Logical Objections
As we have seen in Sect. 2, there were numerous attempts to prove the TRL theory logically faulty. I will first confront my solution with the most natural worry and then I will show that it is to a large extent immune to this sort of trouble.
A Simple Case
One might claim that the TRL theory presented here is just a version of the TRL
1 discussed in Sect. 2.2.1, that is, a theory that singles out a unique history and closely ties the interpretation of the “It will be the case” (F) connective with this very history. It is partly right, so let me reconsider the criticism of this particular version of the TRL theory presented in Sect. 2.2.2.
The objection naturally splits into two, the more abstract one and the more specific which asks for interpretation of particular sentences of natural language. The latter derives from the former which I will confront first. It is phrased as follows: “Branching + TRL has the defect that it gives no account of the future tense relative to moments that do not lie on TRL” (Belnap and Green 1994, p. 379).
In the framework, we are working with, the question is ambiguous since it contains the word “moments”. We need to carefully distinguish moments of context from moments of evaluation. These two notions are co-extensive for branching concretist, but might diverge in other presemantics. In general, the strength of the argument crucially depends on which of the meanings we have in mind. If “moments” refers to moments of evaluation, it is a fair objection; any respectable semantics should provide some reasonable interpretation of a language at every point of evaluation; TRL
1 does not do it, hence it is not acceptable. Observe that my version of the TRL theory does not face this difficulty since it utilizes a decent Ockhamist semantics which provide a (history-dependent) treatment of every sentence at every point of evaluation. However, if the word “moments” in the quote refers to moments of context of speech acts, it is by far an objection that begs the question since it presupposes branching concretism (all possible moments are accessible contexts of use) and then demands a proper analysis of speech acts being made at all possible moments.Footnote 19 This is exactly where the TRL-ist should object by saying that speech acts are concrete events and they happen in our world only and our world is not a branching structure.
Now, let us turn to the examples of sentences which TRL theory is supposed to be unable to interpret:
The coin will come up heads. It is possible, though, that it will come up tails, and then later (*) it will come up tails again (though at that moment it could come up heads), and then, inevitably, still later it will come up tails yet again. (Belnap and Green 1994).
The sentence might be naturally translated into the formal language to \(Fp\wedge\diamondsuit F(q\wedge\diamondsuit Fp\wedge F(q\wedge \square Fq))\) (where p means “The coin is landing heads” and q means “The coin is landing tails”). Now, we can apply our TRL theory to evaluate this particular prediction made at a given context. The TRL postsemantics proposed in Definition 14 (p. 23), the presemantic assumptions made at page 21, and semantic transformations described in Definition 2 (p. 5) guarantee a very non-controversial result. (I leave to the inquisitive reader the straightforward, but rather laborious computation.) I should stress that the procedure provides very natural truth conditions for sentences evaluated in or out of TRL
h
. Sentences are used in the actual world only, but the content of these sentences might appeal to various possible circumstances and we have a natural way to ascribe truth values to sentences in these circumstances.
Let us now consider the second example:
Had things gone otherwise, Jack would have asserted the following ‘It will (eventually) rain.’ Given the context of Jack’s assertion, the TRL is no longer able to guide us in understanding his reference to his future. (Belnap et al. 2001, p. 162)
This one is much more demanding. The authors use a counterfactual construction to move the point of evaluation away from the context at which the initial sentence was used. Then, they quote Jack’s (possible) assertion made at the switched evaluation point and ask about the interpretation of this assertion at the switched point. To give an appropriate semantic treatment of such examples of direct speech, Belnap et al. (2001, p. 174) and Belnap (2002, p. 44) devised the operators which shift the context of use in the process of evaluation (the operators they propose are “α asserts ‘A’” and “Truly utters(t
1, t
2, ‘A’)” respectively).Footnote 20 In particular, the context of use might be shifted to some unactualized circumstance. This idea is perfectly consistent with branching concretism, but it conflicts with the TRL theory since the latter states that there are no contexts besides those which initialize moments on TRL
h
.
Especially interesting cases of direct speech, in context of our discussion, are possible predictions. For example, even though I have not just used the sentence “I will eat dinner soon,” I really could, as the TRL-ist is happy to admit, have used the sentence just now (e.g. to inform my wife who is sitting next to me). So far, we understand when a use of a sentence is true at a context, but what does it mean for a possible use to be true? Formally speaking, we need to provide some analysis of the construction “α truly uses ‘ϕ’ ” or similar. A part of the task of such analysis is to provide the truth conditions for conditionals like: “Had things gone otherwise, I would have truly used the sentence ‘I will eat dinner soon.’”Footnote 21
Belnap et al.’s objection suggests that a correct analysis is out of range of the TRL-ist due to the shortage of contexts. It seems to me, however, that the authors’ context-shifting technique is not the only way to understand such phrases. An alternative idea is to construct a translation ϕ
t of a possibly-used sentence ϕ such that any sentence ψ
t, used at TRL-acceptable context c has the same truth conditions as ψ, where ψ
t is a sentence in which every appearance of “α truly uses ‘ϕ’ ” is replaced by ϕ
t. One needs to be very careful in this process, especially coping with indexical expressions in ϕ. However, I believe this procedure to be available even for such indexically loaded, possibly used sentences as: “I will actually talk to you tomorrow.” Consequently, the project which would give account of the truth values of possible uses of sentences within the TRL setting is not, in principle, doomed to failure. I acknowledge that it is a very important open problem and it needs to be solved for the TRL theory, as presented here, to be adequate. However, I am not going to tackle it at this juncture.
The General Case
The aim of this section is to show that the TRL solution that I propose is in fact semantically equivalent to the conservative proposal of Belnap et al. (2001). It means that the set of truths generated by both these semantics are exactly the same. As a result, no logical objection of the sort presented in Sect. 2 applies to the TRL theory proposed by me. The theory is simply immune to the “logical attacks.”
To proceed with the process of comparison (and differentiation) of the theories, I need to introduce a few general semantic definitions. So far, the only notions we were concerned with were the truth of a sentence at a point of evaluation and the truth of a use of a sentence at a context. Since we use two parallel notions of truth (\(\vDash\) and \(\Vdash\)), the process of generalization is also twofold. We have already seen what it means for a sentence to be true at a point of evaluation \({\mathfrak{M}},m/h\) in Definition 2. Let me introduce some generalizations of \(\vDash\):
Definition 15
(Truth in a model
\({\mathfrak{M}}\)). Let \({\mathfrak{M}}=\langle M,<,V,C,I\rangle\) be a model and ϕ a sentence of our language. The sentence ϕ is true in the model \({\mathfrak{M}}\,(\mathfrak{M}\,\vDash\,\phi)\) iff \({\mathfrak{M}},m/h\,\vDash\,\phi\) for arbitrary \(m\in M\) and \(h\in Hist\) such that \(m\in h\).
Belnap et al. (2001, p. 236) calls this notion validity in model \({\mathfrak{M}}\).
Definition 16
(Truth in a structure
\({\mathfrak{F}}\)). Let \({\mathfrak{F}}=\langle M,<\rangle\) be a BT structure and ϕ a sentence. Then, we say that ϕ is true in \({\mathfrak{F}\,(\mathfrak{F}\,\vDash\,\phi)}\) iff for every model \({\mathfrak{M}}\) based on \({\mathfrak{F},\,\mathfrak{M}\,\vDash\,\phi}\).
Belnap et al. (2001, p. 236) calls this notion \({\mathfrak{F}}\)-validity.
Definition 17
(Ockhamist truth). A sentence ϕ is Ockhamist true iff \({\mathfrak{F}\,\vDash\,\phi}\) for arbitrary BT structure \({\mathfrak{F}}\).
Very closely related notions are called LD-validity by Kaplan (1989b, p. 547) and logical necessity by MacFarlane (2008, p.84).
We can now introduce the parallel generalizations associated with the concept of truth at a context. I already defined what it means for a use of sentence ϕ to be true at context c, in model \({\mathfrak{M}}\) (Definitions 11–14). Even though the predictions differ depending on the accepted postsemantics, we can wave these differences aside in the unified postsemantic definitions:
Definition 18
(Valid in a model
\({\mathfrak{M}}\)). Sentence ϕ is valid in model \({\mathfrak{M}}=\langle M,<,V,C,I\rangle\) iff \({\mathfrak{M}},c\,\Vdash\,\phi\) for every \(c\in C\).
Belnap et al. (2001, p. 237) call it in-context validity in \({\mathfrak{M}}\). Keep in mind though, that Belnap et al.’s postsemantics require them to add a history parameter to every moment of a context of use.
Definition 19
(Valid in a
BT-structure \({\mathfrak{F}}\)). We say that sentence ϕ is valid in structure \({\mathfrak{F}}=\langle M,<\rangle\,({\mathfrak{F}}\,\Vdash\,\phi)\) iff \({\mathfrak{M}\,\Vdash\,\phi}\) for any \({\mathfrak{M}}\) based on \({\mathfrak{F}}\).
Belnap et al. (2001, p. 237) call it in-context validity in \({\mathfrak{F}}\). The definition can be naturally generalized to classes of frames and all BT frames (MacFarlane 2008, p. 84, describes it as the logical truth).
I follow in my notation Kaplan (1989b) and generalize validity from truth at a context. By the same token, I diverge from the common usage of a concept of validity as a generalization of a notion of truth at an evaluation point. Observe, however, that the distinction between truth at a context and truth at a point of evaluation is not, for the most part, introduced in formal modal logic.
With this apparatus at hand, we can begin a comparison of logics of different theories. The first, simple but important, observation is that C and I have no impact on our semantics proper so far, they reveal their effect only in postsemantics. Consequently, we can state the following fact:
Fact 1
For any two models
\({\mathfrak{M}}=\langle M,<,V,C,I\rangle\)
and
\({\mathfrak{N}}=\langle M,<,V,C',I'\rangle\), any sentence
ϕ, any
\(m\in M\), and any
\(h\in Hist\):
$$ {\mathfrak{M}},m/h\,\vDash\,\phi\quad{\rm iff }\quad{\mathfrak{N}},m/h\,\vDash\,\phi $$
Proof
By straightforward induction on complexity of ϕ. □
This observation basically says that no matter which presemantics and postsemantics you choose, you will not notice any difference in truth values of the sentences at points of evaluation. This fact easily generalizes to a higher level of logical truth.
Corollary 2
For any two models
\({\mathfrak{M}}=\langle M,<,V,C,I\rangle\)
and
\({\mathfrak{N}}=\langle M,<,V,C',I'\rangle\)
and any sentence
ϕ:
$$ {\mathfrak{M}}\,\vDash\,\phi\quad{\rm iff}\quad{\mathfrak{N}}\,\vDash\,\phi $$
The analogous equivalence holds for frames and classes of frames. It is an important result in context of the debate of the TRL since it might be seen as a peace treaty putting an end to the logical war I reconstructed in Sect. 2 The branching concretist model differs from the TRL model (as well as from growing universe and presentist models) only with respect to what is an accessible context of use of a sentence. As a result, the logic of all these models is just the same or at least a fragment of logic described in the language we are using so far. Things change when we introduce context dependent expression to our language, especially the “actually” operator (cf. Sect. 4.4). However, the fact 1 will still hold for the indexical-free part of the language.
The perspective shifts dramatically if we switch from the truth of a sentence at a point of evaluation to the truth of a use of a sentence at a context since different models might differ in how to relate contexts to elements of the model:
-
First, they may disagree with respect to which possible moments are initialized by contexts. Let us say that we have three different models based on the same “core” \(\langle M,<,V\rangle\): (a) a branching concretist one \({\mathfrak{M}}_{BC}\), where I[C] = M = C, (b) a TRL-ist model \({\mathfrak{M}}_{TRL}\), where I
TRL
[C] = TRL
h
, (c) and a presentist \({\mathfrak{M}}_{PRES}\), where I
PRES
[C] = {m}. Now, let us think about a use of a sentence ϕ in a branching concretist model \({\mathfrak{M}}_{BC}\) at a moment c such that I
PRES
(c) ≠ m and \(I_{TRL}(c)\notin TRL_h\). Let us assume that \({\mathfrak{M}}_{BC},c/h\,\Vdash^{CON}\phi\)—a use of sentence ϕ at context c is true at history h. The TRL-ist and presentist, if they do not want to accept branching concretist metaphysics can do nothing but object that it is not an accessible context. The presentist would say that there are no accessible contexts other than the present one and the TRL-ist that there are no others than those in our world (which initializes moments on TRL
h
). Hence there is no way to judge the truth of ϕ as used at c.
-
Second, due to the differences in postsemantics, different theories might assess the very same use of a sentence at the very same context differently. If the sentence ϕ at stake is a future contingent used at c in a model \({\mathfrak{M}}\), then:
-
supervaluationism will simply assess both ϕ and \(\neg\,\phi\) not true (\({\mathfrak{M}},c\,\not\Vdash\,^{SUP}\,\phi\) and \({\mathfrak{M}},c\,\not\Vdash\,^{SUP}\,\neg\,\phi\));
-
according to relativism, it depends on a context of assessment, at some future context c
a1, it is true \({(\mathfrak{M}},c,c_{a1}\,\Vdash^{REL}\,\phi)\) and at some other c
a2, it is not \({(\mathfrak{M}},c,c_{a2}\,\not\Vdash^{REL}\,\phi)\), it is most certainly not true as assessed at \({c_a=c;\,(\mathfrak{M}},c,c\,\not\Vdash^{REL}\phi)\);
-
the conservative believes that there is no point asking whether a use of a future contingent at a context is true until we specify in respect of which history the truth is stated. And there are histories h
1 and h
2 such that \({\mathfrak{M}},c/h_1\,\Vdash^{CON}\,\phi\) and \({\mathfrak{M}},c/h_2\,\Vdash^{CON}\neg\,\phi\);
-
while the TRL-ist will simply claim that either a use of ϕ or its negation is true (\({\mathfrak{M}},c\,\Vdash^{TRL}\,\phi\) or \({\mathfrak{M}},c\,\Vdash^{TRL}\,\neg\,\phi\)). Moreover, it is assessed with resort to neither the history nor the context of assessment. Note that it is exactly what TRL theory was meant to achieve!
The concept of validity in a model (Definition 18) needs a few words of comment. Thanks to the introduction of the set of accessible contexts C and the initialization function I, this notion becomes a very interesting one and it might be instructive to examine it. First of all, the whole range of sentences might be valid in a model \({\mathfrak{M}}\) even though they are not true in it. It is not an unknown phenomenon in the philosophy of indexicals. After all, “I am here now” is true whenever used (it is called valid by Kaplan and me), but it is by no means true in every model. After all, there are lots of places and moments, such that I am not there and then. This kind of sentence might be regarded as valid due to its linguistic properties.
Kaplan (1989a, p. 597) distinguishes another class of sentences that might be regarded as valid due to their relation to the contexts of use. The example he gives is “Something exists” which is valid even though there are no indexicals in the sentence. It is valid because, for something to be a context of use of a sentence, it needs to exist.
The introduction of parameters C and I brings another interesting aspect to the notion of validity. The sentence might be valid in a model if it contingently happens to be true in every accessible context of its use. The presentist case provides the most evident example. Let us consider a presentist model \({\mathfrak{M}}\) such that C = {c}. If c is in the year 2012AD, then it is true to say at c that it is 2012AD. Since for the presentist c is the only accessible context, the sentence “It is 2012AD” is simply valid (with no further specification) in this very model or, if it is clear which model we have in mind, we can simply say that the sentence “It is 2012AD” is valid, full stop. There is no need to specify the context or the moment of evaluation. It is valid simpliciter.
The divergence between different presemantic theories can be observed primarily on the level of truth at a context and validity in a model. However, it is enough to generalize the notion only one level further—to the level of validity in a structure (Definition 19)—and all the differences disappear. If a sentence is valid in a structure \({\mathfrak{F}}\), then it is truly used at every context, in every model \({\mathfrak{M}}\) based on \({\mathfrak{F}}\). So the factor of “contingent” content of accessible contexts which influences validity in a model is canceled out if we take all the models into account.
Objective Complaint
What in the structure of our world could determine a single possibility from among all the others to be “actual”? (Belnap et al. 2001, p. 162).
The question has at least two natural readings. If it is to be read as, “What in the structure of our world makes it necessary that TRL
h
rather than some other possible history represents the world as it actually is?”, then the answer is: “Nothing!”. The world is indeterministic and it can develop along any of the possible ways. It simply develops along one of them which we call TRL
h
.
However, if the question is to be read: “What in the structure of the world makes it necessary that only one of the histories represents the world as it actually is?”, then the answer is, “The structure of the world itself.” According to the TRL view, the physical, concrete universe simply does not branch in time. It resembles something like a single spacetime and it does not contain incompatible events (e.g. if a coin is tossed in our world, then there is the coin which shows heads in our world, or there is the coin which shows tails in our world, but there is no place in our world for both of these results to occur). The histories represent all the ways the world might develop, but the world develops in one way only. Therefore, one (and only one) of the histories must represent the world as it actually develops. There is nothing deterministic about this result. The physical world “determines” the TRL
h
in the very same way in which the “complex physical entity (‘the dice,’ thought of a a single object) \((\dots)\) and its actual position determines the actual state of the (two) dice.” (Kripke 1980, p. 17, emphasis mine)
To see that it is no mystery, compare Belnap et al.’s objective complaint with another puzzle. Let there be a fair lottery in which only one of the tickets is drawn and the drawn ticket wins, then we might ask:
What in the structure of the lottery could determine a single ticket from among all the others to be “the winning one.”
If it means to ask, “What makes it necessary that ticket a rather than some other ticket wins?”, the answer is “Nothing!”; after all, the lottery is fair. If, on the other hand, the question is “What makes it necessary that only one of the tickets wins?”, the answer is, “The structure of the lottery itself”—we draw the tickets just once so one (and only one) of the tickets must win. This fact does not make the lottery deterministic.
Is it just a misunderstanding then that leads us to think that TRL theory supports determinism? I think that the issue is deeper than that. Notice that in the answer to the objective complaint presented above, I appealed to the distinction between our world and possibilities of this world. It is legitimate since the TRL theory, as I present it, is based upon actualist modal realism. The accuracy of some such understanding is assumed by the TRL theory outlined here.Footnote 22
For the branching concretist, however, our world is identical to the collection of concretely existing possibilities. Given this perspective, it is difficult to think of any feature of this world which would ontologically distinguish a single branch indeed. There are qualitative differences between histories; nonetheless, it is hard to see how any such difference might make one of the histories in our world “the actual one.” It seems, from the concretist perspective, that the only way to existentially distinguish one of the histories is to make it the only history and, by the same token, to accept determinism. No wonder then that the concretist treats the concept of the Thin Red Line with such distrust.
The dispute between the branching concretist and the TRL theorist might be seen as another manifestation of an argument between genuine (or extreme) modal realism à la David Lewis and actualist modal realism à la Alvin Plantinga. As Divers (2002, p. 300, n. 5) notices, Lewis (1986, ch. 3) argues that only genuine realism is The Realism about possibility and accuses actualist realists of being merely ‘ersatzers’. Analogously, branching concretists seem to believe that only their notion of possibility is good enough to capture the Real Indeterminism. However, some actualists (Plantinga 1987) argued quite the opposite. In their opinion, it is only the actualist realism that can provide the True Realist notion of possibility and David Lewis’ concretist view is some sort of ‘reductionism’. The TRL-ist can argue along the similar lines.
Epistemological Complaint
[H]ow we could know whether we are on TRL
abs
. How could we find out? (Belnap et al. 2001, p. 163)
If one asks oneself anything, one does it at some context. However, if our world is in fact a huge, 4-dimensional, non-branching object, then every context \(c\in C\) is mapped on TRL
h
(which Belnap et al. call TRL
abs
). Consequently, whenever we ask ourselves a question whether we are on TRL
h
or not, the answer is affirmative. We do not even need to investigate. At the same time, it is contingent that TRL
h
rather than some other history corresponds to our world. As a result, the sentence “I am on TRL
h
” is like the sentence “I exist”; they are both contingent but true whenever used, so they are known, in a sense, a priori (cf. Kripke 1980, pp. 54ff.).
Actuality Complaint
The TRL theory also has troubles with actuality. \((\dots)\) [T]his world’s being the actual world does not favor it over any others, but is just a reflection of the fact that this is the world at which we are conversing. To suppose that there is one from among the histories in Our World that is the absolutely actual history is rather like purporting to stand outside Lewis’s realm of concrete possibilia and pointing to the one that is actual. But this is wrong in both cases. (Belnap et al. 2001, p. 163).
There are two worries that might be extracted from this quote. The first one is rather similar to the objective complaint. It expresses the view that TRL
h
is no different from other histories and it cannot be distinguished on the basis of being actual. I refer to Sect. 4.2 for discussion of this argument. I think that this quote also supports my claim that there is some affinity between David Lewis’ and Belnap et al.’s notions of possibility.
Another worry lurking in the quoted fragment originates in the observation that the actual world is the world at which we are conversing. The authors allude to the indexical theory of actuality. Adams (1974, p. 214) traces the origins of this idea back to Leibniz. In the 20th century it was discussed and rejected by Arthur Prior (see Lewis 1970, p. 185, n. 6), then articulated and ably defended by Lewis (1970) and later formally developed by Kaplan (1989b) and applied to Ockhamism by Belnap et al. (2001). The core of the idea is that words like ‘actually’ or ‘actual’ are structurally similar to indexical expressions like ‘now’, ‘here’, ‘I’, etc. The distinctive feature of these words and expressions containing them is that their reference is not fixed once and for all, but changes from one context of use to another. Just as ‘here’ refers to different places on different occasions of use, ‘actual’ refers to different possible circumstances depending on the context in which it is used. This linguistic idea is quite naturally combined with the philosophical picture presented by branching concretism (and Lewisian modal realism) since according to this position, the possible circumstances have the same metaphysical status as the actual ones. The only way to distinguish the latter is by using the phrase “The circumstances I am actually in.” A use of the word “actually” indicates your exact position on the tree of possibilities (or in the space of possible worlds). This account of actuality might seem to be at odds with the TRL ideology since the latter suggests that the actual world is metaphysically distinguished from any possible history (including TRL
h
for that matter). However, I am going to show that my version of the TRL theory is compatible with the indexical notion of actuality. It shows that the indexical nature of the word ‘actually’ is partly independent of the accepted metaphysics of possibility. In fact, my approach generates much better and more intuitive predictions than any other account of indexical “actually” available for BT.
Proceeding formally, I adopt Kaplan’s (1989b) treatment of indexicals appropriately modified by Belnap et al. (2001) to fit the branching framework. The lesson from Kaplan is that to deal with the semantics of indexicals we need to take into account the context of use of a sentence in which an indexical appears. More precisely, we need to somehow “store” the information at which context sentence ϕ was used, so we can utilize this information evaluating the sub-sentence ψ of ϕ which contains an indexical expression. So far, our point of evaluation has a form \({\langle {\mathfrak{M}},m/h\rangle}\) where \({\mathfrak{M}}=\langle M,<,V,I,C\rangle\). The initialization function dictates where to start the process of evaluation of the sentence used at \({\langle{\mathfrak{M}},c\rangle}\), namely, we start at I(c), but this fact is not stored in our point of evaluation. Since the used sentence may contain modal and tense operators, the sub-sentences of it might well be evaluated at a moment different from the context-initialized one. However, as Kaplan makes clear, we need to keep the context of use fixed and utilize it when the embedded indexical connective is being interpreted. For this purpose, let me add another parameter to the point of evaluation, I will just call it a context parameter. As a result, the new point of evaluation has the following form: \({\langle{\mathfrak{M}},c,m/h\rangle}\) where \(c\in C\).
Let me now phrase my semantic definition of the connective “it is actually the case that” (@) and its competitors present in the arena. Notice that all the truth clauses below do refer to the piece of information stored in the context parameter. My proposal is a very simple and natural one:
-
1.
TRL
definition of
Actually:
$$ {\mathfrak{M}}_{TRL},c,m/h \,\vDash\, @ \phi \quad {\text{iff}} \quad {\mathfrak{M}}_{TRL},c,I(c)/TRL_h \,\vDash\, \phi $$
The presemantic structure of \({\mathfrak{M}}\) guarantees that for each \(c\in C,\, I(c)\in TRL_h\) so the operator @ is well-defined. Let me now present the alternative definitions available at the BT market. All of them were designed for the branching concretist model \({\mathfrak{M}}\):
-
2.
Conservative definition of
Actually
1: \({\mathfrak{M}},c,m/h\,\vDash\,@_1\phi \,\rm{ iff}\,{\mathfrak{M}},c,I(c)/h'\,\vDash\,\phi\) for every h′ such that \(I(c)\in h'\). (Belnap et al. 2001, p. 246).
-
3.
Conservative definition of
Actually
2: \({\mathfrak{M}},c,m/h\,\vDash\,@_2\,\phi\,\rm{iff\,either (a)}\,I(c)\in h\) and \({\mathfrak{M}},c,I(c)/h\,\vDash\,\phi\) or (b) \(I(c)\notin h\) and \({\mathfrak{M}},c,I(c)/h'\,\vDash\,\phi\) for every h′ such that \(I(c)\in h'\). Belnap et al. (2001, p. 246).
-
4.
Supervaluationist definition of
Actually: \({\mathfrak{M}},c,m/h\,\vDash\, @\phi\) iff \({\mathfrak{M}},c,I(c)/h'\,\vDash\,\phi\) for every h′ such that \(I(c)\in h'\). (MacFarlane 2008, p. 99).
-
5.
Relativist definition of
Actually: \({\mathfrak{M}},c,c_a,m/h\,\vDash\, @\phi\) iff \({\mathfrak{M}},c,c_a,I(c)/h'\,\vDash\, \phi\), for every \(h'\in H_{I(c)|I(c_a)}\). (MacFarlane 2008, p. 99).
The litmus paper that I am going to use to test the definitions is the initial-redundancy requirement for the actuality operator proposed by MacFarlane (2008). The requirement, appropriately modified for our notation, should be understood as a demand that for any model \({\mathfrak{M}}\), any sentence ϕ, and any context c:
$$ {\mathfrak{M}},c\,\Vdash\,\phi\quad{\rm iff}\quad{\mathfrak{M}},c\,\Vdash\, @\phi $$
The conservative believes that the context c is not sufficient to judge the truth value of a use of a sentence. He additionally demands to specify the history h such that \(I(c)\in h\). Hence the initial redundancy test for the conservative is slightly different:
$$ {\mathfrak{M}},c/h\,\Vdash\,\phi\quad{\rm iff}\quad{\mathfrak{M}},c/h\,\Vdash\, @\phi $$
The initial-redundancy seems to be a reasonable demand. If it is true to say that ϕ it should be equally true to say that @ϕ (and vice versa). The addition or removal of operator @ simply makes no difference as far as uses of sentences are concerned.
The first victim of the test is the conservative operator @1. Let us say that a sentence ϕ is contingent at moment m if there are histories h
1, h
2 such that \(m\in h_1\cap h_2\) and \({\mathfrak{M}},c,m/h_1\,\vDash\, \phi\) and \({\mathfrak{M}},c,m/h_2\,\vDash\, \neg \phi\). Let us now consider a sentence Fp contingent at I(c). Let h
1 be such that \({\mathfrak{M}},c,I(c)/h_1\,\vDash\, Fp\), then by conservative postsemantics \({\mathfrak{M}},c/h_1\,\Vdash\, Fp\). However, since ϕ is contingent at I(c), there is h
2 such that \({\mathfrak{M}},c,I(c)/h_2\,\vDash\, \neg Fp\) which means that, by definition of \({@_1,\,\mathfrak{M}},c,I(c)/h_1\,\vDash\, \neg @_1 Fp\) which implies in turn, by conservative postsemantics, that \({\mathfrak{M}},c/h_1\,\Vdash\, \neg @ Fp\). Consequently, there is model \({\mathfrak{M}}\), sentence ϕ and context c such that \({\mathfrak{M}},c/h\,\Vdash\,\phi\) and \({\mathfrak{M}},c/h\,\Vdash\,\neg @_1 \phi\) which is equivalent to the failure of the initial redundancy test. In fact, it is the case for every future contingent. Therefore, with respect to some histories, we can truly use the sentence, “There will be a sea battle tomorrow even though actually there will be none.”
Belnap et al.’s (2001) alternative proposal (@2) does pass the initial redundancy test but is not acceptable for independent reasons. It generates a number of counter-intuitive results, for example:
-
If sentence ϕ is evaluated at a point m/h such that \(I(c)\in h\), then @2 loses a part of its indexical nature. MacFarlane (2008, p. 99) even claims that Actually
2 is simply redundant at such points. This is not quite accurate since Actuality
2 retains a part of its indexical function even at these points, it just functions as another indexical—Now (as defined by Belnap et al. 2001, p. 246). However, MacFarlane is partially right: at such points Actually
2 loses its indexical nature as a modal operator. It is particularly visible at context-initialized moments of evaluation where the sentence, “The future might be different from what it actually will be” \((\diamondsuit(Fp\wedge \neg @_2 Fp))\) is always false. Similarly, the sentence “Necessarily, there will be a sea battle if and only if there actually will be a sea battle” \((\square(Fp\leftrightarrow @_2 Fp))\) is always true at such points.
-
For some model \({\mathfrak{M}}\), there is a sentence ϕ and a context-initialized point of evaluation \({\langle\mathfrak{M}},c,I(c)/h\rangle\) such that the following is true \({\mathfrak{M}},c,I(c)/h\,\vDash\, P\diamondsuit F (\phi\wedge \neg @_2\phi)\wedge @_2\phi\). So we can truly say “It might have been the case that there would be a sea battle and actually there would be none (even though there actually will be one).” Such oddities result from the fact that even if we use @2 twice in one sentence, it might behave as a modal indexical for the first time and not as one on the other occasion.
I find these reasons decisive in abandoning @2 as a candidate for a proper analysis of the word ‘actually’. Interestingly, there is a straightforward and formally neat way to solve all the problems of the conservative—it is to include “the history of a context of use” (h
c
) as an element of a context and bind the interpretation of the operator @ with this aspect of the context in the following way: \({\mathfrak{M}},c,h_c,m/h\,\vDash\, @\phi\) iff \({\mathfrak{M}},c,h_c,I(c)/I(h_c)\,\vDash\,\phi\). However, Belnap et al. (2001, p. 151) very strongly object to the idea of the history of the context of use. I find their arguments persuasive (contrary to e.g., Borghini and Giuliano 2011) and it is important to stress at this point that my TRL
h
should not be thought of as the history of the context of use which changes from one context to another. The history TRL
h
is “initialized” by the world itself and its structure and content rather than the context.
We have eliminated both the conservative definitions of the operator @ as candidates for an analysis of the indexical meaning of “actually”. However, the remaining three proposals seem to be on a par. To differentiate between them, we need to devise a test stronger than the initial redundancy. One reasonable strengthening is a demand that not only uses of ϕ and @ϕ should be co-true at every context (as MacFarlane 2008 insists), but uses of their negations also should. Formally, the stronger requirement is that both equivalences (\({\mathfrak{M}},c\,\Vdash\,\phi\) iff \({\mathfrak{M}},c\,\Vdash\, @\phi\)) and (\({\mathfrak{M}},c\,\Vdash\,\neg \phi\) iff \({\mathfrak{M}},c\,\Vdash\, \neg @\phi\)) are satisfied. It sounds reasonable since “actually” is a modal indexical, so it should be not only initially redundant but redundant also in scope of extensional connectives such as negation. It amounts to the demand that at a given context c, we can truly say that it is not the case that there will be a sea battle if and only if we can truly say that it is not the case that there actually will be a sea battle.
This stronger test is failed by some sentences at some points of evaluation under every definition of “actually” in the BT setting that I am aware of, except my TRL definition of @ presented above. In particular, the supervaluationist and relativist definitions of @ fail:
Supervaluationism For every contingent sentence ϕ used at \(c{:}\,{\mathfrak{M}},c\,\Vdash\, \neg @\phi\) and \({\mathfrak{M}},c\,\not\Vdash\,\neg\phi\) (even though it is not true either, i.e. \({\mathfrak{M}},c\,\not\Vdash\,\phi\));
Relativism For every sentence ϕ used at c and still contingent while assessed at \(c_a{:}\, {\mathfrak{M}},c,c_a\,\Vdash\, \neg @\phi\) and \({\mathfrak{M}},c,c_a\,\not\Vdash\, \neg\phi\) (even though it is not true either, \({\mathfrak{M}},c,c_a\,\not\Vdash\, \phi\)).
The TRL definition of ‘actually’ that I proposed satisfies this stronger test in full generality. In fact, operators \(\neg\) and @ are mutually “transparent,” i.e. the equivalence \(\neg @\phi\leftrightarrow @\neg\phi\) is true (not only valid) in every \({\mathfrak{M}}_{TRL}\) model.
We can propose an even stronger, and yet still quite natural version of the initial-redundancy requirement, i.e. we can demand for every context c and every sentence ϕ that \({\mathfrak{M}},c\,\Vdash\, \phi\leftrightarrow @\phi\). We just express the metalinguistic version of the initial-redundancy test in the object language. It simply means that at any context, one is guaranteed to say the truth, claiming that there will be a sea battle if and only if there actually will be a sea battle. MacFarlane (2008) explicitly rejects this strengthening but he agrees that we need to “get over our qualms” to do so (p. 99 , n. 22). Again, all the treatments of ‘actually’ discussed in the literature, besides otherwise faulty conservative @2, would falsify this equivalence whenever ϕ is a contingent sentence. At the same time, the equivalence \(\phi\leftrightarrow @\phi\) is valid (but not true) in every TRL-model.
It is easy to understand why the TRL model constitutes such a friendly environment for the operator ‘actually’, while branching concretism is so hostile to it. In the TRL theory, there is exactly one point of evaluation initialized by any use of a sentence. Importantly, this point contains a specific history (TRL
h
) as its element. It is quite evident that this very point should be utilized for interpretation of the operator @. The TRL-ist sharply distinguishes the actual from the possible so the interpretation of @ is quite straightforward.
The branching concretist on the other hand, denies distinguishing any ‘actual’ history. However, to retain the indexical meaning of @, he needs to tie it to some feature of the context. The only available item seems to be “\((\dots)\) a unique causal past, and a unique future of possibilities, the whole of which is summed up by the moment of use” (Belnap et al. 2001, p. 226). As a result, he tends to identify actuality with necessity. Belnap et al. (2001) even propose an intended reading of @1
A to be: “It is settled true at this actual moment that A” (2001, p. 153). Consequently, the concretist (no matter whether conservative, supervaluationist, or relativist) usually takes the sentence \(\square\phi\leftrightarrow @\phi\) to be valid—true whenever used. In particular, we are semantically guaranteed to be right when saying: “If only it actually will rain tomorrow, it is settled that it will.” Such observations further reinforce the concretist conviction that actuality is only a camouflaged form of necessity and whoever talks about the actual future is a determinist in disguise.
The TRL-ist intends to disentangle the notions of actuality and necessity and he seems to be successful in his attempt. In particular, it is not difficult to find a sentence ϕ and a context at which a use of \(@\phi\leftrightarrow \square\phi\) is false. A similar implication, \(@\phi\rightarrow\square @\phi\), is true but it does not doom us to determinism. (Just as the truth of, “If it is raining now, then it will always be the case that it was raining now,” does not doom us to a flood.) It just witnesses to the indexical nature of the operator @.
To sum up, it appears that the TRL combination of presemantics, semantics, and postsemantics presented here generates the most intuitive predictions for the behavior of the indexical operator “actually”. I take this fact as another argument for the more general thesis that the TRL theory is metaphysically underpinned by some actualist notion of possibility.