1

Niko Strobach

Modal Logic for Relativistic Space-Time

with Spatial Operators

2

just being

is bewildering

Moloko, Things to make and do

31. Introduction

Research into the foundations of multi-modal logics has been intensive during recent years

(starting esp. with Venema 1991) and has yielded encouraging results as well as shown reasons

for caution (cf. Wolter 1999, Gabbay / Wolter / Kracht / Zakharishev 2001).

At the same time, the reconstruction of the theory of relativity in terms of modal logic is still

an unsettled matter. Although already the founding father of tense logic, Arthur Prior, in his

classic „Past, Present, and Future“ (Prior 1967) developed a – somewhat biased (cf. Prior in

Copeland 1996) – sketch for relativistic tense logic, and although Robert Goldblatt had worked

out this approach to an impressive degree (Goldblatt 1980), the results could, on the whole, still

be called „sparse“ for long (Øhrstrøm / Hasle 1995). This was regrettable, since a theory as

revolutionary for our thinking about space and time as Einstein’s should be a most prominent

object of research for a logic which, if applied in philosophy, is supposed to clarify the basic

intuitions of spatio-temporal reasoning. Is not the whole project of tense logic anachronistic for

its failure to do justice to the theory of relativity (cf. Massey 1969)? For a philosophical study of

the theory of relativity, a suitable modal logic would be especially interesting in order to see

decide the question whether already the spatio-temporal core of this theory leads to a „rigorous

proof of determinism“ (Rietdijk 1966, fiercely rejected by Stein 1968, 1991, systematically

discussed in Rakic 1997).

The situation has changed through recent work, in particular Belnap (1992), Rakic (1997) and

Müller (2000). The aim of this paper is to sketch a modal logic for relativistic space-time which

considerably differs from Prior’s and Goldblatt’s, overcomes the limitations of their strategy and

sums up the progress made by the authors mentioned above while taking a metaphysical position

that might be different from each of theirs. The main idea is to approach the special theory of

relativity not with the resources of tense-logic alone (as Prior and Goldblatt did) but to use a

pretty complex application of multi-modal logic for this purpose (an idea hinted to in, e.g.,

Strobach 1997). From the semantic side, the sketch is detailed. From the axiomatic side, there

remain open questions esp. concerning the completeness of some of the systems discussed in

this paper, which I hope to be able to answer soon.

After some preliminaries, the paper splits up into two main parts: chapter 3 is concerned with

developing a multi-modal logic for relativistic space-time that contains – to my knowledge a

novelty – spatial operators. Chapter 4 aims at a modalization of the logic developed in chapter 3.

Ch. 3 was developed independently from Müller (2000), ch.4. The convergence on certain

technical details is considerable and will be discussed where this is appropriate. The technical

proximity may be seen as some encouraging mutual corroboration of being on a promising track.

It is just the more surprising as Müller’s and my philosophical intuitions on relativity differ

considerably: to put it very shortly, Müller, following Prior, favours absolute simultaneity, a

Lorentz-style “optical illusion” interpretation of relativity, and a strong emphasis on the A-series

and its expression in the object-language of tense logic. I would rather opt for relative

4simultaneity, an Einstein-style “hard fact” interpretation of relativity and a model-theoretic B-

series approach, with some affinity to Mellor’s theory of time.

Ch. 4 has some common intuitive ground with Belnap (1992) but differs in still finding

problems where Belnap thinks them all solved.

The basic overall idea is to combine several modal logics into a modal logic for relativistic

space-time and so

1. to use different operators for temporal relations (H, G, F, P as usual), spatial relations (¤ =

everywhere, ¡ = somewhere), alethic modalities (N = necessary, M = possible) and relations

between coordinate frame (× = for all coordinate frame, + = for some coordinate frame);

2. to evaluate formulae with regard to multiple evaluation contexts, i.e. in relativistic tempo-

spatial logic for a coordinate frame, a space-point and an instant (relative to that frame), or in modalized relativistic tempo-spatial logic for a coordinate frame, a space-point, an instant

(relative to that frame) and a factual filling of space-time (instead of a „possible world“).

This leads to a more fine-grained expressivity of the new approach in comparison with the old

one.

52. Overview of the systems discussed

2.1. Axiomatics and intended interpretation

a) Systems without alethic modalities

Kr (PC) + (Kt) + transitivity, infinity, continuity, linearity rich tense logic

SP5 (PC) + (¤-S5) S5 as spatial logic

KrxSP5 (PC) + (Kr) + (¤-S5) + (Com) + (Church-Rosser) tempo-spatial logic

KrxSP5e (PC) + (Kr) + (¤-S5) + (Com) + (Church-Rosser) eventisized t.-sp. l.

KrxSP5xF (PC) + (Kr) + (¤-S5) + (×-S5) logic for rel. space-time

b) systems with alethic modalities added

S5 (PC) + (N-S5) alethic modalities

Kr x S5 (PC) + (Kr) + (PN)

Kr x SP5e x S5 (PC) + (Kr) + (¤-S5) eventisized t.-sp. l. with

+ (Com / Church-Rosser) + (PN) alethic modalities

KrxSP5x F x S5 (KrxSP5xF) + (¤-S5) + (×-S5) + (N1-S5) logic for rel. sp.-t. with

+ (N2-S5) + (PN2) + (PN1) + (N2/N1) two sorts of alethic modalities added

2.2. The operators

Definitions of the weak modal operators and intuitive interpretation

M = ~ N ~ N = necessary M = possible

¡ = ~ ¤ ~ ¤ = everywhere ¡ = somewhere

F = ~ G ~ G = always in the future F = at some time in the future

P = ~ H ~ H = always in the past P = at some time in the past + = ~ × ~ × = in all coordinate frames + = in some coordinate frame

M1 = ~ N1 ~ N1 = unpreventable M1 = effectable M2 = ~ N2 ~ N2 = determined, knowable M2 = unexcluded

62.3. Models and truth-conditions

All models are based on the same typical structure for a multi-modal logic (cf. Wölfl 1999 and

the review Strobach 2001) as consisiting of an ordered pair which of some class K providing

evaluation contexts, and a function which assigns an accesibility relation to each modal operator

plus an interpretation function which assigns truth-values to sentential constant s.

M = ⟨ ⟨K, {⟨O1, R ⟩, ...} ⟩, I ⟩ V( ....)=1

a)

Kr M = ⟨ ⟨T, {⟨G,<⟩, ⟨H,<⟩} ⟩, I ⟩ V(α,t)=1

SP5 M = ⟨ ⟨S, {⟨¤, R⟩} ⟩, I ⟩ V(α,s)=1

KrxSP5 M = ⟨⟨{T, S}, {⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩} ⟩, I ⟩ V(α,⟨t,s⟩)=1

KrxSP5e M = ⟨⟨{E,T,S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩} ⟩, I ⟩ V(α,⟨t,I⟩)=1

KrxSP5xF M = ⟨⟨{E,F},{⟨G,AF⟩, ⟨H, A’F⟩,⟨¤,RF⟩,⟨×, A×⟩} ⟩, I ⟩ V(α, t,s,f )=1

b)

S5 M = ⟨ ⟨INT, { ⟨N, A⟩} ⟩, J ⟩ V(α,I)=1

Kr x S5 M = ⟨ ⟨{INT,T} {⟨G,<⟩, ⟨H,<-1⟩,⟨N, A⟩} ⟩, J ⟩ V(α,⟨t,I⟩)=1

Kr x SP5e x S5 M = ⟨ ⟨{INT, E} {⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩,⟨N, A⟩} ⟩, J ⟩ V(α,t,s,I)=1

KrxSP5exFxS5 M = ⟨⟨{E,F,INT},

{⟨G,AF⟩, ⟨H, A’F⟩,⟨¤,RF⟩,⟨×, A×⟩,⟨N1,A1⟩, ⟨N2,A2⟩}⟩,J⟩ V(α, t,s,f,I)=1

73. The systems

3.1. General Basics

The formation rules for wffs of all of the following logics are as usual for languages of

propositional modal logic: alphabets consist of p, q, r, s, *, ), (, v and ~ plus any of the strong

modal operators mentioned above where introduced. The weak modal operators are defined as

mentioned above. p, q, r and s are sentential constants, and if α is a sentential constant, so is α*.

All sentential constants are well-formed formulas. So is (α v β) if α and β are. So is ~α, and so is

� α if α is, where α and β are wffs and where � is a strong modal operator.

Deduction rules presupposed throughout are: modus ponens and NECs for all strong modal

operators used, i.e. if α → β and a are deducible, so is β, and if α is deducible, so is � α . The

antisymmetry, and thus irreflexivity of the earlier-later relation is presupposed, but is left

unaxiomatized. An axiomatization of irreflexivity using the Gabbay-rule (cf., e.g. Wölfl 1999) is

possible and may be added ad lib.

3.2. The basic building blocks: PC, Kr and SP5

3.2.1. PC

PC is a standard two-valued propositional logic. Sentential junctors other than ~ and v are

defined as usual: (α→ β) =df für ~ α v β, (α & β) =df ~ ( α → ~ β), (α ≡ β) =df ( α → β) &

( β → α). Brackets are dropped as usual.The only slightly unusual point is the separation of the

interpretation function for sentential constants and the valuation function for wffs in general. PC

is, thus, defined as follows:

Axioms

(PC-1) α → (β → α )

(PC-2) (α → (β → γ)) → ((α → β ) → (α → γ ))

(PC-3) (~α → ~ β ) → (β → α )

Semantics

(1) A PC-interpretation function is a function which assigns to every sentential constant exactly

one of the values 0 or 1.

(2) Every well-formed formula α of PC is assigned the truth value 1 (true) or 0 (false) with

respect to any PC- interpretation function I - shortly VI(α) = 1 or VI (α) = 0 resp.- subject to the

following conditions:

(i) VI(α) = I(α) if α is a sentential constant

(ii) VI( ~ α) = 1 iff VI(α) = 0

(iii) VI( (α v β) ) = 1 iff VI(α) = 1 or VI( β) = 1 or both.

8

3.2.2. The rich tense logic Kr

Kr is supposed to be a standard refinement of the well-known minimal tense logic Kt, comprising

suitable axioms for PC and axioms which postulate all realistic features of linear time (even

continuity – which cannot be motivated before section 3.6.2., though) and including the

corresponding semantic constraints on models. Kr can, thus, be characterised as follows:

Axioms

(PC)

(G-K) G (α → β) → (Gα → Gβ) G-distribution

(H-K) H (α → β) → (Hα → Hβ) H-distribution

(FH) FH α → α FH combination axiom

(PG) PG α → α PG combination axiom

(Tr-G) Gα → GGα transitivity

(Infin-1) Gα → Fα future infinity

(Infin-2) Hα → Pα past infinity

(D) GGα → Gα denseness

(Cocchiarella) Gα → ( HG (Gα → PGα) → HGα) continuity

(Lin-P) (Pα & Pβ) → (P(α&β) v P(α & Pβ) v P(Pα&β)) past linearity

(Lin-F) (Fα & Fβ) → (F(α&β) v F(α & Fβ) v F(Fα&β)) future linerarity

Semantics

(1) A time-order for Kr is an ordered pair ⟨T, {⟨G,<⟩, ⟨H,<-1⟩}⟩, where T is an nonempty set [of

instants], < is a relation on T, and <-1 is the converse relation to < such that

(i) For all t, t’: if t < t’, then not t’ < t antisymmetry

(ii) For all t, t‘, t“ from T: if t < t‘ and t‘< t“ then t<t“ transitivity

(iii) For all t, t‘ from T: t < t‘ or t = t‘ or t‘ < t linearity

(iv) For all t, t‘ from T: if t < t‘ then there is a t“ such that t < t“ and t“ < t‘ denseness

(v) For every nonempty subset M, M‘ of T: if M and M‘ are disjoint continuity

and T = M∪M‘ and for every t from M, t‘ from M holds t < t‘ then there is

some t“ from T such that either: t“ ∈ M and there is no t“‘ from M such that

t“ < t“‘, or t“ ∈ M‘ and there is no t““ from M such that t““ < t“

(vi) For every t from T there is some t‘ from T such that t < t‘ future infinity

(vii) For every t from T there is some t‘ from T such that t‘ < t past infinity

Kt

9

(2) A model for Kr is an ordered pair ⟨⟨T, {⟨G,<⟩, ⟨H,<-1⟩}⟩, I ⟩, consisting of some time-order

⟨T, {⟨G,<⟩, ⟨H,<-1⟩}⟩ and some two-place interpretation function I which assigns to every

sentential constant with respect to every element of T exactly one of the values 0 or 1.

(3) For any model M for Kr with M = ⟨⟨T,{⟨G,<⟩, ⟨H,<-1⟩}, I⟩ any wff α of Kr is assigned exactly

one of the values „true“ (1) or „false“ (0) with respect to any t from T – shortly: VM(α,t) = 1 or

VM(α,t) = 0 - subject to the following conditions:

(i) VM(α,t) = IM(α,t) , if α is a sentential constant

(ii) VM( ~ α, t) = 1 iff VM( α, t) = 0

(iii) VM( (α v β) , t) = 1 iff VM( α, t) = 1 or VM( β, t) = 1 or both

(iv) VM( Gα, t) = 1 iff for all t‘ with t < t‘: VM( α, t‘) = 1

(v) VM( Hα, t) = 1 iff for all t‘ with t <-1 t‘: VM( α, t‘) = 1

3.2.2. The spatial logic SP5

SP5 is a simple S5 intended to serve as a logic of space. SP5 is not very expressive as a logic of

space: there is no way to formally render the 3D-structure, the infinity or the continuity of space.

Neither would be a spatial logic with irreflexive operators for „everywhere else“ and

„somewhere else“ (like the system proposed by v. Wright as early as 1951 (cf. Wright 1983) and

further discussed from 1976 on by Segerberg (cf. Segerberg 1980)) any more expressive in this respect. A more fine-grained spatial logic which distinguishes several dimensions would have

more of the desired expressivity, but its axiomatization seems to have met unexpectedly severe

difficulties (Gabbay et al. 2001). For the present purpose of combination a rather not too fine-

grained but well-studied system such as a spatial S5 is just what is needed. SP5 is characterized

as follows:

Axioms

(PC)

(¤-K) ¤ (α → β) → (¤ α → ¤ β )

(¤-T) ¤ α → α

(¤-S5) ¡ α → ¤ ¡ α

Semantics

(1) A spatial order is an ordered pair ⟨T,{ ⟨¤,R⟩}⟩, where

(i) S is a nonempty set [of places]

(ii) R is a two-place [accessibility] relation on S such that for all s, s’ from S: s R s’.

10

(2) A model for SP5 is an ordered pair ⟨⟨S,{⟨¤,R⟩}⟩, I ⟩ consisting of some spatial order

⟨S,{⟨¤,R⟩}⟩ and a two-place interpretation function I which assigns to every sentential constant

with respect to every place from S exactly one of the values 0 or 1.

(3) For any Kr x SP5-model M = ⟨⟨S,{ ⟨¤,R⟩}⟩, I ⟩ any wff α of SP5 is assigned exactly one of

the values „true“ (1) or „false“ (0) with respect to any s from S – shortly: VM(α,s) = 1 or

VM(α,s) = 0 - subject to the following conditions:

(i) VM(α,s) = IM(α,s) , if α is a sentential constant

(ii) VM( ~ α, s) = 1 iff VM( α, s) = 0

(iii) VM( (α v β) , s) = 1 iff VM( α, s) = 1 or VM( β, s) = 1 or both

(iv) VM(¤α,s) = 1 iff for all s‘ with s‘Rs: VM( α, s‘) = 1.

3.3. A starting point: the tempo-spatial system KrxSP5

KrxSP5 is an example for by now rather well-known bi-modal logics (for a general discussion

see Gabbay et al. 2001). Its most interesting feature is the interaction of temporal and spatial

operators which is very satisfactorily axiomatized by specialisations of the commutativity

axioms and the axiom for the Church/Rosser property (studied ibid. p.183ff). Formulae are

evaluated with regard to an ordered pair from T x S, where T is a set of instants and S a set of

space-points. The most important semantic constraint is that all space-points are mutually

accessible. Together with the decision for a unique set of space-points in the model (rather than a

different set for each instant, which would theoretically be possible) this leads to the

philosophically interesting constraint that space can neither grow nor shrink nor end in mutually

inaccessible fringes, a feature correspondingly postulated by the combination axioms. The tense-

operators are now of course relativized to space-points (Fp = it will right here be the case that p) and the spatial operators are relativized to instants (¤p = it is right now everywhere the case that

p).

Axioms

(PC), (Kr), (S5 for ¤) and

(com-P¡) P¡α ≡ ¡Pα „chessboard“-axioms

(com-F¡) F¡α ≡ ¡Fα

(chr-F¤) F¤α → ¤Fα Church/Rosser-axioms

(chr-P¤) P¤α → ¤Pα

(chr-¡G) ¡Gα → G¡α

(chr-¡H) ¡Hα → H¡α

11

An example for a very strong theorem which can be deduced from these axioms would be

(1) ¡Fp → ¤¡Fp S5

(2) ¡Fp ≡ ¡¡Fp S5

(3) ¡¡Fp → ¤¡Fp 1., 2.

(4) ¡¡Fp ≡ ¡F¡p with (com-F¡)

(5) ¡F¡p → ¤¡Fp 3., 4.

(6) ¤¡Fp ≡ ¤F¡p with (com-F¡)

(7) ¡F¡p → ¤F¡p 5., 6.

(Simple) Semantics

(1)Tempo-spatial coordinate frame: A tempo-spatial coordinate frame is an ordered pair

⟨{T, S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩}⟩, where

(i) T is a nonempty set [of instants]

(ii) S is a nonempty set [of space-points] which is disjoint from T

(iii) < and <-1 are two-place [accessibility] relations on T

(iv) R is a two-place [accessibility] relation on S.

Usual constraints ensure temporal antisymmetry (and thus irreflexivity), transitivity, linearity and

continuity, and we have: For all s, s’ from S: s R s’.

(2) A Kr x SP5-model is an ordered pair ⟨⟨{T, S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩}⟩, I ⟩ consisting of some

tempo-spatial coordinate frame ⟨{T, S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩}⟩ and a two-place interpretation

function I which assigns to every sentential constant with respect to every ordered pair from T x

S exactly one of the values 0 or 1.

(3) For any Kr x SP5-model M = ⟨⟨{T, S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩}⟩, I ⟩ any wff α of Kr x SP5 is

assigned exactly one of the values „true“ (1) or „false“ (0) with respect to any ordered pair ⟨t,s⟩ from T x S – shortly: VM(α, ⟨t,s⟩) = 1 or VM(α, ⟨t,s⟩) = 0 - subject to the following conditions:

(i) VM(α, ⟨t,s⟩) = IM(α, ⟨t,s⟩) , if α is a sentential constant

(ii) VM( ~ α, ⟨t,s⟩) = 1 iff VM( α, ⟨t,s⟩) = 0

(iii) VM( (α v β) , ⟨t,s⟩) = 1 iff VM( α, ⟨t,s⟩) = 1 or VM( β, ⟨t,s⟩) = 1 or both

(iv) VM( Gα, ⟨t,s⟩) = 1 iff for all t‘ with t < t‘: VM( α, ⟨t‘,s⟩) = 1

(v) VM( Hα, ⟨t,s⟩) = 1 iff for all t‘ with t <-1 t‘: VM( α, ⟨t‘,s⟩) = 1

(vi) VM(¤α, ⟨t,s⟩) = 1 iff for all s‘ with s‘Rs: VM( α, ⟨t,s‘⟩) = 1

12

3.4. Eventisizing Kr x SP5 as Kr x SP5e

The simple semantics for Kr x SP5 given in 3.3. are not the only kind of semantics that would fit

the axioms of Kr x SP5. There is a way of stating different semantics which, at first sight, may

look somewhat pointless and redundant. However, they provide the decisive building units for

the relativistic tempo-spatial system to be introduced in 3.5. The idea is not to construe

topological positions in space-time as ordered pairs of space-points and instants but to work just

the other way around, i.e. to take topological positions in space-time as basic and to construe

space-points and instants from them. Since these topological positions have been called „events“

by Einstein and Minkowski one may say that in this way Kr x SP5 is changed into a system Kr x

SP5e which differs from Kr x SP5 by containing eventisized semantics. Note, however, that wffs

are still evaluated with regard to ordered pairs from T x S and not with regard to events.

Eventisized semantics for Kr x SP5e

1) Eventisized tempo-spatial coordinate frame: an eventisized tempo-spatial coordinate frame is

an ordered pair ⟨{E,T,S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩}⟩, where

(i) E is a nonempty set [of tempo-spatial positions (events)]

(ii) T is a partition of E, i.e. (1) { x : ∃ t [ t ∈ T & x ∈ t] } = E

(2) ∀ t, t’ ∈ T [ t ≠ t’ → t ∩ t‘ = ∅ ]

(3) ∀ t ∈ T [ t ≠ ∅ ]

(iii) S is a partition of E, i.e. (1) { x : ∃ s [ s ∈ S & x ∈ s] } = E

(2) ∀ s, s’ ∈ S [s ≠ s’ → s ∩ s‘ = ∅ ]

(3) ∀ s ∈ S [ s ≠ ∅ ]

(iv) S and T are disjoint

(v) < and <-1 are two-place [accessibility] relations on T

(vi) R is a two-place [accessibility] relation on S.

Usual constraints ensure temporal antisymmetry (and thus irreflexivity), transitivity, linearity and

continuity; and we have the following constraints:

(a) For all s, s’ from S: s R s’.

(b) For every t from T and every s from S there is exactly one e from E such that: t ∩ s = {e}

(c) For every e from E there is exactly one t from T and s from S such that t ∩ s = {e}.

Clearly, constraints (b) and (c) produce a structure which is isomorphic to T x S, since,

analogously to (b) and (c), for every t from T and s from S there is exactly one ordered pair ⟨t,s⟩ in T x S and that for every ordered pair ⟨t,s⟩ from T x S there is exactly one t in T and s in S

which constitute ⟨t,s⟩. And just as we have, by definition of the ordered pair,

13(id-pair) For all t, t‘ from T, s, s‘ from S: t = t’ & s = s’ iff ⟨t,s⟩ = ⟨t‘,s‘⟩

we also have

(id-e) For all t, t‘ from T, s, s‘ from S: t = t’ & s = s’ iff t ∩ s = t‘ ∩ s‘.

For if t ∩ s ≠ t‘ ∩ s‘ although t = t’ and s = s’ there would have to exist some element of t ∩ s

which is not an element of t‘ ∩ s‘ or vice versa, which means that either one of the intersections

would have to be empty, which contradicts (b); or that one of the intersections would have to

contain more elements than the other which likewise contradicts (b). And if t ≠ t‘ or s ≠ s‘

although t ∩ s = t‘ ∩ s‘, there would have to be more than one t from T or s from S such that

they intersect on the same event, which contradicts (c).

An alternative formulation for (id-e) is

(id-e*) For all t, t‘ from T, s, s‘ from S: t ≠ t’ or s ≠ s’ iff t ∩ s ≠ t‘ ∩ s‘,

which is easily seen from the fact that ((p & q ) ≡ r) ≡ ((~ p v ~ q ) ≡ ~ r) is PC-valid.

(2) An eventisized Kr x SP5e-model is an ordered pair ⟨⟨{E, T, S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩}⟩, I ⟩ consisting of some eventisized tempo-spatial coordinate frame ⟨{E, T, S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩}⟩ and a one-place interpretation function I which assigns to every sentential constant with respect

to every e from E exactly one of the values 0 or 1.

(3) For any Kr x SP5e-model M = ⟨⟨{E, T, S},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩}⟩, I ⟩ any wff α of Kr x SP5

is assigned exactly one of the values „true“ (1) or „false“ (0) with respect to any ordered pair ⟨t,s⟩ from T x S – shortly: VM(α, ⟨t,s⟩) = 1 or VM(α, ⟨t,s⟩) = 0 - subject to the following conditions:

(i) VM(α, ⟨t,s⟩) = IM(α,e) , if α is a sentential constant and t ∩ s = {e}

(ii) VM( ~ α, ⟨t,s⟩) = 1 iff VM( α, ⟨t,s⟩) = 0

(iii) VM( (α v β) , ⟨t,s⟩) = 1 iff VM( α, ⟨t,s⟩) = 1 or VM( β, ⟨t,s⟩) = 1 or both

(iv) VM( Gα, ⟨t,s⟩) = 1 iff for all t‘ with t < t‘: VM( α, ⟨t‘,s⟩) = 1

(v) VM( Hα, ⟨t,s⟩) = 1 iff for all t‘ with t <-1 t‘: VM( α, ⟨t‘,s⟩) = 1

(vi) VM(¤α, ⟨t,s⟩) = 1 iff for all s‘ with s‘Rs: VM( α, ⟨t,s‘⟩) = 1.

The transformation of Kr x SP5 into Kr x SP5e may be visualized as follows:

14

T T

t1 t2 t3 t4 t5 t1 t2 t3 t4 t5

s1 s1 e1 e2 e3 e4 e5

s2 s2 e6 e7 e8 e9 e10

S s3 S s3 e11 e12 e13 e14 e15

s4 s4 e16 e17 e18 e19 e20

s5 s5 e21 e22 e23 e24 e25

⟨t4,s5⟩

3.5. The minimal relativistic tempo-spatial logic Kr x SP5 x F

We are now in a position to define a multi-modal logic Kr x SP5 x F whose intended models can

be visualized as Minkowski-style space-time diagrams. Its models are basically superpositions of

Kr x SP5e coordinate frames which are all based on the same set E of tempo-spatial positions and

share one interpretation function, but each of which may partition the members of E into very

different sets of spatial and temporal coordinates and, so-to-say, represent different possible

veinings of space-time:

etc.

etc.

15All frames are mutually accessible which determines the strong frame operator × (supposed to

remind a future and a past light-cone, while the weak frame operator + may be reminiscent of the

maxes of some coordinate system) as an S5-operator. Kr x SP5 x F is characterised as follows:

Axioms (PC), (Kr), (S5 for ¤), (com) and (chr)-axioms plus (S5 for ×), i.e.: (×-K) × (α → β) → (×α → ×β )

(×-T) ×α → α

(×-S5) + α → × + α

Semantics of Kr x SP5 x F

(1) Space-time: A space-time is an ordered pair

⟨{E,F},{⟨G,<F⟩, ⟨H, <-1F⟩,⟨¤,RF⟩, ⟨×, A×⟩ }⟩, where

(i) E is a nonempty set of tempo-spatial positions (events) (ii) F is a non-empty class of eventisized tempo-spatial coordinate frames {f1, f2 ...} all of which

contain E as a component

(iii) <F is a function which assigns to every fi from F just that accesibility relation for G which is

assigned to G in fi (i.e. { ⟨f1,<f1⟩, ⟨f2,<f2⟩... } )

(iv) <-1F is a function which assigns to every fi from F just that accesibility relation for H which

is assigned to H in fi (i.e. { ⟨f1,<-1

f1⟩, ⟨f2,< -1f2⟩... } )

(v) RF is a function which assigns to every fi from F just that accesibility relation for ¤ which is

assigned to ¤ in fi (i.e. { ⟨f1,Rf1⟩, ⟨f2, Rf2⟩... } )

(vi) A× is a two-place relation on F,

subject to the following constraint:

(S5-constraint) For all f, f’ from F: f A×f’.

As Ludger Jansen has pointed out to me, one might also more simply define a space-time as a set

of eventisized coordinate frames plus an accessibility relation: ⟨F, ⟨×, A×⟩ ⟩. The advantage of

this would be that, on a meta-level the very simple structure of a basic Kripke frame reappears. A

disadvantage of this would be to leave at this point the strategy of creating ever more complex

structures which fits the general scheme obvious in 2.3. where all modal operators of the

language can be clearly seen in definition of the model. At the end of the day, this is just a matter

of how to state things - either more elegant or more perspicuous. I shall here proceed with the –

hopefully – more perspicuous version.

16

(2) A Kr x SP5 x F-model is an ordered pair

⟨⟨{E,F},{⟨G,AF⟩, ⟨H, A’F⟩,⟨¤,RF⟩}, ⟨×, A×⟩⟩, I ⟩ consisting of some space-time

⟨{E,F},{⟨G,AF⟩, ⟨H, A’F⟩,⟨¤,RF⟩, ⟨×, A×⟩}⟩

and an interpretation function I which assigns to every sentential constant with respect to every e

from E exactly one of the values 0 or 1.

(3) For any Kr x SP5 x F model M = ⟨⟨{E,F},{⟨G,AF⟩, ⟨H, A’F⟩,⟨¤,RF⟩, ⟨×, A×⟩}⟩, I ⟩ any wff α

of Kr x SP5 x F is assigned exactly one of the values „true“ (1) or „false“ (0) with respect to any f

from F, any t from Tf and s from aus Sf – shortly: VM(α,t,s,f) = 1 or VM(α,t,s,f) = 0 - subject to

the following conditions:

(i) VM(α,t,s,f) = IM(α,e) , if α is a sentential constant and t ∩ s = {e}

(ii) VM( ~ α, t,s,f) = 1 iff VM( α,t,s,f) = 0

(iii) VM( (α v β) ,t,s,f) = 1 iff VM( α, ⟨t,s⟩) = 1 or VM( β, t,s,f) = 1 or both

(iv) VM( Gα, t,s,f) = 1 iff for all t‘ with t <f t‘: VM( α, t‘,s,f) = 1

(v) VM( Hα, t,s,f) = 1 iff for all t‘ with t <-1f t‘: VM( α, t‘,s,f) = 1

(vi) VM(¤α ,t,s,f) = 1 iff for all s‘ with s‘R f s: VM( α, t,s‘,f) = 1

(vii) VM(×α ,t,s,f) = 1 iff for all f‘ with f‘ A× f and all t‘, s‘ with t‘∈Tf‘ and s‘∈Sf‘:

if t∩s = t‘∩s‘ then VM( α, t‘,s‘,f‘) = 1.

Of course, some explanation is required why, of all clauses, the one for the strong frame operator

contains one additional detail which makes it different from the clauses for the other strong

modal operators. This is perhaps best explained by asking why one could not have defined, in

perfect analogy to the other operators, something like:

(vii-wrong) VM(×α ,t,s,f) = 1 iff for all f‘ with f‘ A× f : VM( α, t,s,f‘) = 1.

Now the problem is that VM(α,t,s,f‘) is not even defined if f‘ differs from f: t and s were

supposed to be from Tf and Sf resp., so they do not yield any truth-value in combination with f‘!

The evaluation space-point and the evaluation instant have to be taken from Tf‘ and Sf‘ resp. if,

together with f‘ they are to yield a truth value. So the question is which t‘ and s‘ one should

choose. Clearly, it should be that t‘ and s‘ which intersect right on e, since e is the topological

position via which one wants to access the other coordinate frame. Note, however, that, although

all four intersect on e, t may differ from t‘ and s from s‘:

17 t t‘

s‘

s

e

Before trying to exclude some of the most clearly unintended models, it may be interesting to see

how the modal operators work in intended models Kr x SP5 x F, since this can shed some light on

the expressivity of Kr x SP5 x F. Such models are of course models in which there are all the

coordinate frames there can possibly be. These are infinetely many, they are all roughly directed

in the same way, and in a Minkowski-style diagram their time- and space axes (coordinates

consisting of all happenings at the same place or happenings at the same time of some coordinate

frame resp.) cannot reach an angle of 45° with respect to the rectangular coordinate frame,

although they can arbitrarily approach 45°.

Now if something is the case right here and right now, for all past, and for all future (and

nowhere else and at no other time) in some coordinate frame, then this means that there is one

possible time axis at a certain angle all along which alone the corresponding statement p is true.

If something is the case throughout all the future light cone of right here and right now (and

nowhere else and at no other time), then this means that p is true all along all future parts of all

possible time axes at any permissable angle. Something analogous is true for the past light cone.

If something is the case throughout all the light-cone’s spacelike complement and nowhere else,

then this means that p is true all along all possible space-axes at any permissable angle. We may,

for those and similar reasons, roughly say that:

(1) + ¤ p hightlights a space-axis

(2) + (H p & p & Gp) highlights a time-axis

(3) × Gp highlights a future light-cone

(4) × Hp highlights a past light-cone

(5) + G p is like a searchlight into the future light-cone

(6) + H p is like a searchlight into the past light-cone

(7) × (Hp & p & Gp) highlights a double cone (all time-like e‘ relative to some given e)

(8) × ¤ p highlights the complement of the double cone (all space-like e‘ )

(9) ¤ (H p & p & Gp) highlights the whole space-time field, but only relative to one

floodlight-stand

(10) פ (H p & p & Gp) highlight the whole space-time field as brightly as possible with all floodlights on.

18(1) (2) (3) Gp Gp Gp Gp Gp Gp Gp

p

¤p

Hp

(4) (5) (6) Gp

Hp Hp Hp Hp Hp Hp

(7) (8) (9) ¤ (H p & p & Gp) Gp Gp Gp ¤p ¤p p p p ¤p ¤p ¤p Hp Hp Hp

The fine-grained expressivity of Kr x SP5 x F which thus becomes clearly visible is due to the

fact that different components are distinguished by different operators. This may be seen as an

advantage over the Prior/Goldblatt approach where the angle of time-axes could not be

represented in isolation, spatial distances are not recognizable as such, and purely space-like

events relative to a given event cannot be accessed at all.

Interestingly, the causal, absolute future of some event (about which Prior cared only, possibly

due to some anti-relativistic bias) and the causal past can be expressed in Kr x SP5 x F, too (cf.,

however, section 3.6.4.2. below):

An irreflexive causal F-operator translates as +F ; An irreflexive causal P-operator translates as +P ; An irreflexive causal G-operator translates as ×G ;

An irreflexive causal H-operator translates as ×H

19(for a corresponding result cf. Müller (2000), 200, for an intuitive proof ibid. footnote 355).

This may be somewhat reminiscent of the relation between „F“ in indeterministic tense logic

with a strong future operator and „N F“ in indeterministic tense logic with a weak future

operator. Nicely, the analogy holds even syntactically, which may be seen as follows:

~ GP/G ~ α ≡ FP/G α ⇔ ∼ × G ~ α ≡ +F α ⇔ ∼ ∼ + ∼ G ~ α ≡ +F α ⇔ ∼ ∼ + ∼ ∼ F ~ ~ α ≡ +F α ⇔

+ F α ≡ +F α

3.6. Possible refinementes of Kr x SP5 x F 3.6.1. Strategic reflections

Although the results from 3.5. are encourageing and although clearly all intended models are in,

Kr x SP5 x F leaves us with an enormous lot of quite bizarre and certainly unintended models: it

has, e.g., not been excluded that someone’s time is someone else’s space, that someone’s time is

directed exactly the other way around as someone else’s, that someone’s time is not the same as

someone else’s although they share the same spatial coordinates or that someone’s space is not

the same as someone else’s although they share the same temporal coordinates. All this can, of

course, not happen according to the theory of relativity, and it may be seen as positive side-effect

of Prior’s bias that he stressed that it cannot – contrary to the impression some popular

introductions to relativity and maybe even contrary to the impression some exaggerated wording

in Einstein 1917 and Minkowski 1923 might leave at first sight. Now is it not a strong point

against Kr x SP5 x F that all the crazy models mentioned above are perfectly respectable models

of Kr x SP5 x F? Perhaps it is not, if one compares the situation to classical tense logic: the

minimal tense logic Kt and its standard semantics leave us with an enormous lot of bizarre and

unintended models, too (just think of circular time!). Nevertheless, Kt is very a good system to

start with. Not the smallest advantage of it is that it is easily demonstrated to be complete

(Rescher / Urqhart 1971) with respect to minimally characterised models. Now if one has a

completely axiomatized basic system (with ever so many unintended models) then this is a good

position to start excluding unintended models by providing additional axioms which postulate

some desired features and exclude unwelcome ones, and by adding semantic constraints that

match the axioms as well as possible. If one follows this strategy, completeness of the refined

systems is not the main aim, as the basic system is complete already and the intended models are

in. Rather, the main job of the additional axioms is to distinguish classes of welcome models

from classes of unwelcome ones. The history of tense logic shows that this strategy is very

fruitful when it comes to realistic applications which soon excede what can easily be

demonstrated to be complete anyway.

20 Kr x SP5 x F is an interesting starting point and might turn out to resemble Kt in this respect

just because the frame operator does not axiomatically interact with any of the other operators in

Kr x SP5 x F. Admittedly, this feature is responsible for a lot of unintended models. But on the

other hand it leaves a good deal of optimism concerning its completeness: if Kr x SP5 is

completely axiomatized by adding the chessboard- and the Church/Rosser-axioms to the axioms

of Kr and to the S5 axioms for „everywhere“ then adding S5 axioms for the frame operator

without any further combination axioms should simply be a fusion of two complete modal logics

(since S5 for the frame operator is, of course, complete with respect to a set of coordinate frames

if they are all mutually accessible); and a mere fusion of two complete modal logics is bound to

be complete, too (as is shown in Gabbay et al. 2001).

It would certainly be most desirable if one could go one step further and provide some simple

combination axioms for the interplay of frame operators and other operators which might be

complete relative to some semantics containing constraints tailor-made for these axioms.

Furthermore, one should endorse the following highly plausible principle (which suggests not

to look for realistic but unperspicuous axioms and constraints too quickly):

Any modal logic with n operators should contain the next less complex logic with n – 1

operators as a limiting case, i.e. should allow for vacuous models in which the nth operator is

trivialized.

As far as I see, all well-studied modal logics with reflexive operators follow this principle, which

provides for a nice kind of unity among different systems by linking a system to the next less

complex one. E.g. the N-operator is trivialized in one-world models: one-world models of

combined tempo-modal systems collapse into tense logic, and one-world models of simple S5

collapse into non-modal propositional logic.

To sum up, good combination axioms should fulfil the following conditions:

1. They should exclude a class of particularly unwelcome models

2. They should allow for a vacuous model where the new operators are trivialized.

3. They should match some well-known semantic feature, or they should at least be one-to-one

translatable into more or less handy semantic constraints in order to optimize chances for a

completeness proof.

4. They should express some typical feature of all intended models.

3.6.2. A suggestion for a pair of basic refinement axioms

It can be argued that the following axioms fulfil the requirements just mentioned:

(Rel-F) +F¡+ α → ×F¡+ α (Rel-P) +P¡+ α → ×P¡+ α

21It is trivial that these axioms hold in vacuous models: if there is just one coordinate frame

available what holds for some frame holds for any.

It requires some explanation, though, what these axioms effect in intended models. The idea is

as follows: If, for some coordinate frame, in the future of some given event e there is another

event from which a certain state is spatially accessible, then for all coordinate frames there is an

event in the future of e such that from that event on the same state is spatially accessible. The

same holds analogously for the past.

Since weak spatial and temporal operators commute, it trivially follows from (Rel-F) and

(Rel-P) that:

(1) +¡F+ α → סF+ α (2) +¡P+ α → סP+ α

Pictorially, this means that whatever combination of a „step“ in space and a „step“ in time you

may choose to take in a rectangular coordinate frame the same kind of combination of „steps“

will lead you to the same target in any coordinate frame no matter how much the axes are

inclined. This provides for a certain equalisation of time direction in all coordinate frames. And

this excludes a lot of unwelcome models and stresses a typical feature of all intended models.

This does, however, not explain yet why the second frame operator in each half of the

formulae is needed, e.g.

(Rel-F) +F¡ + α → ×F¡ + α

The following diagram shows why: take α to be the formula Fp. Then without the second frame

operator in the axioms one would at once obtain the following formula as a thereom:

(3) +F¡Fp → ×F¡Fp

But this is not at all plausible: it may well be that for some coordinate frame f there is an event e‘

in the future of a given event e such that another event e“ is spatially connected with e‘ such that

in the future of e“ relative to f p holds, but that for some other coordinate frame f‘ – due to a

different inclination – the time axis of f‘ through e“ misses p:

22

s f11 s f2

1 s f12 s f2

2

tf13 e1 I(p, e1) = 1

f1 = rectangular V(p, t f13, s f1

2, f1) = 1

f2 = rhomboid

V(Fp, t f12, s f1

2, f1) = 1

tf12 V(¡+Fp, t f1

2, s f11, f1) = 1 V(Fp, t f2

2, s f22, f2) = 0 (!)

V(+Fp, t f22, sf2

2, f2) = 1

V(¡+Fp, t f22, sf2

1, f2) = 1

V(F¡+Fp, t f11, s f1

1, f1) = 1

V(+F¡+Fp, t f11, s f1

1, f1) = 1 V(F+Fp, t f21, sf2

2, f2) = 1

t f22 V(F¡+Fp, t f2

2, sf21, f2) = 1 V(F+Fp, t f1

1, s f11, f1) = 1

t f11 V(¡F+Fp, t f2

2, sf21, f2) = 1

V(¡F+Fp, t f11, s f1

1, f1) = 1

t f21

analogously with P

e2

A special case in which the two axioms hold and which is of particular philosophical interest is

one in which the spatial „steps“ are empty. This is possible since the spatial operators are –

unlike the temporal ones – reflexive:

23 s f1

1 s f21 s f2

2

e1 mit I(p, e1) = 1

tf13 V(p, t f1

3, s f12, f1) = 1

V(Fp, t f12, s f1

1, f1) = 1

V(+Fp, t f12, s f1

1, f1) = 1

V(¡+Fp, t f12, s f1

1, f1) = 1 V(¡+Fp, t f22, s f2

2, f2) = 1 tf1

2

V(+Fp, t f22, s f2

1, f2) = 1

t f22

V(F¡+Fp, t f11, s f1

1, f1) = 1

V(+F¡+Fp, t f11, s f1

1, f1)=1

t f11

V(F¡+Fp, t f21, s f2

2, f2) = 1

t f21 V(F+Fp, t f2

1, s f21, f2) = 1 V(¡F+Fp, t f2

1, s f22, f2) = 1

The philosophical importance of this special case is that it accounts for the irreversibility of

causally related happenings (happenings on the same world-line, possibly constituting the

biography of a person): if some event lies here in my future such that I can simply wait for it then

it is possible to wait for it to happen somewhere in any future. As this holds for some arbitrary

accessible coordinate frame, and as two happenings which are related by a possible time axis of some coordinate frame are just such that they may be causally related, one may also say that any

two events which can be causally connected in one coordinate frame can be so in all coordinate

frames. This is just what constitutes the causal futures (and analogously the causal pasts) of the

theory of relativity. And they are absolute futures or pasts in exactly the way that has just been

described.

Besides, the triangular diagram shows why it was sensible to include an enourmously strong

feature such as continuity in the rich tense logic Kr that accounts for the temporal part of

Kr x SP5 x F. The idea is best conveyed by imagining that some metric has already been defined

for Kr x SP5 x F. Now if the distance between t f11∩ s f1

1 and t f12 ∩ s f2

2 were 3 units and the

distance between t f12∩ s f1

1 and t f12 ∩ s f2

2 were, say, 1 tempo-spatial distance unit (of f1), then,

by Pythagoras‘ theorem, the distance between t f11∩ s f1

1 and t f12∩ s f1

1 would have to be √10

such units on the time axis. But in order to be there some event at √10 units from t f11∩ s f1

1 on

the time axis, time has to be not merely dense but continuous, since √10 is irrational.

24If one wants to do without postulating continuity one should ensure meeting points by adding

the following constraint (which does not follow from the definition of the model yet):

Meeting-point constraint

For any f, f’ from F and any tf from Tf, sf’ from Sf’:

If f ≠ f’ then there is exactly one e from E such that tf ∩ sf’ = {e}.

As a side effect, this excludes serpentine models where one time axis winds around another like a

snake around a tree and intersects it more than once. It is tempting to try whether one of the two

axioms, say (Rel-P) might be syntactically redundant in the same way as, in classical tense logic,

either

(Tr-F) FFα → Fα or

(Tr-P) PPα → Pα

(or, equivalently, either Gα → GGα or Hα → HHα) can be shown to be redundant by using the

Kt-axioms FHα → α or PGα → α resp. (for a proof cf. e.g. Wölfl 1999 p.73 applied to p.128).

So far, I have not been able to find out whether this is the case.

It is, however, clear that, as to a semantic constraint, it suffices to translate one of the two

axioms into semantics. The semantic constraint corresponding to both (Rel-F) and (Rel-P) then

turns out to be, as a one-to-one translation of (Rel-F):

equalisation-constraint

For any e from E and any f from F:

if there is some f‘ from F with f‘ Aá

f +...

such that there is some e‘ from E such that t f’e <f‘ tf’

e‘, ...F...

such that there is some e“ from E such that sf’e“ Rf‘ sf’

e‘; ........ ¡

then it holds for any f“ from F that ............. ×

there is some e“‘ from E such that tf“e <f“ tf“

e‘“, .................. F such that sf“

e“‘ Rf“ sf’e. ...................... ¡

This looks forbidding at first sight, but if you read it off from a part of the diagram given above

then it becomes clear that this constraint matches (Rel-F) intuitively, too:

25

f‘ = rectangular

f“ = rhomboid

e‘ e“

e“‘

e

The same constraint applies to (Rel-P) because (Rel-P) corresponds just to the same sort of

situation from a different point of view (i.e. the event which has now been labelled e“).

Admittedly, a one-to-one translation of this sort is not a very elegant procedure, but it works.

And it may turn out to pay off: Firstly, we can say that (Rel-F) and (Rel-P) (or, in fact, either of both) singles out just those models for Kr x SP5 x F in which the equalisation-constraint holds.

But perhaps something stronger is the case, i.e. that the axiomatics of Kr x SP5 x F + (Rel-F) and

(Rel-P) provide a complete axiomatization for temporally equalized Kr x SP5 x F models. There is no guarantee for that, since the equalisation-constraint may lead to all sorts of valid formulae

which are not deducible from Kr x SP5 x F + (Rel-F) + (Rel-P). But there is a reasonable chance, and I leave the answer to the question whether that is so to future work.

3.6.3. Why someone’s owl should rather be someone else’s nightingale

In Platt, the „flatland“ dialect once spoken all over the North of Germany, and still today in some

regions, there exists a nice proverb in oder to express aesthetic relativity:

Den een‘ sien Uhl is den annern sien Nachtigal

Someone’s owl is someone else’s nightingale.

It is clear that for the special theory of relativity, some analogous statements hold if one reads

„someone“ as „someone who has chosen one particular coordinate frame“ and „someone else“ as

„someone who has chosen some different coordinate frame“ as the basis for his tempo-spatial

talk:

26(a) Someone’s time is not someone else’s time

(b) Someone’s space is not someone else’s space

(c) Someone’s time is not someone else’s space

(d) Someone’s space is not someone else’s time

(e) Someone’s future is not someone else’s past

(f) Someone’s past is not someone else’s future

These are interesting constraints on Kr x SP5 x F models, since the basic semantics for

Kr x SP5 x F allows for counterexamples to all of these principles, all of which are clearly very

unwelcome models.

The formal constraints which should be added to the semantics for Kr x SP5 x F for (a) to (f)

are easy to formulate:

Owl / nightingale-constraint

For all f, f’ from F: if f ≠ f’ then both (a) Tf and Tf’ as well as

(b) Sf and Sf’ as well as

(c) Tf and Sf’ are all disjoint.

From (c) and from the fact that being disjoint is a symmetrical relation it follows immediately

that in that case also (d) Sf and Tf’ are disjoint. And from the fact that the mentioned classes are

all disjoint it also follows immediately that there cannot hold Sf ≠ Sf’ and Tf = Tf’ and neither

Tf ≠ Tf’ and Sf = Sf’. Furthermore, if, in that case, Tf and Tf’ are disjoint then so must be <f and

<f’, as they are defined on Tf and Tf’ respectively – which is what (e) and (f) requires.

It is much more difficult, however, to find axioms that correspond to these constraints, and

one might well wonder if Kr x SP5 x F is expressive enough to provide object-language

postulates to match these constraints. In fact it is, although the candidates for axioms in this case

which I have been able to find are suprisingly complicated. It is true that simpler ones would be

welcome. On the other hand, if one follows the strategy outlined in 3.6.1. it often happens that

quite complicated axioms are nevertheless informative and do their job. This is another lesson to

be learnt from classical tense logic. A good examples for a very complicated axiom which is

motivated by thinking of one specific counter-example to a desired semantic feature is the

Cocchiarella axiom which postulates continuity (cf. for its motivation Prior 1967, p.72):

Gp → ( HG (Gp → PGp) → HGp ).

Here are suggestions for formulae which could serve to axiomatically postulate the semantic

features expressed by the owl/nightingale-constraint. They are explained in order of

complication:

27

(ad e) Someone’s future is not someone else’s past:

¤ (Gα & α & H~α) → ~ + (G~ α & α & Hα)

Suppose that everywhere α is true and that this will remain so during all future in f, but that it

had never been anywhere at any time before (relative to f). Then there is no coordinate frame f‘

such that it is just the other way around in f‘ (note that in all examples that follow future and past are relative to the frame and not just causal futures or pasts, i.e. light cones!).

(ad f) Someone’s past is not someone else’s future

¤ (Hα & α & G~α) → ~ + (H~ α & α & Gα)

Suppose that everywhere α is true and that it has been so for all past in f, but that it won’t ever be

again (relative to f). Then there is no coordinate frame f‘ such that it is just the other way around.

So (e) specifically excludes a situation like this:

p ~ p p + p

~ p p

And so does (f) (substitute ~ p for α).

(c) Someone’s time is not someone else’s space:

¤ (Gα & α & H ~ α) → ~ + ¡ (Gα & α & Hα)

Suppose that α will remain true everywhere during all future in f, but that it had never been

anywhere at any time before (relative to f). Then clearly F¤p is true. If you rotate the whole diagram by 90° you have a situation from the point of view of a coordinate frame f‘ where space

and time are just swapped with regard to the original diagram. If such a frame is accessible from

the original frame, then what was formerly described as F¤p must now be described as

¡(Gp & p & Hp): the future operator „it will be the case that“ translates as the weak space

operator „it is somewhere the case that“, and „everywhere“ translates to „always“. Now the

axiom says that no such f‘ exists. So (c) excludes a situation like this:

28

p

¤p + (Gp & p & Hp)

p ~ p p

~ p

The mirror image ¤ (Hα & G ~ α) → ~ + ¡ (Hα & α & Gα) works just as well and should be

added, too, as should happen in all other cases.

(ad c) Someone’s space is not someone else’s time:

¡ (Gp & p & Hp) & ¡ (G~p & ~p & H~p) → ~ + ¤ (Pp & F~ p).

There is one place at which the state described by p holds throughout all time of f:

¡(Gp & p & Hp). And there is another place where it fails to hold, likewise throughout all time

of f: ¡(G~p & ~p & H~p). What you want to exclude is a 90° rotation. In that case only you

could have a situation in which p holds everywhere at some time in the past and ~p holds

everywhere at some time in the future (all other angles would lead to some intersection with both

the p- and the ~p-axis!). So what you want to exclude is that in the situation described you can

have + ¤ (Pp & F~ p):

~ p

p ~p + ¤ (Pp & F~ p)

p

¡ (Gp & p & Hp) & ¡ (G~p & ~p & H~p)

(ad b) Someone’s space is not someone else’s space:

¤ (Gα & α & H ~ α) & (Gβ & β & Hβ) → × ( ~ (Gβ & β & Hβ) → ~ F¤ α )

Think of two states which are described by p and q resp. The state described by p holds

everywhere on the space axis in the middle and everywhere throughout its future (in f), and it

fails to hold throughout the lower half: ¤ (Gp & p & H ~p ). Clearly, there is, then, a future time

(in f) when p is true everywhere: F¤p. The state described by q holds all along the time axis (in

f); it could be a place-proposition: (q & Gq & Hq). Assume that if you incline the time axis this

will not be so, though, so then you will have: ~ (q & Gq & Hq). If you have oblique space axes,

too, this is fine. But then every space axis will have to intersect the p/~p-border somewhere, so it

will never be all p. What you want to exclude is that you have an oblique time axis, but your

29spatial coordinates remain the same. So what you want to exclude is that you have a

coordinate frame in which both ~ (Gq & q & Hq) and F¤p are true. So for all f‘, if ~ (Gq & q &

Hq) is true in f‘, then F¤p won’t: × ( ~ (Gp & p & Hp) → ~ F¤p).

p p ¤p + ~ (Gq & q & Hq)

(Gq & q & Hq)

~ p ~ p

(ad d) Someone’s time is not someone else’s time:

¡(Gα & α & Hα) & ¡(G~α & ~α & H~α) & ¤(G~β & β & H~β) → ...

... × ( ~ ¤ ((G~β&β&H~β) → ~ ¡ (Gα & α & Hα)) )

This looks frightening at first sight. But it is not so difficult to motivate. Think of two states

which are described by p and q resp. There is one place at which the state described by p holds

throughout all time of f: ¡(Gp & p & Hp). And there is another place where it fails to hold,

likewise throughout all time of f: ¡(G~p & ~p & H~p). Furthermore, your present space axis (in

f) is the only time when the state described by q holds, but now it does everywhere (q might be a

date proposition in f): ¤ (G~q & q & H~q). Now what you want exclude is that it is possible to have an oblique space axis while keeping the upward time axis. On the oblique space axis, it

would have to be true that ~ ¤ (G~q & q & H~q), as all the future and past part (in f) is ~q. That is fine as long as your time axis is inclined, too. But in that case it has to intersect the ~p axis

somewhere, so it cannot all be p, i.e. (Gp & p & Hp). So you can say that in all coordinate

frames f‘, if their space axis is oblique (compared to f) then it will not happen that you have an

all-p time axis somewhere: ×(~ ¤ (G~q & q & H~q) → ~ ¡ (Gp & p & Hp) ).

(G~p & ~p & H~p) (Gp & p & Hp)

¤ (G~q & q & H~q)

+ ~ ¤ (G~q & q & H~q)

¡ (Gp & p & Hp)

To sum up, if one adds the owl / nightingale-constraint to the semantics of Kr x SP5 x F then one

should at least add the owl/nightingale-postulates to the axiomatic base in order to distinguish

owl / nightingale-models from others. Completeness, however, is hardly to be hoped for with

formulae as complicated as these.

303.6.4. Postulates requiring models with more than one coordinate frame

3.6.4.1. At least two coordinate frames: the Rietdijk situation

Rietdijk’s argument that already the definitional tempo-spatial core of the special theory of

relativity implies determinism (Rietdijk 1966) is highly contested – to say the least (Stein 1968,

1991). In my view (see the last section below) the price that must be paid in order to escape

Rietdijk’s conclusion is somewhat higher than is usually thought, but I agree with most

philosophers involved that, at the end of the day, Rietdijk’s argument turns out not to be sound.

However, Rietdijk tries to exploit a situation which exhibits a philosophically interesting and

highly characteristic feature of special relativity. And he even states it in a rather moderate

version:

(Rietdijk’s argument): For every coordinate frame f and and any e’ in the causal future of

some given event e in f there is some event e” which is simultaneous with e in f such that

there is some other coordinate frame f’ such that e” is simultaneous with e’ in f’. So in that

sense every event in the causal future of e is already present and thus unalterable.

Never mind here whether the second sentence really follows from the first one. Here is a stronger

version of Rietdijk’s argument:

(Rietdijk’s argument*): For every coordinate frame f and and any e’ in the causal future of

some given event e in f there is some event e”’ which is simultaneous with e in f such that

there is some other coordinate frame f’ such that e”’ lies in the past of e’ in f’. So in that

sense every event in the causal future of e is already past and so sure enough unalterable.

The situation Rietdijk imagines (and which implies also the stronger version of the argument) is

this:

e‘: p

e: Fp, +Fp, ¡+¡p, ¡+¡+p, ¡+¡+Pp

e“: +¡p, +¡ + p

e“‘: +¡ + P p

31Interestingly, the situation can be created with any ever so slight inclination of the space axis

and any event in the ever so distant future (in f): if you choose the spatial distance large enough

you can always reach or even pass by the „future“ event.

In terms of refinement of Kr x SP5 x F one may therefore consider adding as axioms in order

to distinguish realistic models for the theory of relativity from unintended models for Kr x SP5 x

F the following formulae:

(Rietdijk) +Fα → ( ¡+¡+α)

(Rietdijk*) +Fα → (¡+¡+Pα).

Pictorially, the frame operator in the antecedent produces some kind of turnabout searchlight in

order to cover the whole causal future of the evaluation event. The second frame operator in the

consequent is, of course, again due to the requirement of shifting back gears to the original frame

if you want to keep α general and if you want to refer to the past in f by using P (cf. 3.6.2.

above). The formulae nicely exhibit the full force of the correct claim Rietdijk tries exploit

(correctly or not): a future operator relative to one coordinate frame simply disappears relative to

another frame or is even “replaced” by a past operator. Their one-to-one translation into semantic

constraints is easily effected (where tfe is that t from Tf which contains e etc. – the definite

description is in order due to Tf being a partition of E whose elements must be disjoint):

Rietdijk-constraint

For any e from E holds:

If there is some f from F and e‘ from E such that t fe <f t f

e‘

then there is some e“ from E such that sfe“ Rf s f

e

and some f‘ from F with f‘ A× f such that sf‘e“ Rf sf’

e‘

Rietdijk*-constraint For any e from E holds:

If there is some f from F and some e‘ from E, such that t fe <f t f

e‘,

then there is some e“ and some e“‘from E such that s fe“ Rf s f

e“‘

and there is some f‘ from F with f‘ A× f such that sf‘e“ Rf sf’

e“‘ and t f’e“‘ <-1

f‘ tf’e‘.

These constraints, as well as the corresponding postulates, are on a somewhat different level than

those discussed before: they do not work for models which contain only one coordinate frame.

So if the Rietdijk postulates are to be added to the axiomatic base then a semantic requirement is

that the model contain more than one coordinate frame.

Actually, this follows from the Rietdijk-constraints: since Tf, Tf’, Sf and Sf’ are all partitions of

E, both belonging to the same t from Tf and belonging to the same t from Tf’ are equivalence

32relations on E, and are thus reflexive, symmetrical and transitive. If there is some t from Tf

such that e and e” both belong to t, there is some t’ from T f’ such that e” and e’ both belong to t’, and if f = f’, then, due to the transitivity of “belongs to the same t from Tf”, there is some t”

from Tf such that both e and e’ belong to t” (so, actually, t = t’ and t’ = t”). But since tfe <f tf

e’, we

have tfe ≠ tf

e’, due to the irreflexivity of <f. Therefore f ≠ f’.

So how many coordinate frames do we need in order to make the Rietdijk axioms plausible?

The answer is: just two, for the reasons stated above: any angle, no matter how small, will do.

3.6.4.2. Infinitely many coordinate frames

(a) How to integrate Prior’s and Goldblatt’s S4.2.

Let us say that M is a rich model for Kr x SP5 x F is a model for Kr x SP5 x F that contains

infinitely many coordinate frames, and there is one coordinate frame available for every

permissable angle of the axes. It is this kind of model where formulae like ×Gp can serve to fill a whole future light-cone.

Now it seems seems that on rich models some very interesting additional axioms and some

very interesting corresponding semantic constraints can be introduced: such models can

incorporate the whole Prior / Goldblatt approach and may in a certain way be regarded as

conservative extensions of what the two authors have achieved. We have already seen that

irreflexive causal „tense“ operators can be mechanically translated into combinations of

operators of Kr x SP5 x F. Prior’s conjecture (Prior 1967, p.202-4), which was proved to be

correct by Goldblatt (Goldblatt 1980), was that the behavior of causal „tense“ operators in

relativistic tense logic can be adequately axiomatized by a tense-logical version of the somewhat

esoteric modal logic S.4.2., which is described by the following axioms (where the box is just

some strong modal operator and the diamond defined as ~ � ~):

(� -distr) � (α → β) → (� α→ � β) � -distribution

(Refl-� ) � α → α � -reflexivity

(Tr-� ) � α → � � α � -transitivity

(S4.2.) ◊ � α → � ◊ α directedness

So one might simply want to add one-to-one Kr x SP5 x F translations of the tense-logical

version of S4.2. as additional axioms for rich models for Kr x SP5 x F and also add semantic

constraints that simulate an S4.2. model for the combined operators in the translations of the

S4.2.-axioms. These semantic constraints could, in turn, serve to precisely distinguish rich

models from other models of Kr x SP5 x F.

As can be seen from the choice of S 4.2., and for a change from standard tense logic, both

Prior and Goldblatt presuppose reflexive „Diodorean“ tense operators. Goldblatt – unlike Prior –

explicitly postulates reflexivity (cf. Prior 1967, p.205, Goldblatt 1980 p.220). Moreover,

33Goldblatt does not consider any past-tense operators and, thus, neither any connection between

past and future tense operators. If one wishes to consider past-tense operators, too, then one

should add mirror images and the usual Kt combination axioms, as Prior in fact suggests.

Postulating infinity, as Prior also does, seems appropriate enough (cf. ibid.).

Luckily, it is not only possible to define „causal“ operators using the operators from Kr x SP5

x F (GP/G α =df. ×Gα, HP/G α =df. ×Hα, cf. the end of section 3.5. above), but also to define

reflexive operators using irreflexive ones:

GP/Grefl α =df. α & GP/G α

HP/Grefl α =df. α & HP/G α

FP/Grefl α =df. ~ GP/G

refl ~ α

PP/Grefl α =df. ~ HP/G

refl ~ α

If, in the way just suggested, one combines Prior’s and Godblatt’s slightly different tense-logical

translations of S4.2. then one obtains the following very strong additional axioms which are

plausible on rich models for Kr x SP5 x F:

(GP/Grefl-K) GP/G

refl (α → β) → (GP/Grefl α→ GP/G

refl β)

(HP/Grefl-K) HP/G

refl (α → β) → (HP/Grefl α → HP/G

refl β)

Kt

(FHP/Grefl) FP/G

refl HP/Grefl α → α

(PGP/Grefl) PP/G

refl GP/Grefl α → α

(Refl-GP/Grefl) GP/G

refl α → α GP/Grefl-Reflexivity

(Refl-HP/Grefl) HP/G

refl α → α HP/Grefl-Reflexivity

(Tr-GP/Grefl) GP/G

refl α → GP/Grefl GP/G

refl α GP/Grefl-transitivity

(Tr-HP/Grefl) HP/G

refl α → HP/Grefl HP/G

refl α HP/Grefl-transitivity (redundant for (FHP/G

refl))

(Infin-1P/Grefl) GP/G

refl α → FP/Grefl α infinity of the causal future

(Infin-2P/Grefl) HP/G

refl α → PP/Grefl α infinity of the causal past

(DP/Grefl) GP/G

refl GP/Grefl α → Gα denseness (optional)

(4.2.-GP/Grefl) FP/G

refl GP/Grefl α → GP/G

refl FP/Grefl α typical S 4.2.-axiom future version

(4.2.-HP/Grefl) PP/G

refl HP/Grefl α → HP/G

refl PP/Grefl α typical S 4.2.-axiom past version

distribution

future/past links

34Now it might seem that it must be very difficult to state the corresponding semantic

constraints for this. Luckily, it is rather easy if one keeps in mind that S4.2. characterizes just

reflexive, transitive, directed frames (Goldblatt 1980, p. 220, Hughes / Cresswell 1996, p. 134,

362). It is therefore possible to to simulate exactly matching semantic constraints in a

straightforward manner by postulating the „causal“ earlier-than-or-simultaneous-with relation as

a kind of ghost relation:

Prior / Goldblatt constraint

There exists a two-place relation R P/G on E such that

(1) for any e, e‘ from E e RP/G e‘ iff

either e = e‘

or there is some f from F and some t and t‘ from Tf such that t <f t‘ and e ∈ t and e‘∈ t‘,

(2) RP/G is transitive

(3) RP/G is upward directed (Goldblatt 1980, 220f), i.e. for all e, e‘ from E

there is some e“ with both e RG e“ and e‘ RG e“

(4) RP/G is downward directed (Rakic 1997, 267), i.e. for all e, e‘ from E

there is some e“ with both e RG–1 e“ and e‘ RG

–1 e“

(5) For every e, e‘ from E with e RP/G e‘ and e ≠ e‘ and there is some e“ from E

such that e RP/G e“ and e“ RP/G e‘ and e“ ≠ e and e“ ≠ e‘ (denseness)

(6) For every e from E there is some e‘ from E with e ≠ e‘ and e RP/G e‘ (future infinity)

and there is some e“ from E with e ≠ e“ and e RP/G–1 e‘ (past infinity).

Rakic adds antisymmetry for the causal earlier-than relation (Rakic 1997, ibid.), which cannot be

expressed in the object-language, though. The idea to postulate directedness is motivated by the

idea that „all the forward light-cones eventually intersect one another“ (Prior 1967, 204), so that

the following should hold in any case (Prior’s sketch with formulae and some filling added):

p

GP/Greflp F FP/G

reflp

FP/GreflGP/G

reflp GP/GreflFP/G

reflp

35Interestingly, Prior suggests the S4.2. axiom precisely to distinguish models for special and for

general relativity: while they hold for the special theory, they do not for the general one „as that

theory allows for the possibility of light ‚cones‘ which so twist away from one another that after

a while they never intersect at all“ (Prior 1967, 205).

Moreover, a result by Rakic which will be discussed in 3.7.3.2. yields that if a few more

additional assumptions are made for the causal relation then this automatically postulates dense

linear time for all the frame-relative earlier-later relations as was presupposed by the Kt

component anyway.

(b) Goldblatt’s slower-than-light axioms

Finally, some remarks at the end of Goldblatt’s paper (Goldblatt 1980, 234) suggest that it is

appropriate to add the following axioms for rich models:

(Goldblatt) FP/G α & FP/G β → FP/G (FP/G α & FP/G β)

(Goldblatt*) PP/G α & PP/G β → PP/G (PP/G α & PP/G β)

Note that the causal operators are in this case irreflexive. Goldblatt sophisticatedly argues that in

this manner one can exclude the possibility of the axes‘ actually reaching the 45° angle: Suppose,

a time axis of some coordinate frame could actually reach the the angle of 45° corresponding to

the speed of light. Suppose there are two events, e‘ and e“, which could just be reached by a

signal travelling at the speed of light in opposite directions from e onwards. Then there is a

coordinate frame that has e‘ lying on the same time axis as e. And there is another coordinate

frame such that e“ lying on the same time axis as e. Now suppose that at e‘ something is the case

which uniquely makes p true; and that at e“ something is the case which uniquely makes q true.

Then for any f from F and any tf that includes e, FP/G p & FP/G q is true. On the other hand,

sending light signals from e onwards is really the very last chance to reach e‘ and e“. So FP/G

(FP/G p & FP/G q) is false (here the irreflexivity of FP/G is crucial). If, however, no time axis can ever reach the 45° angle then this cannot happen „since a slower-than-light journey can always

be made to go faster, so we could wait some time and then travel at a greater speed to [e‘] and

[e“]“ (234). So FP/G p & FP/G q → F(FP/G p & FP/G q) is valid (Goldblatt’s sketches with e’s and

formulae added):

e‘: p e“: q e‘: p e“: q

e

e: FP/G p & FP/G q & ~ FP/G (FP/G p & FP/G q) e: FP/G p & FP/G q → F(FP/G p & FP/G q)

36I suggest that Goldblatt’s axiom and its mirror image should be added for rich models of

Kr x SP5 x F. In basic Kr x SP5 x F notation they appear as:

(Goldblatt) +Fα & +Fβ → +F(+Fα & +Fβ)

(Goldblatt*) +Pα & +Pβ → +P(+Pα & +Pβ)

No extra semantic constraint is needed, as a time axis at an angle of 45° would lead to a total

collapse of the coordinate frame and is therefore no semantic option anyway: any two t and t‘ from Tf would be identical, so would be any two s and s‘ from Sf, and, on the top of all that Tf

and Sf would be identical. All this is ruled out by the definition of the Kr x SP5 x F-model where

for any f, Tf and Sf are required to be disjoint and are defined as partitions! Note, however, that

Goldblatt’s axioms are only valid on rich models where every permissable angle of time axes is

ready at hand, and that it is definitely not valid on one-frame-models. So the plausibility of

Goldblatt’s axioms, although they are certainly not derivable from the axioms of Kr x SP5 x F,

does not threaten the completeness conjecture from 3.5. which extended to Kr x SP5 x F in

general and not to realistic, rich models only.

3.7. Kr x SP5 x F and Müller’s ASL – a comparison

3.7.1. General remarks

As mentioned in the introduction, among the approaches I know of what comes closest to

Kr x SP5 x F is a semantic sketch for relativistic tense logic by Thomas Müller (Müller 2000,

ch.4) which forms part of the first German monograph on Arthur Prior.

Müller discusses two systems which were inspired by some remarks of Prior’s (Prior 1968,

133f): ASL (“absolutist logic of standpoints”, 195-200) and ISL (“idealist logic of standpoints”,

200-209).

3.7.2. Müller’s ISL

ISL is an attempt at a general reduction of spatio-temporal logic to pure A-series expressions:

against this background, translations from one reference frame to another are treated as special

cases of a more general class of translations of points of view which comprise “here-there” and

“now-then” translations, too, and which ultimately always involve someone’s own point of view

(this is the idealistic bit of ISL). A particularly telling illustration of this approach is given on

p.204 in footnote 365, where Müller comments on the formula [T] [T-1]φ as follows:

37“If someone is facing me at a distance of 1m and rotated to the left by 90° ([T] = 1m

forward shift of point of view, then rotation by 90°), then [T-1] corresponds to a 90° rotation to

the right and a consequent 1m backward shift. Now if φ says that there is a stone under my

feet then this has to be expressed from the point of view of the person opposite as ‘after a 90°

rotation to the right and a 1m backward shift there is a stone under my feet’ “ (my tr., my it.).

In connection with ISL, Müller, alongside tense operators, suggests the introduction of particular

spatial operators “there1”, “there2” etc. which point to individual places. Frame operators are not

introduced explicitly and, in my view, although its basic idea is definitely fascinating, ISL

remains rather sketchy on the whole. I must admit that I am not sure whether Müller can do with

as little model-theory as he seems to intend to (cf. the pretty sudden “B-theoretic semantics”,

207, or the definition of “model-theoretic satisfaction”, 202, which he needs as a first step in

order to derive some allegedly “direct universalist [i.e. A-series] correspondence”).

3.7.3. Müllers ASL

3.7.3.1. General idea and model of ASL

Although Müller seems to regard ASL only as a preliminary step in direction of ISL, to me ASL

looks much more interesting. The absolutist bit of ASL is a strong semantic emphasis on one

particular reference frame. One might, of course, interpret this frame as a frame that is arbitrarily

chosen as one’s system of rest. However, Müller prefers to interpret it as system of absolute rest

(he has argued before (186-189) from cosmology and from Quantum theory that a system of

absolute rest should reasonably be assumed).

The most striking difference between Müller’s approach and the approach presented here is

that while here Kripkean structures are built from the scratch in order to be identified with

models for relativistic space-time at some relatively late stage, Müller, following Rakic 1997,

takes complete Minkowski space-times as models right away. So a model for ASL is an enriched

Minkowski-spacetime ⟨ 4, η, ↑, G⟩ (196), where (302, footnote 172) 4 is a class of events, η is

some Minkowski-metric and ↑ some orientation and where G is an absolute simultaneity relation

(inspired by Rakic’s relation PRES in Rakic (1997), cf. Müller (2000), 184, footnote 326). G partitions 4 into equivalence classes of absolutely simultaneous events which are ordered

linearly by some absolute earlier-than-relation p (which plays the same role < plays with respect

to E and T in models for Kr x SP5e). One could identify the frame of absolute rest S0 as just that

partition of 4 which is the class of all simultaneity classes via G. Müller identifies it as a 10

parameter Poincaré group (197, 170f). Events are usually described as ordered pairs of a time

and a place coordinate relative to S0. So “(t,X)” is a definite description of an event as happening

at time t and place X according to the time and space coordinates of S0. Müller’s atomic

formulae for ASL are sentential constants with inbuilt place-suffixes, like p(X), meaning “p at

place X”. A model for ASL is defined as follows (197, taking footnote 347 into account):

38

A model for ASL is an ordered pair M = ⟨⟨ 4, η, ↑, G⟩, β⟩ consisting of some enriched

Minkowski-spacetime ⟨ 4, η, ↑, G⟩ and a two-place interpretation function β which assigns to

every sentential constant and every element of 4 one of the values 1 or 0.

3.7.3.2. Müller’s relation p defined by Rakic’s theorem 8

Müller claims that p is “derived” from G (196, footnote 345), which is not at all easy to see: in

general, a bunch of equivalence classes does not tell you in which (if any) order they follow upon

another.

But firstly it would be no problem to add p to the model. And secondly, a very interesting

result by Rakic (theorem 8, Rakic 1997, 269 + proofs 279f) indirectly shows how p might in fact

be derived from a Minkowski space-time boiled down to its essentials.

The idea is to identify Müller’s G with Rakic’s very similar relation PRES and to identify

Müller’s p with Rakic’s obviously corresponding relation ∝. It is clear that the relation ≤

(causally earlier-than-or-simultaneous-with) is indeed easily extracted from some enriched

Minkowski space-time ⟨ 4, η, ↑, G⟩ via η and ↑ as a relation on 4.

Firstly, Rakic reasonably presupposes (Rakic 1997, 267) that the relation ≤ between events e,

e’ etc. from 4 of any Minkowski spacetime satisfies the following requirements:

(A1)

(i) ∀ e [ e ≤ e] reflexivity

(ii) ∀ e, e’, e” [e ≤ e’ & e’ ≤ e” → e ≤ e”] transitivity

(iii) ∀ e, e’ ∃ e” [e ≤ e” & e’ ≤ e”] upward directedness

(iv) ∀ e, e’ ∃ e” [e” ≤ e & e” ≤ e’] downward directedness

(v) ∀ e, e’ [ e ≤ e’ & e’ ≤ e → e = e’] antisymmetry of <

(vi) ∀ e, e’ [ e < e’ → ∃ e” [ e < e” & e” < e’]] density

(vii) ∀ e ∃ e’ [ ~ e ≤ e’ & ~ e’ ≤ e]

(viii) ∀ e, e’ [~ e ≤ e’ & ~ e’ ≤ e → ∃ e” [ e < e” & ~ e’ ≤ e” & ~ e” ≤ e’]]

(ix) ∀ e, e’ [~ e ≤ e’ & ~ e’ ≤ e → ∃ e” [ e” < e & ~ e’ ≤ e” & ~ e” ≤ e’]].

Secondly, she formulates the following postulates for some relation R (Rakic 1997, 268) about

whose intuitive interpretation we need not worry for the resent purpose:

(A2) ∀ e, e’ [ e ≤ e’ → e’ R e] causal past implies realizedness

(A3) ∀ e, e’ [ e < e’ → ~ e R e’] causal future excludes realizedness

(A4) ∀ e, e’, e” [e R e’ & e’ R e” → e R e”] transitivity of R

(A5) ∀ e, e’, e” [ e’ < e” & e R e’ & ~ e R e” → ∃ e”’ [ e’ ≤ e”’ & e”’ < e” & e R e”’ & e”’ R e]]

39Thirdly, Rakic notes that it is possible to reformulate R in terms of PRES as follows (Rakic

1997, 271, footnote 9):

e R e’ iff ∃ e” [ e PRES e” & e’ ≤ e”].

If we identify PRES and G we can, thus, reformulate

(A2’) ∀ e, e’ [ e ≤ e’ → ∃ e” [ e’ G e” & e ≤ e”]]

(A3’) ∀ e, e’ [ e < e’ → ~ ∃ e” [ e G e” & e’ ≤ e”]]

(A4’) ∀e,e’,e” [∃e”’[eG e”’&e’≤e”’] & ∃e””[e’Ge””&e”≤e””] → ∃e””’[eGe””’ & e”≤e””’]

(A5’) ∀e,e’,e”[e’<e” & ∃e”’[e’Ge”’& e≤e”’] & ~ ∃ e”” [ eGe”” & e”≤ e””] → ...

... ∃e””’[e’≤e””’&e””’<e”] & ∃e”””[e≤e””” & e”’<e”””] & ∃e”””’ [e”’≤e”””’ & e<e”””]]

Now if R is defined in terms of PRES, ∝ is defined as follows (cf. Rakic 1997, 271, Def.6):

[e] ∝ [e’] iff ~ ∃ e” [ e PRES e” & e’ ≤ e”].

PRES is presupposed to be an equivalence relation and “[e]” means “the class of all e’ with e’

PRES” etc. We may write te, te’ etc. instead of [e], [e’] if we identify instants with equivalence

classes generated by G and reformulate:

te p t e’ iff ~ ∃ e” [ e G e” & e’ ≤ e”]

This definition yields a very precise and intuitive connection between the temporal relation p,

the simultaneity relation G and the causal relation ≤ (interestingly, it equally holds for all frame-

relative temporal relations and for frame-relative simultaneities). What is desired is, of course,

that just if t e p t e’ then t e neither coincides with t e’ nor that t e’ comes to lie below t e in the kind

of diagram used so far.

Now according to the definition, t e’ cannot coincide with te since in that case we have

(1) e G e’ Hyp

(2) e’ = e’ law of identity

(3) e’ ≤ e’ 2.

(4) e G e’ & e’ ≤ e’ 1., 3. I&

(5) ∃ e” [ e G e” & e’ ≤ e”] 4., EG.

40But neither can t e’ come to lie below t e. For if it did there would always be some member e”

of te such that e’ would come to lie in the causal past of e”, which violates the definition:

e

e”

te

te’

e’

Now Rakic’s theorem 8 says that, assuming (A1) – (A5), ∝ has, among other properties, the

properties of irreflexivity, transitivity, linearity and density.

We may therefore conclude that if any enriched Minkowski à la Müller spacetime satisfies

(A1) plus (A2’) to (A5’), as there is no reason to doubt it does, then p must be irreflexive,

transitive, linear and dense, just as Müller claims.

3.7.4. Details and truth-conditions of ASL

If S0 is given (via G and p) the Lorentz transformations enable us to calculate coordinates for

every event for some direction of motion and velocity relative to S0. Adding up all events under

their new coordinates for a certain direction and velocity yields a new reference frame S,

possibly different from S0. S, in turn, determines a new simultaneity relation GS and a new linear

order on GS, i.e. pS. In this way, every possible combination of direction and velocity gives us a

different frame, and we end up a whole bunch of reference frames S, S’, S” etc.

Now Müller adds to the usual tense-logical vocabulary an infinity of concrete frame-operators

AS, AS’, AS” etc. Sentential junctors aside, his truth conditions are as follows (cf. 198):

(1) If φ is an atomic formula then φ is true with respect to some ASL-model M, some S (over

M) and some event (t,X) (from 4M), shortly VM(φ,S,(t,X))=1, iff there is some t’ with

(t’,Y)GS(t,X) and βM(t’,Y)= 1 …

(2) If φ = Fψ then VM(φ,S,(t,X))=1 iff there is some (t’,X’) such that (t,X) pS (t’,X’) and

VM(ψ,S,(t’,X’))=1

(3) If φ = Pψ then VM(φ,S,(t,X))=1 iff there is some (t’,X’) such that (t’,X’) pS (t,X) and

VM(ψ,S,(t’,X’))=1

(4) If φ = AS’ψ then VM(φ,S,(t,X))=1 iff VM(φ,S’,(t,X))=1

41(5) φ is true absolute with respect to M and (t,X) iff VM(φ,S0,(t,X))=1.

Clause (5) clearly prefers S0, and easily yields that for S0 we have:

φ ≡ AS0 φ

More spectacularly, while, via G and p, S0 is a substantial part of the model all other frames are

not: they are, so to say, just huge tables of results of calculations with the aid of the Lorentz

transformations based on G and p, S0 (cf. also 197, footnote 349).

In contrast to Kr x SP5 x F, ASL has no spatial operators and lacks the expressivity of Kr x

SP5 x F for spatial relations as well as their systematization as S5. Moreover, although Müller

quantifies over frames metalinguistically, ASL has an infinity of particular frame operators as

opposed to the more common strong and weak modal operators as frame operators in Kr x SP5 x

F.

However, the most important difference between the two systems is their different

metaphysical bias: in Kr x SP5 x F, all frames are ontologically on a par and form part of the

model; Kr x SP5 x F is, thus, decidedly relativistic, using Priorian techniques against Priorian

intuitions. Müller, on the other hand, basically agrees with Prior’s intuitions, and shows how to

build an absolutist tense logic for special relativity. Both systems in comparison nicely show that

there are not just different metaphysical interpretations of the same metaphysically neutral logic,

but that systems of formal logic themselves may be precise expressions of competing

metaphysics.

424. Kr x SP5 x F x S5: Alethic Modalities for Kr x SP5 x F

4.1. General idea

When it comes to discussing the philosophical strength of Rietdijk-style arguments for a

determinism allegedly implied by the special theory of relativity, notions such as necessity and

possibility are involved. A multi-modal logic which helps to appreciate such arguments should

therefore contain operators for alethic modalities.

Combining tense logic with systems for alethic modalities has been a history of success

concerning semantic refinement and philosophical application. Axiomatization has proved to be

somewhat more difficult. Typically (GAMUT 1991, Kutschera 1997, Wölfl 1999, Strobach

1998), models for such systems are bundles of (possible) world histories branching into the

future (Prior 1967 and, e.g., Harada 1994 prefer branching time-trees, but on the whole, bundle

theories seem to be more successful and are, in any case, highly intuitive). Models contain a set

T of instants and a set W of possible worlds, and formulae are evaluated with respect to ordered

pairs from T x W by some interpretation function I for sentential constants. The accessibility

relation < for G is the usual linear earlier-than relation on T. The accessibility relation for the

necessity operator N is a relation which assigns to every t from T an equivalence class of ordered

pairs of worlds from W. The branching is best effected by a constraint which may be called

„historicity“:

w At w‘ iff for every t’ with t’ ≤ t and every sentential constant α: I(α,⟨t’,w⟩) = I(α,⟨t’,w‘⟩).

So only such worlds are mutually accessible at t which share the same history up to and

including t. This corresponds to the intuition that what is past or present is necessary insofar as it

is now unalterable or determined (cf. Aristotle, De interpretatione ch.9), so the necessity operator

acquires a temporalized reading.

Kutschera has shown (Kutschera 1997) that a logic with semantics of this kind should be

axiomatized by at least (PC), tense-logical axioms for linear tense-logic, S5 for N and the

combination axiom:

(PN) PN α → N P α.

In a way, this axiom excludes backward branching by ruling out situations such as this one:

p, Np PN p & M ~ Pp

H ~ p

43In fact, Kutschera (1997) gives a completeness proof for a tempo-modal system, but

unfortunately for one where the semantics do not presuppose historicity. Therefore wffs such as

p → Np, Pp → PNp, q → Nq , Pp → PNp

are not derivable in his system although they are valid if historicity is presupposed (while their

generalisations with α for p are not, due to possible substitution instances like Fp for α). Wölfl

1999, on this point somewhat reminiscent of the fox who declares all the grapes which are out of

reach as too sour anyway, tries to find some motivation against the very historicity principle

which motivated the whole enterprise and which is, by the way, automatically presupposed in

Prior-style models. Moreover, the axiomatics require the use of an additional operator in order to

be complete which, although it has a nice interpretation as crossworld simultaneity, comes as a

bit of a surprise. So in what follows, no completeness claim is made for any of the axiomatics

beyond simple S5.

The option of applying the idea of indeterministic branching not only to one dimensional

world histories but to whole space-times has been discussed in a beautiful paper by Belnap

(Belnap 1992), which is technically rather in the tradition of Prior’s branching instant trees than

in the tradition of combined temporal and modal logic. A philosophically as well as formally

thorough discussion of Rietdijk-style arguments based on causal accessibility can be found in

Rakic 1997. Both authors have, however, not defined a full-fledged formal language, and their

intuitions deserve further philosophical discussion which would exceed the scope of this paper. A

question that should be adressed, though, is: Can the idea of branching space-time be

implemented into the framework of Kr x SP5 x F?

This chapter has the aim to show that it can. So far, only coordinate frames could vary, but the

interpretation function could not. Different coordinate frames were merely applied on, or helped

to structure, that one big happening called world or universe. For alethic modalities, it should be

permissable that the interpretation function vary, too, allowing for different possible factual

fillings of space-time (this metaphor is a little dangerous, since it suggests some sort of container

theory, or, rather, some sort of hook-theory with Minkowski events as little hooks on which to

hang happenings, but it is intuitive). Possible worlds as extra entities are not needed (as Bertram

Kienzle has pointed out to me, the idea of using interpretation functions instead seems already to

be hinted to in Montague 1974, p.75). As the structure of the resulting logic is not exactly easy to

grasp at a glance, it shall, in what follows be introduced and motivated via a number of

preparatory steps, beginning with some very simple alternative semantics for alethic S5.

444.2. S5 without possible worlds

In fact, it is possible to state alternative semantics for S5 already where possible worlds just are

PC interpretation functions. S5 for alethic modalities is, in this case, defined as follows:

Axioms

(PC)

(N-K) N (α → β) → (N α → N β ) N-distribution

(N -T) N α → α ab necesse ad esse valet consequentia

(N -S5) M α → N M α equivalence axiom

Semantics

(1) A modal order is an ordered pair ⟨INT,{ ⟨N,A⟩}⟩, where

(i) INT is a nonempty set of PC-interpretation functions

(ii) A is a two-place [accessibility] relation on INT such that for all I, I’ from INT: I R I’.

(2) A model for S5 is an ordered pair ⟨⟨INT,{⟨N,A⟩}⟩, J ⟩ consisting of some modal order

⟨INT ,{⟨N,A⟩}⟩ and a two-place interpretation function J which assigns to every sentential

constant with respect to every I from INT exactly one of the values 0 or 1 such that J(α,I) =

I(α).

(3) For any S5-model M = ⟨⟨INT,{ ⟨N,A⟩}⟩, J⟩ any wff α of S5 is assigned exactly one of the

values „true“ (1) or „false“ (0) with respect to any I from INT – shortly: VM(α,I) = 1 or

VM(α,I) = 0 - subject to the following conditions:

(i) VM(α, I) = JM(α, I) , if α is a sentential constant

(ii) VM( ~ α, I) = 1 iff VM( α, I) = 0

(iii) VM( (α v β), I) = 1 iff VM( α, I) = 1 or VM( β, I) = 1 or both

(iv) VM(Nα, I) = 1 iff for all I‘ with I’ A I: VM( α, I‘) = 1.

4.3. The tempo-modal system Kr x S5

It is no problem to define a tempo-modal system based on the same idea, thus, to modalize Kr

and so to obtain a system Kr x S5. Only, INT now consists of two-place function and the

accessibility relation has a time parameter:

45Axioms

at least (PC), (Kr), (S5 for N), (PN)

Semantics

(1) A tempo-modal tree is an ordered pair

⟨{T, INT},{⟨G,<⟩, ⟨H,<-1⟩,⟨N,A⟩}⟩, where

(i) T is a nonempty set [of instants]

(ii) INT is a nonempty set of two-place interpretation functions which assign to every sentential

constant exactly one of the values 0 or 1 with respect to every t from T.

(iii) < and <-1 are two-place [accessibility] relations on T

(iv) A is a function which assigns to every t from T some ordered pair of elements from INT.

Usual constraints ensure temporal asymmetry (and thus irreflexivity), transitivity, linearity and

continuity.

(2) A Kr x S5-model is an ordered pair ⟨⟨{T, INT},{⟨G,<⟩, ⟨H,<-1⟩,⟨N,A⟩}⟩, J ⟩ consisting of some

tempo-modal tree ⟨{T, INT},{⟨G,<⟩, ⟨H,<-1⟩,⟨N,A⟩}⟩ and a two-place interpretation function J

which assigns to every sentential constant with respect to every ordered pair from T x INT

exactly one of the values 0 or 1 such that

(i) J(α,⟨t,I⟩) = I(α,t).

(ii) for every I, I‘ from INT: I At I‘ iff for every t’ with t’ ≤ t and every sentential constant α:

J(α,⟨t‘,I⟩) = J(α,⟨t‘,I‘⟩) historicity

(3) For any Kr x S5-model M = ⟨⟨{T, INT},{⟨G,<⟩, ⟨H,<-1⟩,⟨N,A⟩}⟩, J⟩ any wff α of Kr x S5 is

assigned exactly one of the values „true“ (1) or „false“ (0) with respect to any ordered pair ⟨t,I⟩ from T x INT – shortly: VM(α, ⟨t,I⟩) = 1 or VM(α, ⟨t,I⟩) = 0 - subject to the following conditions:

(i) VM(α, ⟨t,I⟩) = JM(α, ⟨t,I⟩) , if α is a sentential constant

(ii) VM( ~ α, ⟨t,I⟩) = 1 iff VM( α, ⟨t,I⟩) = 0

(iii) VM( (α v β) , ⟨t,I⟩) = 1 iff VM( α, ⟨t,I⟩) = 1 or VM( β, ⟨t,I⟩) = 1 or both

(iv) VM( Gα, ⟨t,I⟩) = 1 iff for all t‘ with t < t‘: VM( α, ⟨t‘,I⟩) = 1

(v) VM( Hα, ⟨t,I⟩) = 1 iff for all t‘ with t <-1 t‘: VM( α, ⟨t‘,I⟩) = 1

(vi) VM(Nα, ⟨t,I⟩) = 1 iff for all I‘ with I’At I: VM( α, ⟨t,I‘⟩) = 1.

Typically, models for Kr x S5 have a two-dimensional tree-structure. A very simple example

would look like this:

46

T I

branching point t I‘

As already mentioned, a striking feature of Kr x S5 is that, as p is a sentential constant, the

following formulae are valid:

p → Np

Pp → NPp.

This is what reflects the Aristotelian intuition on the level of object language. Of course, one will

say, N may both be read as „it is in principle unalterable that“ and „it is determined that“. The

concepts of unalterability and determinateness are not distinguished on this level, and one hardly

sees how they could be.

An interesting feature of Kr x S5 is that of the tempo-modal versions of the com- and the

Church-Rosser formulae some are valid and some are not:

I) (com-PM) PMα ≡ MPα is not valid, because

PMα → MPα is not valid;

II) (com-FM) FMα ≡ MFα is not valid, because

MFα→ FMα is not valid.

III) (chr-FN) FNα → NFα not valid

(chr-PN) PNα → NPα = (PN)

IV) (chr-MH) MHα → HMα valid, as derivable from (PN)

47

I) counterexample

~ p, PMFp & ~MPFp

~p, Fp

p

II) counterexample

p

I: MFp & ~ FM p

~ p I

III) counterexample

~ p

FNp & ~ NFp ~p

p

IV)

1. PNα → NPα (PN)

2. PN ~ α → NP~ α 1. ∼ α / α 3. ∼ NP~ α→ ~ PN ~ α 2. contrapos.

4. ∼ ∼ M ~ P~ α → ~ ~ H ~ N ~ α 3., Def. M, Def. P 5. M ~ P~ α → H ~ N ~ α 4. DN

6. M H α → H M α 5., Def. M, Def. P

484.4. The modalized tempo-spatial system Kr x SP5e x S5

The next step ist to add a spatial dimension to Kr x S5 and, at the same time, to eventisize the

semantics. The result is very similar to Kr x S5 concerning the structure of the semantics.

Semantics

(1) World-book: a world-book is an ordered pair ⟨{E,T,S, INT},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩,⟨N,A⟩}⟩, where

(i) E is a nonempty set [of tempo-spatial positions (events)]

(ii) T is a partition of E

(iii) S is a partition of E

(iv) S and T are disjoint

(v) < and <-1 are two-place [accessibility] relations on T

(vi) R is a two-place [accessibility] relation on S.

(vii) INT is a nonempty set of two-place interpretation functions which assign to every sentential

constant exactly one of the values 0 or 1 with respect to every e from E.

(viii) A is a function which assigns to every t from T some ordered pair of elements from INT.

Usual constraints ensure temporal antisymmetry (and thus irreflexivity), transitivity, linearity and

continuity, and we have the following constraints:

(a) For all s, s’ from S: s R s’.

(b) For every t from T and every s from S there is exactly one e from E such that: t ∩ s = {e}

(c) For every e from E there is exactly one t from T and s from S such that t ∩ s = {e}.

(2) A model for Kr x SP5e x S5 is an ordered pair

⟨⟨{E,T,S, INT},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩,⟨N,A⟩}⟩, J⟩ consisting of some world-book

⟨{E,T,S, INT},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩,⟨N,A⟩}⟩ and a three-place interpretation function J such that

(i) J assigns to every sentential constant with respect to every t from T, s from S and I from INT

from exactly one of the values 0 or 1 such that J(α,t,s,I) = I(α,e) if t ∩ s = {e}

(ii) for every I, I‘ from INT: I At I‘ iff for every t’ with t’ ≤ t, every s from S and every sentential

constant α: J(α,t‘,s,I) = J(α,t‘,s,I‘) historicity

49(3) For any Kr x SP5e x S5-model M = ⟨⟨{E,T,S, INT},{⟨G,<⟩, ⟨H,<-1⟩,⟨¤,R⟩,⟨N,A⟩}⟩, J⟩ any

wff α of Kr x SP5 x S5 is assigned exactly one of the values „true“ (1) or „false“ (0) with respect

to any t from T, s from S and I from INT – shortly: VM(α,t,s,I) = 1 or VM(α,t,s,I) = 0 - subject to

the following conditions:

(i) VM(α,t,s,I) = JM(α,t,s,I) , if α is a sentential constant

(ii) VM( ~ α, t,s,I) = 1 iff VM( α, t,s,I) = 0

(iii) VM( (α v β) , t,s,I) = 1 iff VM( α, t,s,I) = 1 or VM( β, t,s,I) = 1 or both

(iv) VM( Gα, t,s,I) = 1 iff for all t‘ with t < t‘: VM( α, t‘,s,I) = 1

(v) VM( Hα, t,s,I) = 1 iff for all t‘ with t <-1 t‘: VM( α, t‘,s,I) = 1

(vi) VM(¤α,t,s,I) = 1 iff for all s‘ with s‘Rs: VM( α, t,s‘,I) = 1

(vii) VM(Nα, t,s,I) = 1 iff for all I‘ with I’At I: VM( α, t,s,I‘) = 1.

A simple model for Kr x SP5e x S5 may be visualized as follows:

T I

E partitioned into T and S

S

time-edge t I‘

So instead of possible worlds, interpretation functions which assign truth values to sentential

constants for events from E are are the entities which are mutually accessible via the accessibility

relation for the necessity operator N. They are indeed possible worlds insofar as every

interpretation function for sentential constants on E may be regarded as a different filling of

space-time with facts at tempo-spatial positions. As tempo-spatial coordinate systems have a

spatial component, branching, or perhaps rather variation of filling between mutually accessible

tempo-spatial coordinate frames, takes places not at branching points but at spatially extended

temporal edges. The historicity constraint now says that two factual fillings of space-time are

mutually accessible along some time-edge t iff they factually coincide up to and including that

time edge. Of course, also here the partial identity requirement for historicity makes A an

equivalence relation and thus, in turn, requires S5 for the axiomatization of N.

Historical necessity is, though of course dependent on time, is completely independent of

space. One might think of therefore introducing a no-axiom-axiom like

50

¡Nα → ¤Nα.

But even this would be too strong to state that necessity is independent of space. For if it is

necessary that something happens right here, it need by no means be necessary that the same

thing happens everywhere. A proper expression of the independence of space and necessity is

rather that if something is necessary right here then it is everywhere necessary to be necessary

right here, i.e.:

¡Nα → ¤¡Nα.

But this trivially follows from the S5-axiom ¡α → ¤¡α already. Note, by the way, that

¡Nα and N¡α are not equivalent and that confusing them would, in terms of semantics, amount

to a kind of quantifier shift fallacy. Claming that there is some place such that in every possible world something is the case at (at least) that place is making a stronger claim than claiming that

for every possible world there is some place at which something is the case, but that it may be a

different one from world to world.

Although modal and spatial operators are independent in the way described above, this does

not mean that there is no rule-governed interplay between them. So the next interesting question

is, naturally, whether the combination of spatial and modal operators produces valid versions of

the com- and the Church-Rosser formulae so that they should be demanded as axioms which

govern the interplay of modal and spatial operators:

(com-M¡) M¡α ≡ ¡Mα „chessboard“-axiom

(chr-M¤) M¤α → ¤M α Church/Rosser-axioms

(chr-¡N) ¡N α → N ¡α

The first formula means: If there is a possible world such that somewhere right now in that world

α is true then there is some place such that right now some possible world is accessible there in

which α is true at that place, and vice versa. This is very plausible indeed, and the proof is

straightforward (notated with quantifiers as meta-language signs):

Let M be some model for Kr x SP5e x S5 and t, s and I members of TM, SM and INTM resp.

VM (M¡α,t,s,I) = 1 iff ∃ I’ ∃ s’ [ I’∈ INTM & I’ At I & & s’ R s & VM (α,t,s’,I’) = 1], and

VM (¡Mα,t,s,I) = 1 iff ∃ s’ ∃ I’ [s’ R s & I’∈ INTM & I’ At I & VM (α,t,s’,I’) = 1], which is

trivially equivalent.

51The first Church-Rosser formula is, in this version, plausible as well: If right now and here a

possible world is accessible where, right now, α is true everywhere then from whereever one

might be right now one can access a possible world such that one will “hit” α.

So is the second Church-Rosser-axiom: if there is some place such that, at that place, α is,

right now, true in all possible worlds, then whatever possible world I switch to there is a place

where α is true.

This leads to the conjecture that the following axioms are sound for Kr x SP5e x S5 with

historicity constraint and sufficient for a system where the historicity constraint is somewhat weakened in order to match a Kutschera-style completeness proof, but is otherwise just like

Kr x SP5e x S5:

Axioms

(PC), (Kr),

(com) + Church / Rosser for temporal and spatial operators

(com) + Church / Rosser for modal and spatial operators

(S5 for ¤)

(S5 for N)

(PN)

It may be noted that using (chr) ¡N¡α → ¤N¡α can be deduced analogously to the deduction

in 3.3.

4.5. The modalized relativistic tempo-spatial system Kr x SP5e x F x S5

4.5.1. General idea

It is now relatively easy to see how alethic modalities can be added to the relativistic logic

Kr x SP5e x F. The problem is not how to introduce some necessity operator, but to introduce

necessity in a way that makes sense. It would be easy simply to import the semantics for N from

Kr x SP5e x S5 into a modalized version of Kr x SP5e x F. N would then be frame-dependent, and

the temporal edges at which the modal branching takes places would be different from frame to

frame. This is intuitively very odd if N is to represent anything like necessity. For whatever

necessity is in relativistic physics, one will certainly hold necessity to be frame-independent. In

fact, Kr x SP5e x F x S5 will incorporate a rather strong metaphysical claim as to what happens

to the concept of necessity in the theory of relativity:

While in „absolute“ space-time unalterability, unpreventability and determinateness coincide

and indiscernably add up to the concept of necessity, in relativistic space-time they part ways.

Determinateness implies unpreventability, but not vice versa. So the N operator splits up into

two different operators N1 and N2, which coincide on models with just one coordinate frame.

52The idea is relatively simple: From the point of view of a given event e, all happenings at

events which do not belong to the causal future of e, are unalterable, and so are all laws of logic

even within the causal future. But, for reasons to be argued below, this should not lead us to think

that everything outside the causal future is determined. Rather – laws of logic aside - , only what

happens at some event e“ in the causal past of e is determined, so that the causal future as well as

the whole realm of space-like events (relative to e) should be regarded as indeterminate – apart

from laws of logic.

Technically, the accessibility relation for the unpreventability operator N1 will, relative to

some event e, hold between (eventisized) interpretation functions which are identical for all

events except those in the causal future of e. And the accessibility relation for the

determinateness operator N2 will, relative to some event e, hold between (eventisized)

interpretation functions which are identical for all events in the causal past of e.

The situation may be visualized like this: Interpretation functions of Kr x SP5e x S5 are like

sheets of paper filled with text (the are pages of a world-book). Two such sheets are mutually

accessible at some line t iff, up to this line they contain exactly the same text.

t

Interpretation functions of Kr x SP5e x S5 are like sheets of paper filled with text, too. But their

mutual accessibility is not defined relative to a line, but relative to a single point. Pictorially, the

branching edge is folded, forward in terms of N1 and backwards in terms of N2 (this corresponds

to a distinction in Belnap (1992), §8, 410-3, who has no modal operators, though, and arrives at

the same diagrams on an entirely different way).

In terms of N1, two sheets are mutually accessible at e if they contain the same text (at least)

where indicated:

e

53

And in terms of N2, two sheets are mutually accessible at e if they contain the same text (at least)

all over here:

e

Interestingly, the two operators can be defined via the coordinate angle of frames. This is clear if

one thinks of rich models: for some event e‘ to be outside the causal future of e is just the same

as for e‘ to be contained in some t from some coordinate frame identical with or earlier than te.

And for some event e‘ to be in the causal past of e is just the same as for e‘ to be contained in

some t from every coordinate frame identical with or earlier than te.

4.5.2. Definitions (minimal axioms, Semantics, Leonardo-model)

Semantically, Kr x SP5e x F x S5 is characterised as follows:

Semantics

(1) Modalized space-time: a modalized space-time is an ordered pair

⟨{E,F, INT},{⟨G,<F⟩, ⟨H, <-1F⟩,⟨¤,RF⟩, ⟨×, A×⟩, ⟨N1, A1⟩, ⟨N2, A2⟩}⟩, where

(i) E is a nonempty set of tempo-spatial positions (events)

(ii) F is a nonempty class of eventisized tempo-spatial coordinate frames {f1, f2 ...} which all

contain E

(iii) <F, <-1F, RF, are defined as for Kr x SP5 x F-models

(iv) A× is a two-place relation on F

(v) INT is a nonempty class of interpretation functions each of which assigns to every sentential

constant with respect to every e from E exactly one of the values 0 or 1

(vi) A1 is a function which assigns to every e from E some set of ordered pairs consisting of

elements of INT;

(vii) A2 is also a function which assigns to every e from E some set of ordered pairs consisting of

elements of INT;

subject to the following constraint:

(S5-constraint for ×) For all f, f’ from F: f A× f’.

54(2) A model for Kr x SP5 x F x S5 is an odered pair

⟨⟨{E,F, INT},{⟨G,<F⟩, ⟨H, <-1F⟩,⟨¤,RF⟩, ⟨×, A×⟩, ⟨N1, A1⟩, ⟨N2, A2⟩ }⟩, J ⟩ consisting of some

modalized space-time ⟨{E,F, INT},{⟨G,<F⟩, ⟨H, <-1F⟩,⟨¤,RF⟩, ⟨×, A×⟩, ⟨N1, A1⟩, ⟨N2, A2⟩ }⟩

and some interpretation function J which assigns to every sentential constant exactly one of the

values 0 or 1 with respect to any ordered pair ⟨e,I⟩ from E x INT such that J(α, ⟨e,I⟩) = I(α,e);

and A1 and A2 are defined as follows:

(i) for every I, I‘ from INT:

I A1e I‘ iff there is some f‘ from F such that

for all e‘ from E and all t’f‘e from Tf‘ with t’f’

e‘ ≤ f‘ tf‘e and for every sentential constant α:

I(α,e‘) = I’(α,e‘);

(ii) for every I, I‘ from INT:

I A1e I‘ iff for all f‘ from F:

for all e‘ from E and all t’f‘e from Tf‘ with t’f’

e‘ ≤ f‘ tf‘e and for every sentential constant α:

I(α,e‘) = I’(α,e‘).

So, quite along the lines J plays in the simpler systems, J is some kind of rearrangement function

which assigns to α with respect to e and I (in its role as a possible world) just that truth value I

(in its role as an interpretation function) assigns to α for e. And A1 and A2 are defined as

motivated above.

(3) For any Kr x SP5 x F x S5 model

M = ⟨⟨{E,F, INT},{⟨G,<F⟩, ⟨H, <-1F⟩,⟨¤,RF⟩, ⟨×, A×⟩, ⟨N, A⟩ }⟩, J ⟩ any wff α of

Kr x SP5 x F-min x S5 is assigned exactly one of the values „true“ (1) or „false“ (0) with respect

to any f from F, any t from Tf and s from Sf and I from INT – shortly: VM(α,t,s,f,I) = 1 or

VM(α,t,s,f,I) = 0 - subject to the following conditions:

(i) VM(α,t,s,f,I) = J(α,⟨e,I⟩) if α is a sentential constant and t ∩ s = {e}

(ii) VM( ~ α, t,s,f,I) = 1 iff VM( α,t,s,f,I) = 0

(iii) VM( (α v β) ,t,s,f,I) = 1 iff VM( α, t,s,f,I) = 1 or VM( β, t,s,f,I) = 1 or both

(iv) VM( Gα, t,s,f,I) = 1 iff for all t‘ with t <f t‘: VM( α, t‘,s,f,I) = 1

(v) VM( Hα, t,s,f,I) = 1 iff for all t‘ with t <-1f t‘: VM( α, t‘,s,f,I) = 1

(vi) VM(¤α ,t,s,f,I) = 1 iff for all s‘ with s‘R f s: VM( α, t,s‘,f,I) = 1

(vii) VM(×α ,t,s,f,I) = 1 iff for all f‘ with f‘ A× f and all t‘, s‘ with t‘∈Tf‘ and s‘∈Sf‘:

if t∩s = t‘∩s‘ then VM( α, t‘,s‘,f‘,I) = 1

(viii) VM(N1α ,t,s,f,I) = 1 iff for all I‘ with I‘ A1e I: VM( α, t,s,f,I‘) = 1 if t∩s = {e}

(ix) VM(N2α ,t,s,f,I) = 1 iff for all I‘ with I‘ A2e I: VM( α, t,s,f,I‘) = 1 if t∩s = {e}.

55Some examples may help to see one’s way through the definitions and to make clear how the

operators work in rich models. Consider, for this purpose, the following rich models with just

two interpretation functions but every permissable angle of coordinate axes in its state of modal

accessibility for N2 at e:

I

e

I’

Aesthetic considerations aside, the situation with respect to e may be reminiscent of Leonardo di

Caprio’s exclamation at the ship’s head at the end of the first part of the film “Titanic” (“I am the

king of the world”), so it might be called Titanic- or Leonardo-model.

4.5.3. Discussion of formulae (historicity, contingentia praesentia and contingentia praeterita)

If a similar model with three interpretation function is unfolded the interpretation functions come

to lie beside each other. For the following discussion let us add some details of the interpretation

functions:

56 I I’ I”

°e2 °e1 °e2 °e1 °e2 °e1

°e3 °e3 °e3

e4 e0 e4 e0 e4 e0

°e5 °e5 °e5

°e6 °e7 °e6 °e7 °e6 °e7

Let I, I’ and I” be such that

I ( p, e0) = 1 I’ ( p, e0) = 1 I” (p, e0) = 1

I ( q, e1) = 1 I’ ( q, e1) = 0 I” (q, e1) = 0

I (r, e2) = 1 I’ (r, e2) = 0 I” (r, e2) = 0

I (s, e3) = 1 I’ (s, e3) = 1 I” (s, e3) = 0

I (p*, e4) = 1 I’ (p*, e4) = 1 I” (p*, e4) = 0

I (q*, e5) = 1 I’ (q*, e5) = 1 I” (q*, e5) = 0

I (r*, e6) = 1 I’ (r*, e6) = 1 I” (r*, e6) = 1

I (s*, e7) = 1 I’ (s*, e7) = 1 I” (s*, e7) = 1

Let f be the rectangular coordinate frame, and let f’ be some coordinate frame very close to the

45° angle whose space axis through e1 nearly coincides with the strong line in the diagrams. Let

{e0} = tf ∩ sf with t from Tf and s from Sf.

The interpretation functions are now defined in such a way that we have:

I A1e0 I’ I’ A1

e0 I

I A2e0 I” I” A2

e0 I

But we do not have:

I A2e0 I’ I’ A2

e0 I

I A1e0 I” I” A1

e0 I.

57We can now ask questions about the truth-values of certain formulae at tf and sf in f and with

respect to I and answer them precisely. The first stack of valuations is no great surprise:

VM( p, tf,sf,f,I) = 1

VM( Fq, tf,sf,f,I) = 1

VM( F¡r, tf,sf,f,I) = 1

VM( F¡s, tf,sf,f,I) = 1 VM( ¡p*, tf,sf,f,I) = 1

VM( P¡q*, tf,sf,f,I) = 1

VM( P¡r*, tf,sf,f,I) = 1

VM( +Pr*, tf,sf,f,I) = 1

VM( Ps*, tf,sf,f,I) = 1.

Still to be expected are the following results where the different modal operators come in:

VM( N1p, tf,sf,f,I) = 1

VM( N2p, tf,sf,f,I) = 1

This is clear as the truth-value of p at e0 (i.e. the meeting point of tf and sf) for any interpretation

function I* has to coincide with the truth value I assigns to p at e0 for I to be accessible from I*

via e0 in both accessibility relations. Intuitively, nothing is to be said against the idea that what is

the case at e0 is both unpreventable and determined at e0 (this does shed some light on the notion

of the causal present though: strictly speaking there is no such thing, because it’s too late to

change the present now; note, by the way, the small difference in meaning between “unalterable”

and “unpreventable” – which is so small that both words coincide in the German

“unabänderlich”: what is unalterable cannot be future, what is unpreventable may be, and it

remains to be discussed philosophically whether this difference of pre-relativistic natural

language still make any sense in connection with relativistic physics).

Furthermore, the truth-value of r* at e6 for any interpretation function I* has to coincide with

the truth value I assigns to r* at e6 for I to be accessible from I* via e0 in both accessibility

relations. This is because the time axis of the coordinate intersection on e6 is earlier than the time

axis of the coordinate intersection on e0 in any coordinate frame. So we have:

VM( N1P¡r*, tf,sf,f,I) = 1

VM( N2P¡r*, tf,sf,f,I) = 1

VM(N1+Pr*, tf,sf,f,I) = 1

VM(N2+Pr*, tf,sf,f,I) = 1

58VM( N1Ps*, tf,sf,f,I) = 1

VM( N2Ps*, tf,sf,f,I) = 1

Intuitively, this reflects the idea that what is causally or absolutely past at e0 is both

unpreventable and determined at e0. Moreover, it is no great surprise that we have:

VM( N1Fq, tf,sf,f,I) = 0

VM( N2Fq, tf,sf,f,I) = 0

VM( N1F¡r, tf,sf,f,I) = 0

VM( N2F¡r, tf,sf,f,I) = 0

VM( M1~ Fq, tf,sf,f,I) = 1

VM( M2 ~ Fq, tf,sf,f,I) = 1

VM( M1~ F¡r, tf,sf,f,I) = 1

VM( M2 ~ F¡r, tf,sf,f,I) = 1

q is, like alle other simple statements rendered by sentential constants, the description of some

state which might as well fail to obtain. If q is true at e1, as is the case in I, then its future truth

may no more than be correctly guessed by asserting Fq at e0. So q’s future truth should turn out to be undetermined, and its falsehood be possible as it does via N2 and M2.

Also, in principle, something can still be done in order to bring about or to prevent what

makes q true at e1 in I. From e0, it is possible to influence reality in such a way that it does not

turn out to be I, but, say, I’ instead. So q’s obtaining at e1 is not unpreventable, and its falsehood

is possible in that sense, too, as is reflected via N1 and M1. The reasoning with respect to ¡r is

analogous.

More interestingly, laws of logic can neither be prevented from holding at any tempo-spatial

position nor is it in any way undetermined whether they hold. Actually, the semantics for the N

operators owe a great deal of their complication to the idea to effect this while keeping the

contingent contingent. Now, just as things should be, we get, e.g.:

VM( N1 F (q v ~ q), tf,sf,f,I) = 1

VM( N2 F (q v ~ q), tf,sf,f,I) = 1

VM( N1F¡(r v ~ r), tf,sf,f,I) = 1

VM( N2F¡(r v ~ r), tf,sf,f,I) = 1.

59The reason for these results is obvious: F (α v ~ α) and F¡(α v ~ α) are true at the coordinate

intersection of any event e’ in any frame if e’ is accessible from e0 via the accessibility relation

for both N1 and N2 just because they are true at the at the coordinate intersection of any event

whatsoever due to the definition of the PC junctors. For the same reason we have:

VM( N1F¡(s v ~ s), tf,sf,f,I) = 1

VM( N2F¡(s v ~ s), tf,sf,f,I) = 1

VM( N1¡(p* v ~ p*), tf,sf,f,I) = 1

VM( N2¡(p* v ~ p*), tf,sf,f,I) = 1

VM( N1 P¡(q* v ~ q*), tf,sf,f,I) = 1

VM( N2 P¡(q* v ~ q*), tf,sf,f,I) = 1

VM( N1 P¡(r* v ~ r*), tf,sf,f,I) = 1

VM( N2 P¡(r* v ~ r*), tf,sf,f,I) = 1.

Now how about s at e3? On the one hand, at tf, it is too late to do something about it from the

standpoint of e0: no signal can still reach e2 in order to prevent s from being made true (assuming

I to be reality). And indeed we have:

VM( N1F¡s, tf,sf,f,I) = 1.

This is because the truth-value of s at e3 for any interpretation function I* has to coincide with

the truth value I assigns to s at e3 for I to be accessible from I* via e0 in terms of the accessibility

relation for N1, as there is some coordinate frame such that the time axis of the coordinate

intersection on e0 is earlier than the time axis of the coordinate intersection on e0 in that coordinate frame (e.g. the one with the space axis in boldface). Note that we do not have to

worry about I”, as I” and I are not accessible via A2 at e0.

On the other hand, e3 lies outside the causal past of e0 and should therefore, from the

standpoint of e0, not be considered as determined (cf. section 5 below for some more

philosophical motivation of this point). This is reflected by the following result:

VM( N 2F¡s, tf,sf,f,I) = 0

VM( ~ N 2F¡s, tf,sf,f,I) = 1

VM( M 2 ~ F¡s, tf,sf,f,I) = 1.

60This is because in order to have

VM( N 2F¡s, tf,sf,f,I) = 1

the truth-value of s at e3 would have had to be “true” for any interpretation function I* accessible

from I via e0 in terms of A2 (assuming, for the sake of illustration, s to be true at e3 only). But I” is accessible from I via A2 at e0, and we have:

VM( F¡s, tf,sf,f,I’) = 0.

What might make Kr x SP5e x F x S5 quite spectacular is that one can reason analogously

concerning e4 and e5. Thus we get:

VM( N 2¡p*, tf,sf,f,I) = 0

VM( ~ N 2¡p*, tf,sf,f,I) = 1

VM( M 2 ~¡p*, tf,sf,f,I) = 1

VM( N 2P¡q*, tf,sf,f,I) = 0

VM( ~ N 2P¡q*, tf,sf,f,I) = 1

VM( M 2 ~ P¡q*, tf,sf,f,I) = 1.

In order to have

VM( N 2¡p*, tf,sf,f,I) = 1

the truth-value of p* at e4 would have had to be “true” for any interpretation function I*

accessible from I via e0 in terms of A2 (assuming p* to be true at e4 only). But I” is accessible

from I via A2 at e0, and we have:

VM(¡p*, tf,sf,f,I’) = 0.

And in order to have

VM( N 2 P¡q*, tf,sf,f,I) = 1

the truth-value of q* at e5 would have had to be “true” for any interpretation function I*

accessible from I via e0 in terms of A2 (assuming q* to be true at e5 only). But I” is accessible

from I via A2 at e0, and we have:

61

VM(P¡q*, tf,sf,f,I’) = 0.

So we have:

VM( ¡p* & M 2 ~ ¡p* , tf,sf,f,I) = 1

VM( P¡q* & M 2 ~ P¡q* , tf,sf,f,I) = 1

This amounts to no less than the claim that, taking f as one’s chosen coordinate frame, there are contingentia praesentia and even contingentia praeterita with respect to f. Although what makes

p* true if I is reality is, in f, present at e0 and although what makes q* true if I is reality is, in f,

even past at e0 both should not be regarded as determined at e0. Interestingly, this result goes

along with strict historicity at the same place and, in fact, on the same world-line. This may be

stated as the easily verified claim that if α is a sentential constant then

α → N1α α → N2α

Pα → N1Pα Pα → N2Pα

+Pα → N1+Pα +Pα → N2+Pα +Pα → N2P¡α

are all valid on Kr x SP5 x F x S5 models while

¡α → ¡N2α

P¡α → N2P¡α

are not. So the ¡ operator is indeed crucial as it may serve to express spatial distance which can

make all the difference if it exceeds the bounds of the past light cone.

4.5.4. Coincidence of the N operators on one-frame-models

Just as S5 has one-world-models as extreme cases, Kr x SP5 x F has one-frame-models as

limiting cases and Kr x SP5 x F x S5 has one-frame-models as limiting cases insofar as tempo-

spatial coordinates are concerned. One-frame-models for Kr x SP5 x F x S5 may still contain

several interpretation functions and so have a modal dimension. Such models exhibit a feature

which is worth mention: although the N operators are not trivialized on such models they

curiously coincide. The reason for this is a very simple fact of predicate logic which, in a very

different context, has been nicely termed “wild quantity” by Fred Sommers (Sommers 1976): If

there is just one F then it makes no difference if one speaks of some F or all F. Now we have I

A1e I‘ iff there is some f‘ from F and I A1

e I‘ iff for all f‘ from F such that for all e‘ from E and

62all t’f‘

e from Tf‘ with t’f’e‘ ≤ f‘ tf‘

e and for every sentential constant α: I(α,e‘) = I’(α,e‘). So if

there is just one f in F then for any e from E just those interpretation function are mutually

accessible via A1 which are also mutually accessible via A2. The truth conditions for N1α and

N2α depend solely on the accessibility of interpretation functions via A1 or A2 resp. Therefore

N1α ≡ N2α is valid on one-frame models for Kr x SP5 x F x S5.

This is intuively plausible if one is aware that one-frame models for Kr x SP5 x F x S5 simply

look like models for Kr x SP5 x S5 as both kinds of modal branching coincide in one straight

time edge per instant. One might say cum grano salis that the two halves of the border of

unpreventability fold backwards again, and the two halves of the border of determinateness fold

forward, thereby swallowing the realm of spacelike events.

So when only one coordinate frame is taken into account unpreventability and determinateness

collapse again into necessity. This provides a nice indirect illustration of the fact that what was

formerly called necessity falls apart into different concepts at the very point where a multiplicity

of coordinate frames comes is taken into account.

634.5.5. Open questions of axiomatization

Complex semantics with five different kinds of operators are not easy to axiomatize, and it is not

clear if any completeness results can reasonably be expected. A minimal set of axioms it is easy

enough to state:

Axioms for Kr x SP5 x F x S5 (minimally) (PC), (Kr),

(com)- and (chr)-axioms for temporal and spatial operators

(S5 for ¤)

(S5 for á)

(S5 for N1), (S5 for N2)

(PN1) PN1 α → N1 P α

(PN2) PN2 α → N2 P α

(N2/N1) N2 α → N1 α

So, backward branching is still excluded for both N operators. This is intuitive if one imagines

the sheets cut into little slices (the operators are restricted to single places in space!) which, then,

look like Kr x SP5e models again. The axiom (N1/N2) is motivated by the idea that

determinateness implies unpreventability, but not necessarily vice versa.

Unfortunately, already historicity, though a very plausible constraint, is very difficult to

axiomatize. Kutschera (1997) yields the result that axioms for some realistic tense logic + S5 for

N + (PN) is a complete axiomatization of a combined tempo-modal semantics which fulfils a somewhat weaker constraint if supplemented by some axioms for a crossworld simultaneity

operator. In terms of accessibility of interpretation functions the weaker constraint instead of

historicity for Kr x S5 is:

(ii*) for every I, I‘ from INT: if I At I‘ then for every t’ with t’ < t: I At’ I‘.

The corresponding weaker constraint for Kr x SP5 x F x S5 would have to be:

(ii*) for every I, I‘ from INT, f from F:

if I A1e I‘ then for every e’ from E, t’f from f: if t’f

e’ < f t fe then I Ae’ I‘

(iii) for every I, I‘ from INT: if I A1e I‘ then I A2

e I‘.

But historicity is not the only problem. The (com) and (chr) axioms from Kr x SP5e x S5 are

some matter of doubt for Kr x SP5e x F x S5, too:

64(com-M1¡) M1¡α ≡ ¡M1α „chessboard“-axiom

(chr-¡N1) ¡N1 α → N1 ¡α Church/Rosser-axioms (chr-M1¤) M1¤α → ¤M1 α

(com-M2¡) M2¡α ≡ ¡M2α „chessboard“-axiom

(chr-¡N2) ¡N2 → N2¡α Church/Rosser-axioms

(chr-M2¤) M2¤α → ¤M2 α

They are, in fact, at least partly invalid for models of Kr x SP5 x F x S5. This is intuitive if one

keeps in mind that M1p, defined as ~ N1 ~ p may be read as “it is (from right here) still be

effectable that p”, and that M2p, defined as ~ N2 ~ p may be read as “it is (from right here)

unexcluded that p”. Here are some counterexamples:

(1) ¡M1α → M1¡α is not valid:

¡M1Fp → M1¡Fp means “if it is still effectable from somewhere that p will be the case

there then it is still effectable from right here that p will be the case

there“

This is not valid as can be seen from the following model:

e‘: p

¡ M1Fp & M1Fp

~ M1¡Fp

65(2) ¡N1α → N1¡α is not valid.

¡N1F¡p → N1¡ F¡p means that “if from somewhere it is unpreventable that p will

somewhere be the case then it is unpreventable from right here that

p will somewhere be the case“

This is, again, not generally true:

e‘: p

N1F¡p ¡N1F¡p & ~ N1¡F¡p

(3) ¡N2 → N2¡α is not valid.

¡N2Pp → N2¡Pp means that “if from somewhere it is determined that p was

the case right there then it is determined right here that p was the

case“

This, too, is not generally true:

¡N2Pp Pp

~N2¡Pp N2Pp

p

66It remains to be seen if any of the axioms of this group hold. Another important question is

whether the corresponding axioms hold between the two N operators, i.e. whether we have:

(com-M1M2) M1M2α ≡ M2 M1α „chessboard“-axiom

(chr- M2 N1) M2N1 α → N1M2α Church/Rosser-axioms (chr-M1N2) M1N2α → N2M1 α

675. Philosophical implications

The philosophical implications of Kr x SP5 x F x S5 might be far-reaching, as can be seen from

the fact that Kr x SP5 x F x S5 postulates contingentia praesentia and contingentia praeterita. In

a certain way, this is a consequence of construing Kr x SP5 x F, a logic for relativistic space-time,

as a superposition of models of a logic for absolute space-time, Kr x SP5, taking spacelike events

and spatial distances seriously (in a way, Kr x SP5 x F is more relativistic than Prior / Goldblatt’s

S4.2. with its absolute earlier-than-relation). So especially a certain problem must, and can, be

faced in a precise manner which Einstein, late in his life, called the worrying “problem of the

now” (without explicitly stating what it consists in).

Perhaps the best approach to the problem is to see why Rietdijk’s argument is an interesting

argument even though it does not demonstrate what it purports to demonstrate. This, in turn is

seen by asking: Why, actually, was it a good idea to call N1 an unpreventability operator? Could

not N1 express determinateness? Belnap is curiously undecisive on this point: he prefers to

replace the question whether “events in the wings” [space-like events relative to some given

event e] are “ontologically definite or indefinite” (as Stein 1991 puts it) by the “sharper”

question whether they belong to the intersection of two history planes splitting at e (Belnap 1992,

400). According to Belnap’s definitions they do (ibid. 410-3). This is plausible since the

definitions involve the notion of some (conscious or unconcious, more or less metaphorical) choice at e whether to take a certain course or another and are, thus, closely related to

effectability and unpreventability from e onward, which is mirrored by A1. This does not answer

the question, though, what should be regarded as “definite” at e, for which A2 might be decisive

instead. So against the background of Kr x SP5 x F x S5 we may say that Belnap does not give a

sharp answer to a sharp question instead of hopelessly trying to answer a muddled question, but

that he gives a sharp answer to a sharp question which is, alas, different from the equally sharp

question the Rietdijk / Stein debate is all about.

It is true that if we read N1 as a determinateness operator then we could just forget about N2

and would not have to buy something as surprising as contingentia praesentia and contingentia praeterita. We should rather not because if one accepted N1 as a determinateness operator, the

following situation would be pretty worrying in the light of Rietdijk’s argument:

68

e‘: p

+N1¡ p ¡+N1¡ p

N1F¡ p ¡N1F¡ p

¡N1¡Fp

If we read N1 as a determinateness operator then for any p in the local future we would not only

obtain in a two-frame model that there is a place such that there is a coordinate frame such that it

is determined that p is somewhere the case (¡+N1¡p), but also that it is somewhere determined

that somewehere p will be the case – coordinate frames aside (¡N1¡Fp). If, however, we read N1 as an unpreventability operator the result is not nearly so worrying, for in that case

¡N1¡Fp

merely says that from somewhere (else) it is unpreventable that somewhere, i.e. here, p will be

the case.

But what does it mean that the plausible determinateness operator in our system is N2? If it is

there is no escape from contingentia praesentia and contingentia praeterita. First of all, this

reveals a striking similarity of spacelike and (causally) future contingents. With regard to both

kinds of happening we are only in a position to guess: with regard to future contingents in the

causal future we guess into time, with regard to present and past contingents we guess into space.

Before reality has turned out to us (right where we are) to be a part of a certain filling of space-

time rather than some other we are neither in a position to know nor to say “thank goodness

that’s over” (cf. Prior 1976, 84f, and the discussion in Müller 2000, 144-152). So seen through

the glasses of relativistic tense-logic the old problem of future contingents appears only as a

special case of a more general problem of guesswork.

However, there are features of spacelike contingents which distinguish them from future

contingents on our own world-line. As to our own world-line, we can always say that what we

guess at is no sooner realized than we may be informed of it and may be regarded open before in

a straightforward way. But in what way can some distant happening be undetermined if (from the

point of view of my coordinate frame, which is not worse than any other) I am informed of it

69later on such that when I guessed at it I guessed correctly (for a very concise argument in this

direction cf. Rakic 1997, 264, where she even hints toward praeterita contingentia by using

Aristotle famous example of a seafight)?

Perhaps, the situation is not quite so bad if one is prepared to treat determinateness (or: at

least one kind of it) as frame-relative and takes counterfactuals into account: there is an

interesting difference between events in my relative past + present and events in my future as to

contingent happenings at them. Once I have decided on a coordinate frame, I may say about

happenings in my future that if I were somewhere else now instead of where I am then I could in

principle prevent them. The same cannot be said about events in my relative present and past.

Luckily, we do not have to introduce a third modal operator with a frame-relative accessibility

relation or even a special counterfactuality operator in order to express this idea of edge-

branching: “it is now preventable from somewhere that α” is straightforwardly expressible as

¡M1α

This might mitigate the idea of contingentia praesentia and contingentia praeterita.

But are we content with frame-relative determinateness? Or does the classic distinction

between epistemic and ontic determinateness make no more sense, and has, more radically, to be

replaced by a new notion of don’t-care?

Even nearly a century after the discovery of the special theory of relativity these questions

seem to await philosophical clarification, and one may hope that the further study of multi-

dimensional modal logic provides a suitable formal tool for this task.

Version of: August 21,2001: http://www.uni-rostock.de/fakult/philf ak/fkw/iph/strobach/demo/project821.doc

Latest changes: Sept. 5, 2001

Niko Strobach

Institut für Philosophie

Universität Rostock

August-Bebel-Str. 28

18051 Rostock, Germany

phone: 0049-(0)381-498 2815 or 497 39 59 fax: 0049-(0)381-498 2817

e-mail: niko.strobach@philfak.uni-rostock.de

homepage: http://www.uni-rostock.de/fakult/philfak/fkw/iph/strobach/strobach.html

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