7 Definition: Modalities
7.1 Deducing Truthmakers of Modal Propositions
In EUO the Universe is a single non-branching sequence of temporal stages (TSUs) which are in a forward directed temporal and causal succession. The past TSUs have been realized in the past, the present TSU is realized now, and future TSUs will be realized
101According to Knuuttila [205], diachronic modality is “the model of antecedent necessities and possi-bilities with respect to a certain moment of time.” See also Knuuttila [203]. See Von Wright’s definition of the diachronic-synchronic dichotomy in p. 166.
one at a time. This portion of EUO suffices in giving an account of the truthmaker TSUs of a modal proposition with an arbitrary combination of aspect and target times. Table A represents nine combinations which are the nine (α,τ) pairs whereα, τ ∈ {past, present, future}. The deduction of the truthmakers is started from the most common propositions (1-3). After this the general rule of deducing the truthmaker for all combinations (1-9) is presented.
aspect time α target time τ
1. present past
2. present present
3. present future
4. past past
5. past present
6. past future
7. future past
8. future present
9. future future
Table A: 9 combinations of aspect and target times.
1. aspect=present; target=future. In perhaps the most common modal propo-sitions the aspect time is the present and the target time is in the future. Consider the first example: It is possible from the aspect of the present time that it will rain tomorrow in Helsinki. In this proposition the aspect time is the present timep, the target time isf tomorrow, O=it rains in HelsinkiandM=possible. The proposition can be repre-sented as (p,O,f)=possible. If the proposition is true what is its truthmaker, and if it is false what is its falsemaker? The answer is: TSUp. For, in EUO the TSUs are in a forward directed temporal and causal succession, and therefore TSUpis the cause of the TSUs that come after it, i.e., TSUpdetermines102 which TSUs arerealizablein the future. Realizability in the future is equivalent with future possibility. For instance, the present TSUp determines which TSUs are realizable at time p+ 1: those TSUs which are realizable atp+ 1 from the aspect ofpare possible atp+ 1 from the aspect ofp.
In the following, whenp ≤f, {p→f}denotes the collection which contains all TSUs that are realizable at the target timef from the aspect ofp. The truth value of (p, O, f)=possibleis determined by{p→f}. If the proposition is true, thenOcorresponds to at least one element of{p →f}. If the proposition is false, then O corresponds to no element of{p→f}. Note that the expression ‘Ocorresponds to an element of{p→f}’
is a figure of speech which means that the proposition (p,O,f)=possiblecorresponds to TSUp, forp determines the contents of {p →f}, i.e.,p is the genuine truthmaker of the proposition. Consider (p,O,f)=M with each modality M.
possibility: (p, O, f)=possible is true if and only if O corresponds to at least one element of {p → f}; (p, O, f)=possible is false if and only if O corresponds to no element of{p→f}.
necessity: (p, O, f)=necessary is true if and only if O corresponds to all elements of{p→ f}; (p, O, f)=necessaryis false if and only if there is at least one element in {p→f}to whichO does not correspond.
impossibility: (p,O,f)=impossibleis true if and only ifO corresponds to no element of{p →f}; (p, O, f)=impossibleis false if and only if there is at least one element in {p→f}to whichO corresponds.
102The meaning of ‘determines’ is further qualified in§7.2: TSUpdetermines the future disregarding if total determinism or partial determinism holds.
contingency: (p, O, f)=contingent is true if and only if O corresponds to one or more but not to all elements of{p→f}; (p,O,f)=contingent is false if and only ifO corresponds either to all elements of{p→f}or to no element of{p→f}.
The above scenario is shown to be an application ofpossible worlds semantics in §7.5, and it is compatible with how Divers [110, pp. 3-4] interrelates modalities:
possibilityrules out impossibility, allows contingency and necessity.
necessityrequires possibility, rules out impossibility and contingency.
impossibilityrules out possibility, necessity and contingency.
contingencyrequires possibility, rules out impossibility and necessity.
Although the truthmaker/falsemaker is known to bep, truth values of modal proposition about the future are often strictly speaking not known, even when it seems to be ex-tremely probable that they are true, for often we cannot predict withabsolute certainty that these are true.
2. & 3. aspect=present; target=past or present. In EUO, the past has already been realized in exactly one way: the past is not realizable any longer and cannot be influenced from the present. Both of the following propositions are equivalent with the proposition that Xwas realizedin the past at timep−1: X was possibly realized in the past at timep−1 from the aspect of the present p; X was necessarily realized in the past at timep−1 from the aspect of the present p. Also the present is necessarily realized from the aspect of the present, i.e., the pastwasand the present isnecessarily realized from the aspect of the present.103 Consider the proposition S=It is possible today that it rained in Helsinki in 1100, which can be represented as (p, It rains in Helsinki, 1100)=possible, and where 1100 denotes an instantial moment in the year 1100. As the past has already been realized, the collection {p → 1100} contains exactly one TSU, 1100, which is the truthmaker/falsemaker of S. If S is true, then it did rain in Helsinki in 1100, and also (p, It rains in Helsinki, 1100)=necessaryis true. If it did not rain in Helsinki in 1100, then S is false, whereas (p, It rains in Helsinki, 1100)=impossibleis true. In this sense nothing about the actual past is contingent.104 Propositions whose aspect and target times are both present are equivalent with propositions whose aspect time is the present and target time is the past, with the exception that the present is realwhile the pastwas real: that object Y is possibly realized atpfrom the aspect ofp is equivalent with that Y is necessarily realized atpfrom the aspect ofp.
In sum, when the aspect time is the presentp and the target timeτ ≤p, it follows that {p→τ}contains exactly one TSU. Whenp=τ, the collection{p→τ}contains exactly
103That past existence is necessary from the present aspect is characterized by Rice [333]: “Is what is true of the past necessary? Well certainly pretty well everyone thinks that what happened in the past cannot be undone. The past cannot now be altered.” But, given partial determinism, the present and the past could have been differently (§7.3). Aristotle’s notion that the present is necessary (On Interpretation, 19a23-7) is congenial with the presentist-diachronic view of possibility, and this does not require total determinism. For, although the present exists necessarily from the present aspectp, this does not entail thatpcould not have been different from the aspect of a time beforep. Compare to Knuuttila [205]: “Another interpretation is that Aristotle wanted to show that the necessity of an event at a certain time does not imply that it would have been antecedently necessary.” The necessity of the past reflects step (1) of theMaster Argumentof Diodorus Cronus, whereas step (3) reflects partial determinism as will be revealed in§7.2: (1) every past truth must be necessary; (2) an impossibility does not follow from a possibility; (3) something is possible which neither is nor will be true. (2) can be interpreted in may ways; see Akama et al. [6] for different guesses of what it may mean.
104Again, in the context of partial determinism some propositions of the formsomething that was not realized in the past could have been realized in the pastare true as they do not concern the actual past, whereas in the context of total determinism all such propositions are false (§7.3).
one TSUτ which is realized. When p > τ, the collection{p→τ}contains exactly one TSUτ whichwas realized. Consider the two cases with all modalities:
possibility: (p, O,τ ≤p)=possibleis true if and only ifO corresponds to at least one element of{p→τ ≤p}, i.e., to its only element.
necessity: (p,O,τ ≤p)=necessaryis true if and only ifO corresponds to all elements of{p→τ ≤p}, i.e., to its only element.
impossibility: (p,O, τ ≤p)=impossibleis true if and only ifO does not corresponds to the single element of{p→τ ≤p}.
contingency: (p, O,τ ≤p)=contingentis true if and only ifO corresponds to one or more but not to all elements of{p→τ ≤p}, i.e., it is never true.
cases 1. - 9. The classification gets slightly more complex. As in combination 4 α andτ are both in the past and as in 9 α andτ are both in the future, there are three sub-combinations in both cases: α > τ; α=τ; α < τ. These combinations are written out in table B, as well as the TSUs which are the truthmakers of the propositions. When partial determinism is supposed, the truth value of some propositions is indeterminate and therefore we are dealing with indeterminate-makers as well as with truthmakers and falsemakers. Accordingly, the term truth-value-makeris used in the remainder of this section. The general rule is straightforward: the truth-value-maker is always the earliest ofα, τ, p. This reminds that in presentism the present is always the ultimate viewpoint.
past present future truth-value-maker
1. τ α τ
2. α=τ α=τ =p
3. α τ α=p
4.1 α < τ α
4.2 α=τ α=τ
4.3 τ < α τ
5. α τ α
6. α τ α
7. τ α τ
8. τ α τ =p
9.1 α < τ p
9.2 α=τ p
9.3 τ < α p
Table B: 13 combinations of aspect and target times and truth-value-makers.
In cases 1, 2, 4.2, 4.3, 7 and 8 the target timeτ is the earliest and it has been realized; as these propositions state that a part of TSUτ has certain properties, τ determines their truth values and is thus their truth-value-maker.
In cases 2, 3, 4.1, 4.2, 5 and 6 the aspect timeαis the earliest and it has been realized;
as these propositions state that TSUα has certain properties —that it makes or made this and that necessary, contingent or impossible—α determines their truth values and is thus their truth-value-maker.
In cases 2, 3, 8, 9.1, 9.2 and 9.3p is the earliest and itisrealized; as these propositions state that TSUphas certain properties —that it makes or made this and that necessary, contingent or impossible—pdetermines their truth values and is thus their truth-value-maker.
In sum, EUO implies which TSU is the truth-value-maker of all 13 types of propositions, including propositions whose truth value is indeterminate. In some of the casesτ = α,
in some cases τ = p, in some cases α = p and in one case α= τ =p, but the general rule stands.
7.2 Partial and Total Determinism. Diachronic vs. Synchronic Modalities
The mutually exclusive versions of determinism are qualifications of the causality axiom which states that the TSUs are in a forward directed causal succession, i.e., qualifications of how the presentpdetermines what is realizable atp+ 1,p+ 2,p+ 3, and so on. Total determinism is depicted on the top of figure 20 and partial determinism on the bottom.105 (The dichotomy of ‘total determinism’ and ‘partial determinism’ is applied instead of e.g.
the dichotomy of ‘determinism’ and ‘indeterminism’ because the total-partial dichotomy characterises the intended meanings very well: in both cases the present clearly affects the future insome degreeand therefore the term ‘indeterminism’ would be misleading.)
Figure 20: Total determinism on the top and partial determinism on the bottom.
future in total determinism:106 the present TSU ptotally determines the unique TSU which will be realized atp+ 1. Asp+ 1 totally determinesp+ 2, asp+ 2 totally determines p+ 3, and so forth, it follows that p totally determines all TSUs that will be realized after p. In total determinism, if TSU x is possibly realized at p+ 1 from the aspect of p, x is necessarily realized at p+ 1. Accordingly, if it is possible that TSUx is not realized at p+ 1 from the aspect of p, then it is necessary that x is not realized atp+ 1, i.e., it is impossible thatxis realized atp+ 1. This means that in total determinism no future possibility is contingent. The future timexcan be equated with the TSU thatwill berealized at timex, and the period of time [p+ 1x] can be equated with the sequencep+ 1, p+ 2, . . ., xof TSUs whichwill berealized in the future. In total determinism, the past and the future are equally determined.
future in partial determinism: the present TSUpdetermines totally thecollection of all TSUs which are realizable at p+ 1. This collection is denoted as{p → p+ 1}.
Similarly for later times: pdetermines totally the collection{p→p+2}of all TSUs which are realizable atp+ 2, i.e.,p determines all realizable chainsor unbroken sequences of causally connected TSUs which start fromp. While the elements of{p→p+2}are single TSUs which are realizable atp+ 2 from the aspect ofp, a separate notation for sequences
105Both options are contemplated because it turned out difficult to make a selection between them by economy. Although total determinism clashes with genuine free will (Honderich [172]), e.g. Wegner [409] notes that free will might be just an illusion, and this would be compatible with total determinism.
On one hand, partial determinism does not require that the experience of free will is an illusion, but on the other hand partial determinism is more complex than total.
106Total determinism is analogous tocausal determinismor “the idea that every event is necessitated by antecedent events and conditions together with the laws of nature” (Hoefer [171]) and it is compatible with the definitions of e.g. James [183], Thomason [395] and Moya [280, p. 130].
is sometimes handy: {[p→p+2]}contains all chains of TSUs which are realizable atp+2 from the aspect ofp. As in partial determinism it is not determined at the present which TSUs will be realized in the future, the future timexcan be equated with the collection {p → x}. The period of time [p x] can be equated with the collection of sequences {[p → x]}. In partial determinism it is necessary that some element of{p → p+ 1}
will be realized at timep+ 1 and in this sense the future is necessary, but no individual element of {p → p+ 1} is necessarily realized at time p+ 1.107 Partial determinism thus allows future contingents (§7.3). If the Universe is partially deterministic, then future possibilities branch. Naturalism implies that the Universe does not branch: only possibilities do, and only if partial determinism holds.108 Kripke’s illustration in figure 21 and his explanation are compatible with diachronic modalities in EUO:
The point 0 (or origin) is the present, and the points 1, 2, and 3 (of rank 2) are the possibilities for the next moment. If the point 1 actually does come to pass, 4, 5, and 6 are its possible successors, and so on. The whole tree then represents the entire set of possibilities for present and future; and every point determines a subtree consisting of its own present and future. Letter from Saul Kripke to A.N. Prior, dated September 3, 1958, kept in the Prior Collection at Bodleian Library, Oxford, Box 4. As quoted in Øoslash and Hasle [299]
Figure 21: Kripke’s illustration of partial determinism.
asymptotical determinism. In asymptotical determinism, the realization of a partic-ular at timetimplies that a particular which is an element of a more or less homogeneous collection of realizable particulars, will be realized at some more or less specific time after t. Earman commits to partial determinism:
A feature can be said to be asymptotically fated if it emerges in the limit as time goes on far enough. . . . I take it, for instance, that the laws of biology dictate that I am naturalistically fated to die; but I also take it that the particular time and manner of my death are not fated by any of the laws of nature. Earman [117, p. 18]
For comparison, Aristotle’s (Metaphysics1027b10-14) statement “it is necessary that he who lives shall one day die” is alone compatible with both partial and total determinism, but it is seen that Aristotle presupposes partial determinism as he states: “But whether he dies by disease or by violence, is not yet determined, but depends on the happening of something else.”
107Another interrelated approach to future necessity is that that what is from the aspect ofpnecessarily realized atp+ 1 is what is common to every element of{p→p+ 1}.
108It is crucial to understand the difference of partial determinism in EUO which directly entails that only future possibilities branch and e.g. Belnap’s [46]branching space-timetheory where the branching of possibilities means that all branches literally exist in some substantive sense (§7.4).
diachronic vs. synchronic modalities. Von Wright’s [419, p. 92-3] characteri-sations of diachronic modalities are largely compatible with the unified theory, such as:
“By a set of possible histories up tot we shall mean all the alternative ways in which the world from its actual state at timet′ in the past might have developed up to time t.” Von Wright uses the diachronic-synchronic dichotomy as follows:
I shall say that≪Mtpt≫expresses asynchronicmodality meaning that the attribution of modal status is for the same time as the possible truth of the proposition to which the modal status is attributed. And I shall say that ≪Mt′pt ≫in the formula expresses diachronic modality because of the temporal difference between the asserted validity of the attribution of the modal status and the possible truth of the proposition whose modal status is involved. Von Wright [418, p. 43]
In the context of the unified theory, the synchronic-diachronic dichotomy is not onto-logical nor onto-logical: in a synchronic modal statement the statement and target times are the same, whereas in a diachronic modal statement they are different, but the logic and ontology of both types of statements is the same. In contrast, Von Wright [418] is deal-ing with two different logical systems, and he does not give clear ontological grounddeal-ings for neither. Modalities in the below equations are complemented as follows in order to distinguish Von Wright’s definitions (1-3) from those of the unified theory:
s: Von Wright’s synchronic expression.
su: synchronic expression in the unified theory.
d: diachronic expression.
(1)s.possibletOt= ∃t′< t(d.possiblet′Ot).
(1’)su.possibletOt =Ot&∃t′< t(d.possiblet′Ot).
In (1), it iss.possible att thatO is realized att, if and only if (iff) some time beforet it wasd.possible thatOwill be realized att. (1) does not fully capture the meaning of (1’). It must be added thatO was or is or will be realized att. Whentis the present, O is realized now. Whent is in the past,O was realized in the past. Whent is in the future, O will be realized in the future, i.e., if (1’) is true and t is in the future, then d.possibletOt= Ot&d.necessaryt′<tOt.
(2)s.strongly necessarytOt= ∀t′< t(d.necessaryt′Ot).
(2’)su.necessarytOt= Ot&∃t′< t(d.possiblet′Ot).
In (2),O is realizeds.strongly-necessarily att, iff at all timest′ < tit wasd.necessary that O will be realized at t. (2) is incompatible with (2’), where su.necessarytOt is equivalent withsu.possibletOt, i.e., as in (1’).
(3)s.weakly necessarytOt= ∃t′< t(d.necessaryt′Ot).
(3’)su.necessarytOt= Ot&∃t′< t(d.possiblet′Ot).
In (3), O is realized s.weakly-necessarily at t, iff there exists at least one time t′ < t when it isd.necessary thatO will be realized att. (3) is incompatible with (3’), where su.necessarytOtis equivalent with su.possibletOt, i.e., as in (1’-2’).
Von Wright (ibid. pp. 46-8) concludes that the logic of his synchronic modalities is S5 and that of diachronic modalities is S4 or S4-like. The unified theory is compatible with the reduction formula of S4 but not with that of S5, wheret1< t2< t3.
S4 reduction: d.possiblet1d.possiblet2Ot3→d.possiblet1Ot3
S5 reduction: d.possiblet1d.impossiblet2Ot3→d.impossiblet2Ot3
The reduction formula of S4 holds in the unified theory, for ifd.possiblet2Ot3 is true, thend.possiblet1Ot3 is true for all timest1< t2. The reduction formula of S5 does not hold in the unified theory, for the truth ofd.possiblet1d.impossiblet2Ot3 does not imply thatd.impossiblet2Ot3 is true.