The lattice of propositions
Living in the quantum world, as we do and as we are described in Sect. 2.1, what can we say about it? Statements about our possible experiences form a Boolean lattice related to the experience state vectors \(|\eta _i\rangle \). In the fiction that there is a countable basis \(|\eta _i\rangle \), reports of the experiences \(\eta _i\) are atoms in this lattice; more generally, it is a Boolean sublattice \(\mathcal {E}\) of the lattice of closed subspaces of the Hilbert space \(\mathcal {H}_S\). To form the lattice \(\mathcal {T}\) of statements that we want to make about our experience we need maps \(N:\mathcal {E}\rightarrow \mathcal {T}\) (to give statements about our present experience), \(P_t:\mathcal {E}\rightarrow \mathcal {T}\) for each positive real number t (to give statements about our experience a time t in the past) and \(F_t:\mathcal {E}\rightarrow \mathcal {T}\) (for the future). The complete lattice \(\mathcal {T}\) is then generated by the images \(N(\mathcal {E})\), \(P_t(\mathcal {E})\) and \(F_t(\mathcal {E})\). This lattice consists of statements that we make, and should conform to the structure of our language; I therefore assume that the lattice has a classical structure, and in particular that \(\wedge \) distributes over \(\vee \). (The non-classical features will enter when we consider the relation of this lattice to the physical world, given by truth values—“everything that is the case”). The map N simply embeds the lattice \(\mathcal {E}\) into \(\mathcal {T}\)—it adds the word “now” to a report of an experience—and therefore respects the structure of these reports, so N is an injective lattice homomorphism. Since we are assuming that memory gives reports of past experience with the same quality of definite truth or falsehood as present experiences, I also take the maps \(P_t\) to be injective homomorphisms.
It is not so clear that the future operators \(F_t\) should be homomorphisms, but it is not clear what “and” and “or” should mean for tensed statements with truth values between 0 and 1. I will assume that each \(F_t\) is a homomorphism in order to give a meaning to \(p\wedge q\) and \(p\vee q\) for all tensed statements p and q, but these meanings might be different from “p and q” and “p or q”. This is discussed further at the end of this section. A major objective of the mathematical development in this section is to delineate the differences between \(\wedge \) and “and”, and between \(\vee \) and “or”; this is the content of Theorem 14.
On the assumption that \(\mathcal {T}\) is a distributive lattice, it follows that every proposition in \(\mathcal {T}\) is a disjunction of histories
\(h_\text {P}\wedge h_0 \wedge h_\text {F}\) where
$$\begin{aligned} h_0 = N(\Pi _0), \quad h_\text {F} = F_{t_1}(\Pi _1)\wedge \ldots \wedge F_{t_n}(\Pi _n) \; \text { with } \; 0< t_1< \cdots < t_n, \end{aligned}$$
(6)
and \(h_\text {P}\) is formed similarly with past operators \(P_t\). Here each \(\Pi _k\) represents an element of the lattice of subspaces \(\mathcal {E}\), being the linear operator of orthogonal projection onto the subspace. I will refer to n as the length of the history \(h_\text {F}\). Since the departures from classical logic in this system occur only in future-tense propositions, the rest of this section will be concerned only with the sublattice \(\mathcal {T}_\text {F}\) generated by \(F_t(\mathcal {E})\) for all t.
The truth of histories: conjunction
Truth values are assigned to elements of \(\mathcal {T}\) from the perspective of a particular experience \(\eta _0\) at a time \(t_0\). Past and present propositions are taken to obey classical logic, so any element \(N(\Pi )\) or \(P_t(\Pi )\) 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 \(F_t(\Pi )\), 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 \(|\eta _0\rangle \) at the time \(t_0\) is \(|E_0\rangle = |\eta _0\rangle |\psi '_0(t_0)\rangle \), which would evolve by the Schrödinger equation to \(\text{ e }^{-iHt/\hslash }|E_0\rangle \) after a time interval t, where H is the universal Hamiltonian. On the other hand, the experience state \(|\eta _j\rangle \) will, after the lapse of time t, be associated with the component
$$\begin{aligned} |\eta _j\rangle |\psi '_j(t_0 + t)\rangle = (\Pi _j\otimes 1)|\Psi (t_0 + t)\rangle = (\Pi _j\otimes 1)\text{ e }^{-iHt/\hslash }|E_0\rangle \end{aligned}$$
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 \(\eta _j\) after a time t” in the context of experience \(\eta _0\) at time \(t_0\) (from now on this context will be understood):
$$\begin{aligned} \tau \left( F_t(\Pi _j)\right) = \langle E_0|\widetilde{\Pi }_j|E_0\rangle \end{aligned}$$
(7)
where \(\tau \) denotes truth value and
$$\begin{aligned} \widetilde{\Pi }_j = \text{ e }^{iHt/\hslash }(\Pi _j\otimes 1)\text{ e }^{-iHt/\hslash }. \end{aligned}$$
Eq. (7) 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 \(h_\text {F}\) given by (6), we adopt the standard extension of the Born rule (Griffiths 2002; Wallace 2012) to the probability of a history \(h_\text {F} = F_{t_1}(\Pi _1)\wedge \ldots \wedge F_{t_n}(\Pi _n)\):
$$\begin{aligned} \tau (h_\text {F})&= \langle E_0|\widetilde{\Pi }_1\ldots \widetilde{\Pi }_{n-1}\widetilde{\Pi }_n\widetilde{\Pi }_{n-1}\ldots \widetilde{\Pi }_1|E_0\rangle \nonumber \\&= \langle E_0|C_h C_h^\dagger |E_0\rangle \end{aligned}$$
(8)
where \(C_h\) is the history operator
$$\begin{aligned} C_h = \widetilde{\Pi }_1\cdots \widetilde{\Pi }_n. \end{aligned}$$
(9)
Note that if \(t_1 = t_2\),
$$\begin{aligned} \tau (h_1\wedge h_2) = \langle E_0|\widetilde{\Pi }_1\widetilde{\Pi }_2|E_0\rangle = \langle E_0|\widetilde{\Pi }_1^2\widetilde{\Pi }_2|E_0\rangle = \langle E_0|\widetilde{\Pi }_1\widetilde{\Pi }_2\widetilde{\Pi }_1|E_0\rangle \end{aligned}$$
since the projectors \(\Pi _1\) and \(\Pi _2\) commute and therefore so do \(\widetilde{\Pi }_1\) and \(\widetilde{\Pi }_2\) if \(t_1 = t_2\). So the formula (8) for \(\tau (h_1\wedge h_2)\) holds for \(t_1 = t_2\) as well as \(t_1 < t_2\).
I will now explore the logical properties of this definition. First we note the elementary fact
Lemma 1
Let \(\Pi \) be a projection operator and \(|\psi \rangle \) any state vector. Then
Proof
$$\begin{aligned} \langle \psi |\psi \rangle - \langle \psi |\Pi |\psi \rangle = \langle \psi |(1 - \Pi )|\psi \rangle = \langle \psi |(1-\Pi )^2|\psi \rangle \ge 0, \end{aligned}$$
with equality if and only if \(\Pi |\psi \rangle = |\psi \rangle \). \(\square \)
Theorem 1
For any future history \(h_{\mathrm{F}}\),
$$\begin{aligned} 0\le \tau (h_{\mathrm{F}}) \le 1. \end{aligned}$$
Proof
By repeated application of Lemma 1(i), using \(\langle E_0|E_0\rangle = 1\). \(\square \)
Theorem 2
For any two future histories \(h_1, h_2\),
$$\begin{aligned} \tau (h_1\wedge h_2) = 1 \;\iff \; \tau (h_1) = \tau (h_2) = 1. \end{aligned}$$
Proof
First note that if \(k_1,\ldots k_n\) are one-time histories \(k_i = F_{t_i}(\Pi _i)\) with \(t_1\le \cdots \le t_n\),
$$\begin{aligned} \tau (k_1\wedge \ldots \wedge k_n) = 1 \;\iff \; \tau (k_1) = \cdots = \tau (k_n). \end{aligned}$$
This is proved by induction on n, using Lemma 1(ii). Now if \(h_1\) and \(h_2\) are any two future histories, \(\tau (h_1\wedge h_2)\), \(\tau (h_1)\) and \(\tau (h_2)\) are all conjunctions of one-time histories, so both sides of the equivalence in the theorem are equivalent to \(\tau (k) = 1\) for all one-time histories occurring in \(h_1\) and \(h_2\). \(\square \)
Corollary
\(\quad \tau (h_1\wedge h_2) \ne 1 \;\Longrightarrow \; \tau (h_1)\ne 1\; \text { or }\; \tau (h_2) \ne 1\).
Theorem 3
For one-time histories \(h_1,h_2\) with \(t_1 < t_2\),
$$\begin{aligned} \tau (h_1) = 0 \; \Longrightarrow \; \tau (h_1\wedge h_2) = 0. \end{aligned}$$
Proof
$$\begin{aligned} \tau (h_1) = 0 \;\Longrightarrow \; \widetilde{\Pi }_1|E_0\rangle = 0 \;\Longrightarrow \; \tau (h_1\wedge h_2) = 0. \end{aligned}$$
\(\square \)
Theorem 4
For one-time histories \(h_1,h_2\) with \(t_1<t_2\),
$$\begin{aligned} \tau (h_1\wedge h_2) + \tau (h_1\wedge \lnot h_2) = \tau (h_1) \end{aligned}$$
Proof
$$\begin{aligned} \tau (h_1\wedge h_2) + \tau (h_1\wedge \lnot h_2)&= \langle E_0|\widetilde{\Pi }_1\widetilde{\Pi }_2\widetilde{\Pi }_1|E_0\rangle + \langle E_0|\widetilde{\Pi }_1\left( 1 - \widetilde{\Pi }_2\right) \widetilde{\Pi }_1|E_0\rangle \\&= \langle E_0|\widetilde{\Pi }_1|E_0\rangle = \tau (h_1). \end{aligned}$$
\(\square \)
Corollary
\(\tau (h_1\wedge h_2) \le \tau (h_1)\).
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 \(h_1\wedge h_2\), 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 \(h = F_{t_1}(\Pi _1)\wedge \cdots \wedge F_{t_n}(\Pi _n)\) be any history in the lattice of temporal propositions, and for any binary sequence \(\alpha = (\alpha _1,\ldots ,\alpha _n)\) (\(\alpha _{i} = 0\) or 1), let
$$\begin{aligned} h_\alpha = F_{t_1}(\Pi _1^{\alpha _1})\wedge \ldots \wedge F_{t_n}(\Pi _n^{\alpha _n}). \end{aligned}$$
where \(\Pi ^0 = \Pi \), \(\Pi ^1 = \lnot \Pi = 1 - \Pi \). Then
$$\begin{aligned} \langle E_0|C_{h_\alpha } C_{h_\beta }^\dagger |E_0\rangle = 0 \;\text { if }\; \alpha \ne \beta . \end{aligned}$$
where \(C_h\) is the history operator of (9).
This is part of a much wider assumption that demarcates the admissible histories in the “consistent histories” formulation of quantum mechanics (Griffiths 2002), and can be justified for macroscopic states like our experience states by decoherence theory (Wallace 2012). We do not need the full strength of the consistent histories formulation.
Lemma 2
If CH holds, the truth value of a history \(h = F_{t_1}(\Pi _1)\wedge \ldots \wedge F_{t_n}(\Pi _n)\) is given by
$$\begin{aligned} \tau (h) = \langle E_0|\widetilde{\Pi }_1\ldots \widetilde{\Pi }_n|E_0\rangle . \end{aligned}$$
(10)
Proof
By CH, for each non-zero \((\alpha _1,\ldots , \alpha _n) \in \{0,1\}^n\) we have
$$\begin{aligned} \langle E_0|\widetilde{\Pi }_1\cdots \widetilde{\Pi }_n\widetilde{\Pi }_n^{\alpha _n}\cdots \widetilde{\Pi }_1^{\alpha _1}|E_0\rangle = 0. \end{aligned}$$
(11)
Taking \(\alpha _n = \delta _{ir}\) for some r gives
$$\begin{aligned} \tau (h) = \langle E_0|\widetilde{\Pi }_1\cdots \widetilde{\Pi }_n\widetilde{\Pi }_n\cdots ]\widetilde{\Pi }_r[\cdots \widetilde{\Pi }_1|E_0\rangle \end{aligned}$$
(12)
where \(]\widetilde{\Pi }_r[\) denotes that \(\widetilde{\Pi }_r\) is omitted from the product. We now prove by downward induction on r that for any subset \(R = \{i_1,\ldots ,i_r\}\) of \(\{1,\ldots ,n\}\),
$$\begin{aligned} \tau (h) = \langle E_0|\widetilde{\Pi }_1\cdots \widetilde{\Pi }_n\widetilde{\Pi }_{i_r}\ldots \widetilde{\Pi }_{i_1}|E_0\rangle . \end{aligned}$$
(13)
Taking \(\alpha \) so that \(\alpha _n = 0\) if \(i\in R\), \(\alpha _n = 1\) if \(i\notin R\), (11) gives the right-hand side of (13) as a sum of terms with \(r+k\) factors to the right of \(\widetilde{\Pi }_n\), with \(\left( {\begin{array}{c}n-r\\ k\end{array}}\right) \) terms if \(k>0\), all multiplied by \((-1)^{k+1}\), and all equal to \(\tau (h)\) by the inductive hypothesis. This sum is
$$\begin{aligned} \sum _{k\ne 0}(-1)^{k+1}\left( {\begin{array}{c}n-r\\ k\end{array}}\right) \tau (h) = \tau (h). \end{aligned}$$
\(\square \)
We can now restore symmetry to Theorems 3 and 4 and extend them to multi-time histories.
Theorem 5
If CH holds,
$$\begin{aligned} \tau (h_1) = 0 \; \Longrightarrow \; \tau (h_1\wedge h_2) = 0 \end{aligned}$$
for any two histories \(h_1\) and \(h_2\).
Proof
First take \(h_2 = F_{t_r}(\Pi _r)\) to be a one-time history and \(h_1\) to be given by
$$\begin{aligned} h_1 = F_{t_1}(\Pi _1)\wedge \cdots \wedge F_{t_{r-1}}(\Pi _{r-1})\wedge F_{t_{r+1}}(\Pi _{r+1})\wedge \cdots \wedge F_{t_n}(\Pi _n). \end{aligned}$$
If CH holds, \(\tau (h_1\wedge h_2)\) is given by (12). But if \(\tau (h_1) = 0\),
$$\begin{aligned} \widetilde{\Pi }_n\cdots ]\widetilde{\Pi }_r[ \cdots \widetilde{\Pi }_1|E_0\rangle = 0 \end{aligned}$$
so that \(\tau (h_1\wedge h_2) = 0\).
Now any \(h_2\) can be written as a conjunction of one-time histories, say \(h_2 = h_{21}\wedge \cdots \wedge h_{2k}\), so
$$\begin{aligned} \tau (h_1) = 0\; \Longrightarrow \; \tau (h_1\wedge h_{21}) = 0\; \Longrightarrow \cdots&\Longrightarrow \tau (h_1\wedge h_{21}\wedge \cdots \wedge h_{2k}) = 0\\&\Longrightarrow \; \tau (h_1\wedge h_2) = 0. \end{aligned}$$
\(\square \)
Theorem 6
If CH holds, and \(h_2\) is a one-time history,
$$\begin{aligned} \tau (h_1\wedge h_2) + \tau (h_1\wedge \lnot h_2) = \tau (h_1). \end{aligned}$$
Note that in general \(\lnot h_1\) is a disjunction of histories, so \(\tau (\lnot h_1\wedge h_2\)) is not yet defined.
Proof
Suppose that \(h_1\) and \(h_2\) are as in the proof of Theorem 5, so that \(\lnot h_2 = F_{t_r}(1 - \Pi _r)\). Then the result follows immediately from Lemma 2. \(\square \)
Theorem 7
If CH holds,
$$\begin{aligned} \tau (h_1\wedge h_2) \le \tau (h_1) \end{aligned}$$
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 \(\wedge \). If \(h_3\) is a one-time history and the inequality holds for \(h_1\) and \(h_2\),
$$\begin{aligned} \tau (h_1\wedge h_2\wedge h_3) \le \tau (h_1\wedge h_2) \le \tau (h_1) \end{aligned}$$
so the inequality holds for \(h_1\) and \(h_2\wedge h_3\). \(\square \)
Theorem 8
If CH holds,
$$\begin{aligned} \tau (h_1) + \tau (h_2) - 1 \le \tau (h_1\wedge h_2) \end{aligned}$$
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
$$\begin{aligned} \tau (h_1\wedge h_2) = \langle E_0|\widetilde{\Pi }_1\widetilde{\Pi }_2\widetilde{\Pi }_1|E_0\rangle = \langle E_0|\widetilde{\Pi }_2\widetilde{\Pi }_1|\mathcal {E}_0\rangle . \end{aligned}$$
(14)
Hence
$$\begin{aligned} 1 - \tau (h_1) - \tau (h_2)&+ \tau (h_1\wedge h_2) = \langle E_0|(1 - \widetilde{\Pi }_1 - \widetilde{\Pi }_2 + \widetilde{\Pi }_2\widetilde{\Pi }_1)|E_0\rangle \\&= \langle E_0|(1-\widetilde{\Pi }_2)(1-\widetilde{\Pi }_1)|E_0\rangle \\&= \tau (\lnot h_1\wedge \lnot h_2) \quad \text { by Lemma 2 again}\\&\ge 0. \end{aligned}$$
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
$$\begin{aligned} \tau (h_1) + {}&\tau (h_2\wedge h_3) - 1 = \tau (h_1) + \tau (h_2) - \tau (h_2\wedge \lnot h_3) - 1 \quad \text { by Theorem 6}\\&\le \tau (h_1\wedge h_2) - \tau (h_2\wedge \lnot h_3) \;\quad \quad \quad \quad \text { by the inductive hypothesis}\\&= \tau (h_1\wedge h_2\wedge h_3) + \tau (h_1\wedge h_2\wedge \lnot h_3) - \tau (h_1\wedge \lnot h_3) \text { by Theorem 6}\\&\le \tau (h_1\wedge h_2\wedge h_3) \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { by Theorem 7} \end{aligned}$$
so the inequality holds for \(h_1\) and \(h_2\wedge h_3\). Hence, by induction it holds for one-time \(h_1\) and any \(h_2\). Now a similar induction on \(h_1\) shows that it holds for all \(h_1,h_2\). \(\square \)
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, which will be a standing assumption for the remainder of this section, we can, as anticipated in Sect. 3 (Eq. (3)), adopt the following definition from probability logic:
$$\begin{aligned} \tau (h_1\vee h_2) = \tau (h_1) + \tau (h_2) - \tau (h_1\wedge h_2) \end{aligned}$$
(15)
since Theorems 7 and 8 assure us that this lies between 0 and 1.
We extend this to disjunctions of any finite number of future histories, i.e. to general elements of the lattice \(\mathcal {T}_\text {F}\), by the following definition, expressing the principle of inclusion and exclusion:
$$\begin{aligned} \tau (h_1\vee \cdots \vee h_n) = H^{(n)}_1 - H^{(n)}_2 + \cdots + (-)^{n-1}H^{(n)}_n \end{aligned}$$
(16)
where
$$\begin{aligned} H^{(n)}_r = \sum \tau (h_{i_1}\wedge \cdots \wedge h_{i_r}) \end{aligned}$$
(17)
in which the sum extends over all r-subsets \(\{i_1,\ldots ,i_r\}\) of \(\{1,\ldots ,n\}\). This satisfies
$$\begin{aligned} \tau (h_1\vee \cdots \vee h_n) = \tau (h_1\vee \cdots \vee h_{n-1}) + \tau (h_n) - \tau \big ( (h_1\vee \cdots \vee h_{n-1})\wedge h_n\big )\nonumber \\ \end{aligned}$$
(18)
where, by the assumed distributivity of the lattice \(\mathcal {T}_\text {F}\), the argument of the last \(\tau \) can be expanded into a disjunction of \(n-1\) histories. However, we do not take it as an inductive definition of \(\tau (h_1\vee \cdots \vee h_n)\) since it would be necessary to show that it is well-defined, i.e. independent of the ordering of \(h_1,\ldots ,h_n\). We must show that the definition (16) gives a result between 0 and 1, and for this we need to extend Theorem 6:
Theorem 9
If p is any disjunction of histories and h is a one-time history, then
$$\begin{aligned} \tau (p) = \tau (p\wedge h) + \tau (p\wedge \lnot h). \end{aligned}$$
Proof
This is a straightforward induction on the number of disjuncts in p. \(\square \)
Theorem 10
If \(\tau (h_1\vee \cdots \vee h_n)\) is given by (16), then
$$\begin{aligned} 0 \le \tau (h_1\vee \cdots \vee h_n) \le 1. \end{aligned}$$
Proof
This follows the same lines as the proof of Theorem 8. If \(h_n\) is a one-time history, Theorem 9 gives
$$\begin{aligned} \tau (h_1\vee \cdots \vee h_{n-1}) \ge \tau \big ((h_1\vee \cdots \vee h_{n-1})\wedge h_n\big ) \end{aligned}$$
and arguing by induction on the length of \(h_n\), as in Theorem 7, we can extend this to general \(h_n\). Hence, by (18),
$$\begin{aligned} \tau (h_1\vee \cdots \vee h_n) \ge \tau (h_n) \ge 0. \end{aligned}$$
For the right-hand inequality, first suppose that \(h_1,\ldots ,h_n\) are one-time histories, with \(\tau (h_n) = \langle E_0|\widetilde{\Pi }_n|E_0\rangle \); then in the definition (16) we have, from Lemma 2,
$$\begin{aligned} H_r^{(n)} = \sum \langle E_0|\widetilde{\Pi }_{i_1}\ldots \widetilde{\Pi }_{i_n}|E_0\rangle \end{aligned}$$
and therefore
$$\begin{aligned} \tau (h_1\vee \cdots \vee h_n)&= 1 - \langle E_0|(1 - \widetilde{\Pi }_n)(1 - \widetilde{\Pi }_{n-1})\cdots (1-\widetilde{\Pi }_1)|E_0\rangle \\&= 1 - \tau (\lnot h_1\wedge \cdots \wedge \lnot h_n). \end{aligned}$$
Since \(\lnot h_1\wedge \cdots \wedge \lnot h_n\) is a single history, this lies between 0 and 1.
The right-hand inequality can be extended to arbitrary histories \(h_1,\ldots ,h_n\) by a similar argument to the last part of the proof of Theorem 8, using successive inductions on the lengths of \(h_1,\ldots ,h_n\). \(\square \)
We can now show that (15) is true in general. This is the replacement for the truth table for \(\vee \) in truth-functional many-valued logic.
Theorem 11
If p and q are any two future propositions (elements of the lattice \(\mathcal {T}_\text {F}\)),
$$\begin{aligned} \tau (p\vee q) = \tau (p) + \tau (q) - \tau (p\wedge q). \end{aligned}$$
Proof
Suppose \(p = h_1\vee \cdots \vee h_m\) and \(q = k_1\vee \cdots \vee k_n\) where \(h_i\) and \(k_j\) are histories. The definition (16) gives expressions for \(\tau (p)\), \(\tau (q)\), \(\tau (p\vee q)\) and \(\tau (p\wedge q)\) which can be rewritten in a factorised form as follows.
For any set X, we write \(2^X\) for the set of subsets of X, as usual. Let U(X) be the commutative ring generated by the subsets of X with union as the ring multiplication (in other words, U(X) is a \(\mathbb {Z}\)-module with basis \(2^X\), with the additional operation of union defined on the basis and extended by \(\mathbb {Z}\)-linearity). We continue to use the symbol \(\cup \) for this extended operation. Then the empty set is a multiplicative identity in U(X), and we will denote it by 1.
Let \(M = \{1,\ldots ,m\}\) and \(N = \{1,\ldots , n\}\). Let \(M\sqcup N\) be the disjoint union of M and N, realised as
$$\begin{aligned} M\sqcup N = \{(1,0),\ldots ,(m,0),(0,1),\ldots ,(0,n)\} \subset \mathbb {Z}\times \mathbb {Z}\end{aligned}$$
(so \(M\sqcup N\) is the union of the projections of \(M\times N\) onto the axes in \(\mathbb {Z}\times \mathbb {Z}\)). Define real-valued functions \(\tau _p: 2^M \rightarrow [0,1]\), \(\tau _q: 2^N\rightarrow [0,1]\), \(\tau _{p\vee q}:2^{M\sqcup N} \rightarrow [0,1]\) and \(\tau _{p\wedge q}:2^{M\times N} \rightarrow [0,1]\) by
$$\begin{aligned} \tau _p(\{i_1,\ldots ,i_r\})&= \tau (h_{i_1}\vee \cdots \vee h_{i_r}),\\ \tau _q(\{j_1,\ldots ,j_s\})&= \tau (k_{j_1}\vee \cdots \vee k_{j_s}),\\ \tau _{p\vee q}\big (\{(i_1,0)\ldots ,(i_r,0),(0,j_1),\ldots ,(0,j_s)\}\big )&= \tau \big (h_{i_1}\vee \cdots \vee h_{i_r}\vee k_{j_1}\vee \cdots \vee k_{j_s}\big ),\\ \tau _{p\wedge q}\big (\{(i_1,j_1),\ldots ,(i_t,j_t)\}\big )&= \tau \big (h_{i_1}\vee k_{j_1}\vee \cdots \vee h_{i_t}\vee k_{j_t}\big ). \end{aligned}$$
Then the definition (16) can be stated as
$$\begin{aligned} \tau (p) = \tau _p\big (1 - (1 - \{1\})\cup \cdots \cup (1 - \{m\})\big ) \end{aligned}$$
with corresponding expressions for \(\tau (q)\), \(\tau (p\vee q)\) and \(\tau (p\wedge q)\). To write the last two, it is convenient to introduce the abbreviations \(e_n = \{(i,0)\}\), \(f_j = \{(0,j)\}\) and \(g_{ij} = \{(i,j)\}\) for the singleton sets in \(2^{M\sqcup N}\) and \(2^{M\times N}\), which generate the rings \(U(M\sqcup N)\) and \(U(M\times N)\). Then
$$\begin{aligned} \tau (p\vee q)&= \tau _{p\vee q}\big (1 - (1 - e_1)\cup \ldots \cup (1-e_m)\cup (1-f_1)\cup \cdots \cup (1-f_n)\big ),\\ \tau (p\wedge q)&= \tau _{p\vee q}\left( 1 - \bigcup _{ij}(1 - g_{ij})\right) . \end{aligned}$$
Define a map \(f:2^{M\times N} \rightarrow 2^{M\sqcup N}\) by projecting \(M\times N \subset \mathbb {Z}\times \mathbb {Z}\) onto the axes:
$$\begin{aligned} f\big (\{(i_1,j_1),\ldots (i_r,j_r)\}\big )&= \{i_1,\ldots ,i_r\}\sqcup \{j_1,\ldots ,j_r\}\\&=\{(i_1,0),\ldots ,(i_r,0),(0,j_1),\ldots ,(0,j_r)\}. \end{aligned}$$
This satisfies \(f(S\cup T) = f(S)\cup f(T)\) and therefore can be extended to a ring homomorphism \(f:U(M\times N) \rightarrow U(M\sqcup N)\). It is completely specified by its action on the generators:
$$\begin{aligned} f(g_{ij}) = e_i\cup f_j. \end{aligned}$$
(19)
Now let \(S = \{(i_1,j_1),\ldots ,(i_r,j_r)\}\) be any subset of \(M\times N\). Then
$$\begin{aligned} \tau _{p\wedge q}(S)&= \tau \left( h_{i_1}\vee k_{j_1}\vee \cdots \vee h_{i_r}\vee k_{j_r}\right) \\&= \tau \left( h_{i_1}\vee \cdots \vee h_{i_r}\vee k_{j_1}\vee \cdots \vee k_{j_r}\right) \\&= \tau _{p\vee q}\left( e_{i_1}\cup \cdots \cup e_{i_r}\cup f_{i_1}\cup \cdots \cup f_{j_r}\right) \\&= \tau _{p\vee q}(f(S)), \end{aligned}$$
so \(\tau _{p\wedge q} = \tau _{p\vee q}\circ f\). Hence
$$\begin{aligned} \tau (p\wedge q)&= \tau _{p\wedge q}\left( 1 - \bigcup _{ij}(1 - g_{ij})\right) = \tau _{p\vee q}\circ f\left( 1 - \bigcup _{ij}(1 - g_{ij})\right) \\&= \tau _{p\vee q}\left( 1 - \bigcup _{ij}\left( 1 - e_i\cup f_j\right) \right) . \end{aligned}$$
We now note that for any \(e, f_1,\ldots ,f_n\in U(M\sqcup N)\) satisfying \(e\cup e = e\), in particular if e is one of the generators of \(U(M\sqcup N)\),
$$\begin{aligned} \bigcup _{j=1}^n (1 - e\cup f_j) = 1 - e + e\cup \bigcup _{j=1}^n(1 - f_j). \end{aligned}$$
which can be proved by a simple induction on n. Hence
$$\begin{aligned} \bigcup _{ij}(1 - e_i\cup f_j) = \bigcup _{i=1}^m(1 - e_i + e_i\cup F) \end{aligned}$$
where
$$\begin{aligned} F = \bigcup _{j=1}^n(1 - f_j). \end{aligned}$$
Note that \((1 - f_n)\cup (1 - f_n) = 1 - f_n\), so \(F\cup F = F\). Induction on m can now be used to show that
$$\begin{aligned} \bigcup _{i=1}^m(1 - e_n + e_n\cup F) = E + F - E\cup F \end{aligned}$$
where
$$\begin{aligned} E = \bigcup _{i=1}^m (1- e_i). \end{aligned}$$
We now have
$$\begin{aligned} \tau (p\wedge q)&= \tau _{p\vee q} (1 - E - F + E\cup F)\nonumber \\&= \tau _{p\vee q}(1 - E) + \tau _{p\vee q}(1 - F) - \tau _{p\vee q}(1 - E\cup F). \end{aligned}$$
(20)
But
$$\begin{aligned} \tau _{p\vee q}(1 - E)&= \tau _{p\vee q}\left( 1 - \bigcup _{i=1}^m(1 - \{i,0\})\right) \\&= \tau _p\left( 1 - \bigcup _{i=1}^m(1 - \{i\})\right) \\&= \tau (p), \end{aligned}$$
and similarly
$$\begin{aligned} \tau _{p\vee q}(1 - F) = \tau (q), \qquad \tau _{p\vee q}(1 - E\cup F) = \tau (p\vee q). \end{aligned}$$
Eq. (20) therefore gives the stated result. \(\square \)
Negation
For a one-time history \(h = F_t(\Pi )\), negation is defined by the usual orthocomplement in the lattice of closed subspaces of Hilbert space:
$$\begin{aligned} \lnot h = F_t(1 - \Pi ) \end{aligned}$$
with truth value
$$\begin{aligned} \tau (\lnot h) = 1 - \tau (h). \end{aligned}$$
(21)
For a general history \(h = h_1\wedge \cdots \wedge h_r\) where \(h_n\) are one-time histories, we take negation to be given by
$$\begin{aligned} \lnot h = \lnot h_1\vee \cdots \vee \lnot h_r, \end{aligned}$$
and for a general proposition \(p = k_1\vee \cdots \vee k_r\), where \(k_n\) are multi-time histories,
$$\begin{aligned} \lnot p = \lnot k_1\wedge \cdots \wedge \lnot k_r. \end{aligned}$$
The truth values of such general h and p are aleady covered by the definition in the previous section. Our aim in this subsection is to prove that (21) still holds.
Lemma 3
If \(h_1\) and \(h_2\) are any two histories,
$$\begin{aligned} \tau (h_1) = \tau (h_1\wedge h_2) + \tau (h_1\wedge \lnot h_2) \end{aligned}$$
Proof
By induction on the length of \(h_2\). By Theorem 6, the theorem holds for all \(h_1\) and all one-time histories \(h_2\). Suppose it holds for all \(h_2\) of length n, and let \(h'_2\) be a history of length \(n+1\), so \(h'_2 = h_2 \wedge h_3\) where \(h_2\) has length n and \(h_3\) has length 1. Then
$$\begin{aligned} \tau (h_1 \wedge h'_2) + \tau (h_1 \wedge \lnot h'_2)&= \tau (h_1 \wedge h_2 \wedge h_3) + \tau \big (h_1 \wedge (\lnot h_2 \vee \lnot h_3)\big )\\&= \tau (h_1 \wedge h_2 \wedge h_3) +\tau \big ((h_1\wedge \lnot h_2)\vee (h_1\wedge \lnot h_3)\big )\\&= \tau (h_1 \wedge h_2 \wedge h_3) +\tau (h_1\wedge \lnot h_2) + \tau (h_1\wedge \lnot h_3)\\&\quad \quad \quad -\tau (h_1\wedge \lnot h_2\wedge \lnot h_3)\\&= \tau (h_1 \wedge h_2 \wedge h_3) + \tau (h_1\wedge \lnot h_2\wedge h_3) + \tau (h_1\wedge \lnot h_3)\\&\quad \quad \quad \text {by Theorem 6}\\&= \tau (h_1\wedge h_3) + \tau (h_1\wedge \lnot h_3)\\&\quad \quad \quad \text {by the inductive hypothesis}\\&= \tau (h_1) \quad \text {by Theorem 6 again}. \end{aligned}$$
\(\square \)
Lemma 4
If h is any history,
$$\begin{aligned} \tau (\lnot h) = 1 - \tau (h) \end{aligned}$$
Proof
By induction on the length of h. Suppose the result holds for all histories of length n, and let \(h'\) be a history of length \(n+1\), so that \(h' = h_0\wedge h\) where \(h_0\) is a one-time history and h has length n. Then
$$\begin{aligned} \tau (\lnot h')&= \tau (\lnot h_0\vee \lnot h)\\&= \tau (\lnot h_0) + \tau (\lnot h) - \tau (\lnot h_0 \wedge \lnot h). \end{aligned}$$
But \(\tau (\lnot h) = 1 - \tau (h)\) by the inductive hypothesis, and
$$\begin{aligned} \tau (\lnot h_0) - \tau (\lnot h_0\wedge \lnot h) = \tau (\lnot h_0\wedge h) \text { by Theorem 3,} \end{aligned}$$
so
$$\begin{aligned} \tau (\lnot h')&= 1 - \tau (h) + \tau (\lnot h_0\wedge h)\\&= 1 - \tau (h_0\wedge h) \quad \text { by Theorem 6}\\&= 1 - \tau (h'). \end{aligned}$$
The result holds for \(n = 1\) by (21), and therefore holds for all n. \(\square \)
Theorem 12
If p is any disjunction of histories,
$$\begin{aligned} \tau (\lnot p) = 1- \tau (p) \end{aligned}$$
Proof
By induction on the number of disjoined histories in p. Suppose the theorem holds for all \(p = h_1\vee \cdots \vee h_n\), and let \(p' = p\vee k\) where k is another history. Then
$$\begin{aligned} \tau (\lnot p')&= \tau (\lnot p \wedge \lnot k)\\&= \tau (\lnot p) + \tau (\lnot k) - \tau (\lnot p\vee \lnot k)\\&= \tau (\lnot p) + \tau (\lnot k) - \tau \big (\lnot (p\wedge k)\big )\\&= 1 - \tau (p) + 1 - \tau (k) - [1 - \tau (p\wedge k)] \end{aligned}$$
by the inductive hypothesis, since \(p\wedge k\) is also a conjunction of n histories \(h_n\wedge k\). Hence
$$\begin{aligned} \tau (\lnot p') = 1 - \tau (p\vee k) = 1 - \tau (p') \end{aligned}$$
and the theorem is established for all p by induction. \(\square \)
A similar double induction can be used to prove the general version of Theorem 6:
Theorem 13
If p and q are any two elements of the lattice \(\mathcal {T}_\text {F}\),
$$\begin{aligned} \tau (p) = \tau (p\wedge q) + \tau (p\wedge \lnot q). \end{aligned}$$
We can now state the general logical properties of conjunction and disjunction in this temporal logic:
Theorem 14
Let p and q be any two temporal propositions.
Proof
(i) If \(\tau (p\wedge q) = 1\), then, by Theorem 13,
$$\begin{aligned} \tau (p) = \tau (p\wedge q) + \tau (p\wedge \lnot q) \ge 1 \end{aligned}$$
and therefore \(\tau (p) = 1\). Conversely, if \(\tau (p) = \tau (q) = 1\), then
$$\begin{aligned} \tau (p\wedge q) = \tau (p) + \tau (q) - \tau (p\vee q) \ge 1 \end{aligned}$$
and therefore \(\tau (p\wedge q) = 1\).
(ii) If \(\tau (p) = 0\), then
$$\begin{aligned} \tau (p\wedge q)&= \tau (p) + \tau (q) - \tau (p\vee q)\\&= \tau \big (\lnot (p\vee q)\big ) - \tau (\lnot q)\\&= \tau (\lnot p \wedge \lnot q) - \tau (\lnot q)\\&= -\tau (\lnot q\wedge p) \qquad \text {by Theorem 13}\\&\le 0, \end{aligned}$$
so \(\tau (p\wedge q) = 0\).
(iii) and (iv) follow from these by writing \(\tau (p\vee q) = 1 - \tau (\lnot p\wedge \lnot q)\). \(\square \)
A possibly disturbing feature of this list of properties is the one-sidedness of the implications (ii) and (iv): \(p\wedge q\) might be false without either p or q being false (though (i) shows that if \(p\wedge q\) is not definitely true, then p and q cannot both be definitely true); and the truth of \(p\vee q\) does not imply the truth of either p or q (though (iii) shows that this implication does hold if “truth” (\(\tau = 1\)) is replaced by “possible truth” (\(\tau \ne 0\))). The first of these has already been discussed, but the second, concerning truth rather than falsity, might be felt to cast doubt on whether \(\vee \) can legitimately be regarded as a generalised form of “or” (a similar objection has been made to quantum logic Sudbery 1986). 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 (1962, p. 244), Aristotle’s assertion “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 sea-battle going on” or “there is no sea-battle going on” ’ will be true tomorrow”; in symbols,
$$\begin{aligned} \text {By } F_t(p) \vee F_t(\lnot p) \text { Aristotle meant } F_t(p\vee \lnot p). \end{aligned}$$
But in the logic proposed here \(F_t\) is taken to be a homorphism, so that
$$\begin{aligned} F_t(p) \vee F_t(q) = F_t(p\vee q), \end{aligned}$$
showing that the meaning of the connective \(\vee \) in this system is in accordance with Prior’s reading of Aristotle. 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 \(\vee \) occurring in this temporal logic is in fact the “or” of common usage in talk about the open future.