The logic of the future in quantum theory

According to quantum mechanics, statements about the future made by sentient beings like us are, in general, neither true nor false; they must satisfy a many-valued logic. I propose that the truth value of such a statement should be identified with the probability that the event it describes will occur. After reviewing the history of related ideas in logic, I argue that it gives an understanding of probability which is particularly satisfactory for use in quantum mechanics. I construct a lattice of future-tense propositions, with truth values in the interval $[0,1]$, and derive logical properties of these truth values given by the usual quantum-mechanical formula for the probability of a history.


Introduction
In the course of trying to understand quantum theory following Everett and Wheeler, I found myself in need of some form of temporal logic, and was therefore led to study the work of Arthur Prior and his followers. Quantum theory being an indeterministic theory, Prior's handling of future contingents would be particularly relevant. Conversely, since quantum theory is the framework for our most fundamental theories of physics, the form of its statements might be of particular interest to philosophers and logicians concerned with future contingents. In this paper I will examine the logical form of statements about the future that are allowed in quantum theory as I understand Everett and Wheeler to understand it.
It follows that if the value of |Ψ(t) is known at some one time, its value at all other times is determined.
There are two problems with this. The first is that the vectors |Ψ(t) produced by the Schrödinger equation are not recognisable as descriptions of the universe we experience: if one starts with a reasonable |Ψ at time t = 0, describing a possible state of the universe we live in, then it will evolve to a vector which is not so reasonable, but instead is the vector sum (or superposition) of a number of terms each describing different, incompatible states of the universe. The second problem is that it is not the right sort of theory for the phenomena which quantum theory is supposed to describe. These phenomena, like the decays of radioactive nuclei, are random and unpredictable; the theory should not give a deterministic account of these events, but should capture their unpredictability.
The conventional resolution of these problems (the "Copenhagen interpretation") is to suppose that the steady development of |Ψ(t) , following the Schrödinger equation, is interrupted by unpredictable "quantum jumps" into one of its sensible components (or eigenstates) whenever a physicist makes a measurement. This, known as the "collapse postulate", is what I referred to as the dirty second apparatus, and I hope it will be clear why I chose this adjective.
Everett's proposal [9] was that there was no need for the collapse postulate; the unmutilated vector |Ψ(t) , even though it is a superposition of states that are incompatible in our experience (live and dead cats, in Schrödinger's famous example), should be seen as the truth about the universe. And that is all. However, this clean proposal has itself acquired extra dirty layers of unnecessary ontology. Following de Witt's 1970 evangelisation [6], Everett's view of the universal state vector |Ψ(t) has come to be seen as one of "many worlds", co-existing with the world we live in; and as time progresses these worlds are supposed to branch into ever-increasing number of worlds (and each of us acquires an ever-increasing number of "descendants"). Questions now arise of how these "worlds" are to be picked out of the universal state vector (the "preferred basis problem"), when branching happens, and what distinguishes branching from the quantum jumps of the Copenhagen interpretation [26]. These questions have been given sophisticated answers by Wallace [29] and others; nevertheless, it seems to me simpler, more economical and ontologically more reasonable to stay with the original vision of Everett [9] 1 , as endorsed by Wheeler [31].
Thus by "quantum theory" I mean a theory of a single universe U de- 1 The history [4] is less simple than this suggests. The ontology of many worlds was indeed part of Everett's original vision, and is affirmed in the full version of his thesis [10]. The shorter published version contains no reference to many worlds and only one mention scribed by a time-varying state vector |Ψ(t) , and nothing else: one world, no branching, no preferred basis, no collapse. But doesn't this mean that our experiments have no results?

Living in the quantum world
To make sense of this, we have to recognise that we ourselves are physical systems which are described by |Ψ(t) , and that the statements we make about what we observe are physical events occurring in |Ψ(t) . Any one of us is a subsystem S of the universe (an observer ) with sufficient structure to have experiences. Then the universe can be divided as U = S + U ′ , where U ′ is the rest of the universe. Mathematically, the vector space of state vectors of the universe is a tensor product of spaces describing the states of the observer and the rest of the universe: H U = H S ⊗ H U ′ . (If it seems solipsistic to make the theory depend on a particular observer, Everett shows that the experiences of different observers are correlated and so S could be taken to be the whole human race.) Now the state space of the observer has an orthonormal basis (assumed, for ease of exposition, to be countable) of states |η i which include definite experiences; so the vector describing the state of the universe at time t can be written as (in this notation, due to Dirac, juxtaposition of | symbols denotes the tensor product of vectors). Note that the |η i , representing all possible experiences, are independent of time. Everett and Wheeler tell us that a statement about the external world must be understood as being relative to a particular state of the observer: if, for any reason, we are entitled to say that the state of the observer at time t is |η i , then we are entitled to assert that the state of the rest of the universe is |ψ ′ (t) . And, of course, as an observer I am entitled to say which |η i is my experience state, precisely because it is my experience. But this looks like equivocation. I seem to be saying both that the state of the universe is |Ψ(t) , obtained from the Schrödinger equation, and that it is |η i |ψ ′ i (t) , obtained from |Ψ(t) by something very like the projection postulate. Which is the truth?
(in a footnote) of branching, Everett having been persuaded by his thesis advisor John Wheeler to remove them. I think Wheeler was right so to persuade him.

Internal and external
This contradiction is of the same type as many familiar contradictions between objective and subjective statements. It can be resolved in the way put forward by Thomas Nagel [18,19]: we must recognise that there are two positions from which we can make statements of fact, and statements made in these two contexts are not commensurable. In the external context (the God's-eye view, or the "view from nowhere") we step outside our own particular situation and talk about the whole universe. In the internal context (the view from now here), we make statements as actors inside the universe. Thus in the external view, |Ψ(t) is the whole truth about the universe; the components |η i |ψ ′ (t) are (unequal) parts of this truth. But in the internal view, from the perspective of some particular experience state |η i , the component |η i |ψ ′ i (t) is the actual truth. I may know what the other components |η j |ψ ′ j (t) are, because I can calculate |Ψ(t) from the Schrödinger equation; but these other components, for me, represent things that might have happened but didn't. The universal state vector, in this perspective, is not a description of the world but a true statement of how the world might change.

Time and chance
Two questions present themselves. First, time: if it is unproblematic that at time t I can identify my experience at time t and the corresponding state |ψ ′ (t) of the rest of the universe, what can I say about other times? For past times, memory provides an answer: it seems that we are constructed in such a way that each experience state that actually occurs at time t (i.e. for which |ψ ′ (t) is non-zero) contains information about a unique experience state at each time before t. But in general, there is nothing in the physics to pick out experiences future to t: there is no "thin red line" stretching into the future. 2 The Schrödinger equation can be applied to |η i to yield a state vector e −iH(t ′ −t) |η i to which it will evolve at a given future time t ′ ; but in general this will not be another experience state.
Second, probability: quantum mechanics is an indeterministic theory, describing chance events. Its empirical success rests on its ability to give probabilities for such events. But how can there be any place for probability in the framework I have described? In the external view, |Ψ(t) develops deterministically according to the Schrödinger equation; nothing is left to chance. In the internal view, as we have just seen, there are no future events and therefore, it seems, no chances. How can "the probability that my experience will be |η j tomorrow" mean anything if "my experience will be |η j tomorrow" has no meaning?

Probability and truth
My solution to these two questions is to propose that they answer each other. If there is no experience that can be identified as the experience I will have at a future time t, then for each η the statement "I will experience η at t" is not true. Nevertheless, if that experience is very likely then this statement is nearly true; in other words, it has a degree of truth less than, but close to, 1. This leads to the suggestion that the probability of a future event E should be identified as the degree of truth of the future-tense statement "E will occur".
This argument reflects the way that probabilities are calculated in quantum mechanics. The vector e −iH(t ′ −t) |η i to which an experience eigenvector |η i evolves between times t and t ′ , according to the Schrödinger equation, will not in general be an experience eigenvector; but it may be near an experience eigenvector, and the nearness is measured by a number (close to 1 if the vectors are near to each other) which is identified with a probability in conventional quantum mechanics.
What exactly is a "degree of truth" for a future-tense statement? In the rest of the paper I will explore the possibility that it can be taken to be a truth value in a temporal logic.

Probabilities as truth values
3.1 Problems 1. One function of truth values is to define the meaning of logical connectives such as and and or by means of truth tables. The existence of such tables is made a prerequisite for many-valued logics in Gottwald's comprehensive treatise [11]. This is not possible with probabilities. Using τ to denote the probability of a proposition, the values of τ (p ∧ q) and τ (p ∨ q) are not determined as functions of τ (p) and τ (q); the two equations which this would require are replaced by a single relation 2. Another use of truth values is to identify logical tautologies, which are defined as formulae, with propositional variables, which have truth value 1 for all assignments of truth values to the variables. If truth values were replaced in this definition by probabilities satisfying (1), it would not yield a useful set of tautologies. Instead, the usual procedure in probability logic [1] is to require that the value of probability should be 1 for all logical tautologies, assuming that these have already been identified.

History
Aristotle's discussion of future contingents in De Interpretatione is often taken to support a three-valued logic for statements about the future. But Aristotle says more than denying truth or falsity to "There will be a sea-battle tomorrow"; he also notes that it may be more or less likely that there will be a sea-battle. Thus if there is a case for regarding Aristotle as a proponent of many-valued logic for future-tense statements, there might also be a case that he would regard the appropriate truth values as probabilities.
Lukasiewicz's first system of many-valued logic [16] had truth values related to probabilities, though of a rather different kind from those considered here (he was concerned with propositions containing variables, giving them truth values equal to the proportion of values of the variable which made the proposition true). Later ( [17]) he was motivated by the problem of future contingents and expressed a preference for many-valued logic in which the truth values could be any real number between 0 and 1; he asked how this was related to probability theory. In his system the truth values do not satisfy (1), but they satisfy (2) with inequality replaced by equality, so that the logical connectives are truth-functional. I will consider the possible relevance of this system to the quantum theory in section 4.3.
The failure of probabilities to be truth-functional led to a reluctance among logicians to accept them as truth values, but this view was defended by Reichenbach [22] (though his frequentist conception of probability was different from that espoused here) and especially by Rescher [23]. Both these authors claimed to prove that the tautologies of ordinary two-valued logic can all be obtained as tautologies of probability logic, but Reichenbach's argument depended on his frequentism and Rescher's, though more formal, required axioms which themselves refer to the classical tautologies, giving his argument an element of circularity.
The notion of degrees of truth also occurs in fuzzy logic [32,8,24], and in this context also Edgington proposed replacing the classical truth tables by the relation (1). More complicated truth values occur in topos theory [15], which has been proposed as a suitable logical formalism for the foundations of physics by Isham [14] and for the discussion of partial truth by Butterfield [3]. Döring and Isham have identified probabilities with truth values in topos theory [7], but the probabilities they discuss seem to be credences rather than chances, occurring in both classical and quantum physics but, in the latter, relevant to mixed states and not to pure states. Isham has also formulated a temporal form of quantum logic [13] in which histories are treated as propositions.
In the subtitle of this paper I have used the term "Everett-Wheeler understanding of quantum theory" to distinguish what is discussed here from the "many-worlds theory" or "Everettian quantum mechanics" developed by the Oxford school of Saunders [25], Deutsch [5] and Wallace [29]. Saunders, in a paper with a similar title to this one, also has a similar view of probability: "events in the future . . . are indeterminate; . . . probabilities . . . express the degrees of this indeterminacy" (emphasis in the original). Deutsch and Wallace have a different concept of probability based on decision theory and are concerned to derive the quantum formula for probability (the Born rule) from decision-theoretic axioms. I am content to take the Born rule as an underived postulate, fitting naturally with the geometric nature of the mathematical objects in the theory. In the next section I will explore how this postulate coheres with what I take to be the logical and temporal nature of probability. 4 The logic of tensed propositions in our quantum world

The lattice of propositions
Living in the quantum world, as we do and as we are described in section 2.2, what can we say about it? Statements about our possible experiences form a Boolean lattice related to the experience state vectors |η i . In the fiction that there is a countable basis |η i , the experiences η i are atoms in this lattice; more generally, it is a Boolean sublattice E of the lattice of closed subspaces of the Hilbert space H S . To form the lattice T of statements that we want to make about our experience we need maps N : E → T (to give statements about our present experience), P t : E → T for each positive real number t (to give statements about our experience a time t in the past) and F t : E → T (for the future). These should be lattice homomorphisms, and the complete lattice T should be generated by the sublattices N(E), P t (E) and F t (E). On the assumption that T is a distributive lattice, it follows that every proposition in T is a disjunction of histories and h P is formed similarly with past operators P t . Here each Π k represents an element of the lattice of subspaces E, being the linear operator of orthogonal projection onto the subspace. I will refer to n as the length of the history h F .

The truth of histories: Conjunction
Truth values are assigned to elements of T from the perspective of a particular experience η 0 at a time t 0 . Past and present propositions are taken to obey classical logic, so any element N(Π) or P t (Π) has a truth value of 0 or 1, and elements of the sublattice generated by these have truth values determined by the usual truth tables. The truth value of a future proposition F t (Π), however, is equated with its probability and could lie anywhere in the closed unit interval [0, 1]. It is determined by quantum mechanics as follows. The component of the universal state vector determined by the experience |η 0 at the time t 0 is |E 0 = |η 0 |ψ ′ 0 (t 0 ) , which would evolve by the Schrödinger equation to e −iHt |E 0 after a time interval t, where H is the universal Hamiltonian. On the other hand, the experience state |η j will, after the lapse of time t, be associated with the component of the universal state vector. The geometrical measure of the closeness of these two vectors is taken to be the truth value of the statement "I will have experience η j after a time t" in the context of experience η 0 at time t 0 (from now on this context will be understood): where τ denotes truth value and This is the usual expression (the Born rule) for the probability in quantum mechanics.
To extend this to a conjunction of future-tense propositions, i.e. to a history h F given by (4), we adopt the standard extension of the Born rule ( [12,29]) to the probability of a history: where C h is the history operator Note that if t 1 = t 2 , since the projectors Π 1 and Π 2 commute and therefore so do Π 1 and Π 2 if t 1 = t 2 . So the formula (6) for τ (h 1 ∧ h 2 ) holds for t 1 = t 2 as well as t 1 < t 2 .
I will now explore the logical properties of this definition. For reasons of space, proofs are omitted; they can be found in an extended version of this paper available online at http://uk.arxiv.org/abs/(number to be determined).
First we note the elementary fact Lemma 1. Let Π be a projection operator and |ψ any state vector. Then Proof.
Theorem 1. For any future history h F , Proof. By repeated application of Lemma 1(i), using E 0 |E 0 = 1.
Theorem 2. For any two future histories h 1 , h 2 , Proof. First note that if k 1 , . . . k n are one-time histories This is proved by induction on n, using Lemma 1(ii). Now if h 1 and h 2 are any two future histories, τ (h 1 ∧ h 2 ), τ (h 1 ) and τ (h 2 ) are all conjunctions of one-time histories, so both sides of the equivalence in the theorem are equivalent to τ (k) = 1 for all one-time histories occurring in h 1 and h 2 .
Theorems 2, 3 and 4 show that the truth values of two-time histories (conjunctions of one-time propositions) have some of the properties that we would expect for the truth and falsity of conjunction. However, the strict falsity of a conjunction (as opposed to its lack of truth) does not imply the strict falsity of one of the conjuncts. This is not surprising, given the probabilistic nature of the truth values. The quantum nature of the truth values becomes apparent in Theorems 3 and 4 which display a lack of symmetry between the conjuncts. The falsity of a proposition h 2 at the later time t 2 does not imply the falsity of the conjunction h 1 ∧ h 2 , because the interposition of a fact h 1 at t 1 affects the truth of h 2 at t 2 . However, this quantum effect should not be visible at the level of our experience. In order to restore the symmetry of conjunction, to make it possible to extend some theorems which would otherwise only apply to one-time histories, and to complete the logic generally, we need the following Consistent Histories assumption: CH Let h = F t 1 (Π 1 ) ∧ · · · ∧ F tn (Π n ) be any history in the lattice of temporal propositions, and for any binary sequence α = (α 1 , . . . , α n ) (α i = 0 or 1), let where C h is the history operator of (7).
This assumption demarcates the admissible histories in the "consistent histories" formulation of quantum mechanics [12], and can be justified for macroscopic states like our experience states by decoherence theory [29].

Lemma 2. If CH holds, the truth value of a history
Proof. By CH, for each non-zero α ∈ {0, 1} n we have Taking α i = δ ir for some r gives τ (h) = E 0 | Π 1 · · · Π n Π n · · · ] Π r [· · · · · · Π n |E 0 where ] Π r [ denotes that Π r is omitted from the product. We now prove by downward induction on r that for any subset R = {i 1 , . . . , i r } of {1, . . . , n}, Taking α so that α i = 0 if i ∈ R, 1 if i / ∈ R, (8) gives the right-hand side of (10) as a sum of terms with r + k factors to the right of Π n , with n−r k terms if k > 0, all multiplied by (−1) k+1 , and all equal to τ (h) by the inductive hypothesis. This sum is r =0 We can now restore symmetry to Theorems 3 and 4 and extend them to multi-time histories.
Theorem 5. If CH holds, for any two histories h 1 and h 2 .
Proof. First take h 2 = F tr (Π r ) to be a one-time history and h 1 to be given by If CH holds, τ (h 1 ∧ h 2 ) is given by (9). But if τ (h 1 ) = 0, Now any h 2 can be written as a conjunction of one-time histories, say Theorem 6. If CH holds, and h 2 is a one-time history, Note that in general ¬h 1 is a disjunction of histories, so τ (¬h 1 ∧ h 2 ) is not yet defined.
Proof. Suppose that h 1 and h 2 are as in the proof of Theorem 5, so that ¬h 2 = F tr (1 − Π r ). Then the result follows immediately from Lemma 2.
Theorem 7. If CH holds, for any two future histories h 1 , h 2 .
Proof. By Theorem 6, the inequality holds if h 2 is a one-time history. Now we argue by induction on the length of h 2 , using the associativity of ∧. If h 3 is a one-time history and the inequality holds for h 1 and h 2 , so the inequality holds for h 1 and h 2 ∧ h 3 .

Theorem 8. If CH holds,
for any two future histories h 1 and h 2 .
Proof. First suppose that h 1 and h 2 are one-time histories. By Lemma 2, CH gives Now we proceed by double induction on the lengths of h 1 and h 2 : if h 3 is a one-time history, and the inequality holds for h 1 and h 2 , then

Disjunction
If it is clear what is meant by a future history and how to assign its truth value in quantum theory (though whether this really is clear will be discussed in the next section), it is not so clear how to approach a disjunction of histories. However, with the assumption CH, we can adopt the following definition from probability logic: since Theorems 7 and 8 assure us that 0 ≤ τ (h 1 ∨ h 2 ) ≤ 1. It can be shown that the assumption CH also guarantees that this can be extended to conjunctions of any finite number of histories in such a way that (12) holds for any two elements of the lattice T , and that it is consistent with negation in the lattice, i.e.
With this definition, Theorem 6 can be extended to hold for any two histories, and analogues of Theorems 2, 5, 6 and 7 for disjunction follow from the results for conjunction: A disturbing feature of this list of properties is the one-sidedness of the implication (ii): the truth of h 1 ∨ h 2 does not imply the truth of either h 1 or h 2 (though (iii) shows that this implication does hold if "truth" (τ = 1) is replaced by "possible truth" (τ = 0)). This may cast doubt on whether ∨ can legitimately be regarded as a generalised form of "or" (a similar objection has been made to quantum logic [26]). Such an objection would be an argument against any identification of probabilities with truth values. However, far from being objectionable, this feature of disjunction seems to be necessary in a temporal logic that can deal with indeterminism. As Prior noted ( [21] p.244), "Either there will be a sea-battle tomorrow or there won't" should be taken to mean, not what it appears to mean, but " 'Either "There is a seabattle going on" or "there is no sea-battle going on" ' will be true tomorrow"; in symbols, But in the logic proposed here F t is taken to be a homorphism, so that Aristotle's usage (or confusion, if that is what it is) is of course very common. I take Prior's elucidation of it as grounds for claiming that the disjunction occurring in this temporal logic is in fact the "or" of common usage in talk about the open future.
Be tht as it may, there are alternative meanings for the binary operation ∨, defined by different truth values, which are not subject to this objection.
Lukasiewicz's rule could be adopted only for the disjunction of histories, replacing (12); this would restore the reverse implication in Theorem 9(ii).
Metalogical disjunction The classical meaning of "or" can be imposed on the operation ∨ by brute force by defining This is tantamount to making h 1 ∨ h 2 a statement in the metalanguage, with the meaning "Either h 1 is (completely) true, or h 2 is". As will be discussed in the next section, there is a case for doing this in quantum theory.

Retrospect
In this final section I will consider to what extent the proposals in this paper meet the objectives set out in Section 2.
The open future The reader may well have observed that there is nothing new or specifically quantum-mechanical about the indeterministic world described in Section 2; this is the classical picture of an open future. The assumption CH made in Section 4.2 simply ensures the classical nature of our logic. Thus the logic developed here is a temporal logic appropriate to a metaphysics of time that incorporates an open future. It can be summarised as follows: 1. The lattice T of tensed propositions is formed from a lattice E of tenseless propositions by means of lattice homomorphisms {P t , N, F t : t ∈ R, t > 0} whose images generate T .
All the theorems of section 4 can be derived from these axioms.

Strictly Quantum
The reader may also have observed that I have not kept to the faith boldly proclaimed in Section 2. I there renounced the devil and all his works, such as the collapse postulate, and pomps, such as a separate classical realm. But the formula I took for the truth value (or probabiity) of a history is taken from a calculation of probability based on the collapse postulate; and the Consistent Histories assumption, which I introduced in order to get a recognisable logic, is just a way of eliminating the characteristic features of quantum mechanics. This assumption can be justified by decoherence theory within quantum mechanics, but only if one restricts the set of possible histories (for example, it will not be true if one wants to include times that are arbitrarily close to each other). Moreover, there is no justification for the definite past propositions P t (p); there is no thin red line between experience states going into the past, any more than there is one going into the future. There are only statements taken from present memory, which are aspects of the present experience states (in other words, statements about the past can only be in the perfect tense). These objections can be met by restricting the logic as follows. The present sublattice N(E) describes present experience, including memory (so statements about the past occur in the perfect tense as elements of N(E)). From the context of a particular experience state η 0 , these elements of N(E) have truth values restricted to {0, 1}. It is possible to make future-tense statements only about experiences at one future time. A statement about a future history must be regarded as an element of F t f (E), where t f is the last time in the history; the other elements of the history are in the future perfect tense, referring to the contents of memory at time t f (this, after all, is the only way we can verify a statement referring to a number of future times: we wait until the last time referred to and then consult records of earlier times).
Thus the set of propositions is not a lattice but is the union of the sublattices N(E) and F t (E). A conjunction or disjunction of elements of different sublattices is not a well-formed proposition in this logic; it must be regarded as a statement in the metalanguage, with metalogical truth values as at the end of Section 4.