. 9
( 11)


For each point of a tree, there is only one backwards route (towards
the past). However, a point of the tree may have many alternative forward
routes (towards the future). In the case of the tree of time, this crucial
feature represents the idea that while the past is unique and determined,
the future is open, i.e. can take many different courses. With regard to
the openness of the future, it is important to be clear that the multiple
alternative forward routes do not represent mere epistemic alternatives: it
is not the case that the multiplicity of the forward routes represents our
ignorance of what the future will be like. Rather, the multiplicity of the
forward routes represents the fact that various alternative future evolutions
are possible. Another important feature is that none of the many forward
routes is privileged in the sense that it represents the future which will be
realised: all forward routes are on a par. Times are conceived of as intimately
tied to the possible world-states obtaining in them: for this reason we speak
Appendix 6 271
of different future possible times rather than of possible future events or
states at the same time.11
The properties of the tree of time reduce to those of its ordering relation
<. This relation, which obtains only between times, is asymmetric and
transitive (and is therefore a strict partial order), has no ˜beginning™ and no
˜end™, and is connected (for any two times there is a further time preceding
both). The property that characterises the tree-structure is that any two
distinct times that are followed by one and the same time are linked by <.
Here is the full formulation:
[112] ∀t ∀s (t < s ’ Xt & Xs).
[113] ∀t ∀s (t < s ’ ¬s < t).
[114] ∀t ∀s ∀u (t < s & s < u ’ t < u).
[115] ∀t ∀s ∀u (t < u & s < u ’ t < s ∨ t = s ∨ s < t).
[116] ∀t ∀s (Xt & Xs ’ ∃u (u < t & u < s)).

Set theory. The expanded semantic theory to be presented relies on a mini-
mal amount of set theory:
[117] ∀x ∃y (My & ∀z (z ∈ y ” z = x)).
[118] ∀x ∀y (Mx & My ’ ∃z (Mz & ∀u (u ∈ z ” u ∈ x ∨ u ∈ y))).
[119] ∀x (Mx & ∀t (t ∈ x ’ Xt) & ∀t ∀s (t ∈ x & s ∈ x ’ t < s ∨ t = s
∨ s < t) ’ ∃u (Mu & ∀t (t ∈ u ’ Xt) & ∀t ∀s (t ∈ u & s ∈ u ’
t < s ∨ t = s ∨ s < t) & ∀y (My & ∀t (t ∈ y ’ Xt) & ∀t ∀s (t ∈ y &
s ∈ y ’ t < s ∨ t = s ∨ s < t) & ∀t (t ∈ u ’ t ∈ y) ’ u = y) & ∀t
(t ∈ x ’ t ∈ u))).
Proposition [117] assumes that for every object there is its singleton, i.e. the
set whose only element is that object. Proposition [118] assumes that for any
two sets there is their union, i.e. the set whose elements are precisely the
elements of those two sets. Proposition [119] makes instead a rather strong
assumption: that every linearly ordered set of times can be extended to a
maximal linearly ordered set of times. This assumption can be regarded as

11 For the representation of time as a tree cf. McCall (1966), 280“1; Prior (1967a), 126“7; Rescher (1968),
209“15; Thomason (1970), 265; Boudot (1973), 454“8; von Wright (1974), 162“6; Code (1976), 184;
McArthur (1976), 37“41; M. J. White (1980), 288; Thomason (1984), 141“2; Yourgrau (1985), 549“50;
Lucas (1989), 99“102, 135“9; Landman (1991), 101“3; Weidemann (1994/2002), 251“4, 285; Le´ n o
(1996), 385“6, 388“9; Mignucci (1996b), 291“2; Strobach (1998), 106“8.
272 The failure of Bivalence
an instance of the Maximal Chain Principle, which is equivalent to Zorn™s

Branches through the tree of time. A branch is a maximal linearly ordered set
of times:
[120] ∀x (Bx ” Mx & ∀t (t ∈ x ’ Xt) & ∀t ∀s (t ∈ x & s ∈ x ’ t < s ∨
t = s ∨ s < t) & ∀y (My & ∀t (t ∈ y ’ Xt) & ∀t ∀s (t ∈ y & s ∈ y ’
t < s ∨ t = s ∨ s < t) & ∀t (t ∈ x ’ t ∈ y) ’ x = y)).
Every element of a branch is a time:
[121] ∀x ∀t (Bx & t ∈ x ’ Xt).
The proof is trivial.
Whatever precedes an element of a branch is also an element of that branch:
[122] ∀x ∀t ∀s (Bx & t ∈ x & s < t ’ s ∈ x).
This is a well-known result of the theory of trees, and I restrict myself to
sketching the proof. Let a, b, and c be such that a is a branch, b is an element
of it, and c precedes b. Let d be the union of a with the singleton of c. d
can be shown to be a linearly ordered set of times. Since a is a subset of d
and (being a branch) is a maximal linearly ordered set of times, a must be
identical to d. Since c is an element of d, it is also an element of a.
Every time lies on some branch or other:
[123] ∀t (Xt ’ ∃x (Bx & t ∈ x)).
I again restrict myself to sketching the (trivial) proof. Let a be a time. Let
b be the singleton of a. Obviously, b is a linearly ordered set of times. By
[119], b can be extended to a maximal linearly ordered set of times c. Then
c is a branch and a is an element of it.
The day after. One straightforward assumption is that every time that lies
within the day after a given time is preceded by that given time:
[124] ∀t ∀s (Wts ’ t < s).
Another assumption is that on every branch every time is followed by a time
within the day that follows its day “ in short, no branch has a ˜last day™:
[125] ∀x ∀t (Bx & t ∈ x ’ ∃s (s ∈ x & Wts)).

12 Cf. Landman (1991), 115“17.
Appendix 6 273
Correctness is relative to times as well as to their possible futures. Consider a
time t with several ˜forward routes™ stemming from it, i.e. several alternative
possible futures. How can a future-tense assertion (i.e. an utterance of an
instance of ˜It will be the case that a™ or ˜Tomorrow it will be the case
that a™) be evaluated at t? One might lack adequate grounds for such an
evaluation: it might be the case that at t the assertion is correct on one of
its several possible futures but not on another. The best one can then do is
to evaluate the assertion as correct, or not correct, at t relatively to one of its
several possible futures. For such a reason the notion of correctness is here
regarded as relative not only to a time t, but also to a possible future of t,
i.e. as relative to t as well as to a branch of which t is an element (a possible
future of t determines a complete branch because t™s past is unique).13

Correctness at a time on a branch. Present-tense predicative assertions are
correct at a time on a branch just in case they are ˜true™ at that time according
to appendix 5™s de¬nition of truth:
[126] ∀x (Sn x ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” Txt))).
A past-tense (future-tense) predicative assertion is correct at a time on a
branch just in case the present-tense predicative assertion of which it is a
past (future) in¬‚ection is correct at some earlier (later) time on the same
branch.14 Analogously, a ˜tomorrow™-assertion is correct at a time t on a
branch just in case the present-tense predicative assertion of which it is a
˜tomorrow™-in¬‚ection is correct on the same branch at some time that lies
within the day after the day in which t lies.
[127] ∀x ∀y (Sn y & Ib xy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” ∃u (u ∈ s &
u < t & Cyus)))).
[128] ∀x ∀y (Sn y & If xy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” ∃u (u ∈ s &
t < u & Cyus)))).
[129] ∀x ∀y (Sn y & It xy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” ∃u (u ∈ s & Wtu
& Cyus)))).
A negative (disjunctive) assertion is correct at a time on a branch just in
case the assertion of which it is the negation is not correct (at least one of
its disjuncts is correct) at that time on that branch:
[130] ∀x ∀y ((Sn y ∨ Sb y ∨ Sf y ∨ St y) & Ic xy ’ ∀s ∀t (Bs & t ∈ s ’
(Cxts ” ¬Cyts))).
13 Cf. Prior (1967a), 122“7; Thomason (1970), 268“70; (1984), 133“4; Weidemann (1994/2002), 256.
14 Cf. n. 82 of ch. 7.
274 The failure of Bivalence
[131] ∀x ∀y ∀z ((Sn y ∨ Sb y ∨ Sf y ∨ St y ∨ Sc y) & (Sn z ∨ Sb z ∨ Sf z ∨ St z ∨
Sc z) & Iv xyz ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” Cyts ∨ Czts))).

Tokens of the same type are either both or neither correct at the same time on the
same branch. The ¬rst result in this area concerns present-tense predicative
[132] ∀x ∀y (Sn x & Sn y & Oxy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” Cyts))).
Let a and b be such that Sn a & Sn b & Oab. Since Sn b, Vb (by [57]). Then
Sn a & Vb & Oab, so that (by [106]) ((Sa a & Sa b) ∨ (Se a & Se b) ∨ (Si a &
Si b) ∨ (So a & So b) ∨ (Sy a & Sy b) ∨ (Sd a & Sd b)) & Ck ab & Ch ab. Since
Sn a & Sn b & Ck ab, ∃z (Kza & Kzb) (by [64]): let then c be such that Kca
& Kcb. Similarly, since Sn a & Sn b & Ch ab, ∃z (Hza & Hzb) (by [63]): let
then d be such that Hda & Hdb. Let e and f be such that Be & f ∈ e. Then
Xf (by [121]). Since (Sa a & Sa b) ∨ (Se a & Se b) ∨ (Si a & Si b) ∨ (So a & So b)
∨ (Sy a & Sy b) ∨ (Sd a & Sd b), distinguish six cases corresponding to these
six disjuncts.
(1) Sa a & Sa b. Since Sn a, ∃y (Kya & ∀z (Kza ’ z = y) & Uy) (by [60]):
let then g be such that Kga & ∀z (Kza ’ z = g) & Ug. Since Sa a, ∃y (Hya
& ∀z (Hza ’ z = y) & Uy) (by [61]): let then h be such that Hha & ∀z
(Hza ’ z = h) & Uh. Analogously, since Sn b, ∃y (Kyb & ∀z (Kzb ’ z = y)
& Uy) (by [60]): let then i be such that Kib & ∀z (Kzb ’ z = i) & Ui.
Since Sa b, ∃y (Hyb & ∀z (Hzb ’ z = y) & Uy) (by [61]): let then j be such
that Hjb & ∀z (Hzb ’ z = j) & Uj. Since Sn a, ∀s ∀t (Bs & t ∈ s ’ (Cats
” Tat)) (by [126]), so that Cafe ” Taf. Similarly, since Sn b, ∀s ∀t (Bs &
t ∈ s ’ (Cbts ” Tbt)) (again by [126]), so that Cbfe ” Tbf. Since Sa a,
∀t (Xt ’ (Tat ” ∃y ∃z (Kya & Hza & Cu yzt))) (by [66]). Since Xf, Taf
” ∃y ∃z (Kya & Hza & Cu yzf ). Analogously, since Sa b, ∀t (Xt ’ (Tbt
” ∃y ∃z (Kyb & Hzb & Cu yzt))) (again by [66]). Since Xf, Tbf ” ∃y ∃z
(Kyb & Hzb & Cu yzf ). Suppose Cafe. Then Taf. Then ∃y ∃z (Kya & Hza
& Cu yzf ). Let then k and l be such that Kka & Hla & Cu klf. Since Kka &
Kca & ∀z (Kza ’ z = g), k = g & c = g, so that k = c; since Hla & Hda
& ∀z (Hza ’ z = h), l = h & d = h, so that l = d. Since Cu klf & k = c
& l = d, Cu cdf. Then Kcb & Hdb & Cu cdf, so that ∃y ∃z (Kyb & Hzb &
Cu yzf ). Then Tbf, so that Cbfe. Thus: Cafe ’ Cbfe. Vice versa, suppose
Cbfe. Then Tbf. Then ∃y ∃z (Kyb & Hzb & Cu yzf ). Let then m and n be
such that Kmb & Hnb & Cu mnf. Since Kmb & Kcb & ∀z (Kzb ’ z = i),
m = i & c = i, so that m = c; since Hnb & Hdb & ∀z (Hzb ’ z = j),
n = j & d = j, so that n = d. Since Cu mnf & m = c & n = d, Cu cdf. Then
Kca & Hda & Cu cdf, so that ∃y ∃z (Kya & Hza & Cu yzf ). Then Taf, so
that Cafe. Thus: Cbfe ’ Cafe. Hence Cafe ” Cbfe.
Appendix 6 275
The remaining cases (2)“(6) are treated similarly: in each of them we
have Cafe ” Cbfe. Thus, in all cases, Cafe ” Cbfe. But e and f were
arbitrary. Then ∀s ∀t (Bs & t ∈ s ’ (Cats ” Cbts)). But a and b were
arbitrary. Then ∀x ∀y (Sn x & Sn y & Oxy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ”
More results of the same type can be proved on the basis of [132]:
[133] ∀x ∀y (Sb x & Sb y & Oxy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” Cyts))).
[134] ∀x ∀y (Sf x & Sf y & Oxy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” Cyts))).
[135] ∀x ∀y (St x & St y & Oxy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” Cyts))).
[136] ∀x ∀y (Sc x & Sc y & Oxy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” Cyts))).
[137] ∀x ∀y (Sv x & Sv y & Oxy ’ ∀s ∀t (Bs & t ∈ s ’ (Cxts ” Cyts))).

Truth-value gaps: (i) quasi-formal considerations:
(1) We want to think of the truth of an assertion as being determined
exclusively by a time. We shall thereby speak of an assertion a as being true
at a time t. Analogously, we want to think of the falsehood of an assertion
as being determined exclusively by a time. We shall thereby speak of a as
being false at t.
(2) In a preceding subsection it was made clear that t is not in general
enough to evaluate a future-tense assertion as true or false: a possible future
of t (i.e. one of the possibly many ˜forward routes™ stemming from t) may
need to be taken into account. Say that a valuation of a based on t is a
valuation of a carried out with reference to t and, moreover, one possible
future of t.
(3) We want to think that whether a is true at t depends on the valuations
of a which are based on t (i.e. those valuations of a each of which is carried
out with reference to t and, moreover, one possible future of t). Analogously,
we want to think that whether a is false at t depends on the valuations of a
which are based on t.
(4) As far as the valuations of a based on t are concerned, what is deter-
mined exclusively by t is only what is common to all valuations of a based
on t. Hence, as far as the valuations of a based on t are concerned, what
is determined exclusively by t can only be either that a is correct at t on
every possible future of t, or else that a is correct at t on no possible future
of t.
These points suggest (i) that a should be true at t if a is correct at t on
every possible future of t; (ii) that a should be false at t if a is correct at t on no
276 The failure of Bivalence
possible future of t; and (iii) that a should be neither true nor false, i.e., indet-
erminate, at t if a is correct at t on some but not all possible futures of t.15

Truth-value gaps: (ii) intuitive considerations. It is worthwhile re¬‚ecting
on the intuitions behind the last subsection™s quasi-formal considerations.
Think of t as the present moment, and think of the forward routes stemming
from t as the possible futures of t: the past of t is unique, but its future
branches out into several alternatives “ this is how the world now really is.
By requiring that the truth or falsehood of an assertion a should be
determined exclusively by t,16 we are requiring that the truth or falsehood
of a should be determined by how the world is at present. At present the
world is such as to be unique in the direction of the past, but branching
out into alternative possible futures. Up to now the world has been, and
is, going in precisely one way, but from now onwards it could go in several
ways. If a is a future-tense assertion, then how the world is at present
can determine a as true (false) only if a is correct on every (no) possible
future development. Otherwise (i.e. if a is correct on some possible future
developments but not on others), how the world is at present is unable
to determine a as true, and is also unable to determine a as false. In such
circumstances, at present a is neither true nor false, i.e. is indeterminate.17

Truth-value gaps: (iii) truth does not enjoy the disquotational property. Clearly,
the indeterminacy of truth-value of a at the present moment (whose past
is unique whereas its future branches out into several possible evolutions)
is compatible with a™s turning out to be correct ˜on the actual evolution
(whatever that may be) of the present™. Hence, a™s truth now is not entailed
by a™s turning out to be correct ˜on the actual evolution (whatever that
may be) of the present™. The converse entailment holds trivially: if a is true
now, it is correct on every possible evolution of the present, and is therefore
correct ˜on the actual evolution (whatever that may be) of the present™. In
this sense, asserting that a is true now is something stronger than asserting
a: it might be the case that by asserting a, one commits oneself (merely)
to a™s turning out to be correct ˜on the actual evolution (whatever that
15 Cf. van Fraassen (1966), 487, 494; Prior (1967a), 136; Thomason (1970), 272“3; Seeskin (1971), 770“3;
Haack (1974/96), 85“6, 90; M. J. White (1980), 288“91; Thomason (1984), 144“6; Bencivenga et al.
(1986/91), 40“5; Sayward (1989), 129; Gaskin (1994), 85; Weidemann (1994/2002), 320; Williamson
(1994), 144, 156“7; Vuillemin (1996), 160“1; Hafemann (1999), 129.
16 Cf. point (1) of the preceding subsection.
17 Cf. Hildebrand (1884), vi; Prior (1953), 322“3; Lenz (1957), 773“4; Ackrill (1963), 140“1; Hintikka
(1964/73a), 172“3; McCall (1966), 275“6; Patzig (1973), 923; von Wright (1974), 166“7; Dickason
(1976), 20“2; Wolf (1979), 114, 417; Weidemann (1980), 420; M. J. White (1981), 236, 240; Seel
(1982a), 241; M. J. White (1985), 66“7; Craig (1988), 23, 58; D. Frede (1990), 202; Weidemann
(1994/2002), 257“8; Inciarte (1999), 146“7.
Appendix 6 277
may be) of the present™, but does not commit oneself to a™s being true
now (because “ the speaker thinks “ the present branches out into several
possible evolutions, and a is correct on some but not all of these possible
evolutions). Thus, the direction ˜from fact to truth™ of the equivalence that
constitutes the disquotational property of truth can fail.18

Truth-value gaps: (iv) objections and answers. One might object: ˜If the claims
made in the preceding subsections are correct, truth lacks the disquotational
property; but truth enjoys the disquotational property;19 hence some of the
claims made in the preceding subsections are not correct.™ The objection
could be rubbed in: consider a man who one day makes a prediction by
uttering the sentence ˜A sea-battle will take place tomorrow™. The day after,
a sea-battle actually takes place. If the preceding subsections™ claims are
correct, this man cannot say ˜The remark I made yesterday was true™.
This objection can be answered by assuming that there are several notions
of truth, and that the claims made in the preceding subsections concern only
one of these notions of truth “ a ˜loaded™ notion of truth. In particular,
the last subsection™s claim about the lack of the disquotational property
concerns only this ˜loaded™ notion of truth. Obviously, this claim about the
˜loaded™ notion of truth is consistent with the claim that other notions of
truth enjoy the disquotational property. The idea here is that apart from
the ˜loaded™ notion of truth, there is also a ˜plain™ notion of truth: while
the ˜loaded™ notion of truth lacks the disquotational property, the ˜plain™
notion enjoys it.20
But what guarantees that the ˜loaded™ notion of truth should be a notion
of truth at all? Does not its lack of the disquotational property show that it
is not a notion of truth? This development of the objection can be answered
by means of two arguments.
(iv.i) One of our most deeply entrenched (albeit hazy) intuitions about
truth is that for an assertion or a belief to be true is to ˜match™ or ˜correspond
to™ the world. Now, the notion discussed in the preceding subsections is
that of an assertion corresponding to as much of the world as there is at a
given time. Hence, the notion discussed in the preceding subsections is a
notion of truth.
(iv.ii) The notion discussed in the preceding subsections matches certain
tensed uses of the phrase ˜. . . is true™. If at time t it is not yet certain whether

18 Cf. McCall (1966), 274“5; Gaskin (1994), 87; (1995), 85.
19 Cf. Tarski (1933), 187“8; Williamson (1994), 162; K. Taylor (1998), 116.
20 Cf. Peter de Rivo apud Baudry (1950), 94“5, 97“9, 221“3; Albritton (1957), 30, 33; McCall (1966),
270“3, 274“7; von Wright (1974), 174“7; (1984b), 5“10; Weidemann (1994/2002), 255“61; Mignucci
(1996b), 295“6.
278 The failure of Bivalence
at the later time t it will be the case that a, at t one can react to a statement
to the effect that at t it will be the case that a by saying ˜That is not yet
true™. Situations of this kind arise when one reacts to the worry that an as
yet uncertain future calamity could come about: e.g. Jim is on the verge
of bankruptcy and asserts that he is going to lose all his money, but Fred,
who optimistically thinks that matters might be mended, answers by saying
˜That is not yet true™.21 Moreover, if at t it is not yet certain whether at the
later time t it will be the case that a but at t , later than t but still earlier
than t , it becomes certain that at t it will be the case that a, at t one
can react to a statement to the effect that at t it will be the case that a
by saying ˜That is now true™ or ˜That has become true™. Situations of this
sort can arise when what one feared might fail to come about becomes
certain to be going to happen: e.g. Jim has been believing for a long time
that he was going to obtain his share of the inheritance, but only after the
legal uncertainties have been sorted out can his lawyer (who had previously
been reacting to Jim™s remarks by expressing doubt) say ˜Your belief is at
last true™. The fact that the notion discussed in the preceding subsections
corresponds to these uses of ˜. . . is true™ suggests that it is a notion of truth.
According to this argument, the ˜loaded™ notion of truth corresponds to
some tensed uses of ˜. . . is true™ while the ˜plain™ notion corresponds to
other (tensed or tenseless) uses of that phrase.22
The ˜loaded™ and the ˜plain™ notions of truth seem to be connected with
two different ways of regarding the world™s history. The ˜loaded™ notion is
connected to a viewpoint ˜inside™ the world™s history, at a certain stage of it
at which the past is uniquely determined while the future branches out in
several possible alternatives: according to the ˜loaded™ notion of truth, an
assertion is true at a certain time just in case it corresponds to the world at
the stage it has reached at that time. The ˜plain™ notion of truth is connected
with a viewpoint ˜outside™ the world™s history, which is ˜given™ in its totality:
according to the ˜plain™ notion, an assertion is true just in case it corresponds
to the whole world™s history.23 Note that the viewpoint corresponding to
the ˜loaded™ notion of truth, i.e. the viewpoint ˜inside™ the world™s history,
is likely to be that of an agent for whom it is important to regard the future
as at least partially ˜open™.
A further possible objection is that the ˜loaded™ notion of truth can be
ignored because the ˜plain™ notion of truth is more important. However,
this objection makes a dubious move: even granting that the ˜plain™ notion
21 Cf. McCall (1966), 273.
22 On the uses of ˜. . . is true™ corresponding to the ˜loaded™ notion of truth see von Wright (1974), 175.
23 Cf. Peter de Rivo apud Baudry (1950), 94“5, 97“9; Burrell (1964), 47“50; Hartshorne (1965), 58;
McCall (1966), 270“3, 274, 279“81; Jeffrey (1979), 253; M. J. White (1981), 232“5.
Appendix 6 279
of truth is more important, it does not follow that the ˜loaded™ notion can
be ignored. In natural language the tensed uses of ˜. . . is true™ that introduce
the ˜loaded™ notion are common: why should one ignore them? Codifying
the rules that govern their behaviour seems a worthwhile enterprise.

Truth-value gaps: (v) formalities. Let me now state rigorously the principles
which in the preceding subsections were only vaguely characterised:
[138] ∀x ∀t (Sx & Xt ’ (T *xt ” ∀s (Bs & t ∈ s ’ Cxts))).
[139] ∀x ∀t (Sx & Xt ’ (F *xt ” ∀s (Bs & t ∈ s ’ ¬Cxts))).

The two properties of truth coincide on present-tense predicative assertions:
[140] ∀x ∀t (Sn x & Xt ’ (Txt ” T *xt)).
Let a and b be such that Sn a & Xb. Since Sn a, ∀s ∀t (Bs & t ∈ s ’
(Cats ” Tat)) (by [126]) and Sa (by [105]). Since Sa & Xb, T *ab ” ∀s
(Bs & b ∈ s ’ Cabs) (by [138]).
Suppose that Tab. Let c be such that Bc & b ∈ c. Since ∀s ∀t (Bs &
t ∈ s ’ (Cats ” Tat)), Cabc ” Tab. Then Cabc. But c was arbitrary. Then
∀s (Bs & b ∈ s ’ Cabs). Hence T *ab. Thus Tab ’ T *ab. Vice versa,
suppose that T *ab. Then ∀s (Bs & b ∈ s ’ Cabs). Since Xb, ∃x (Bx &
b ∈ x) (by [123]). Let then d be such that Bd & b ∈ d. Then Cabd and Cabd
” Tab. Then Tab. Thus T *ab ’ Tab. Hence Tab ” T *ab. But a and b
were arbitrary. Then ∀x ∀t (Sn x & Xt ’ (Txt ” T *xt)), i.e. [140] holds.

Bivalence fails for ˜tomorrow™-assertions. Consider the tree of time partially
illustrated by the following diagram: m


Part of the information encoded by this diagram is that Xl & Xm & Xn &
Dlm & Dln & Dmn (i.e. l, m, and n are distinct times) & l < m & l < n.
Let j and k be such that Bj & l ∈ j & m ∈ j & Bk & l ∈ k & n ∈ k (i.e. j
is a branch through l and m, and k is a branch through l and n).
Let Wlm (i.e. m lies within the day after the day within which l lies).
Let c and d be such that Pcdm & ∀t (t ∈ k ’ ¬Pcdt) (i.e. c is predicated
of d at m, but at no time on branch k is c predicated of d ).
Let a and b be such that Sy b & It ab & Kcb & Hdb (i.e. a is a ˜tomorrow™-
assertion constructed from the singular af¬rmative present-tense predicative
280 The failure of Bivalence
assertion b, c is signi¬ed by the predicate of b, and d is signi¬ed by the subject
of b). Since Sy b, Sn b (by [58]). Then Sn b & It ab, so that ∃y (Sn y & It ay).
Then (by [101]) St a, so that (by [105]) Sa.
Since Sy b, ∃y (Hyb & ∀z (Hzb ’ z = y) & Iy) (by [62]). Let then e be
such that Heb & ∀z (Hzb ’ z = e) & Ie. Since Sn b, ∃y (Kyb & ∀z (Kzb
’ z = y) & Uy) (by [60]). Let then f be such that Kfb & ∀z (Kzb ’
z = f ) & Uf.
Suppose T *al. Since Sa & Xl, T *al ” ∀s (Bs & l ∈ s ’ Cals) (by [138]).
Then ∀s (Bs & l ∈ s ’ Cals). Since Bk & l ∈ k, Calk. Since Sn b & It ab, ∀s
∀t (Bs & t ∈ s ’ (Cats ” ∃u (u ∈ s & Wtu & Cbus))) (by [129]). Since
Bk & l ∈ k, Calk ” ∃u (u ∈ k & Wlu & Cbuk). Then ∃u (u ∈ k & Wlu
& Cbuk). Let then g be such that g ∈ k & Wlg & Cbgk. Since Sn b, ∀s ∀t
(Bs & t ∈ s ’ (Cbts ” Tbt)) (by [126]). Since Bk & g ∈ k, Cbgk ” Tbg.
Hence Tbg. Since Bk & g ∈ k, Xg (by [121]). Since Sy b and Xg, Tbg ” ∃y
∃z (Kyb & Hzb & Cs yzg) (by [70]). Hence ∃y ∃z (Kyb & Hzb & Cs yzg).
Let then h and i be such that Khb & Hib & Cs hig. Since Khb & Kcb & ∀z
(Kzb ’ z = f ), h = f & c = f, so that h = c; similarly, since Hib & Hdb
& ∀z (Hzb ’ z = e), i = e & d = e, so that i = d. Since Cs hig & h = c
& i = d, Cs cdg. Then (by [76]) Pcdg. Since g ∈ k & ∀t (t ∈ k ’ ¬Pcdt),
¬Pcdg, which yields a contradiction. Hence ¬T *al.
Suppose F *al. Since Sa & Xl, F *al ” ∀s (Bs & l ∈ s ’ ¬Cals) (by
[139]). Then ∀s (Bs & l ∈ s ’ ¬Cals). Since Bj & l ∈ j, ¬Calj. Since Pcdm,
Cs cdm (by [76]). Then Kcb & Hdb & Cs cdm, so that ∃y ∃z (Kyb & Hzb &
Cs yzm). Since Sy b and Xm, Tbm ” ∃y ∃z (Kyb & Hzb & Cs yzm) (by [70]).
Hence Tbm. Since Sn b, ∀s ∀t (Bs & t ∈ s ’ (Cbts ” Tbt)) (by [126]).
Since Bj & m ∈ j, Cbmj ” Tbm. Hence Cbmj. Since m ∈ j & Wlm &
Cbmj, ∃u (u ∈ j & Wlu & Cbuj). Since Sn b & It ab, ∀s ∀t (Bs & t ∈ s ’
(Cats ” ∃u (u ∈ s & Wtu & Cbus))) (by [129]). Since Bj & l ∈ j, Calj ” ∃u
(u ∈ j & Wlu & Cbuj). Then Calj, which contradicts a previous result.
Hence ¬F *al.
Thus: St a & Xl & ¬T *al & ¬F *al. In other words, Bivalence fails for
some ˜tomorrow™-assertion.
A similar argument, based on the same diagram, establishes that Biva-
lence fails for some future-tense predicative assertion: take a and b to be
such that Sy b & If ab & Kcb & Hdb, and establish that Sf a & Xl & ¬T *al
& ¬F *al.

Excluded Middle is valid: (i) general considerations. In what sense does Int. 9
endorse Excluded Middle? Surely not in the sense of claiming that every dis-
junctive assertion with contradictory disjuncts is always true, or necessarily
Appendix 6 281
true. Again, not in the sense of claiming, with regard to certain disjunctive
assertions with contradictory disjuncts, that they are always true, or neces-
sarily true “ e.g. Aristotle does not claim that utterances of ˜A sea-battle will
take place tomorrow or a sea-battle will not take place tomorrow™ are always
true, or necessarily true. In general, Int. 9™s endorsement of Excluded Mid-
dle does not involve mentioning disjunctive assertions with contradictory
Int. 9™s endorsement of Excluded Middle involves using some disjunctive
assertion whose disjuncts are future in¬‚ections of contradictory assertions
to state that something is necessary. Here is what Aristotle says:
It is necessary that everything should either be or not be, and either be going to
be or not [. . .]. For example, I mean that it is necessary that either there will be a
sea-battle tomorrow or there will not be one, but it is not necessary that tomorrow
a sea-battle should come to be nor that it should not come to be: however, it is
necessary that one should either come to be or not come to be. (19a 28“32 < T53)
Hence, to establish the coherence of Int. 9, it is not necessary to show
that every disjunctive assertion with contradictory disjuncts is always true,
or that every disjunctive assertion whose disjuncts are future in¬‚ections
of contradictory assertions is always true.24 However, showing that every
disjunctive assertion of the kinds described is always true suf¬ces to establish
the coherence of Int. 9. For, if every such disjunctive assertion is always true,
then in particular it is true before any given time. Then, on the basis of the
deterministic argument whose validity he accepts, Aristotle is justi¬ed in
using the disjunctive assertion to state antecedent necessity (ineluctability).
For instance, if an utterance of ˜Tomorrow a sea-battle will take place or
tomorrow a sea-battle will not take place™ is true now, Aristotle is justi¬ed
in saying that it is now necessary that either tomorrow a sea-battle will take
place or tomorrow a sea-battle will not take place. The present semantic
theory can show that every disjunctive assertion of the kinds described is
always true.

Excluded Middle is valid: (ii) formulation with ˜internal negation™. Aristotle
does not recognise ˜external negation™, i.e. an operator ˜it is not the case
that . . .™ whose utterances attach to complete sentences.25 Hence, although
the present semantic theory does have ˜external negation™, appealing to it
would hardly count as a vindication of Aristotle. However, in the present

24 Cf. Linsky (1954), 250“1; D. Frede (1970), 75“6; Sorabji (1980), 94; D. Frede (1985), 77; Weidemann
(1994/2002), 319.
25 However he does sometimes use ˜external negation™: see Int. 11, 20b 35“6.
282 The failure of Bivalence
semantic theory it can be shown that every disjunction whose disjuncts
are ˜tomorrow™-assertions in¬‚ected from contradictory present-tense pred-
icative assertions is always true (this includes utterances of ˜Tomorrow a
sea-battle will take place or tomorrow a sea-battle will not take place™). Here
are the precise formulation and the proof:
[141] ∀x ∀y ∀z ∀u ∀v (Iv xyz & It yu & It zv & ((Sa u & So v) ∨ (Se u & Si v) ∨
(Si u & Se v) ∨ (So u & Sa v) ∨ (Sy u & Sd v) ∨ (Sd u & Sy v)) & Ck uv &
Ch uv ’ ∀t (Xt ’ T *xt)).
Let a, b, c, d, and e be such that Iv abc & It bd & It ce & ((Sa d & So e) ∨ (Se d
& Si e) ∨ (Si d & Se e) ∨ (So d & Sa e) ∨ (Sy d & Sd e) ∨ (Sd d & Sy e)) & Ck de &
Ch de. Since (Sa d & So e) ∨ (Se d & Si e) ∨ (Si d & Se e) ∨ (So d & Sa e) ∨ (Sy d &
Sd e) ∨ (Sd d & Sy e), it follows that (Sa d ∨ Se d ∨ Si d ∨ So d ∨ Sy d ∨ Sd d ) &
(Sa e ∨ Se e ∨ Si e ∨ So e ∨ Sy e ∨ Sd e). Then Sn d & Sn e (by [58]). Then Sn d
& It bd, so that ∃y (Sn y & It by), whence St b (by [101]). Similarly, we have
Sn e & It ce, so that ∃y (Sn y & It cy), whence St c (again by [101]). Then (Sn b
∨ Sb b ∨ Sf b ∨ St b ∨ Sc b) & (Sn c ∨ Sb c ∨ Sf c ∨ St c ∨ Sc c) & Iv abc, so that
∃y ∃z ((Sn y ∨ Sb y ∨ Sf y ∨ St y ∨ Sc y) & (Sn z ∨ Sb z ∨ Sf z ∨ St z ∨ Sc z) &
Iv ayz). Then (by [103]) Sv a. Then (by [105]) Sa. Since Sn d & Sn e & Ck de &
Ch de, Ck ed & Ch ed (by [83]).
Let Xf. Let g be such that Bg & f ∈ g. Then ∃s (s ∈ g & Wfs) (by [125]).
Let then h be such that h ∈ g & Wf h. Since Bg & h ∈ g, Xh (by [121]).
Distinguish two cases.
(1) Suppose that Cehg. Then h ∈ g & Wf h & Cehg, so that ∃u (u ∈ g &
Wfu & Ceug). Since Sn e & It ce, ∀s ∀t (Bs & t ∈ s ’ (Ccts ” ∃u (u ∈ s &
Wtu & Ceus))) (by [129]). Since Bg & f ∈ g, Ccfg ” ∃u (u ∈ g & Wfu &
Ceug). Hence Ccfg, so that Cbfg ∨ Ccfg.
(2) Suppose that ¬Cehg. Since Sn e, ∀s ∀t (Bs & t ∈ s ’ (Cets ” Tet))
(by [126]). Since Bg & h ∈ g, Cehg ” Teh. Hence ¬Teh. Since (Sa d & So e)
∨ (Se d & Si e) ∨ (Si d & Se e) ∨ (So d & Sa e) ∨ (Sy d & Sd e) ∨ (Sd d & Sy e),
distinguish four subordinate cases. (2.1) If (Sa d & So e) ∨ (Se d & Si e), then
((Sa d & So e) ∨ (Se d & Si e)) & Ck de & Ch de. Then (by [90]) ∀t (Xt ’
(Tdt ” ¬Tet)). Since Xh, Tdh ” ¬Teh. Hence Tdh. (2.2) If (Si d
& Se e) ∨ (So d & Sa e), then ((Sa e & So d ) ∨ (Se e & Si d )) &
Ck ed & Ch ed. Then (again by [90]) ∀t (Xt ’ (Tet ” ¬Tdt)).
Since Xh, Teh ” ¬Tdh. Then ¬¬Tdh, whence Tdh. (2.3) If Sy d
& Sd e, then Sy d & Sd e & Ck de & Ch de. Then (by [95]) ∀t
(Xt ’ (Tdt ” ¬Tet)). Since Xh, Tdh ” ¬Teh. Hence Tdh. (2.4) If Sd d &
Sy e, then Sy e & Sd d & Ck ed & Ch ed. Then (again by [95]) ∀t (Xt ’ (Tet
” ¬Tdt)). Since Xh, Teh ” ¬Tdh. Then ¬¬Tdh, whence Tdh. Thus, in
Appendix 6 283
each of the four subordinate cases, Tdh. Since Sn d, ∀s ∀t (Bs & t ∈ s ’
(Cdts ” Tdt)) (by [126]). Since Bg & h ∈ g, Cdhg ” Tdh. Hence Cdhg.
Then h ∈ g & Wf h & Cdhg, so that ∃u (u ∈ g & Wfu & Cdug). Since Sn d
& It bd, ∀s ∀t (Bs & t ∈ s ’ (Cbts ” ∃u (u ∈ s & Wtu & Cdus))) (by [129]).
Since Bg & f ∈ g, Cbfg ” ∃u (u ∈ g & Wfu & Cdug). Hence Cbfg, so that
Cbfg ∨ Ccfg.
Thus, in all cases, Cbfg ∨ Ccfg. Since (Sn b ∨ Sb b ∨ Sf b ∨ St b ∨ Sc b) &
(Sn c ∨ Sb c ∨ Sf c ∨ St c ∨ Sc c) & Iv abc, it follows (by [131]) that ∀s ∀t (Bs
& t ∈ s ’ (Cats ” Cbts ∨ Ccts)). Since Bg & f ∈ g, Cafg ” Cbfg ∨ Ccfg.
Hence Cafg. But g was arbitrary. Then ∀s (Bs & f ∈ s ’ Cafs). Since Sa
& Xf, T *af ” ∀s (Bs & f ∈ s ’ Cafs) (by [138]). Then T *af. But f was
arbitrary. Then ∀t (Xt ’ T *at). But a, b, c, d, and e were arbitrary. Then
∀x ∀y ∀z ∀u ∀v (Iv xyz & It yu & It zv & ((Sa u & So v) ∨ (Se u & Si v) ∨ (Si u
& Se v) ∨ (So u & Sa v) ∨ (Sy u & Sd v) ∨ (Sd u & Sy v)) & Ck uv & Ch uv ’
∀t (Xt ’ T *xt)), i.e. [141] holds.
Similarly, every disjunction whose disjuncts are future-tense predicative
assertions in¬‚ected from contradictory present-tense predicative assertions
is always true:
[142] ∀x ∀y ∀z ∀u ∀v (Iv xyz & If yu & If zv & ((Sa u & So v) ∨ (Se u & Si v)
∨ (Si u & Se v) ∨ (So u & Sa v) ∨ (Sy u & Sd v) ∨ (Sd u & Sy v)) & Ck uv
& Ch uv ’ ∀t (Xt ’ T *xt)).
The proof of [142] resembles that of [141].

Excluded Middle is valid: (iii) formulation with ˜external negation™. Since
Aristotle does not recognise ˜external negation™, proving the validity of
Excluded Middle for assertions formulated with ˜external negation™ goes
beyond the range of tasks which Aristotle can be taken to set for himself.
It is, none the less, worthwhile stating such a result:
[143] ∀x ∀y ∀z ∀u ((Sn y ∨ Sb y ∨ Sf y ∨ St y) & (Sn u ∨ Sb u ∨ Sf u ∨ St u) &
Oyu & Ic zu & Iv xyz ’ ∀t (Xt ’ T *xt)).
The proof, which is straightforward, is omitted.

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