## 1. The Discussion of Future Contingency and Its Background in Ancient and Medieval Logic



Future contingents appear to hold a strange quality when compared with
present or past tense statements, such as "it is raining"
or "Napoleon lost at Waterloo", whose truth-value does not
depend on future states or events. For it seems straightforward to
claim that the latter two sentences are true if and only if the states
or events in question correspond with reality. But can it make sense
to claim that the truth or falsity of a contingent future statement,
such as "The first human being to set foot on Mars will be a
woman", depends on the future reality in a similar manner?
Clearly, this can only make sense if we can meaningfully refer to the
future reality in the same way as we can refer to the past reality.
However, if the future is open such a reference may be rather
problematic.



The philosophical and logical challenge to which the future
contingency discussion gives rise is two-fold. First of all, anyone
who wants to maintain some kind of indeterminism regarding the future,
may be confronted with some standard arguments in favor of logical
determinism, i.e., arguments designed to demonstrate that there are no
future contingents at all. In addition, anyone who holds that there
are future contingents can be challenged to establish a reasonable
truth-theory compatible with the idea of an open future. Such a theory
should provide answers to questions like: Can one meaningfully regard
future contingents as true or false now, if the future is open? And if
so, how? Can assertions about the contingent future make any sense at
all? And if so, how? Some logicians have held that no future
contingent is true. However, other logicians have found that this is
unacceptable. Instead, they have looked for a theoretical basis on
which we might hold that a future contingent is true (or false).



Already Aristotle (384-322 B.C.E.) was aware of the problem of
future contingents. Chapter IX of his work, *On
Interpretation*, is without doubt the philosophical text which has
had the greatest impact on the debate about the relations between
time, truth, and possibility. The discussion in this text certainly
bears witness to the fact that Ancient philosophy was highly conscious
of tense-logical problems. Central to the discussion in this famous
Aristotelian text is the question of how to interpret the following
two statements:



"Tomorrow there will be a sea-battle"
  

"Tomorrow there will not be a sea-battle" 



Aristotle considered questions like: Should we say that one of these
statements is true today and the other false? How can we make a clear
distinction between what is going to happen tomorrow and what must
happen tomorrow? (See *On Interpretation*, 18 b 23 ff.).



The interpretative problems regarding Aristotle's logical
problem about the sea-battle tomorrow are by no means simple. Over the
centuries, many philosophers and logicians have formulated their
interpretations of the Aristotelian text (see Ohrstrom
and Hasle 1995, p. 10 ff.). In the following we shall present an
interpretation of the text from the scholastic period and a modern
interpretation based on a three-valued semantics.



In the generation after Aristotle, Diodorus Cronus (ca. 340-280
B.C.E.) analysed similar problems using his so-called Master Argument.
This argument was a trilemma. According to Epictetus, Diodorus argued
that the following three propositions cannot all be true:



(D1)
Every proposition true about the past is necessary.
(D2)
An impossible proposition cannot follow from a possible
 one.[2]
(D3)
There is a proposition which is possible, but which neither is nor
will be true.


Diodorus used this incompatibility combined with the plausibility of
(D1) and (D2) to argue that (D3) is false. Assuming (D1) and (D2) he
went on to define the possible as "that which either is or will
be true" and the necessary as "that which, being true,
will not be false". In this way his argument seems to have been
designed to demonstrate that there cannot be any future contingency at
all. However, little is known about the way in which Diodorus used his
premises in order to reach the conclusion. The reconstruction of the
Master Argument certainly constitutes a genuine problem within the
history of logic. Various philosophers and logicians have tried to
reconstruct the argument as it might have been. It is very likely that
the main structure of the argument was close to the argument presented
in the next section. (See (Ohrstrom and Hasle 1995, p.
15 ff.) and (Gaskin 1995) for references to the literature on the
Master Argument.)



The discussion took on a particularly interesting form in the Middle
Ages. During the Middle Ages logicians related their discipline to
theology. One of the most important theological questions was the
problem of the contingent future in relation to Christian doctrine.
According to Christian tradition, divine foreknowledge comprises
knowledge of the future choices to be made by men and women. But this
assumption apparently gives rise to a straightforward argument from
divine foreknowledge to the necessity of the future: if God already
now knows which decision I will make tomorrow, then a
now-unpreventable truth about my choice tomorrow is already given. My
choice, then, appears to be necessary, not free. Hence, there seems to
be no basis for the claim that I have a free choice between genuine
alternatives. This conclusion, however, violates the idea of human
freedom and moral accountability presupposed in much theology (though
not all).



The medieval discussion regarding the logic of divine foreknowledge
is, from a formal point of view, very close to the classical
discussion concerning future contingency. If we add the assumption
that necessarily, something is true if and only if it is known to God,
then it is easy to see how the discussion regarding the logic of
divine foreknowledge is, from a formal point of view, essentially the
same discussion as the classical discussion concerning future
contingency. This was clearly realised by the medieval logicians.



In his treatise *De eventu futurorum*, Lavenham (c. 1380) gave
a succinct overview over the basic approaches to the problem within
scholasticism (see Ohrstrom 1983, Tuggy 1999). Lavenham
considered a central argument leading from God's foreknowledge
to the necessity of the future and the lack of proper human freedom.
In fact, the various positions on future contingency may be presented
as possible reactions to this argument. The main structure of this
argument is very close to what is believed to have been the Master
Argument of Diodorus Cronus (see Gaskin 1995). It is clear from
Lavenham's text that he had some knowledge of this old Stoic or
Megaric argument, probably through his reading of Cicero's
*De Fato*. The main idea is to transfer the assumed necessity
of the past to the future. In order to make things clearer one might
state the argument in terms of yesterday and tomorrow, instead of past
and future in general (as Lavenham tends to do). A non-theological
version of the argument can be presented in the following way. In this
sequence E is some event, which may or may not take place tomorrow
(e.g. a sea-battle). Non-E is just a state of affairs without E
occurring. E and non-E are supposed to be mutually exclusive.


1. Either E is going to take place tomorrow or non-E is going to take
place tomorrow. (Assumption).
2. If a proposition about the past is true, then it is now necessary,
i.e., inescapable or unpreventable. (Assumption).
3. If E is going to take place tomorrow, then it is true that
yesterday it was the case that E would take place in two days.
(Assumption).
4. If E is going to take place tomorrow, then it is now necessary
that yesterday E would take place in two days. (Follows from 2. and
3.).
5. If it is now necessary that yesterday E would take place in two
days, then it is now necessary that E is going to take place tomorrow.
(Assumption).
6. If E is going to take place tomorrow, then E is necessarily going
to take place tomorrow. (Follows from 4. and 5.).
7. If non-E is going to take place tomorrow, then non-E is
necessarily going to take place tomorrow. (Follows by the same kind of
reasoning as 6.).
8. Either E is necessarily going to take place tomorrow or non-E is
necessarily going to take place tomorrow. (Follows from 1., 6. and
7.).
9. Therefore, what is going to happen tomorrow is going to happen
with necessity. (Follows from 8.).



Lavenham accepted the validity of this argument, and he pointed out
that one should consider four possible reactions to it. He presented
this classification in a theological context, but it can be translated
into non-theological language. Assuming that necessarily, something is
true if and only if it is known to God, the four possible reactions in
Lavenham's analysis can be listed in the following way:


1. Accept the above argument (including its premises). Grant that
there are no future contingents, i.e., statements about the future are
either impossible or necessary.
2. Deny that if a certain event is going to occur, then it is true
that it has always been the case that it would occur.
3. Deny the following: For any possible event, which might occur at a
certain time in the future, either it will be that the event takes
place at that future time, or it will be that the event does not take
place at that time.
4. Deny that the past in general is necessary.



Clearly, if we don't want to accept the deterministic conclusion
of the above argument, and if the argument is accepted as valid, then
we have to question at least one of the premises. Not taking premise 5
into consideration, this leaves us with the premises 1, 2, and 3.
According to the reaction (b), premise 3 is rejected. Reaction (c)
implies the rejection of premise 1 in the argument. Reaction (d)
implies the rejection of premise 2.



Lavenham took option (a) to imply that there is no human freedom. In
his understanding (b) would mean that God does not know future
contingents. He rejected both (a) and (b) as contrary to the Christian
faith.



It seems that Lavenham, like William of Ockham (c. 1287-1347),
took Aristotle to hold that propositions about the contingent future
are neither true nor false. A number of scholastic logicians favored
this Aristotelian view (c), for instance Peter Aureole
(c.1280-1322). Lavenham, however, rejected this view. He
insisted that future contingents are either true or false now, and
that God knows the truth-values of all future contingents. He
preferred (d), and he argued that by rejecting the necessity of the
past as a general principle the doctrines of free will and God's
foreknowledge of the contingent future can be united in a consistent
manner. This solution was first formulated by Ockham, although some of
its elements can already be found in Anselm of Canterbury
(1033-1109). It is also interesting that Leibniz
(1646-1711) much later worked with a similar idea as a part of
his metaphysics. (See Ohrstrom 1984.)



The most characteristic feature of Lavenham's solution is the
concept of the true future. The view is that God possesses certain
knowledge not only of the necessary future, but also of the contingent
future. This means that among the possible contingent futures there
must be one which has a special status, namely that it corresponds to
the course of events which is going to happen or take place in the
future. This line of thinking may be called the medieval solution,
even though other approaches certainly existed. Its justification is
partly the observation that the notion of the true future is the
specifically medieval contribution to the discussion, and partly that
leading medieval logicians regarded this solution as the best one.
Lavenham himself called it "opinio modernorum", i.e., the
opinion of the modern people. Lavenham argued that the notion of the
true future can be maintained together with indeterminism, if the
assumption of the necessity of the past is rejected. This will be
explained in more detail in sections 2 and 5.



A later contribution by the Jesuit Luis Molina (1535-1600) is
relevant for a modern interpretation of the concept of the true
future. Molina's ideas have been thoroughly discussed in (Craig
1988). Molina's special contribution is the idea of
(God's) middle knowledge, "by which, in virtue of the most
profound and inscrutable comprehension of each free will, He saw in
His own essence what each such will would do with its innate freedom
were it to be placed in this or that or indeed in infinitely many
orders of things -- even though it would really be able, if it so
willed, to do the opposite" (quoted from Craig 1988, p. 175).
Craig goes on to explain it as follows: "... whereas by His
natural knowledge God knows that, say, Peter when placed in a certain
set of circumstances *could* either betray Christ or not betray
Christ, being free to do either under identical circumstances, by His
middle knowledge God knows what Peter *would* do if placed
under those circumstances" (Craig 1988, p. 175).



As Lavenham knew, William of Ockham had discussed the problem of
divine foreknowledge and human freedom in his work *Tractatus de
praedestinatione et de futuris contingentibus*. (See William of
Ockham 1983.) Ockham asserted that God knows the truth or falsity of
all future contingents, but he also maintained that human beings can
choose between alternative possibilities. In his *Tractatus* he
argued that the doctrines of divine foreknowledge and human freedom
are compatible. Richard of Lavenham made a remarkable effort to
capture and clearly present the logical features of Ockham's
system as opposed to (what was assumed to be) Aristotle's
solution, i.e., (c).



In the following section a formal version of the medieval argument for
determinism will be presented without theological references. It will
be demonstrated that at least two of the premises used in the argument
may be questioned. In section 3 we shall present a particularly
important framework for the discussion of future contingents known as
branching time and its semantics. In sections 4 and 5 we shall see how
these possible reactions to the classical argument may be turned into
modern truth-theories corresponding to the medieval positions listed
above.


## 2. A Formalisation of the Classical Argument



The logic of the expressions "tomorrow" and
"yesterday" used in section 1 was in fact formalized by
A.N. Prior (1914-69) who suggested that the two terms should be
treated as propositional operators, \(T\) and \(Y\) (1967, p. 67 ff.).
This means that \(T(p)\) and \(Y(p)\) should stand for the
propositions "tomorrow it will be the case that \(p\)" and
"yesterday it was the case that \(p\)", respectively.
Prior even suggested some basic axioms for the two operators 


1. \(p \supset Y(T(p))\)
2. \(p \supset T(Y(p))\)



By iterative uses of the \(Y\)- and \(T\)-operators we may define
operators corresponding to the past and the future as \(Y^n\) and
\(T^n\), respectively, where \(n\) is an arbitrary natural number.
Obviously, this may be done with an arbitrary choice of time unit,
i.e., \(T(p)\) and \(Y(p)\) could simply stand for "at the next
moment it will be the case that \(p\)" and "at the moment
just before now it was the case that \(p\)", respectively. A
logic of this kind appears to be based on time conceived as a linear
series of discrete moments, but Prior pointed out that we may find
this sort of discrete tense-logic attractive without committing
ourselves to a strong metaphysical stance regarding time as such: 



> 
> The usefulness of systems of this sort does not depend on any serious
> metaphysical assumption that time *is* discrete; they are
> applicable in limited fields of discourse in which we are concerned
> only with what happens next in a sequence of discrete states, e.g. in
> the workings of a digital computer. (Prior 1967, p. 67)
> 



It should also be mentioned that it is generally assumed that all
philosophical and logical aspects of the topic of future contingency
can be satisfactory discussed in terms of discrete tense-logic. This
means that regarding the topic of future contingency, we may choose to
limit the discussion to the discrete case. On the other hand, in some
discussions there is no reason to make a limitation of this kind.
Prior based the metric version of his general tense-logical system on
the operators: 

 

\[\begin{align}
F(x) \quad & \ldquo\text{in } x \text{ time units it will be the case that } \ldots\rdquo \\
P(x) \quad & \ldquo x \text{ time units ago it was the case that } \ldots\rdquo \\
\Box \quad & \ldquo\text{it is necessary that } \ldots\rdquo
\end{align}\]


Here the \(x\) in the tense operators, \(F(x)\) and \(P(x)\), does not
have to be a natural number, but can take any positive value, allowing
for the case that time is non-discrete. Prior showed how the argument
from section 1 can be formalized in terms of this general
tense-logic.



It should, however, be noted that Prior also often used
tense-operators without any reference to time units. He used \(F\) for
"it will be the case that ..." and \(P\) for
"it has been the case that ...". In terms of these
non-metric tense-operators he defined the operators, \(G\) and \(H\),
as \({\sim}F{\sim}\) and \({\sim}P{\sim}\) respectively. \(G\) may be
read "it will always be the case that ...", and \(H\)
may be read "it has always been the case that ...".
Using these non-metric tense-operators Prior (1967, p. 32 ff.) even
formulated a reconstruction of the Diodorean Master Argument which
comes rather close to the classical argument which we shall present in
the following.



It is essential to notice that the necessity at stake in the classical
argument is a historical necessity. This means that what is not
necessary at one moment may become necessary at another moment.
Instead of speaking about what is necessary we might - as
already hinted at - talk about what is now settled, inevitable,
inescapable, or unpreventable.



The argument may be understood as based on the following five
principles, where \(p\) and \(q\) represent arbitrary well-formed
statements within the logic:



(P1)
\(F(y)p \supset P(x)F(x)F(y)p\)
(P2)
\(\Box(P(x)F(x)p \supset p)\)
(P3)
\(P(x)p \supset \Box P(x)p\)
(P4)
\((\Box(p \supset q) \wedge \Box p) \supset \Box q\)
(P5)
\(F(x)p \vee F(x){\sim}p\)


(P1) and (P2) are basic tense-logical claims which can serve as
crucial elements in a formalization of the argument mentioned in
section 1. (P3) may be labeled the 'necessity of the
past'. (P4) is a theorem well-known from standard modal logic.
(P5) may be read as a version of the principle of the excluded middle
("tertium non datur"), although it does not take the exact
form of \(p \vee{\sim}p\). In order to avoid confusion, we shall use
the modified name, 'future excluded middle', for (P5).



Let \(q\) stand for some atomic statement such that \(F(y)q\) is a
statement about the contingent future. Formally, then, the argument
goes as follows:




|  |  |  |
| --- | --- | --- |
| (1) | \(F(y)q \supset P(x)F(x)F(y)q\) | (P1) |
| (2) | \(P(x)F(x)F(y)q \supset \Box P(x)F(x)F(y)q\) | (from (P3)) |
| (3) | \(F(y)q \supset \Box P(x)F(x)F(y)q\) | (from (1) & (2)) |
| (4) | \(\Box(P(x)F(x)F(y)q \supset F(y)q)\) | (from (P2)) |
| (5) | \(F(y)q \supset \Box F(y)q\) | (from (3), (4), (P4)) |



Similarly, it is possible to prove




|  |  |  |
| --- | --- | --- |
| (6) | \(F(y){\sim}q \supset \Box F(y){\sim}q\) |  |



The second part of the main proof is carried out in the following
way:




|  |  |  |
| --- | --- | --- |
| (7) | \(F(y)q \vee F(y){\sim}q\) | (from (P5)) |
| (8) | \(\Box F(y)q \vee \Box F(y){\sim}q\) | (from (5), (6), (7)) |



Remember now that \(q\) may stand for any atomic proposition,
including statements about human actions. Therefore, (8) is equivalent
to a claim of determinism, i.e., that there are no future contingents.
So if one wants to preserve indeterminism, at least one of the above
principles (P1-5) has to be rejected.



A.N. Prior constructed two systems showing how that can be done,
namely the Peircean system (in which (P1) and (P5) are rejected) and
the Ockhamist system (in which (P3) is rejected). Formally, each of
these systems offers a basis for the rejection of the claim of
determinism as expressed in (8). As we shall see in the next section
the same can be said about Jan Lukasiewicz's three-valued
semantics, which Prior himself investigated further (see Prior 1953),
and which also involves a rejection of (P5). Since Prior, several
philosophers have discussed which one of these systems should be
accepted, or whether other and more attractive systems dealing with
the problem can be constructed. In sections 4 and 5 we shall see how
the various solutions to the problem of future contingents can be
grouped according to their consequences with respect to (P3) and
(P5).


## 3. Branching Time Semantics



Since Prior's time, it has become a standard to study
tense-logical systems in terms of semantical models based on the idea
of branching time. This idea was not realised in Prior's early
works on temporal logic. Indeed it had not yet been formulated in his
*Time and Modality* (1957), which otherwise marked the major
breakthrough of the new logic of time. As an explicit (or formalised)
idea, branching time was first suggested to Prior in a letter from
Saul Kripke in September 1958. This letter contains an initial version
of the idea and a system of branching time, although it was not worked
out in details. Kripke, who was then only 17 years old, suggested that
we may consider the present as a point of Rank 1, and possible events
or states at the next moment as points of Rank 2; for every such
possible state in turn, there would be various possible future states
at the next moment from Rank 3, the set of which could be labelled
Rank 4, and so forth. Kripke wrote:



> 
> 
> Now in an indetermined system, we perhaps should not regard time as a
> linear series, as you have done. Given the present moment, there are
> several possibilities for what the next moment may be like - and
> for each possible next moment, there are several possibilities for the
> next moment after that. Thus the situation takes the form, not of a
> linear sequence, but of a "tree"... [Letter from Saul
> Kripke to A.N. Prior, dated September 3, 1958, kept in the Prior
> Collection at Bodleian Library, Oxford, Box 4. See also Ploug et al.
> 2012.]
> 
> 
> 



In this way it is possible to form a tree structure representing the
entire set of possible futures expanding from the present (Rank 1)
- indeed a set of possible futures can be said to be identified
for any state, or node in the tree. In this structure every point
determines a subtree consisting of its own present and possible
futures. Kripke illustrated this idea in the following way:



![Figure 1](fc_figure_1.png)


In the letter Kripke wrote:



> 
> The point 0 (or origin) is the present, and the points 1, 2, and 3 (of
> rank 2) are the possibilities for the next moment. If the point 1
> actually does come to pass, 4, 5, and 6 are its possible successors,
> and so on. The whole tree then represents the entire set of
> possibilities for present and future; and every point determines a
> *subtree* consisting of its own present and future.
> 



In Prior's opinion the notion of branching time is certainly not
unproblematic. After all it is a representation of time in terms of
space. The notion seems to involve the idea that the "Now"
is moving through the system. Several authors have argued that the
picture of a moving point within the branching time system is rather
problematic. In fact, this problem of the "Now" as a
moving point goes back to Jack Smart (1949). Later it has been debated
by Storrs McCall (1976, p. 348, 1995) and Graham Nerlich (1995).
Recently MacFarlane has pointed out that there is nothing such a
motion could represent, since we have already represented time as one
of the spatial dimensions of the tree (MacFarlane 2008, p. 86).



It seems that Prior right from the beginning was aware of the basic
conceptual problems involved in the notion of branching time. However,
he certainly found this notion useful as long as it is applied
carefully. During the 1960s he substantially developed the idea. He
worked out the formal details of several different systems, which
constitute different and even competing interpretations of the idea of
branching time, as we shall see below.



A tense-logical model (TIME,\(\le\),C,\(\TRUE\)) based on a branching
time system is a structure, where (TIME,\(\le)\) is a partially
ordered set of moments of time, and C is a set of so-called histories
or chronicles i.e., maximally ordered linear subsets in
(TIME,\(\le)\). It is standard procedure how to define "="
and "\(\lt\)" in terms of "\(\le\)". The
before/after relation, \(\lt\), is supposed to be irreflexive,
asymmetric, transitive and backwards linear. Backwards linearity means
"no backwards branching" i.e.

 

\[
(t\_1 \lt t\_0 \wedge t\_2 \lt t\_0) \supset (t\_1 \lt t\_2 \vee t\_2 \lt t\_1 \vee t\_2 = t\_1)
\]


for all moments of time \(t\_0, t\_1\), and \(t\_2\). This means that the
past relative to \(t\_0\), PAST\((t\_0)\), i.e., the set \(\{t \mid t
\lt t\_0\}\), is linear. In consequence, PAST\((t\_0)\) will be a subset
of any chronicle through \(t\_0\). Much more on the mathematics of
partially ordered sets and their use in temporal logic can be found in
(van Benthem 1991, in particular in Part I). 



In addition, historical connectedness may be considered as an axiom,
i.e., it may be assumed that \(c\_1 \cap c\_2 \ne \varnothing,\) for any
two chronicles \(c\_1\) and \(c\_2\) in the branching time system.



In many branching time models C will just be the set of all maximally
ordered linear subsets in (TIME,\(\le)\). In such cases C will not be
an independent parameter of the model. In other cases, however, there
will be some additional restrictions on C, i.e., it will be a proper
subset of the set of all maximally ordered linear subsets in
(TIME,\(\le)\). In some branching time models there will also be
introduced a relation of (counterfactually) co-temporaneous moments.
Given that such a relation is an equivalence relation, it may give
rise to the definition of instants as equivalence classes of
co-temporaneous moments.



For any propositional constant, \(p\), and any moment in TIME, \(t\),
there is a truth-value, \(\TRUE(p,t)\). This means that the
truth-value of a propositional constant does not vary with the
chronicles. The truth-value of a propositional constant depends only
on the moment. On this basis the truth-value of any well formed
formula (wff) has to be defined recursively. In the following sections
we shall see that this can be done in several different ways.



It may, however, be objected that it is problematic to operate with
two different kinds of propositions: 1) propositional constants with
truth-values that do not vary with the chronicles and 2) other wffs
with truth-values that may vary with the chronicles. Thomason (1970,
p. 280) has pointed out that this distinction means that substitution
of propositions in the system will have to be restricted, since we
will not be allowed to substitute a propositional constant with an
arbitrary wff. Prior was aware of this, but he argued that it is in
fact possible to handle a system with restrictions on the
substitution-rules. (See Prior 1967, p. 122 ff.)


## 4. The Solutions Based on Rejection of the Principle of Future Excluded Middle



A truth-theory may involve the rejection of the principle of future
excluded middle, (P5), for at least two different reasons:


1. The theory may imply that future contingents are neither true nor
false, but undetermined (often conceived as a third truth-value).
2. The theory may be based on the idea that all future contingents
are false.



A possible third position would be to maintain that all future
contingents are true. Strictly speaking, such a claim does not
contradict (P5) as mentioned in section 2, although it does in fact
contradict a version of (P5) formulated in terms of an exclusive
disjunction. However, from a philosophical point of view, such a claim
has had no serious role to play in the debate, even though the
assumption of all future contingents being true is in fact what holds
in the early tense-logical systems, \(\mathrm{K}\_t\) and
\(\mathrm{K}\_b\), introduced in (Prior 1967, p. 187) and in (Rescher
and Urquhart, p. 68 ff.). The problem is that it is highly
counter-intuitive to accept both "Tomorrow there will be a
sea-battle" and "Tomorrow there will not be a
sea-battle" as true now. It seems that if one of these
propositions is true now, then the other has to be false. On these
grounds, we shall not consider this possibility any further in this
context.



In the two following subsections, we shall briefly consider some
solutions corresponding to the possibilities 1 and 2 mentioned
above.


### 4.1 Jan Lukasiewicz's Three-valued Semantics



In a series of articles during the 1920s and 30s the Polish logician
Jan Lukasiewicz (1878-1956) advocated a particular
interpretation of Aristotle's discussion of the status of
sentences about the contingent future, as developed in his sea-battle
example. Lukasiewicz's interpretation crucially rests on a
rejection of the principle of bivalence. In fact, this kind of
interpretation was not new, but had been formulated already by the
Epicureans. However, Lukasiewicz presented this position more
clearly than had ever been done before, and developed it with the aid
of modern symbolic logic. He used his interpretation of Aristotle and
the status of sentences about the contingent future as an argument
against logical determinism and in favor of logical indeterminism, for
which he declared his wholehearted support. In order to avoid
determinism, he found it necessary to restrict the validity of
bivalence by introducing a third truth-value. This truth-value,
"undetermined", is applied to contingent propositions
regarding the future (McCall 1967, p. 64). For instance, a proposition
stating that there will be a sea-battle tomorrow can be assigned the
truth-value undetermined today. This is because today it is not given
or definitely determined whether the sea-battle is actually going to
take place tomorrow or not.



It is an important property of Lukasiewicz's three-valued
logic that the truth-value of the disjunction of two undetermined
propositions is undetermined, i.e., \((p \vee q)\) is undetermined for
\(p\) undetermined and \(q\) undetermined. This may be based on the
observation that since \(p \equiv(p \vee p)\), a disjunction of two
undetermined propositions has to be undetermined. If \(p\) is
undetermined, \({\sim}p\) is also undetermined. It follows that \((p
\vee{\sim}p)\) is undetermined for \(p\) undetermined. This problem
also occurs for future contingents such as \(F(x)q\) and
\(F(x){\sim}q\). According to Lukasiewicz's trivalent logic:
if \(F(x)q\) and \(F(x){\sim}q\) are two future contingents, i.e., if
they are both undetermined, then the same will be the case for the
disjunction of the two statements, \(F(x)q \vee F(x){\sim}q\). This
means that the theory leads to the rejection of the principle of
(P5).



In general, it does not seem possible to solve the problem of future
contingents in terms of three-valued semantics in a satisfactory
manner if the logic is truth-functional, i.e., if the truth-value of
any proposition used in the logic is determined by the truth-values of
its parts. As argued by Prior (1953, p. 326) it will not help to
change the truth-tables to something different from
Lukasiewicz's model. As long as the model or theory is
truth-functional, it is obvious that the two disjunctions \((F(x)q
\vee{\sim}F(x)q)\) and \((F(x)q \vee F(x)q)\) will have the same
truth-value. From an intuitive and common sense point of view, this is
not satisfactory, since \((F(x)q \vee{\sim}F(x)q)\) is clearly true,
whereas \((F(x)q \vee F(x)q)\) is undetermined, given that \(F(x)q\)
is undetermined.



Lukasiewicz's interpretation of the Aristotelian text was
disputed by Prior (1962, p.240 ff.), who pointed out that there is a
significant difference between Lukasiewicz's trivalent logic
and Aristotle's text. Prior pointed out that according to
Aristotle it is true already today, that *either* there will
\(or\) there will not be a sea-battle tomorrow, whereas this
disjunction, as just mentioned, is undetermined according to
Lukasiewicz's trivalent logic.


### 4.2 The Peircean Solution



The solution Prior himself favored is based on so-called Peircean
models. Prior demonstrated that the semantics of these models can be
presented in two different ways. In the following we shall concentrate
on the first of these possibilities, but also comment briefly on the
other possible approach to the Peircean solution.



In order to define the Peircean models according to Prior's
first attempt, it is assumed that there is a valuation function,
*TRUE*, which gives a truth-value (0 or 1) for any
propositional constant at any moment in TIME. On this basis, the
valuation function of a Peircean model, \(\Peirce(t,c,p)\) can be
defined recursively for any wff \(p\), any moment of time \(t\) and
any chronicle \(c\) with \(t \in c\):




|  |  |  |  |
| --- | --- | --- | --- |
| (a) | \(\Peirce(t,c, p)=1\) | iff | \(\TRUE(p,t)=1\), where \(p\) is any propositional
constant. |
| (b) | \(\Peirce(t,c, p\wedge q)=1\) | iff | both \(\Peirce(t,c,p)=1\) and \(\Peirce(t,c,q) =1\) |
| (c) | \(\Peirce(t,c,{\sim}p)=1\) | iff | not \(\Peirce(t,c,p)=1\) |
| (d) | \(\Peirce(t,c,Fp) =1\) | iff |  for all \(c'\) with \(t \in c'\) there is some \(t' \in c'\)
with \(t \lt t'\) such that \(\Peirce(t',c',p) = 1\) |
| (e) | \(\Peirce(t,c,Pp)=1\) | iff | \(\Peirce(t',c,p)=1\) for some \(t' \in c\) with \(t' \lt
t\) |
| (f) | \(\Peirce(t,c,\Diamond p)=1\) | iff | \(\Peirce(t,c',p) =1\) for some \(c'\) with \(t \in c'\) |



Strictly speaking, (a)-(f) do not define the function
\(Peirce\). These conditions only explain when \(Peirce\) has the
value 1. However, here and in all models below we assume that the
valuation function has the range \(\{0,1\}\). The value is 0, if it
does not follow from the recursive definition that it is 1.



In the Peircean system another future operator corresponding to the
notion of "possible future" may also be defined, i.e.,




|  |  |  |  |
| --- | --- | --- | --- |
| (g) | \(\Peirce(t,c,fp) = 1\) | iff | \(\Peirce(t',c',p) = 1\) for some \(c'\) with \(t \in c'\) and
some \(t' \in c'\) with \(t \lt t'\) |



In addition, \(G\) may be defined as \({\sim}f{\sim}\) and \(g\) as
\({\sim}F{\sim}\). In this way the Peircean system comprises four
different future-oriented operators \((f, g, F, G)\).



It should also be mentioned that we can define the necessity operator,
\(\Box\), in the usual manner, i.e., as \({\sim} \Diamond{\sim}\).



\(\Peirce(t,c,q) = 1\) can be read '\(q\) is true at \(t\) in
the chronicle \(c\)'. A formula \(q\) is said to be Peirce-valid
if and only if \(\Peirce(t,c,q) = 1\) for any \(t\) in any \(c\) in
any branching time structure (TIME,\(\le\),C) and any valuation
function *TRUE*.



To obtain a metric version of the Peircean system, a duration function
has to be added. Let \(\dur(t\_1,t\_2,x)\) stand for the statement
'\(t\_1\) is \(x\) time units before \(t\_2\)', where
\(t\_1\) and \(t\_2\) belong to the same chronicle, and where \(x\) is a
positive
 number[3].
 Using this function (d) and (e) above are replaced by:




|  |  |  |  |
| --- | --- | --- | --- |
| \((\text{d}')\) | \(\Peirce(t,c,F(x)p) = 1\) | iff | for all \(c'\) with \(t \in c'\) there is some \(t' \in c'\)
with \(\dur(t,t',x)\) such that \(\Peirce(t',c',p) = 1\) |
| \((\text{e}')\) | \(\Peirce(t,c,P(x)p) = 1\) | iff | \(\Peirce(t',c,p) = 1\) for some \(t' \in c\) with \(\dur(t',t,x)\) |



Given the truth clauses (a)-(e), the modality introduced in (f)
is rather trivial. For instance it follows that

 

\[
F(x)q \supset \Box F(x)q
\]


is a Peirce-valid formula. This means that a statement about the
future is true in the Peircean sense only if it is true in all
possible futures, i.e., only if it *has* to be the case. It
follows that if \(F(x)q\) is a future contingent, it will be false
according to the theory. The same will be the case for
\(F(x){\sim}q\). For this reason 

 

\[
F(x)q \vee F(x){\sim}q
\]


will also be false. So the principle of future excluded middle, (P5),
is not a thesis in the system.



It may be objected that the use of the parameter \(c\) is not really
needed in the definition of the Peirce-function. Clearly, it has no
role to play neither in (a)-(e) nor in (g). The parameter is in
fact used in (f), but this may be said to be rather unimportant, since
as indicated above necessity is in fact incorporated in the Peircean
notion of future. Based on such considerations Prior (1967, p. 132
ff.) showed that the Peircean models may in fact be defined in terms
of a simpler Peirce-function without any reference to the parameter
\(c\) (i.e. the chronicles), if it is assumed that (f) can be left out
of the Peircean system in question. The main advantage of keeping the
extended formalism, is that it facilitates its comparison with the
system to be presented in section 5.1.



According to the Peircean system the future should simply be
identified with the necessary future. More precisely, to say something
about the future is to say something about the necessary future.
Although the identification of the future with the necessary future
makes the position counter-intuitive, A.N. Prior and many of his
followers favored this possibility. The reason is that Prior strongly
believed in free choice and held that this freedom is essential for
the understanding of the very notion of future. According to Prior
nobody (not even God) can know what a person will freely choose,
before the person has made his or her choice. So whatever could make a
statement about a future choice by some free agent true now? From
Prior's point of view, nothing. Hence, such statements must be
false. In his *Some Free Thinking about Time*, Prior maintained
that "if something is the work of a free agent, then it
wasn't going to be the case until that agent decided that it
was" (Copeland 1996, p.48).



Consider the consequences of (d\(')\) and (e\(')\) when applied to the
following model:



![Figure 2](fc_figure_2.png)


In this case the Peircean position implies that \(F(y)p\) is true at
\(t\_2\), whereas \(F(x)F(y)p\) is false at \(t\_1\) and
\(P(x)F(x)F(y)p\) is consequently false at \(t\_2\). This means that
(P1) must be rejected in this system.



In general the formula

 

\[
q \supset P(z)F(z)q
\]


is not Peirce-valid.



It can be concluded that in the Peircean system both (P1) and (P5)
must be rejected.



Many researchers have studied the formalities of the Peircean system.
Axiomatizations of the non-metrical version of the system can be found
in (Burgess 1980) and in (Zanardo 1990).


## 5. The Solutions Based on the Rejection of the Necessity of the Past



As argued for instance in (Gabbay *et al*. 2000, p. 65), the
Peircean system has some obvious weaknesses, which make the system
problematic as a satisfactory candidate for a theory of future
contingency. First of all, the system fails to represent many
common-sense notions of time, which are arguably reasonable. This is
due to the fact that the idea of a plain future as a 'middle
ground' between possible future and necessary future cannot be
expressed in the Peircean system. Suppose I say:



> 
> "It will be sunny in London tomorrow"
> 



I do not mean that tomorrow *perhaps* will be sunny in London,
or maybe not; I mean that indeed it will be the case; but on the other
hand I do not mean that there is no other option, or that it must be
so. One should be aware that in adopting the Peircean system, one
would have to consider this everyday intuition illusory -- there
really is only the "possibly", or the
"necessarily" (corresponding to \(fp\) and \(Fp\)
mentioned in the (g) and (d) clauses in section 4.2). In fact,
logically speaking, in this system "it will be sunny in London
tomorrow" would have to be considered as equivalent to
either



> 
> "Possibly, it will be sunny in London tomorrow"
> 



or



> 
> "Necessarily, it will be sunny in London tomorrow".
> 



In addition, it should be noticed that it is a crucial feature of the
Pericean system that the expressions \(F(x){\sim}q\) and
\({\sim}F(x)q\) are non-equivalent. This certainly gives rise to a
serious challenge when confronted with everyday intuition. In fact, it
is rather difficult to make a clear distinction between the two
expressions in terms of natural language. E.g. it is doubtable whether
a distinction between "tomorrow it will not be sunny in
London" and "it is not the case that tomorrow will be
sunny in London" will be accepted as sufficiently clear.



For such reasons many scholars have found it rather problematic to
reject (P5). Instead they have focused on systems accepting (P5) but
rejecting (P3). In the following we shall consider five such
theories.


### 5.1 Prior's Ockhamist Solution



In *Past, Present and Future* Prior presented his so-called
Ockhamist system, which accepts (P5) but rejects (P3) (see Prior 1967,
p. 126 ff.). This system is inspired by some of the ideas formulated
by William of Ockham.



As with the Peircean semantics, it is assumed that there is a
truth-function, *TRUE*, which gives a truth-value (0 or 1) for
any propositional constant at any moment in TIME. On this basis, the
valuation function of an Ockhamist model, \(\Ock(t,c,p)\) can be
defined recursively for any wff \(p\), any moment of time \(t\), and
any chronicle \(c\) with \(t \in c\):




|  |  |  |  |
| --- | --- | --- | --- |
| (a) | \(\Ock(t,c,p)=1\) | iff | \(\TRUE(p,t)=1\), where \(p\) is any propositional constant. |
| (b) | \(\Ock(t,c,p\wedge q)=1\) | iff | both \(\Ock(t,c,p)=1\) and \(\Ock(t,c,q)=1\) |
| (c) | \(\Ock(t,c,{\sim}p)=1\) | iff | not \(\Ock(t,c,p)=1\) |
| (d) | \(\Ock(t,c,Fp)=1\) | iff | \(\Ock(t',c,p)=1\) for some \(t' \in c\) with \(t \lt t'\) |
| (e) | \(\Ock(t,c,Pp) = 1\) | iff | \(\Ock(t',c,p)=1\) for some \(t' \in c\) with \(t' \lt t\) |
| (f) | \(\Ock(t,c,\Diamond p) = 1\) | iff | \(\Ock(t,c',p)=1\) for some \(c' \in C(t)\) |



Here \(C(t)\) is defined as the set of chronicles through \(t\), i.e.,
\(C(t) = \{c\mid t \in c\}\).



We define the dual operators, \(H, G\), and \(\Box\) in the usual
manner as \({\sim}P{\sim}, {\sim}F{\sim}\), and \({\sim}
\Diamond{\sim}\) respectively.



\(\Ock(t,c,p) = 1\) can be read '\(p\) is true at \(t\) in the
chronicle \(c\)'. A formula \(p\) is said to be Ockham-valid if
and only if \(\Ock(t,c,p) = 1\) for any \(t\) in any \(c\) in any
branching time structure, (TIME,\(\le\),C) and any valuation function
*TRUE*. Here C should not be taken as an independent parameter.
In this case C is just the set of all maximally ordered linear subsets
in (TIME,\(\le)\). Furthermore, it should be noted that relative to a
single chronicle, (a) - (e) are exactly the same
definitions as those used in linear tense-logic (i.e. the tense-logic
which follows if (TIME,\(\le)\) is a linear structure).



Prior himself did not accept the view represented in the Ockhamist
system, but as many later researchers he was interested in the
exploration of the system. It should be mentioned that the basic views
held by Belnap *et al*. (2001) are in fact rather close to
Priorean Ockhamism, although there are certainly many further
developments of the theory in Belnap's philosophical writings on
the subject (Belnap 1992, 2002, 2003, 2005). Belnap has strongly
emphasized the distinction between what he calls 'plain
truth' and "settled truth". Whereas plain truth
corresponds to the branch-dependent truth used in the Ockhamistic
model, settled truth will be branch-independent, i.e., truth at a
moment of time. It should also be pointed out that in the definition
of \(Ock\), only (d) differs from the corresponding Peircean
definition. In fact, Prior (1967, p.130) has pointed out that the
Peircean system may be seen as a fragment of the Ockhamistic system in
which \(F\) does not occur except as immediately preceded by an
necessity operator.



To obtain a metric version of the Ockhamist system, a duration
function has to be added. Let \(\dur(t\_1,t\_2,x)\) stand for the
statement '\(t\_1\) is \(x\) time units before \(t\_2\)'.
Using this formalism, (d) and (e) are replaced by:




|  |  |  |  |
| --- | --- | --- | --- |
| \((\text{d}')\) | \(\Ock(t,c,F(x)p) = 1\) | iff | \(\Ock(t',c,p) = 1\) for some \(t' \in c\) with \(\dur(t,t',x)\) |
| \((\text{e}')\) | \(\Ock(t,c,P(x)p) = 1\) | iff | \(\Ock(t',c,p) = 1\) for some \(t' \in c\) with \(\dur(t',t,x)\) |



It can be verified that neither \(P(x)q \supset \Box P(x)q\) nor \(Pq
\supset \Box Pq\) are Ockham-valid for all \(q\). Let for instance
\(q\) stand for \(F(y)p\). It is easy to verify that \(P(x)F(y)p
\supset \Box P(x)F(y)p\) will not in general hold in an Ockhamistic
branching time model. This may be illustrated using the following
diagram, in which it is easily seen that \(\Ock(t, c\_1, P(x)F(y)p) =
1\), whereas \(\Ock(t, c\_1, \Box P(x)F(y)p) = 0\) since \(\Ock(t, c\_2,
P(x)F(y)p) = 0\).



![Figure 3](fc_figure_3.png)


This does away with (P3) in the formal version of the medieval
argument discussed above. Still, both formulas, \(P(x)q \supset \Box
P(x)q\) and \(Pq \supset \Box Pq\), will hold if the truth of \(q\)
does not depend on what the future brings.



If (P3) does not hold in general, one may reject (2) in the argument
in section 2. According to Ockham, (P3) (that is, its verbal analogue
as he could formulate it with the means then available) should only be
accepted for statements which are genuinely about the past, i.e.,
which do not depend on the future. According to this view, (P3) may be
denied precisely because the truth of statements like
\(P(x)F(x)F(y)q\) has not been settled yet -- since they depend
on the future.



In this way, one can make a distinction between "soft
facts" and "hard facts" regarding the past (see
Plantinga 1986; de Florio and Frigerio 2016). Following the Ockhamist
position, a statement like \(P(x)q\) would correspond to a hard fact,
if \(q\) does not depend on the future, whereas statements like
\(P(x)F(x)F(y)q\) would represent soft facts. Critics of the Ockhamist
position, however, may still say that if \(F(x)F(y)q\) was true \(x\)
time units ago, then there must have been something making it true at
that time, and that something must have been a hard fact. On the other
hand, supporters of the position hold that it is fully conceivable and
acceptable that what makes a statement true could also be a soft fact,
i.e., something which depends on the future.



The rationality of Ockham's suggestion according to which future
happenings can (in a very limited sense) influence the past, has been
defended by Alvin Plantinga (1986). It should also be mentioned that
Ockham's theory is relevant for the conceptual analysis of the
idea of prophecy (see the entry on
 prophecy).



However, it may be disputed that Prior's Ockhamist system fits
the ideas formulated by William of Ockham completely. Although many of
Ockham's original ideas are satisfactorily modelled in
Prior's Ockhamist system, Prior's system (1967) lacks a
proper representation of the notion of 'the true
 future'.[4]
 This was in fact one of the most basic ideas in Ockham's world
view. Ockham believed that there is truth (or falsity) also of
statements about the contingent future, which human beings cannot
know, but which God knows. Prior's Ockhamist system cannot be
said to include more than the idea of a proposition being true
relative to a moment of time and a chronicle. A proper theory in
accordance with William of Ockham's ideas would have to include
the idea of a proposition being true relative to a moment of time
(without any specification of a chronicle). Let us therefore
investigate a truth-theory which includes the idea of a true future in
this sense.


### 5.2 The Leibnizian Theory



An alternative approach to the semantics for future contingents is
inspired by the works of Leibniz and has been called a Leibnizian
semantics (see Ohrstrom and Hasle 1995). According to
this view the set of possible histories is not seen as a traditional
tree structure, but as a system of "parallel lines". On
the set of "parallel lines" a relation corresponding to
*qualitative identity up to a certain instant* is defined. In
such a model it will be straightforward to introduce truth-values for
future contingents.



The idea can be introduced in terms of Prior's Ockhamistic
model. As mentioned above any maximally ordered linear subset in
(TIME,\(\le)\) will be accepted as a chronicle in the Ockhamistic
model. However, in the Leibnizian model only some of these subsets
will be accepted as chronicles, although the union of all chronicles
will still be the full set TIME, i.e., any moment will belong to at
least one Leibnizian chronicle. The set of "parallel
lines" in the system may just be a subset of the set of all
chronicles considered in the Ockhamistic model. Formally, each
temporal moment in the Leibnizian semantics corresponds to a pair of a
moment of time, \(m\), and a chronicle, \(c\), with \(m \in c\). This
means that any Leibnizian time can be written as a structured formal
object \(\temp(m,c)\), where \(m \in c\). The Leibnizian valuation
function can be defined in terms of Prior's Ockhamistic model in
the following way:

 

\[
\textit{Leib}(\textit{temp}(m,c),p) = \Ock(m,c,p)
\]


Formally, this means that in the Leibnizian semantics the truth-value
of a proposition only depends on the Leibnizian time. According to
this semantics (P3) is obviously not valid in general.



A semantics introduced in this manner also fits with models defined in
terms of so-called bundled trees (see Zanardo 2003), and it is similar
to the approach taken by David K. Lewis in his *On the Plurality of
Worlds* (1986).



On the Leibnizian view \(p \supset P(x)F(x)p\) holds, whereas \(p
\supset P(x)\Box F(x)p\) does not hold. This may be illustrated in the
following way:




|  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- |
| Figure 4 | 

|  |
| --- |
| t\(\_1 ' =\) temp(m\(\_1\),c\(')\) |
| t\(\_1 =\) temp(m\(\_1\),c) |
| t\(\_2 ' =\) temp(m\(\_2 '\),c\(')\) |
| t\(\_2 =\) temp(m\(\_2\),c) |

 |



This diagram illustrates that chronicles may be represented as
parallel lines up to a certain temporal instant (containing both \(t\_1
= \temp(m\_1,c)\) and \(t\_1 ' =\temp(m\_1,c'))\), from where they
diverge. Until the "branching point" the chronicles are
indistinguishable.



According to a Leibnizian semantics propositions without modal
operators, such as \(p \supset P(x)F(x)p\), will have to be evaluated
within the sub-model defined by the chronicle (i.e., in fact a linear
model). The point is that to determine the truth-value of a formula
without modals at a Leibnizian time defined as \(t =\temp(m,c)\), one
need not look at other chronicles than \(c\) if the evaluation is
going to take place on the basis of Leibnizian semantics. However, in
the above model the proposition \(p \supset P(x)\Box F(x)p\) will not
be true at \(t\_2 =\temp(m\_2,c)\) since even if \(p\) is true at
\(t\_2\) and \(t\_1 =\temp(m\_1,c)\) is a time \(x\) time units earlier
earlier than \(t\_2\), the proposition \(\Box F(x)p\) will be false at
\(t\_1\), since there is a co-temporal moment \(t\_1 '\) at which
\(F(x)p\) is false.



From a formal point of view the semantics of the Leibnizian theory may
be seen as an alternative interpretation of the semantics of the
Ockhamist theory with the only difference that in the Leibnizian
theory not all maximal linear subsets have to be accepted as proper
chronicles in the model.



Some philosophers have argued that the Leibnizian theory at least in
some cases seem to be more plausible from an intuitive point of view
than the Ockhamistic theory. The reason is that there exist some
rather intricate propositions which some hold to be intuitively
invalid, which they are according to the Leibnizian theory, although
they are valid according to the Ockhamist theory. One such example can
be given in terms of these two statements:




|  |  |
| --- | --- |
| \(p\_1{:}\) | "Inevitably, if today there is life on earth, then either
this is the last day (of life on earth), or the last day will
come." |
| \(p\_2{:}\) | "At any possible day on which there is life on earth, it
is possible that there will be life on earth the following
day." |



Hirokazu Nishimura (1979) has argued that if time is assumed to be
discrete, then an Ockhamist cannot consistently accept the conjunction
of \(p\_1\) and \(p\_2\), whereas a Leibnizian can maintain such a view
without contradicting himself. The purpose of the following figure is
to clarify the difference between these two views.



![Figure 5](fc_figure_5.png)


As indicated in the above figure, an infinite number of ovals named
i\(\_1\), i\(\_2\), i\(\_3\), i\(\_4,\ldots\) represent a series of
instants i.e., equivalence classes of co-temporaneous moments, as
mentioned in section 3. The cronicles are named c\(\_1\), c\(\_2\),
c\(\_3\), c\(\_4,\ldots\). For \(n\in \{1,2,3,4,\ldots \}\) the moment
corresponding to i\(\_n\) on c\(\_n\) will be the last day of life on
earth. The totality of this infinity of chronicles represents the
acceptance of \(p\_1\). At the last day on each of these chronicles,
c\(\_j\), it would in fact be possible that life on earth could have
continued yet another day. This is evident because of the existence in
the model of c\(\_{j+1}\). Taken together this means that the statement
\(p\_2\) holds at any possible day in the model. This is what a
Leibnizian would say. However, an Ockhamist would say, that given this
model it would be possible to construct a chronicle c\(\_\*\), as shown
on the above figure for which the last day of life on earth would
never come. According to the Leibnizian, this construction of c\(\_\*\)
may not be permitted at all, since c\(\_\*\) may in fact be a maximal
linear subset which does not belong to C.



The point is that in the Ockhamist semantics, any maximal linear
subset of TIME is accepted as a chronicle. In the Leibnizian
semantics, the set of chronicles is an independent parameter. In a
Leibnizian model, any subset of the set of all maximal linear subsets
could be accepted as the set of chronicles, C, as long as all moments
in TIME belong to at least one chronicle.



Belnap *et al*. have argued that it is implausible to assume
that there could be some property which could "justify treating
some maximal chains as real possibilities and others as not"
(Belnap *et al*. 2001, p. 205). On the other hand,
Nishimura's example is in fact a rather remarkable argument
suggesting that not all maximal chains have to be accepted as
chronicles in the semantics for future contingents. The example also
speaks in favor of the view that the Leibnizian theory is more
plausible than Prior's Ockhamistic theory, see
(Ohrstrom and Hasle 1995, p. 268).



The Leibnizian and Prior's Ockhamistic views seem to be very
similar, and most differences between them seem to be nuances of
metaphysical interpretation. In fact, the Leibnizian way of
introducing truth-values for future contingents seems somewhat tricky.
It should also be mentioned that if the idea of chronicles as
'parallel lines' is taken seriously, then there is no
proper branching in the Leibnizian model. For this reason, it can be
argued that this model is incompatible with objective indeterminism,
since the alternative lines should not be counted as proper
possibilities, see e.g. (MacFarlane 2003, p. 325). On the other hand,
it may be argued that all the conceivable chronicles in the Leibnizian
model represent logical possibilities. Obviously, only some of them
are chosen, but from a logical point of view any of them could in
principle have been chosen. Still, it may be objected that the
Leibnizian model is rather sophisticated and speculative, and that it
may be more attractive to look for other ways of defining "truth
at a moment of time".


### 5.3 The True Futurist Theory: The Thin Red Line



The medieval assumption of the true future can in terms of modern
logic and a branching time model be rendered as meaning that there is
a privileged branch (i.e., a specific chronicle) in the model. If
\(b\) is this privileged branch, then the truth-value of a
proposition, \(p\), at a moment of time, \(t\), may be defined in
terms of the Ockhamistic valuation function as \(\Ock(t,b,p)\). We may
speak of "the actual future" (Lucas 1989). This solution
has been studied in (Ohrstrom 1981) and further
elaborated and discussed in (Malpass et al. 2012) and (Borghini et al.
2013). Consider, for instance, the following model, in which the
arrows indicate the true future at any moment.



![Figure 6](fc_figure_6.png)


In this model, \(F(x)q\) is true at \(t\_2\) and \(F(x+y)q\) is true at
\(t\_1\), although none of the propositions are necessary, since
\(F(x){\sim}q\) is possible at \(t\_2\). The reason why \(F(x)q\) is
true at \(t\_2\) is just that the evaluation of a proposition according
to the true futurist theory should be based on the specified branch
through \(t\_2\) representing "the future" at \(t\_2\)
within the model. However, as we shall see in the following, it turns
out that the idea of a specified branch at every moment can be
integrated into the semantics in several ways. But first of all some
comments on the very idea of a specified branch.



What makes the specified branch privileged? Is it just that it
represents what is going to happen? Is there anything in the present
situation, \(t\_2\), which makes one branch ontologically special as
opposed to the other branches? It might be tempting to refer to some
sort of a "wait-and-see" status of the privileged branch,
since we have no way of knowing which branch is the specified one
representing "the future" except by waiting.



Some authors have held that the idea of a privileged branch is
incompatible with indeterminism. Hence Rich Thomason (1970, 1984) has
argued that from an indeterministic point of view no special branch
deserves to be called the true future. Of course, the problem is what
exactly the idea of indeterminism implies. According to MacFarlane, it
is problematic to give one future branch a special status, if we want
to hang on to objective indeterminism regarding the future (MacFarlane
2003, p. 325). On the other hand, although the true futurist theory
does contain some intricate notions, it has not been shown to be
inconsistent, and a supporter of the theory may still hold that the
theory correctly explains what reality is like. It should be borne in
mind that true futurist theory was introduced exactly to avoid what
many have held to be counter-intuitive tenets, e.g. that all future
contingents are false now (the Peircean view), or that they have no
chronicle-independent truth-values now (the Ockhamistic view).
Therefore, it should be carefully considered which approach ultimately
leads to the fewest problems.



According to Belnap and Green a true futurist theory should include
the idea that at any moment of time - including any
counterfactual moment - there is a true future, a so-called
"thin red line" (Belnap and Green 1994), passing through
that moment. Formally, this means that there must be a function,
\(\TRL\), which gives the true future for any moment of time, \(t\).
More precisely, \(\TRL(t)\) yields the linear past as well as the true
future of \(t\), extended to a maximal
 set.[5]



In fact, the idea of adding a function like \(\TRL\) to the semantical
model had earlier been suggested by (McKim and Davis 1976) and by
(Thomason and Gupta 1980). But unlike Belnap and Green these authors
did not name the function in any spectacular way.



It would of course be fatal for the true futurist theory if it could
be demonstrated that it contradicts assumptions which we for other
reasons should accept. Belnap and Green (1994) have argued that there
are in fact such fundamental problems related to the true futurist
picture. They have argued that it is not sufficient for the model to
specify a preferred branch corresponding to the true history (past,
present, and future): it must be assumed that there is a preferred
branch at *every* counterfactual moment. They have illustrated
their view using the following statement:



> 
> "The coin will come up heads. It is possible, though that it
> will come up tails, and then later it will come up tails again (though
> at this moment it could come up heads), and then, inevitably, still
> later it will come up tails yet again." (Belnap and Green 1994,
> p. 379)
> 



This statement may be represented in terms of tense logic with
\(\tau\) representing tails and \(\eta\) heads, respectively:



\[
F(1)\eta \wedge \Diamond F(1)(\tau \wedge \Diamond F(1)\eta \wedge F(1)(\tau \wedge \Box F(1)\tau))
\]


and in terms of the following branching time structure:



![Figure 7](fc_figure_7.png)


The example shows that if we want to take this kind of everyday
reasoning into account, we need to be able to speak not only about the
future, but also about what would be the future at any counterfactual
moment. As mentioned above this is formally done in terms of the
TRL-function. But what are the constraints on this function? Belnap
and Green have argued that:



(TRL1)
\(t \in \TRL(t)\)


should hold in general. Moreover, they have also maintained that:



(TRL2)
\(t\_1 \lt t\_2 \supset \TRL(t\_1) = \TRL(t\_2)\)


should hold for the TRL-function. On the other hand, they have argued
that the combination of (TRL1) and (TRL2) is inconsistent with the
very idea of branching time. The reason is that if (TRL1) and (TRL2)
are both accepted, it follows from \(t\_1 \lt t\_2\) that \(t\_2 \in
\TRL(t\_1)\) i.e., that all moments of time after \(t\_1\) would have to
belong to the thin red line through \(t\_1\), which means that there
will in fact be no branching at all. However, it is very hard to see
why a true futurist would have to accept (TRL2), which seems to be too
strong a requirement. Rather than (TRL2), the weaker condition
(TRL2') can be employed:



(TRL2')
\((t\_1 \lt t\_2 \wedge t\_2 \in \TRL(t\_1)) \supset \TRL(t\_1) =
\TRL(t\_2)\)


This seems to be much more natural in relation to the notion of a true
futurist branching time logic. Belnap has later accepted that
(TRL2') is a plausible alternative to (TRL2) (see Belnap et al.
2001, p. 169).



(TRL1) and (TRL2') are essential in a branching time structure
with a TRL-function. This may be illustrated using a discrete
tense-logic like the one mentioned in section 2. We shall assume that
there are two functions, \(\next\) and \(back\), from TIME to TIME
such that \(\next(t)\) is the moment just after \(t\) on \(\TRL(t)\),
and \(back(t)\) is the moment just before \(t\) on \(\TRL(t)\). The
similarity to the "tomorrow" \(T(p)\) and
"yesterday" \(Y(p)\) of section 2 - and hence, to
Prior's considerations in (Prior 1967) - is worth noting,
even though the functions ranged over propositions rather than times.
Starting from an arbitrary moment \(t\_0\) we may define \(t\_n =
\next^n(t\_0)\) for \(n = 1,2,3,\ldots.\) It follows from (TRL1) and
(TRL2') that \(\TRL(t\_0) = \TRL(t\_1) = \TRL(t\_2) =\TRL(t\_3) =
\ldots\) and that \(t\_n \in \TRL(t\_0)\) for any \(n\). In other words,
we might have started from the functions \(\next\) and \(\back\) in
order to define the function, \(\TRL\). Taking \(t\_0\) as now the
function \(\next\) would by iterative use constitute the elements of
the future part of \(\TRL(t\_0)\). Similarly, the past part of
\(\TRL(t\_0)\), i.e., PAST\((t\_0)\) could be established by iterative
use of \(back\), given that the branching time structure is backwards
linear (as mentioned in section 3). In consequence, it turns out that
there is a nice correspondence between a discrete branching time
system defined in terms of \(\TRL\) and \(\lt\) on the one hand, and
on the other hand the same system defined in terms of the functions,
\(\next\) and \(\back\). In this way, a study of the discrete case
leads to an interesting illustration of the idea of the thin red line
as well as to an emphasis of the importance of (TRL1) and
(TRL2').



In general, we can inductively define a chronicle-independent
valuation function, using *TRUE*, which as mentioned in section
4.2 gives a truth-value (0 or 1) for any propositional constant at any
moment in TIME, and using the TRL-function. On this basis, the
valuation function, \(T(t,p)\) can be defined recursively for any wff
\(p\), and any moment of time \(t\):




|  |  |  |  |
| --- | --- | --- | --- |
| (a) | \(T(t, p) = 1\) | iff | \(\TRUE(p,t) = 1\), where \(p\) is any propositional
constant. |
| (b) | \(T(t, p\wedge q) = 1\) | iff | both \(T(t,p) = 1\) and \(T(t,q) = 1\) |
| (c) | \(T(t,{\sim}p) = 1\) | iff | not \(T(t,p) = 1\) |
| (d) | \(T(t,Pq) = 1\) | iff | there is some \(t'\) with \(t'\lt t\) and \(T(t',q) = 1\) |
| (e) | \(T(t,Fq) = 1\) | iff | there is some \(t' \in \TRL(t)\) with \(t\lt t'\) and \(T(t',q)
= 1\) |



\(T(t,q) = 1\) can be read '\(q\) is true at \(t\)'. As in
section 4.2 the valuation function has the range \(\{0,1\}\). A
formula \(q\) is said to be T-valid if and only if \(T(t,q) = 1\) for
any \(t\) in any branching time structure (TIME,\(\le\),C), any
valuation function *TRUE*, and any TRL-function defined on
TIME.



This means that sentences only involving tenses are what Belnap (in
honor of Carnap) has called "moment-determinate" (Belnap
1991, p. 163), indicating that their truth-value doesn\('\)t vary with
the chronicle. The advantage of this view is that it corresponds with
everyday reasoning and natural language understanding as it is most
commonly conceived.



As in sections 4.2 and 5.1, it is possible to extend the language in
order to take metrical notions into consideration:




|  |  |  |
| --- | --- | --- |
| \(T(t,P(x)q) = 1\) | iff | \(\exists t'{:} \dur(t',t,x) \amp T(t',q) = 1\) |
| \(T(t,F(x)q) = 1\) | iff | \(\exists t'{:} \dur(t,t',x) \amp t' \in \TRL(t) \amp T(t',q) =
1\) |



Belnap and Green have argued that the constraints on the TRL-function
should give rise to a logic in which the following theorems hold:



(T1)
\(PPq \supset Pq\)
(T2)
\(FFq \supset Fq\)
(T3)
\(q \supset PFq\)


If we accept the constraints (TRL1) and (TRL2'), and use the
above recursive definition of the valuation function \(T(t,p)\), we
obtain a semantics according to which (T1) and (T2) are valid.



However, with the semantics presented above, (T3) will not be valid.
To see why this is the case, consider a situation with a moment of
time \(t\_1\) such that \(t\_1 \not\in \TRL(t\_0)\) for any \(t\_0\lt
t\_1\). Assume that \(t\_1\) is the only moment at which \(q\) is true.
Then \(PFq\), hence also \(q \supset PFq\), will be false at
\(t\_1\).



Even the formula



(T3')
\(q \supset P(x)F(x)q\)


is false when evaluated with semantics of this kind.



The rejection of (T3') can be illustrated by the following
diagram, in which the arrow on the upper branch indicates the thin red
line. (The vertical line in this diagram represents a set of
co-temporaneous moments, i.e., what is sometimes called an
instant.)



![Figure 8](fc_figure_8.png)


According to this diagram \(q\) holds at the counterfactual moment of
time, \(t\). However, as indicated in the diagram \(F(x)q\) was false
\(x\) time units earlier than \(t\), since at that time \(t'\) would
be the true future \(x\) time units later.



The rejection of (T3) and (T3') is not the only problem related
to a TRL-semantics defined in this way. It should also be pointed out
that it is somewhat complicated to state the semantics of modal
expressions if we follow this procedure, since it may involve the
quantification over possible TRL-functions. This approach has been
further investigated in (Brauner *et al*. 1998).



There is, however, a simpler strategy which makes it possible to
ensure the validity of (T3) and (T3') even if one wants to
insist on the assumption of the thin red line. This can be done by
defining "true at time \(t\)" in terms of "true at
time \(t\) and chronicle \(c\)", as it is defined in the
Ockhamist semantics:

 

\[
T(t,p) = \Ock(t, \TRL(t),p)
\]


where \(p\) is an arbitrary propositional expression. \(T(t,p) = 1\)
can be read "\(p\) is true at \(t\)". This idea of
obtaining a thin red line semantics by introducing a unique historical
parameter has been discussed in (MacFarlane 2003, 330-331, cf.
n. 10).



As in Prior's Ockhamistic model it is straightforward to
introduce metrical tense operators in this system.



As in the Leibniz-theory we should not necessarily assume that all
maximal linear subsets in the branching time structure should be taken
into account as chronicles in the semantics. It may be reasonable to
assume various restrictions regarding the set of chronicles and its
use in the semantical model. In fact, it has turned out to be
interesting to consider the possibilities of modifying the definition
of \(C(t)\) used in (f) of the \(Ock\) definition in section 5.1.



The following validity definition may be suggested:



(V)
A formula \(p\) is said to be TRL-valid if and only if
\(\Ock(t,\TRL(t),p) = 1\) for any \(t\) in any branching time
structure, (TIME,\(\le\),C), any valuation function *TRUE*, any
definition of \(C(t)\) with \(\TRL(t) \in C(t)\) and \(C(t) \subseteq
\{c \in C\mid t \in c\}\) for all \(t\), and any TRL-function for which
(TRL1) and (TRL2') hold.


Given this definition it is easily seen that (T1-3) and
(T3') are all TRL-valid. Regarding the interplay between the
tense operators and the modal operator, it is straightforward to
verify that the following is TRL-valid:



(T4)
\(F(x)p \supset \Diamond F(x)p\)


whereas (P3) in section 2 will not be TRL-valid for propositions
depending on the future. However, the notion of validity suggested
above may also allow for the following definition of \(C(t)\), which
has been discussed in (Brauner *et al*. 2000):



\(C(t) = \{c \mid t \in c \amp \TRL(t')=c\), for all \(t' \in c\) with
\(t \lt t'\}\) 



Note that with this definition (TRL1) and (TRL2') together say
exactly that \(\TRL(t) \in C(t)\). Also note that \(C(t)\) may contain
more branches than just \(\TRL(t)\).



However, it should be mentioned that the possibility operator in this
model is somewhat surprising. In the obvious metrical extension of the
system the following formula is invalidated:



(T5)
\(F(x)\Diamond F(y)p \supset \Diamond F(x)F(y)p\)


According to the usual Ockhamist semantics (T5) is valid. The
rejection of (T5) in the system presented in (Brauner et al.
2000) may be illustrated with reference to the following model:



![Figure 9](fc_figure_9.png)


Here it is assumed that \(\TRL(t)=c\_2\) for all \(t\) on \(c\_2\) after
\(t\_2\) and that \(\TRL(t)=c\_3\) for all \(t\) on \(c\_3\) after
\(t\_2\). Clearly, this means that \(C(t\_2)=\{c\_2,c\_3\}\). In
consequence, the proposition \(\Diamond F(y)p\) holds at \(t\_2\). This
means that \(F(x)\Diamond F(y)p\) is true at
\(t\_1\). However, the proposition \(\Diamond F(x)F(y)p\) is false at
\(t\_1\), since \(c\_2\) is not included in \(C(t\_1)\). According to the
definition, \(C(t\_1)\) should include exactly the chronicles which
pass through \(t\_1\) and which immediately after \(t\_1\) are specified
by the TRL-function. This means that \(C(t\_1)=\{c\_1,c\_3\}\), and then
(T5) turns out to be false at \(t\_1\).



This rejection of (T5) amounts to the following idea: a chronicle may
not be available as a possibility now, although it may later become
available. That is, new possibilities may show up.



This example illustrates that true futurist logics satisfying
requirements which correspond to Belnap's and Green's
semantical criteria may differ significantly from Prior's
Ockhamism. Even if we assume that (T1-4) should be valid and
that (TRL1) and (TRL2') should hold, we cannot be sure that (T5)
is valid. On the other hand, some might of course intuitively find
(T5) just as plausible as (T1-4), for which reason they would
insist on a definition of \(C(t)\) according to which (T5) is valid.
This, of course, means that the validity notion in (V) should be
modified introducing further restrictions on the acceptable
definitions of \(C(t)\). In order to ensure the validity of (T5) one
might require that \(C(t') \subseteq C(t)\) for all \(t\) and \(t'\)
with \(t \lt t'\). Another possibility is of course to insist on the
Ockhamistic definition of \(C(t)\), i.e., \(C(t) = \{c \mid t \in c\}\),
in which case TRL-validity would give the same result as
Ockham-validity.



It is still an open question which definition of TRL-validity we
should prefer, if we want it to include a semantical definition of
what it means for a future contingent to be true, and if we want it to
ensure the validity of (T3) and (T3'). In addition to the two
solutions mentioned above, we may consider a use of a TRL-function
that appears to bring us very close to Molina's notion of middle
knowledge mentioned in section 1 (see Ohrstrom 2014).
The idea is to let the evaluation of the truth-value of a proposition
depend on two moments, \(s\) and \(t\), instead of just one (see
MacFarlane 2003; MacFarlane 2014; De Florio and Frigerio 2020). We may
speak of the truth-value of a proposition at the moment \(t\) from the
perspective of moment \(s\). Whereas \(t\) is the moment of
evaluation, \(s\) is the "context of utterance" or at
least a moment at which the statement in question is imagined to be
considered.



Given the TRL-function, \(\TRL\), and a moment \(s\), we introduce an
adjusted thin red line function, \(\TRL\_s\), by defining that
\(\TRL\_s\) and \(TRL\) differ only for moments that belong to the past
of \(s\), in which case \(\TRL\_s\) gives the chronicle \(TRL(s)\). In
other words, \(\TRL\_s(t) = \TRL(s)\) for \(t \lt s\), and otherwise
\(\TRL\_s(t) = \TRL(t)\). Intuitively, using \(\TRL\_s\) means that
\(s\) is conceived as the perspective of the statement in question,
and it is actual in the sense that at any moment before it would
belong to the true future according to the evaluation currently used.
It is easy to verify that \(\TRL\_s\), like \(TRL\), satisfy the
constraints (TRL1) and (TRL2'). This means that \(\TRL\_s\)
formally qualifies as a thin red line function for an arbitrary
moment, \(s\), in the same sense as \(\TRL\).



We may illustrate the idea using discrete time. Assuming that every
chronicle in TIME is isomorphic with the set of integers, there will
for each chronicle be a function, \(back\), from TIME to TIME such
that \(back(t)\) is the moment on the chronicle that is just before
the moment \(t\). Furthermore, there is a function \(\next\_s\) such
that \(\next\_s(t)\) is the moment following immediately after \(t\) on
\(\TRL\_s(t)\).



It should be noted that it follows from (TRL1-2) that
\(\TRL\_s(t) = \TRL\_s(\next\_s(t)) = \TRL\_s(\next\_{s}^2(t)) = \ldots\)
and in general that \(\next\_{s}^n(t) \in \TRL\_s(t)\) for any natural
number \(n\). In addition, it should be mentioned that since the
branching-time system is backwards linear, there is no need to specify
the chronicle in \(back(t)\).



Assuming that there is a basic assignment function giving a
truth-value for each propositional letter at each instant, we may then
introduce the following recursive definition of the truth-value of a
tense-logical proposition at the moment \(t\) seen from the
perspective of the moment, \(s\):




|  |  |  |
| --- | --- | --- |
| \(T(t,s,p)=1\) | iff | \(p\) is a propositional letter assigned with the truth-value 1
at the moment \(t\). |
| \(T(t,s,P(n)p)=1\) | iff | \(T(\back^n(t),s,p)=1\) |
| \(T(t,s,F(n)p)=1\) | iff | \(T(\next\_{s}^n(t),s,p)=1\) |
| \(T(t,s,\Diamond p)=1\) | iff | \(\exists t': t\lt t' \amp T(t,t',p)=1\) |



Negation and the propositional connectives are treated in the
traditional manner. The necessity operator, \(\Box\), is defined as
\(\sim \Diamond \sim\).



A tense-logical proposition, \(p\), is said to be valid in general iff
for any branching-time system with a valuation function, \(T\),
defined in this manner it holds that \(T(t,t,p)=1\) at any instant
\(t\) in the branching-time system.



If we identify t and s in this way, it is obvious that we obtain:



(O1)
\(\next\_{t}^n(\back^n(t))=t\)
(O2)
\(\back^n(\next\_{t}^n (t))=t\)


It follows from (O1-2) that \(P(n)F(n)p \equiv p\) and
\(F(n)P(n)p \equiv p\) are valid theses in the semantics. Furthermore,
it is easily verified that (T3'), (T4), and (T5) are valid as
well in the system. It is also obvious that \(P(n)F(n+m)p \supset \Box
P(n)F(n+m)p\) is not valid in general according to this semantics. In
consequence, the crucial principle (P3) should be rejected on this
view. It seems that the semantics sketched here will have all the
properties, which one would expect in Ockhamistic (and Molinistic)
semantics (see Ohrstrom 2014; Ohrstrom and
Jakobsen 2018; De Florio and Frigerio 2019).


### 5.4 The Supervaluationist Theory



Some logicians have argued that the notion of true future is
unacceptable on philosophical grounds or that it is at least
unnecessary, since it is possible to establish a semantic model
accepting (P5) but rejecting (P3) without involving any idea of a true
future. Richmond H. Thomason (1970, 1981, 1984) has formulated a
theory based on so-called supervaluations. According to this theory a
proposition \(p\) is true at a moment \(t\) if and only if it is true
at \(t\) for every chronicle \(c\) passing through \(t\), and a
proposition \(p\) is false if and only if it is false for every
chronicle \(c\) passing through \(t\). Formally speaking, we may again
use the Ock-function to recursively define truth at a moment \(t\) and
a chronicle \(c\). Then we may define truth at a moment of time by
supervaluating. This means that \(p\) is true at \(t\) if and only if
\(\Ock(t,c,p) = 1\) for all \(c\) with \(t \in c\). Future contingent
propositions will not meet that condition, nor will their negations,
so they are considered neither true nor false. They are
"indeterminate" in the sense that they lack
truth-values.



This theory allows the supervaluationist to reject (P3) and to accept
future excluded middle, (P5), without accepting the idea of 'the
thin red line' or "the true future". It should be
pointed out, however, that although this theory implies a rejection of
(P3), it does in fact accept a related inference principle i.e., if
\(P(x)p\) is true at a certain moment of time, \(t\), then \(\Box
P(x)p\) will also be true at \(t\).



Thomason has shown that the supervaluation theory can in fact meet
some of the basic challenges related to the future contingents. He has
also shown that the theory can be extended in such a way that it may
also incorporate deontic logic i.e., the logic of moral obligation
(Thomason 1981, pp. 165 ff.). A crucial question for this approach is,
however, whether the idea of truth-value gaps for future contingents
is philosophically acceptable. In other words, is it acceptable that
some well-formed propositions simply lack truth-values?



A distinctive feature of Thomason's theory is that the usual
truth-functional technique breaks down. For instance, if \(F(1)p\) is
a future contingent, then \(F(1)p\) and \(F(1){\sim}p\) are both
"indeterminate", but the conjunction \(F(1)p
\wedge F(1){\sim}p\) will be false and the
disjunction \(F(1)p \vee F(1){\sim}p\) will be
true. It may be objected that it seems odd that a disjunction could be
true when neither of the disjuncts is true, and a conjunction false
when neither of the conjuncts is false.


### 5.5 Relativist Theory



Recently John MacFarlane (2003, 2008) has suggested a new approach to
the problem of future contingents. Like Nuel Belnap *et al*.
(2001), MacFarlane challenges the classical notion that the
truth-value of a statement or proposition should be determined solely
with respect to the context (including the moment) of utterance. He
also agrees with Belnap and Green in rejecting the idea of the true
future (i.e., the thin red line). But unlike Belnap, MacFarlane does
not want to give up all talk of truth in a context. In this way
MacFarlane accepts the same definition as Belnap, Green and Thomason
of \(\Ock(t,c,p)\), where \(t\) is a moment of time and \(c\) is a
history (or a chronicle), but differs on "truth at a
moment" or "truth at a context".



According to MacFarlane's theory the truth of a statement should
be relativised to both a context of utterance and a context of
assessment. The context of utterance is the context in which the
speech act is made. The context of assessment is the context in which
we assess the speech act. According to the theory, a statement \(S\)
is true as used in the context \(C\) and assessed in the context
\(C'\), if and only if \(S\) is true at \(m\) for every chronicle
\(c\) passing through \(m\) and \(m'\), where \(m\) and \(m'\) are
moments within the contexts \(C\) and \(C'\) with \(m \le m'\). This
means that a statement like "it will be sunny tomorrow" is
true as uttered yesterday and assessed from today, given that it is in
fact sunny today. But it will not be true assessed from yesterday,
given that it was not settled yesterday that today would be sunny.



It turns out that MacFarlane's theory implies an acceptance of
the principle of future excluded middle, (P5), and a rejection of the
principle of the necessity of the past, (P3). An advocate of the
theory (i.e., a relativist) will clearly also agree with
supervaluationism in allowing for truth-value gaps. However, in
addition a relativist will accept the view that future contingents can
be true as assessed from a future context. For criticism of
MacFarlane's position see Heck 2006.


## 6. Conclusion



As we have seen, Lavenham's medieval attempt at systematising
the various possible responses to the problem of future contingents
gives rise to a classification. This classification is based on the
observation that in order to reject the logical argument for
determinism we have to reject at least one of the principles, (P3)
(necessity of the past) and (P5) (the principle of future excluded
middle). It is also clear from Lavenham's analysis that he
wanted to focus on the question whether future contingents have
determinate truth-values. When formulated in terms of the modern
debate, the question is whether future contingents have
branch-independent truth-values. Or using Belnap's vocabulary:
Is there settled truth about future contingents? Using the reactions
on the classical argument mentioned in section 2 and the views on
truth-values of future contingents as two classification principles,
we may group the solutions considered in the following diagram:




|  |  |  |
| --- | --- | --- |
|  | *Some future contingents are neither true nor false* | *All future contingents are either true or
false* |
| *Rejection of the principle of the necessity of the past
(P3)* |  Priorean Ockhamism

(Nuel Belnap)


Supervaluationism

(Richmond Thomason)


Relativism

(John MacFarlane)  |  Leibnizianism

(Davis Lewis, Alberto Zanardo)


True Futurism

(classical Ockhamism, Molinism)  |
| *Rejection of the principle of future excluded middle
(P5)* | Three valued logic

(Jan Lukasiewicz)  | Priorean Peirceanism

(A.N. Prior) |



The classification of Belnap's view in the above diagram is
based on the assumption that his "settled true" refers to
"true" as it is used in the present context.



Regarding the classification represented in the diagram it should be
noted that from a logical point of view the two rows are not mutually
exclusive. However, although it would in principle be possible to
reject both (P3) and (P5), we do not know any important theory of that
kind.



It should also be noted that the theories in the first row all
formally make use of the Ockhamist recursive semantics, although they
differ in how or whether truth at a moment of time is defined in terms
of \(\Ock(t,c,p)\).



As we have seen, there are ongoing philosophical debates regarding
future contingents. There is still a focus on the questions
represented in the diagram, but other problems are also discussed. One
problem which has attracted much attention is the study of future
contingents as seen in relation to branching space-time and various
ideas within physics. Here Belnap (1992, 2003, 2005), Muller
(2007, Muller *et al*. 2008), and Placek (2000) have
contributed significantly.