c\). But, MacFarlane concludes, “that seemswrong; the assessor has only to feel the rain on her skin to know thatJake’s assertion was inaccurate” (2014: 210). InMacFarlane’s view, TRL theorists cannot address this problem bymaking TRL truth sensitive to contexts of assessment withoutcollapsing their view into a form of relativism (see Wawer &Malpass 2020: 7.1, for a discussion). Convincing or not as thisobjection may be, it highlights the complexity of the notion ofactuality inherent in Molinist positions.

3.3 Branching time semantics in computer science

The execution of a computer program consists of a sequence of states,and this sequence occurs in a temporal order. Thus, in the context ofcomputer science (CS), time itself can be conceived as a set ofsuccessive states, which play the role of moments. These basicobservations lie at the foundation of the pioneering work Pnueli(1977), where a temporal logic is defined with the goal of describingand verifying properties of programs.

Many variants and extensions of Pnueli (1977)’s logic weresubsequently defined. A common aspect of these logics is that they arepropositional logics and that the semantical structures for them arebased on pairs \((S,R)\), where \(S\) is a set (of states) and \(R\)is a binary relation on \(S\). The intended reading of \(s R s'\) isthat \(s'\) is a successor state of \(s\), or that the program underconsideration transforms state \(s\) into \(s'\). A further element ofthis semantics is a labeling function \(L\), which assigns to eachstate the set of propositional variables that are assumed to be trueat that state.

In this framework, a (possibly endless) execution of a program isrepresented by an \(R\)-sequence, that is, a sequence \(s_0, s_1,\dots\) in which, for every \(i\), \(s_i R s_{i+1}\) holds. This meansthat \(R\)-sequences play the role that histories have in synchronizedBT frames. The controversy “tree vs bundled tree” can alsobe transferred to structures \((S,R,L)\): one may consider the set ofall \(R\)-sequences or just a set \(\Sigma\) of sequenceswith suitable closure properties (see, e.g., Stirling 1992). Thus, themost general structures have the form \((S,R, L , \Sigma)\).

There is a natural ordering between the elements of \(\Sigma\), which,according to the observations above, can be thought as a temporalordering. Given the sequences \(\sigma = s_0, s_1, \dots\) and\(\sigma' = s'_0, s'_1, \dots\), we write \(\sigma \lhd \sigma'\) ifthere exists \(i \gt 0\) such that, for all \(n\), \(s'_n = s_{n+i}\).That is, \(\sigma'\) can be obtained from \(\sigma\) by deleting thefirst \(i\) states. The relation \(\lhd\) is trivially transitive.Other possible properties of \(\lhd\) depend on the relation \(R\) andon the sequences in \(\Sigma\).

Also the languages for temporal logics of programs have many variantsin the literature. The most common language is apparently apropositional language with the binary operator \(\mathsf{U}\)(Until). Write \(\mathcal{L}_{\mathsf{U}}\) for thislanguage. In ordinary (Priorean) temporal logic, the formula\(\alpha\, \mathsf{U} \, \beta\) is true at the moment \(m\) whenever,for some moment \(m' \succ m\), \(\beta\) is true at \(m'\), and\(\alpha\) is true at every \(m''\) such that \(m \prec m'' \prec m'\).^[12]

Given a structure \((S, R, L, \Sigma)\), the formulas of\(\mathcal{L}_{\mathsf{U}}\) are recursively evaluated at elements of\(\Sigma\). A propositional variable is true or false at \(\sigma =s_0, \dots\) according to whether it belongs to \(L(s_0)\) or not.Boolean connectives are interpreted in the usual way. The truth valueof \(\alpha\, \mathsf{U} \beta\) at \(\sigma\) is determined on thebasis of the semantics for \(\mathsf{U}\) considered in the previousparagraph, with the relation \(\prec\) replaced by \(\lhd\). Thus, inorder to make sense of this, an obvious closure property for\(\Sigma\) is that: if \(\sigma, \sigma' \in \Sigma\) and \(\sigma\lhd \sigma'' \lhd \sigma'\), then \(\sigma'' \in \Sigma\). The logicbased on this or similar semantics is known asLinear TemporalLogic (LTL). In fact, the truth value of a formula at a given\(\sigma\) depends only on the truth values of other formulas atsubsequences of \(\sigma\), which are linearly ordered by \(\lhd\).LTL has proven useful for describing and verifying properties ofprograms (Emerson 1990).

In order to deal with different computations simultaneously, othertemporal logics have been defined, which are also based on structures\((S, R, L, \Sigma)\). Among these, the most popular ones are theComputational Tree Logics CTL and CTL\(^*\) (Emerson &Clarke 1982), which can be viewed as the computer science counterpartsof Peircean and Ockhamist logics. In particular, the language forCTL\(^*\) includes a unary operator, denoted by \(E\) or \(\exists\),corresponding to the Ockhamist \(\Diamond\) discussed above. Both LTLand CTL turn out to be fragments of CTL\(^*\). There is, instead, noinclusion relation between LTL and CTL. There are properties ofprograms that can be expressed in LTL but not in CTL, and, conversely,there are CTL-expressible properties that cannot be expressed inLTL.

4. Philosophical views on branching time

This section deals with philosophical debates concerning branchingtime and the BT conception. In§4.1, we classify versions of the BT conception depending on theirassumptions about time, modality, and the future; moreover, we brieflydiscuss the connections between these positions and the semantic viewsoutlined in§3. In§4.2, we introduce some general challenges to the BT conception.

4.1 Varieties of the branching time conception

The philosophical debate on branching time is strictly connected totraditional debates on the nature of time and modality (see theentries ontime andpossible worlds for further information). The debate about time is traditionallyframed in terms oftimes and not ofmoments.However, when a specific history is at stake, times can be identifiedwith moments. In this way, different views in the philosophy of timecan be made to correspond to different interpretations of BTframes.

There are two main positions in the debate, known as (standard)A-theory (ortense realism), andB-theory.For our purposes, we may represent their disagreement as concerningthe following principles:

• Temporal orientation. the present time is eithermetaphysically privileged over all the other times (past or futuretimes) or it is the only time that exists.
• Temporal neutrality. all times, present, past andfuture, are metaphysically on a par (i.e., they are entities of thesame kind and none of them plays an objectively prominent role).

A-theorists subscribe to temporal orientation.^[13] As a consequence, they hold that tensed notions such aspresent,past andfuture can be used todenote objective, non-perspectival (and non-indexical) features ofreality. When we say that an event is present in this sense, we areattributing to it an objective feature, just as when we say that it istemporally extended. The present time is not just the temporalcoordinate where we are located but it is also the correct vantagepoint on the whole of reality: what is the case from our presentperspective is what isabsolutely the case.

In some philosophical quarters, the A-theory is also indicated as“presentism”. In contemporary philosophy of time, however,presentism is an ontological position: the view that onlywhat is present exists (see the entry onpresentism). The main ontological alternatives to presentism areeternalism, the view that past, present and future things allexist (see Williamson 2013; Cameron 2015; Deasy 2015), and thegrowing-block theory, the view that only past and presentthings exist (see Broad 1923; Tooley 1997; Briggs & Forbes 2012;Correia & Rosenkranz 2018). Presentism and the growing blocktheory are different forms ofno-futurism, the view thatfuture things do not exist.

B-theorists are temporally neutral. As a consequence, they regardtensed notions as purely perspectival: there is nothing like beingabsolutely present, or future, for the same reason that there isnothing like being absolutely here, or three miles ahead. When we saythat an event is present, we are just saying that it is (roughly)simultaneous with our utterance. There is no privileged vantage pointwithin the timeline, and reality is best represented from a tenselessviewpoint, blind to the distinction between (absolute) present, pastand future. Temporal neutrality is only consistent with an eternalistontology.

The debate about modality can be characterized in a similar way. Itincludes two main positions, which are traditionally characterized interms of (possible)worlds (and not histories):

• Modal orientation. the actual world is eithermetaphysically privileged over all the others or it is the only onethat exists.
• Modal neutrality. all worlds are metaphysicallyon a par.

The distinction between modal orientation and modal neutrality, unlikeits temporal counterpart, cannot be immediately applied to the debateabout BT, for it is not obvious how worlds relate to histories. Moreprecisely, while B-theorists can safely equate worlds with histories,many A-theorists would resist the identification. For instance,presentists think of the actual world as consisting of (what we havecalled) a moment and not a whole history. Thus, it is best tointegrate the distinction between modal orientation and modalneutrality with a further distinction, which is specific to the debateon BT:

• Forward orientation. the actual future is eithermetaphysically privileged over all the others or it is the only onethat exists.
• Forward neutrality. all futures aremetaphysically on a par.

Most philosophers subscribe to modal orientation. As a consequence,they think that modal words like “actual” are not justindexical expressions and can also be used to indicate objective,non-perspectival features of reality. Modally oriented philosopherswho subscribe to temporal neutrality (B-theory) are also committed toforward orientation, for the (tenseless) existence of a unique,complete actual history entails the existence of a unique actualfuture. Conversely, in the debate on BT, modally oriented philosopherswho subscribe to temporal orientation (A-theory) generally subscribeto forward neutrality (although their view is also consistent withforward orientation, see§4.1.2 below).

A few philosophers—most famously, David Lewis—are modallyneutral and regard modal notions such asactual andpossible as purely perspectival in nature. Among BTtheorists, the most prominent modally neutral philosopher is NuelBelnap.

From an ontological viewpoint, modal orientation is consistent withtwo views:actualism (only actual things/worlds exist) andpossibilism (all possible things exist). Modal neutrality isonly consistent with a possibilist ontology. Similar ontologicalconsiderations apply to forward orientation and forwardneutrality.

To summarize, there are two opposite stances concerning time,modality, and possible futures in the debate on branching time:oriented and neutral approaches. In general, oriented approaches canbe further classified on the basis of their ontology. To illustratethese distinctions, consider the contrast between the two treediagrams inFig. 8.

Legend

(a) left tree

(b) right tree

Figure 8: Two possible views (modal BTrealism and “pruning”). [Anextended description of figure 8 is in the supplement.]

The diagram on the left represents a view consistent with temporal,modal and forward neutrality: all moments in the tree aremetaphysically on a par, and what we can call present, or actual,entirely depends on our “local” experiential perspective.The diagram on the right represents a view consistent with modalorientation, temporal orientation, and forward neutrality, in whichpast actuality and presentness are absolute (not only perspectival),and all future branches exist and are on a par, but past branches goout of existence as time goes by. We shall discuss both views in a fewlines.

4.1.1 Branching time realism and Branching Space-Times realism

We shall speak ofbranching time realism (BT realism) toindicate modally neutral variants of the BT conception. BT realism isconsistent with the idea that all futures departing from the presentmoment obtain (i.e., are absolutely actual) and each single thing,including ourselves, is constantly splitting into countless duplicatesof itself. Conceptions of this sort are popularly associated with theEverettian ormany-worlds interpretation of quantum mechanics(on Everettian quantum mechanics, see the entries onEverettian quantum mechanics andthe many-worlds interpretation of quantum mechanics). We collectively refer to them asstrong BT realisms. StrongBT realisms have never been rigorously developed or seriously defendedby analytic philosophers. In the literature on BT, strong BT realismis generally discussed in connection with the B-theory, underdifferent labels:B-theoretic branching (Barnes & Cameron2011: 10);‘block multiverse’ (Pooley 2013: 339);realism about many futures (Meyer 2016: 206);naïveBT realism (Wawer 2016: 8);many worlds view (Spolaore& Gallina 2020: 108);actual branching (Ninan 2023: 478).However, strong BT realism can also be joined with temporalorientation (A-theory), resulting in a tensed variant of strong BTrealism (this is, arguably, the position outlined in Abruzzese 2001).In strong tensed BT realism, each history is actual and is endowedwith a privileged, “locally present” moment. If we thinkof all these “locally present” moments as lying on thesame instant (see above,§2.1), we can illustrate this view as inFig. 9.

(a) first moment

(b) next moment

Figure 9: Strong tensed BT realism (atsuccessive moments). [Anextended description of figure 9 is in the supplement.]

The only version of BT realism that has been clearly defined anddefended so far in the literature is due to Belnap and his co-authors.We may call itmodal BT realism (Fig. 8, left diagram). One of its key assumptions is the Aristotelian notionthat “talk of possibilities only makes sense before a contrastbetween possibility and actuality” (Belnap, Müller, &Placek 2022: 3). By modal neutrality, this means that we can onlyspeak of possibility and actuality in aperspectival sense,that is, from a perspective internal to the tree. More precisely, talkof (historical) possibility only makes sense when amoment isfixed, and talk of actuality, when both a momentand ahistory are fixed. What is possible istemporally local (itis relative to a moment) and what is actual is alsohistorically local (it is relative to a moment/history pair).If we identify actuality with the realm of whatobtains (asopposed to whatcan possibly obtain), we can also say thatthere is no absolute obtainment: facts and events obtain only relativeto moment-history pairs. From a global or “external”perspective, it makes no sense to distinguish actuality andpossibility, that is, the things thatdo obtain from thosethatcan obtain. Again from a global perspective, we can alsosay that all moments in the tree are equally concrete, for all playsimilar causal roles and there is no intrinsic difference betweenthose that are (perspectivally) actual and those that are merelypossible.

As seen before (§2.3), in a series of works spanning from Belnap (1992) to Belnap,Müller, and Placek (2022), Belnap and his coauthors havedeveloped Branching Space-Times (BST) frames. Just like BT frames, BSTframes allow for different interpretations, including a realistconception akin to BT realism (“BST realism”). The onlyvariant of BST realism discussed in the literature is due to Belnapand his coauthors, and we shall call itmodal BST realism. Inthis view, all that BT realists say about moments also holds for pointevents (and other spatially limited events). From a globalperspective, again, it makes no sense to distinguish what point eventsare actual and what are merely possible, and all point events can beregarded as equally concrete.

In what follows I will try to avoid indexical language. In particular,I will not draw a distinction (inevitably indexical when notrelational) between the actual and the possible except in motivatingor giving examples. ‘Possible point events’ are thus just‘point events’. These point events are to be taken not asmere spatiotemporal positions open for alternate concrete fillings,but as themselves concrete particulars. (Belnap 1992: 388)

There are several objections to (some versions of) BT realism in theliterature. Some of these objections point to conflicts between BTrealism and common sense presuppositions. Here is David Lewis:

The trouble with branching […] is that it conflicts with ourordinary presupposition that we have a single future. If two futuresare equally mine, one with a sea fight tomorrow and one without, it isnonsense to wonder which way it will be—it will be bothways—and yet I do wonder. (Lewis 1986: 207–208)

Lewis’ objection has different bite against strong BT realismand against Belnap’s modal BT realism. Strong BT realists haveno reason to worry about the objection: they can simply agree that thefuture will be both ways, although only one of them will be part ofour experience. On the other hand, Belnap and other modal BT realistsreplied to Lewis’s objection that it rests on a misguidedunderstanding of their position. They grant that, if tomorrow’ssea fight is contingent, the present situation will evolve in one wayin some history and in the otherin some differenthistory, but deny that we can simply drop the relativization tohistories and conclude that “it will be both ways”.

Lewis misdescribes the theory of branching time in saying of such asituation that “it will be both ways.” Branching time isentirely clear that “Tomorrow there will be a sea fight andtomorrow there will not be a sea fight” is a contradiction.[…]Given indeterminism, it does not suffice to think oftruth (or denotation, etc.) as relative only to moments.[…] One must relativize truth to the history parameter as well.(Belnap, Perloff, & Xu 2001:225)^[14]

This reply is technically correct: modal BT realism does not entailthat there will both be and will not be a sea fight tomorrow. However,Lewis has a point when he contends that BT realism clashes withordinary presuppositions. Consider again the issue whether tomorrowthere will be a sea fight. We commonly presuppose that as of tomorrow,this issue will be settled in asingle andabsoluteway, not just in different ways relative to different futures.However, this is a presupposition BT realists are bound to reject (butsee Belnap, Perloff, & Xu 2001: 205–206). For discussions ofother putative counter-intuitive consequences of BT realism, seeBelnap, Perloff, and Xu (2001: 7B.2A), and Cameron (2015: 5.2).

According to another family of objections, BT realism entails that ouruniverse is (in some metaphysical sense) entirely determinate, ordeterministic. For instance, Barnes and Cameron (2009) argue that, inB-theoretical BT realism, “it’s perfectly settled howthings will be, you just don’t know whereabouts you’ll bewithin reality”. In the same vein, Benovsky (2013) contendsthat, in BT realism, “the futures are all there, they are allfixed because they ‘already’ exist”. Somethingsimilar to what just said about Lewis’s objection also holdshere. These objections have no bite against strong BT realistsinspired by the Everettian interpretation, which is fundamentallydeterministic (see below,§4.2). As formodal BT realists, they would reply that there is nosense of “settled” or “already” in which, intheir view, the future issettled or all futuresalready exist.

For other objections against BT and BST realism see Earman (2008) andthe replies in Placek and Belnap (2012).

4.1.2 Actual futurism

Let us speak ofactual futurism to indicate modally andforwardly oriented variants of the BT conception.

Actual futurism is the most common B-theoretical position (seeFig. 10). B-theoretical actual futurism is usually combined with actualism:only the course of events corresponding to the actual history exists(Wawer 2014; Wawer & Malpass 2020). Actual futurists who subscribeto actualism regard BT frames as useful tools for representing themodal properties of actual individuals and events, but do not regardthem as literal representations of reality. In their view, thinking ofthe actual history and merely possible histories as entities of thesame kind is a little like taking Sherlock Holmes for a real person.Alternatively, one can subscribe to a possibilist version of actualfuturism, modeled on Bricker (2006)’s position in themetaphysics of modality: all courses of events represented in the treeexist, but some of them, those corresponding to the actual history,are metaphysically privileged (Borghini & Torrengo 2013). Let usemphasize that all these actual futurists regard the actual history asunique or privileged only from a tenseless, “end of time”perspective, and they agree that,as of the present time,many futures are possible and are all on a par. B-theoretical actualfuturism is by far the most debated variant of actual futurism. Itscritics from both the A-theoretical and the BT realist camp oftencontent that it does not capture a deeper, metaphysical sense in whichthe future is indeterminate or “open” (see, e.g., Barnes& Williams 2011; Barnes & Cameron 2011; Cameron 2015; andBelnap, Perloff, & Xu 2001: 6).

(a) actualist

(b) possibilist

Figure 10: B-theoretical actualfuturism: actualist (left) and possibilist (right) variants. [Anextended description of figure 10 is in the supplement.]

Forward orientation can also be joined with temporal orientation,resulting in atensed version of actual futurism. Accordingto tensed actual futurism, one history (the actual one) ispresently privileged over all the others, or it is the onlyone that exists. This view is internally consistent, but is in tensionwith indeterminism, and it has never been extensively defended in theliterature (see De Florio & Frigerio 2022 for a discussion).Tensed actual futurism also admits of a “dynamic” variant,known asmutable futurism, in which the privileged future canchange across time. Mutable futurism is clearly consistentwith indeterminism, but it requires us to make sense of the puzzlingidea of a changing future (for discussions, see Todd 2016b andAndreoletti & Spolaore 2021).

4.1.3 Asymmetry views

Let us speak ofasymmetry views to indicate the variants ofthe BT conception that combine forward neutrality with modalorientation. In asymmetry views, there exists a unique, absolutelyactual past, but no actual future. All asymmetry views areA-theoretical in character, and can be distinguished on the basis oftheir ontology.

Inno-futurist asymmetry views, possible futures areconceived as abstract or general entities (e.g., Briggs & Forbes2012; Todd 2021; Rosenkranz 2012; Rumberg 2016b). This was arguablyPrior’s (1967a) own position (see, e.g., Hasle &Øhrstrøm 2016: 3411). Like actual futurists,no-futurists tend not to regard BT frames as literal representationsof reality. A growing-block variant of this view is represented inFig. 11.

(a) first moment

(b) next moment

Figure 11: A growing-block asymmetryview (at successive moments). [Anextended description of figure 11 is in the supplement.]

Ineternalist asymmetry views, futures are concrete coursesof events. The most debated eternalist asymmetry view is the so-calledpruning conception of branching time (McCall 1994, seeFig. 12). In the pruning conception, the only moments representing existingevents are the unique present moment and those that are\(\prec\)-connected with it. Intuitively, the idea is that the wholetree changes over time: as presentness “moves” towards thefuture, all the moments that do not lie on its path go out ofexistence:

Of all the possible futures […] one and only one becomes“actual”, i.e. becomes part of the past. The otherbranches vanish. The universe model is a tree that “grows”or ages by losing branches. (McCall 1994: 3)

(a) first moment

(a) next moment

Figure 12: Pruning (at successivemoments). [Anextended description of figure 12 is in the supplement.]

Inpossibilist asymmetry views, all possible moments exist,all past moments are actual, but only the unique present moment andits past are absolutely actual. Intuitively, the idea is that of a‘moving dot’ of presentness that travels along the treetowards the future, leaving a trace of actuality (for discussions, seeBarnes & Cameron 2011; MacFarlane 2014: 212); Wawer 2016:170–172).

The main objections to asymmetry views essentially coincide with themain objections to the A-theory of time: both would introducepostulates, such as the existence of an absolute present and of verystrong temporal asymmetries, which lack scientific justification inlight of present-day physics. On their side, the advocates ofasymmetry have a variety of traditional A-theoretical replies to theseobjections at their disposal, spanning from strong rebuttals (e.g.,insist that the first-person evidence for their view that comes fromour conscious experience outweighs any scientific and methodologicalconsideration one may label against it) to more modest defenses (e.g.,suggest that for all we know, future developments in physics couldultimately favor the A-theory). For overviews of the debate on thephysical respectability of the A-theory, see the entries onpresentism (§8) andtime (§11).

4.1.4 Semantics and metaphysics

Let us conclude this subsection by outlining some natural connectionsbetween the versions of the BT conception discussed so far (see Table1) and the branching time semantics introduced in the previous section.Recognizing these connections helps to better understand and assessmetaphysical or semantical choices, but natural connections should notbe confused with logical entailments: the choice of a semantics doesnot need to determine the choice of a metaphysic, and vice versa (seeWawer 2016: 6.5, for a discussion of this point). Looking back at thesemantics introduced in§3, we can distinguish between two general strands. On the one hand, wehave semantics in which bivalence is preserved and there are truefuture contingents. In these semantics, we can say that one future isalethically privileged over all the others, i.e., the futurewhere all true future contingents are true. The advocates of thesesemantics are generally calledtrue futurists. Standard TRLviews are true futurist semantics in this sense. Philosophers whooppose true futurism, and think that all possible futures arealethically on a par, are calledopen futurists. Paradigmaticopen futurist semantics include supervaluationism, relativism andPeirceanism.

• True futurism. One future is alethicallyprivileged over all the others.
• Open futurism. All futures are alethically on apar.

Forward neutrality, the view that no future is metaphysicallyprivileged over all the others, tends to correlate with open futurism.If there is no unique actual future, one is naturally led to think,then there is nothing against which future contingents can be assessedas true. As just seen, forward neutrality is consistent with two kindsof positions: BT realism and asymmetry views. Let us start with thelatter.

Most advocates of asymmetry views subscribe to both future neutralityand open futurism (e.g., Prior 1967a; Rhoda 2011; Todd 2016a, 2021).However, a few A-theorists disagree on this point and combine futureneutrality with true futurism (Westphal 2006; Rosenkranz 2012; seealso Baia 2012; Wawer 2016; and Correia & Rosenkranz 2018).“Deviant” combinations like these, in which semantics andmetaphysics seem to pull in different directions, are generally basedon non-standard views concerning truth. More specifically, bothWestphal and Rosenkranz give up the traditional principle that truthis grounded in (or supervenes on) reality.

BT realists tend to be open futurists, but unlike the advocates ofasymmetry views, they reject the very notion of absolute actuality asmeaningless, or empty. For this reason, they tend to favor differentsemantic approaches. From a BT realist perspective, as Belnap,Perloff, and Xu (2001: 6B.7) have stressed, future contingents aresemantically similar to (contingent) open formulae: just like onecannot evaluate “\(x\) is red” without specifying anassignment of value to \(x\), so one cannot sensibly assess(1) relative to a moment alone. Thus, future contingents can only besensibly assessed relative to points of evaluation where theirtruth-value is settled. For this reason, Belnap, Perloff, and Xu(2001) adopt Ockhamist semantics and insist that future contingentscan only be evaluated when both a momentand a historyparameter are provided. However, this is not the only choice availablefor BT realists—for instance, they can also subscribe to someform of relativist semantics (MacFarlane 2003, 2008).

Generally, actual futurists adopt some form of true futurist semantics(e.g., Malpass & Wawer 2012; Wawer 2014; see also Andreoletti& Spolaore 2021). This is a natural choice: if one holds that theactual future exists, one is led to think that future-tensedstatements are to be assessed against it, and vice versa. But it isnot a forced one, and some actual futurist favors Ockhamist semanticsinstead (Iacona 2014).

4.2 General challenges to the branching time conception

So far, we have discussed the main variants of the BT conception andsome objections that may be labeled against each of them. In thissubsection, we briefly introduce a few, very general objections, whichapply to most or all variants of the BT conception. These are generalchallenges to the very idea that trees, however understood, are bettersuited to represent our indeterministic universe than alternative,non-tree-like structures. Here, we focus on challenges concerning twokey tenets of the BT conception:forwardbranchingandNo backward branching.

Forward branching is the view that our indeterministicuniverse and its tense-modal features are best represented in terms ofa plurality of possible futures following a certain specificmoment/event. This conception of indeterminism has been calledAristotelian in Placek and Belnap (2012) (see alsoMüller 2012). In the Aristotelian conception, indeterminism is amodal notion, having to do with future-directedpossibilities. The Aristotelian conception can be contrasted withother approaches, in which indeterminism is characterized in terms ofsimilarity and divergence between different models (see, e.g.,Montague 1974; Earman 1986) or different worlds (see above,§2.2; see also Lewis 1986), or by resorting to a primitive notion ofindeterminacy (see, e.g., Barnes & Cameron 2009). Let us note thatforward branching presupposes that, in some sense, differentpossibilities can share certain objects or events. This view is partof the standard Kripkean (1963, 1980) approach to modality, in whichthe domains of different worlds may include one and the same entity.In fact, BT frames have also been used by philosophers such as Prior(1960) and J. L. Mackie (1974) to vindicate the standard approach, byoffering reasonably clear sufficient conditions for an entity toparticipate in alternative courses of events (see P. Mackie 2006 for adiscussion of these attempts; see also Placek & Belnap 2012 andthe entry ontransworld identity). The idea that different worlds may share parts (in a literal, realistsense) has been famously criticized by Lewis (1986: 4.2), whomaintained that distinct possible worlds have no object or event incommon, that is, are all pairwise mereologically disjoint (seemereology). This Lewisian conception of possible worlds is calleddivergence.

A metaphysical objection to forward branching is due to Barnes &Cameron (2011), who contend that, by resorting to trees, we cannotmodel contingent “doomsday” situations, where it isindeterminatewhether anything will happen at all.

The problem is that the absence of further branches from a nodedoesn’t represent the further open possibility that nothing willhappen beyond that node, it simply represents the absence of furtheropen possibilities. (2011: 15)

Let us call this thedoomsday objection to branching time.From a technical viewpoint, the objection is based on the fact thathistories, as standardly defined, are maximal sequences of moments(see above,§2.1), and thus no history can properly include another history. Standard asthis definition may be, however, it is unclear whether is forced uponBT theorists. Arguably, the doomsday objection has weight only againstBT theorists who subscribe to amoment-based conception ofhistories, in which histories are completely determined by theirconstituent moments (see Andreoletti 2022), as opposed to views inwhich histories are not defined in terms of moments, such as, forinstance, in van Benthem’s (1999) geometrical approach to BT(see above,§2.1). Moreover, the objection assumes that philosophers have a univocal,shared notion of what a contingent doomsday would consist in, which iscontentious (see Todd forthcoming).

Many BT theorists look to quantum mechanics as a core motivation forsubscribing to indeterminism and, consequently, to forward branching(see, e.g., Placek & Belnap 2012; Belnap, Müller, &Plackek 2022). The underlying reasoning is that quantum indeterminacyentails indeterminism, and indeterminism is best modeled by resortingto forward branching. This train of thought may be called thequantum argument for forward branching. Initially plausibleas it may be, the quantum argument has at least three problems.Although neither of these problems is untreatable (and arguably,mature BT conceptions such as modal BT/BST realism have the resourcesto address them), they should be taken into account.

The first problem is that not every interpretation of quantummechanics is indeterministic. More specifically, the many-worldsinterpretation, in which the forward branching process is taken atface value, is deterministic at the fundamental level, and all theindeterminacy (if any) posited by the theory is restricted toderivative aspects of the ontology (see, e.g., Wilson 2020). Moreover,according to some philosophers, the ontology of the many-worldsinterpretation is also consistent with a divergence (rather thanbranching) conception of worlds (Saunders 2010; Wilson 2012). Thus, ingeneral, philosophers who subscribe to the quantum argument should beclear as to what interpretation of quantum mechanics they arepresupposing and how it accounts for the branching process.

Secondly, one can doubt that quantum indeterminacy can be interpretedas unsettledness between pairs of determinate (classical) states ofaffairs, and some authors (e.g., Skow 2010; Darby 2010) haveindependently argued that it cannot (but see Mariani, Michels, &Torrengo 2024 for a different view).

Thirdly, the relation between quantum indeterminacy and dynamicalindeterminism is not obvious, and most scholars tend to agree that, atleast in principle, they are independent notions (see Calosi &Mariani 2021 for a recent review of the debate on quantumindeterminacy). Thus, the advocates of the quantum argument should bevery clear as to the connections they posit between quantumindeterminacy and indeterminism.

Let us turn to the second potentially problematic aspect of the BTconception, i.e., No backward branching, according to which eachmoment has a unique (possible) past. As mentioned above, the mainmotivations for the principle are philosophical: keeping as clear andneat as possible the distinctions between (i) particular events andrepeatable (qualitative or informational) states, and (ii) historical(objective) modality and epistemic modality.

Against No backward branching, one can argue that the time asymmetryit introduces lacks physical justification, for at least two reasons(see Earman 2008; Farr 2012). Firstly, because under certain naturalassumptions, all known fundamental laws of physics are time-reversalinvariant. Thus, in a very standard sense of indeterminism, ouruniverse is indeterministic in both temporal directions: assuming allpresent events are kept fixed, it is indeterminate what events willhappen in the futureand also what events had happened in thepast. Secondly, because the time asymmetries that we observe in ourexperience and in science are commonly explained in terms of anasymmetry in the boundary conditions of the universe and not of afundamental asymmetry in the laws of nature (see Albert 2000; Loewer2012; Kutach 2013; Farr 2022). If so, in the words of Farr (2012:114):

there is insufficient reason to think that [the observableasymmetries] constitute, or provide evidence of the existence of, atime-asymmetric modal structure of the world.

BT theorists can reply to both these objections. On the one hand, theycan insist that time reversibility is relevant only when the focus ison qualitative or epistemic states as opposed to particular, nonrepeatable events. In their view, the very idea of two alternativeparticular events temporally or causally followed by a uniqueparticular event is absurd.

No sense can be made of two alternative possible evolutions, separatebefore some event and combining into a single evolution after it.(Placek & Belnap 2012: 465)

On the other hand, BT theorists can subscribe to a primitivistconception of the direction of time, according to which observabletime asymmetries depend on a fundamental asymmetry in the laws ofnature (see Maudlin 2002, 2007).

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Acknowledgments

This entry is dedicated to the memory of Nuel Belnap.

We are grateful to an anonymous reviewer and to Giacomo Andreoletti,Fabrizio Cariani, Ciro De Florio, Andrea Iacona, Aldo Frigerio,Cristian Mariani, Angelo Montanari, Thomas Müller, ElisaPaganini, Tomasz Placek, Antje Rumberg, Patrick Todd, GiulianoTorrengo, and Jacek Wawer for useful comments and suggestions.