Future Actuality and Truth Ascriptions

Abstract

One question that arises in connection with Ockhamism, and that perhaps has not yet received the attention it deserves, is how a coherent formal account of truth ascriptions can be provided by using a suitable truth predicate in the object language. We address this question and show its implications for some semantic issues that have been discussed in the literature on future contingents. Arguably, understanding how truth ascriptions work at the formal level helps to gain a deeper insight into Ockhamism itself.

1. Overview

According to Ockhamism, every future contingent is either true or false, because either it is true in the actual future or it is false in the actual future. That is, for any future contingent $\alpha$ and any time t, truth is defined as follows:

Definition 1.

α is true at t iff α is true at t in the actual future.

For example, the following sentence as uttered today is either true or false:

(1)

Tomorrow will be sunny

Whether it is true or false depends on whether it will actually be sunny tomorrow. This idea goes back to Ockham’s doctrine of divine foreknowledge. Ockham famously claimed that the truth value of every future contingent is known to God, as it depends on what happens in the “true” future. The “true” future is nothing but the actual future, that is, the future part of the actual course of events (See [1], pp. 97–102).

A key feature of Ockhamism, which sets it apart from other views of future contingents, is that it draws a distinction between truth and determinate truth: while the former is understood as truth in the actual future, the latter is understood as truth in all possible futures. According to Definition 1, $\alpha$ is true at t if it is true at t in the actual future. But the actual future is just one among the many futures that are possible at t, so the truth of $\alpha$ at t is consistent with its falsity at t in some non-actual future. For example, even if (1) is true, this does not mean that it is determinately true, because there are possible futures in which it is false.

Ockhamism has been widely discussed in the last three decades, and some attempts have been made to provide a rigorous formulation of Definition 1 in order to elucidate its logical implications. The formal apparatus that has been adopted for this purpose may roughly be described as follows: the language is a propositional language endowed with temporal operators, and its semantics is some version of branching time semantics in that it is based on a set M of moments and a partial order < on M that determines a set of histories, that is, maximal linearly ordered sets of moments. Formulas are thus evaluated relative to moment–history pairs: a moment–history pair $m/h$ is formed by a moment m and a history h such that $m\in h$. In particular, discussions of Ockhamism have typically employed a version of branching time semantics that may be called TRL semantics, by defining models where one of the histories is marked as the actual history, or the Thin Red Line.1

The distinction between standard branching time semantics and TRL semantics can be illustrated by means of the following two figures (Figure 1 and Figure 2).

Figure 1 displays a simple tree diagram in which two histories, $h_1$ and $h_2$, split after the moment $m_0$, which may be taken to represent the present moment. Figure 2 differs from Figure 1 in that one of the two histories is marked as the actual history.

The formal apparatus just illustrated has one obvious advantage, that is, it makes Ockhamism easily comparable with other views of future contingents which are formulated by using the same kind of language and the same kind of semantics. Notably, Thomason’s supervaluationism, Belnap’s double-time reference approach, and MacFarlane’s relativism are formulated by using the same kind of language and by adopting branching time semantics [3,4,6].

However, this formal apparatus also has some limitations, which make it potentially misleading in some respects. One of them is that its language can only represent truth-free sentences, that is, it is unable to represent sentences in which truth is explicitly ascribed to other sentences. There are at least two distinct categories of such ascriptions. The first includes simultaneous ascriptions like the following:

(2)

Sentence (1) as uttered today is true

These are sentences in which truth is ascribed to a future contingent $\alpha$ as uttered at a time t from a vantage point that is t itself or in any case precedes the time of the event described in $\alpha$. The second includes retrospective ascriptions, that is, sentences in which truth is ascribed to a future contingent $\alpha$ as uttered at a time t from a vantage point that is simultaneous with or later than the time of the event described in $\alpha$. For example, if (1) is uttered today, and tomorrow is sunny, tomorrow one can assert what follows:

(3)

Sentence (1) as uttered yesterday was true

Note that in a branching time perspective, the word ‘tomorrow’ as used above does not pick up a single moment, because there are different courses of events that overlap until today and lead to different tomorrows. For example, the moments $m_1$ and $m_2$ in Figure 1 and Figure 2 represent two distinct referents for ‘tomorrow’. More generally, retrospective ascriptions can be made from vantage points located in different courses of events, that is, moments belonging to different histories.

In the literature on future contingents, truth ascriptions of the two kinds just illustrated have often been used as an intuitive test for formal accounts of truth-free sentences such as (1). However, they have been discussed mainly at the metalinguistic level, that is, without addressing the question of how truth ascriptions can be represented in the same object language in which truth-free sentences are formalized. One thing is to ask how the truth conditions of a formula that represents (1) can be defined in accordance with the claims expressed by (2) or (3), quite another thing is to ask how (2) and (3) themselves can adequately be formalized.

This paper focuses precisely on the formalization of truth ascriptions. Its aim is to show how Ockhamism can provide a coherent formal account of truth ascriptions by means of a suitable truth predicate in the object language and to explain how such account helps to gain a deeper insight into Ockhamism itself. We believe that the formal apparatus adopted so far, due to its expressive limitations, may leave room for confusions and misconceptions that could be avoided if truth ascriptions were adequately formalized.

The structure of the paper is as follows. Section 2 presents two objections against Ockhamism that suffer from the expressive limitations mentioned above. Section 3 outlines some basic principles about truth that we regard as highly plausible from an Ockhamist perspective. Section 4 and Section 5 set out the syntax and semantics of a first-order language in which truth ascriptions can be formalized. Section 7 shows how the principles about truth outlined can be expressed in this language. Finally, Section 8 completes the discussion of the two objections and provides some conclusive remarks.

2. Two Objections Against Ockhamism

In order to appreciate the relevance of an object-linguistic treatment of truth ascriptions, it may be helpful to consider two objections that have been raised against Ockhamism and that are based on a purely metalinguistic discussion of a formal apparatus of the kind mentioned above.

The first objection, due to Belnap and Green, focuses on a version of TRL semantics that may be called absolute TRL, as it assumes that truth at a moment is defined in terms of a distinguished history that is fixed once and for all in the model. Let $\langle M,<,A,V\rangle$ be a model where M is a set of moments, < is a partial order on M, A is a constant function from M to the set of histories, and V is a valuation function that assigns values to atomic formulas relative to moment–history pairs. Let h the unique value of A. In such a model, truth at a moment may be defined as follows: $\alpha$ is true at m if and only if $\alpha$ is true at $m/\mathbf{h}$. Consider Figure 2, where $h_1=\mathbf{h}$. Let us assume that $m_0$ represents today and that $m_1$ and $m_2$ represent two alternative versions of tomorrow such that it is sunny at $m_1$ and it is rainy at $m_2$. As long as one considers (1) as uttered today, this definition yields the expected result. That is, (1) as uttered today turns out to be true. However, Belnap and Green contend that absolute TRL wreaks havoc when it comes to evaluating formulas at moments that do not belong to h. Consider the following sentence:

(4)

Today is rainy

Clearly, we want (4) to be true at $m_2$. But the definition considered entails that (4) is true at a moment m only if $m\in \mathbf{h}$, for otherwise there is no such thing as the moment–history pair $m/\mathbf{h}$. On the assumption that falsity amounts to absence of truth, we get that (4) is false at $m_2$, which seems unacceptable ([2], p. 379, [3], pp. 162–163).

The second objection, due to MacFarlane, targets a different variant of TRL semantics which may be called relative TRL, as it leaves room for the possibility that distinct moments are associated with distinct Thin Red Lines. Let $\langle M,<,R,V\rangle$ be a model where M, <, and V are as above, and R is a function from moments to histories that associates each moment m to the corresponding Thin Red Line $R\left(m\right)$. In relative TRL, truth at a moment is defined as follows: $\alpha$ is true at m if and only if $\alpha$ is true at $m/R\left(m\right)$. Let us consider again Figure 2 as interpreted above and let us assume that $R\left(m_0\right)=h_1$. Suppose that today Jake asserts (1). MacFarlane asks us to imagine two assessors, one located at $m_1$ and the other located at $m_2$. The first assessor regards Jake’s assertion as accurate. But what about the second?

As before, the assessor should take Jake to have spoken accurately just in case (1) is true at $m_0$. Since, according to the Thin Red Line view, (1) is true at $m_0$, the assessor should take Jake to have spoken accurately. But that seems wrong; the assessor has only to feel the rain on her skin to know that Jake’s assertion was inaccurate.2

In a nutshell, MacFarlane’s problem is that the definition considered entails that an assessor located at $m_2$ should take (1) to be true as uttered the day before, at $m_0$, against strong semantic intuitions.

The two objections just presented rely on the assumption that TRL semantics is the most plausible way, if not the only way, to make sense of Ockhamism. This assumption is questionable. Ockhamism is not committed to branching time semantics as a formal representation of indeterminism, and can be combined equally well with alternative semantics.3 Moreover, even assuming that a branching time model should be employed, it not essential to Ockhamism that the actual history is represented in the model as a distinguished history, for all is needed to make sense of the right-hand side of Definition 1 is that the context in which a future contingent is uttered picks up one of the histories represented in the model as the actual history.4

Independently of any issue concerning the relation between Ockhamism and TRL semantics, there is a second feature that the two objections share and that is directly relevant to our main point: in both cases, the language considered lacks the expressive resources to formalize truth ascriptions. So, although truth-free sentences such as (1) can be formalized in the language, the ascriptions of truth to such sentences cannot. This means that, strictly speaking, from the objections discussed so far, nothing follows about the truth value of the formal counterparts of (2) and (3) in that semantics, given that there is no such counterpart.

The objectors might think that it is enough for them to show that TRL semantics, when applied to a simple propositional language with temporal operators, yields wrong predictions about plain future contingents, so that it is not necessary to engage with a more complex language. However, we do not agree with them on such predictions, and we believe that dealing with a more complex language may help to spell out the implications of Ockhamism. The aim of the following sections is to show that, once a truth predicate is defined in the object language in accordance with some plausible constraints, it turns out that Ockhamism yields plausible predictions about future contingents and the ascriptions of truth to them.

3. Basic Properties of Truth

It is generally taken for granted that sentences can be assessed as true or false relative to contexts, in that they express truth-evaluable contents—that is, propositions—relative to contexts. When a sentence $\alpha$ is uttered in a context c, it expresses a proposition in c, that is, it describes things as being a certain way. Therefore, $\alpha$ is true in c if and only if things are that way. To say that $\alpha$ is true in c is to say that $\alpha$ as uttered in c is true, or equivalently that the utterance constituted by $\alpha$ and c is true.

A context is a set of parameters, which may include a variety of elements. Since in our case the relevant parameters are times and worlds, contexts can be identified with time–world pairs. According to Ockhamism, to say that a sentence $\alpha$ as uttered in a context c is true is to say that $\alpha$ is true at the time of c in the world of c, that is, in the course of event that is fixed by c as the actual course of events. For example, (1) as uttered today is true if and only if it is true today in the actual future, that is, in the future of the actual world.

This does not mean that truth ascriptions must include expressions that refer to time–world pairs. In ordinary language, truth ascriptions almost never contain explicit reference to the world of the context, and in this respect they behave exactly like truth-free sentences. Examples (2) and (3) provide a clear illustration of this fact, for they contain only temporal adverbs, just like (1). As far as the syntax of truth ascriptions is concerned, it is plausible to assume that time is the only explicit contextual parameter, so that truth ascriptions are sentences of the form ‘$\alpha$ is true at t’. A sentence of this form can be evaluated relative to contexts just like any truth-free sentence.

As long as truth ascriptions are phrased in the way just suggested, a principle about truth can be formulated as a general statement that holds for every sentence $\alpha$ at every time t. In particular, the following three principles are highly plausible from an Ockhamist perspective:

P1 

Either $\alpha$ is true at t or $\neg \alpha$ is true at t;

P2 

$\alpha$ is true at t iff it is true at t that $\alpha$ is true at t;

P3 

$\neg \alpha$ is true at t iff it is true at t that $\alpha$ is not true at t.

P1 conveys the idea that bivalence holds for any sentence at any time, as it says that either $\alpha$ of its negation $\neg \alpha$ is true at t. Since the truth of $\neg \alpha$ can be equated with the falsity of $\alpha$, P1 can be rephrased as follows: either $\alpha$ is true at t or $\alpha$ is false at t, which means that there are no truth value gaps. From P1 we get that every future contingent is either true or false, the claim that characterizes Ockhamism as distinct from other views of future contingents.

Note, among other things, that as long as bivalence holds for any sentence at any time, we get that truth differs from determinate truth, provided that the latter is understood as truth in all possible futures. For if $\alpha$ is a future contingent, neither $\alpha$ nor $\neg \alpha$ is determinately true at t, while P1 implies that at least one of them is true in t.

P2 and P3 express a property of truth that is widely regarded as fundamental and plays a key role in theories of truth, namely transparency: every sentence $\alpha$ is equivalent to a sentence in which truth is ascribed to $\alpha$ itself. In particular, P2 says that ascribing truth to $\alpha$ at t is tantamount to ascribing truth to the statement that $\alpha$ is true at t. Similarly, P3 says that ascribing truth to $\neg \alpha$ at t is tantamount to ascribing truth to the negation of the statement that $\alpha$ is true at t, or equivalently that ascribing falsity to $\alpha$ at t is tantamount to ascribing falsity to the statement that $\alpha$ is true at t.

In the current debate on future contingents, P2 and P3 can hardly be regarded as uncontentious, since transparency does not hold for some views of future contingents. For example, if truth is understood as truth in all possible futures, as implied by supervaluationism, the truth conditions of a sentence $\alpha$ at a context c turn out to differ from the truth conditions of ‘$\alpha$ is true at t’ at c, where t is the time of c. Ockhamism instead preserves transparency, as it implies that ‘$\alpha$ is true at t’ is true at c just in case $\alpha$ itself is true at c. We regard this as a desirable feature of Ockhamism.

The next two sections set out a first-order language in which P1–P3 can be expressed by means of a truth predicate that applies to formulas relative to times. The semantics of this language is a first-order version of branching time semantics where contexts are identified with moment–history pairs, which amount to time–world pairs. In other words, to say that $\alpha$ is true in c is to say that, for some moment m and some history h, m represents the time of c, h represents the world of c, and $\alpha$ is true at $m/h$.

4. The Language $\mathsf{L}$: Syntax

The languages usually adopted in theories of truth are first-order languages in which a monadic truth predicate is applied to terms that denote closed formulas through some sort of coding. Here, instead, in order to take into account the contextual variation explained above, we use a dyadic predicate T that applies to pair of terms such that one denotes a formula and the other denotes a moment. That is, if $⌜\alpha ⌝$ denotes a formula $\alpha$ and t denotes a moment m, $T(⌜\alpha ⌝,t)$ expresses the statement that $\alpha$ is true at m. For the rest, the language we need is like those usually adopted in theories of truth, that is, it is assumed to have at least the expressive resources of the language of Peano Arithmetic (PA).

Definition 2.

$\mathsf{L}$ is a first-order language that satisfies the following conditions:

1. 

The vocabulary of $\mathsf{L}$ includes the symbols of PA;

2. 

$\mathsf{L}$ has two functions symbols s and p;

3. 

$\mathsf{L}$ has a binary predicate T;

4. 

$\mathsf{L}$ has a closed term for any formula.

Condition 1 implies that the vocabulary of $\mathsf{L}$ includes the symbols of PA. More specifically, we assume that the logical constants of $\mathsf{L}$ are $\neg ,\wedge ,\vee ,\forall ,\exists$ plus the symbol of identity.

Condition 2 adds to the symbols of PA two function symbols s and p that can be combined with terms that refer to moments. More specifically, s is used to express a temporal version of the successor operation in arithmetic in that it assigns to each moment m the moment that lies one unit after m along a given history. Inversely, p is used to express a temporal version of the predecessor operation in arithmetic in that it assigns to each moment m the moment that lies one unit before m along a given history. Just as the successor symbol in PA, s and p can be iterated any number of times, as in $ss\left(t\right)$, $sss\left(t\right)$, and so on.

Condition 3 requires that $\mathsf{L}$ has the predicate T in addition to the logical constants specified in Condition 1. This is the symbol that enables us to construct formulas that express truth predications. Although a falsity predicate turns out to be definable in terms of T along the lines suggested in Section 3, for the sake of simplicity here we only use T.

Condition 4 guarantees that $\mathsf{L}$ contains names of formulas that can be used in the formulas containing T. One way to state this condition is the following: there is a function $⌜⌝$ that applies to formulas of $\mathsf{L}$ in such a way that, for every formula $\alpha$, $⌜\alpha ⌝$ is a closed term of $\mathsf{L}$, so it can occur as the first term in a formula containing T.

The formation rules of $\mathsf{L}$ are as follows:

Definition 3.

1 

If P is a n-place predicate and $t_1,\cdots ,t_n$ are terms, $P{t}_1,\cdots ,t_n$ is a formula;

2 

If α is a formula, $\neg \alpha$ is a formula;

3 

If α and β are formulas, $\alpha \wedge \beta$ and $\alpha \vee \beta$ are formulas;

4 

If α is a formula and x is a variable, $\forall x\alpha$ and $\exists x\alpha$ are formulas;

5 

If α is a fomula and t is a term, $T(⌜\alpha ⌝,t)$ is a formula.

Clauses 1–4 imply that $\mathsf{L}$ includes the language of PA. As usual, ⊃ and ≡ are definable in terms of ¬ and ∨. Clause 5 provides the formation rule for the formulas of the form $T(⌜\alpha ⌝,t)$, which represent truth ascriptions. Note that t can denote any object, while in order to express a truth ascription, t must denote a moment. This distinction, however, can be drawn only at the semantic level when the satisfaction conditions of $T(⌜\alpha ⌝,t)$ are specified.

A final remark about Definitions 2 and 3 is that they do not include modal or temporal operators. Modal operators are not strictly necessary for the point we want to make about truth ascriptions, although they could be added to $\mathsf{L}$ so as to obtain a richer language capable of expressing modal notions such as determinate truth. Temporal operators are a different issue. In a first-order language, it is possible to formalize tensed sentences by using formulas that involve quantification over times or function symbols such as s and p. For example, (1) can be paraphrased as ‘For some t such that t is one day later than the present time, it is sunny at t’. According to this analysis, call it the extensional analysis, tensed sentences are adequately represented by using the quantificational apparatus of first-order logic, without using temporal operators.5 Although the extensional analysis is not universally accepted, it will do no harm to adopt it here as a background hypothesis. In any case, if one were unwilling to grant it, one could simply consider an augmented version of $\mathsf{L}$ obtained by adding the usual temporal operators.6

5. The Language $\mathsf{L}$: Semantics

In order to provide an interpretation for $\mathsf{L}$, we take for granted the standard interpretation of the language of PA, which implies that $\mathsf{L}$ has at least one countable acceptable model with domain U.7 We assume that U is extended by adding to it a denumerable set of moments M, and that a partial order is defined on M in the usual way so as to obtain a set of histories. Let us start with the following definition:

Definition 4.

A frame $\mathcal{F}$ for $\mathsf{L}$ is a pair $\langle M,<\rangle$, where M is a denumerable set and < is a partial order on M.

Since we want to represent possible courses of events that have an infinite extension, we restrict consideration to frames in which each history is itself denumerable. In such frames, each history h satisfies the following condition: for each $m\in h$, there is $m^{\prime }\in h$ such that $m<m^{\prime }$ and for every $m^{″}\in h$ such that $m^{″}\ne m^{\prime }$ and $m<m^{″}$, $m^{\prime }<m^{″}$. That is, the set of moments that follow m in h has a least element. We call it the successor of m in h. Similarly, for each $m\in h$, there is $m^{\prime }\in h$ such that $m^{\prime }<m$ and for every $m^{″}\in h$ such that $m^{″}\ne m^{\prime }$ and $m^{″}<m$, $m^{″}<m^{\prime }$. That is, the set of moments that precede m in h has a greater element. We call it the predecessor of m in h.

Definition 5.

A model $\mathcal{M}$ for $\mathsf{L}$ is a triple $\langle \mathcal{F},D,I\rangle$, where

• $\mathcal{F}$ is a frame $\langle M,<\rangle$;
• $D=U\cup M$;
• I is a function such that

  – 

  for each individual constant t, $I\left(t\right)\in D$;

  – 

  for each n-place function symbol f other than s and p, $I\left(f\right)$ is a function F from $D^n$ to D;

  – 

  $I\left(s\right)$ is a function that assigns to each history h a function $s_h$ such that, for each $m\in h$, $s_h\left(m\right)$ is the successor of m in h;

  – 

  $I\left(p\right)$ is a function that assigns to each history h a function $p_h$ such that, for each $m\in h$, $p_h\left(m\right)$ is the predecessor of m in h;

  – 

  for each n-place predicate P, $I\left(P\right)$ is a function that assigns to each history h a set $I{\left(P\right)}_h$ of n-tuples of elements of D.

D is the domain of $\mathcal{M}$. This set is denumerable, given that U and M are both denumerable. I is the interpretation function of $\mathcal{M}$. This function assigns objects of D to the individual constants and operations on D to the standard function symbols, as usual. Moreover, I interprets the function symbols s and p in the way suggested above: for each history h, $I\left(s\right)$ yields a function that assigns to each moment its successor, and $I\left(p\right)$ yields a function that assigns to each moment its predecessor. Finally, for each history, I assigns to each n-place predicate a set $I{\left(P\right)}_h$ of n-tuples of elements of D. When P stands for a predicate that does not contain any ‘trace of futurity’, one can assume that $I{\left(P\right)}_h=I{\left(P\right)}_{h^{\prime }}$ for any $h^{\prime }$. For example, if P stands for ‘is red’, the set of objects that satisfy P is the same for any history.

Definition 6.

For any model $\mathcal{M}=\langle \mathcal{F},D,I\rangle$, an assignment σ is a function such that, for each variable x, $\sigma \left(x\right)\in D$.

An assignment so defined provides a denotation for the variables of $\mathsf{L}$. Since the denotation of the individual constants of $\mathsf{L}$ is fixed by I, we get that every simple term of $\mathsf{L}$ has a denotation relative to an assignment, and the same goes for every complex term that contains function symbols other than s and p. However, the denotation of the complex terms that contain occurrences of s and p remains indeterminate, for these terms can have denotation only relative to histories. To see this, it suffices to think that one and the same moment m can lie in two distinct histories $h_1$ and $h_2$ such that successor of m in $h_1$ differs from the successor of m in $h_2$. In order to fix a denotation for every term of $\mathsf{L}$, we need to consider relativity to histories in addition to relativity to assignments.

Definition 7.

For any model $\mathcal{M}=\langle \mathcal{F},D,I\rangle$, the denotation of a term t relative to an assignment $\sigma$ in a history h, indicated as ${\left[t\right]}_{\sigma ,h}$, is as follows:

• if t is an individual constant, ${\left[t\right]}_{\sigma ,h}=I\left(t\right)$;
• if t is a variable, ${\left[t\right]}_{\sigma ,h}=\sigma \left(t\right)$;
• if t is a complex term $f(t_1,\dots ,t_n)$, ${\left[t\right]}_{\sigma ,h}=F({\left[t_1\right]}_{\sigma ,h},\dots ,{\left[t_n\right]}_{\sigma ,h})$;
• if t is a complex term $s\left(t^{\prime }\right)$, ${\left[t\right]}_{\sigma ,h}=s_h\left(\left[t^{\prime }\right]\right)$ if $\left[t^{\prime }\right]\in M$, and ${\left[t\right]}_{\sigma ,h}={\left[t^{\prime }\right]}_{\sigma ,h}$ otherwise;
• if t is a complex term $p\left(t^{\prime }\right)$, ${\left[t\right]}_{\sigma ,h}=p_h\left(\left[t^{\prime }\right]\right)$ if $\left[t^{\prime }\right]\in M$, and ${\left[t\right]}_{\sigma ,h}={\left[t^{\prime }\right]}_{\sigma ,h}$ otherwise;

Individual constants, variables, and complex terms that do not contain s or p are treated in the usual way, with the only difference that their denotation is relativized to h. As to s and p, their denotation is fixed relative to h in terms of $s_h$ and $p_h$. When $t^{\prime }$ does not denote a moment, the occurrence of s and p is inert, in that it has no effect on the denotation of the term. For example, if x denotes an object that is not a moment and P stands for ‘is red’, there is no relevant semantic difference between $Px$ and $Ps\left(x\right)$: to say that the successor of x is red is just another way to say that x is red. The idea is that the notion of temporal order can apply only vacuously to objects that are not temporal units.

Now we can provide a definition of satisfaction of a formula by an assignment at a moment–history pair in a model:

Definition 8.

1 

 σ satisfies $P{t}_1,\dots ,t_n$ at $m/h$ if ${\left[t_1\right]}_{\sigma ,h},\cdots {\left[t_n\right]}_{\sigma ,h}\in I{\left(P\right)}_h$;

2 

 σ satisfies $\neg \alpha$ at $m/h$ if it does not satisfy α at $m/h$;

3 

 σ satisfies $\alpha \wedge \beta$ at $m/h$ if it satisfies α and β at $m/h$;

4 

 σ satisfies $\alpha \vee \beta$ at $m/h$ if it satisfies α or β at $m/h$;

5 

 σ satisfies $\forall x\alpha$ at $m/h$ if every x-variant of σ satisfies α at $m/h$;

6 

 σ satisfies $\exists x\alpha$ at $m/h$ if some x-variant of σ satisfies α at $m/h$;

7 

 σ satisfies $T(⌜\alpha ⌝,t)$ at $m/h$ if ${\left[t\right]}_{\sigma ,h}\in M$ and σ satisfies α at ${\left[t\right]}_{\sigma ,h}/h$.

Clause 1 defines satisfaction at $m/h$ for a truth-free atomic formula $P{t}_1,\cdots ,t_n$. For example, if P is a monadic predicate that stands for ‘is red’, $\sigma$ satisfies $Px$ at $m/h$ if the object it assigns to x belongs to the extension that I assigns to P relative to h, that is, the set of objects that are red in h.

Clauses 2–6 are standard, as they define in the usual way the satisfaction conditions for the formulas that contain the connectives $\neg ,\wedge ,\vee ,\forall ,\exists$ using moment–history pairs as parameters.

Clause 7 is the crucial one as it concerns the formulas of the form $T(⌜\alpha ⌝,t)$. Provided that t refers to a moment $m^{\prime }$, this clause says that the satisfaction condition of $T(⌜\alpha ⌝,t)$ at $m/h$ is the same as the satisfaction condition of $\alpha$ at $m^{\prime }/h$. This includes the case in which $m^{\prime }=m$, but such identity is not required: it can happen that $m^{\prime }\ne m$. Informally speaking, the equivalence at a given context between $\alpha$ and the ascription of truth to $\alpha$ holds at any context along the same history.

The definition of truth of a formula at a moment–history pair in a model is as follows, using the notation ${\left[\alpha \right]}_{m/h}$ to indicate the value of $\alpha$ at $m/h$:

Definition 9.

${\left[\alpha \right]}_{m/h}=1$ iff every assignment satisfies α at $m/h$.

Falsity is defined in similar way:

Definition 10.

${\left[\alpha \right]}_{m/h}=0$ iff no assignment satisfies α at $m/h$.

Definitions 9 and 10 yield that, for any closed formula $\alpha$ and any pair $m/h$, either every assignment satisfies $\alpha$ at $m/h$ or no assignment satisfies $\alpha$ at $m/h$, which means that either ${\left[\alpha \right]}_{m/h}=1$ or ${\left[\alpha \right]}_{m/h}=0$. In particular, for any formula $T(⌜\alpha ⌝,t)$ such that $\alpha$ is a closed formula and t is a closed term, we have that either every assignment $\sigma$ satisfies $\alpha$ at ${\left[t\right]}_{\sigma ,h}/h$ or that no assignment satisfies $\alpha$ at ${\left[t\right]}_{\sigma ,h}/h$, hence that the same goes for $T(⌜\alpha ⌝,t)$.

Finally, logical consequence, indicated by the symbol ⊧, is defined in the usual way in terms of satisfaction:

Definition 11.

For any set of formulas Γ and any formula α, we have Γ $\vDash \alpha$ if for every assignment σ and every moment–history pair $m/h$ in any model, if σ satisfies all the formulas in Γ at $m/h$, then σ satisfies α at $m/h$.

Validity turns out to be a special case of logical consequence: a formula $\alpha$ is valid, that is, $\vDash \alpha$ just in case for every assignment $\sigma$ and every moment–history pair $m/h$ in any model, $\sigma$ satisfies $\alpha$ at $m/h$.

6. Semantic Paradoxes

As is well known, any attempt to define a truth predicate for a language with the expressive resources of $\mathsf{L}$ has to face the problem of semantic paradoxes. In particular, $\mathsf{L}$ licenses the construction of a temporal version of the liar, that is, a sentence $\lambda$ that turns out to be true at t if and only if it is not true at t. Among the strategies that might be adopted to deal with this problem, one that seems especially suited for our purposes is the definition method developed by Kripke.

Kripke suggests a way of defining an extension and an anti-extension for a modadic truth predicate that satisfies the intuitive condition of transparency. He constructs the two sets in stages, starting from the truth-free atomic formulas that are satisfied, or not satisfied, in the base model, and then defining each stage as the result of applying a chosen valuation schema to the formulas in the previous stage. The process of building more and more stages eventually stabilizes because it reaches a fixed point, that is, it defines a set such that any further application of the valuation schema leaves it unchanged.

A similar definition can be provided for T, with two crucial differences. First, it must be taken into account that T is dyadic rather than monadic. Second, a plurality of points of evaluation must be contemplated instead of a single one, given that the models of $\mathsf{L}$ include a denumerable set of moments. The semantics needed is essentially a generalization of the original Kripkean fixed-point semantics: instead of there being a single fixed point, there is a plurality of fixed points that serve as moments. Once a model for $\mathsf{L}$ is appropriately defined by using moments so constructed, we obtain the desired result that $\lambda$ belongs neither to the extension nor to the anti-extension of T. In Kripke’s terminology, the sentences in the extension and anti-extension of T are grounded, while $\lambda$ is ungrounded.

As in Kripke’s original construction, we expect that the resulting logic is a non-classical trivalent logic where $\lambda$ always takes the third value, but that classical logic is preserved in the realm of grounded sentences. In particular, the inductive definition that generates the fixed points should be such that the notions of satisfaction, truth, and logical consequence that hold for grounded sentences turn out to be exactly as in Definitions 8–11.

Since our discussion focuses on future contingents, which are represented by formulas of $\mathsf{L}$ that belong to the realm of grounded sentences, the issue of paradoxes can be left aside. In the rest of the paper, we restrict consideration to grounded sentences, assuming that their semantic properties are those outlined in Section 5.

7. P1–P3 Formalized

In order to show that $\mathsf{L}$ is adequate for the purpose of formally representing truth ascriptions in an Ockhamism framework, we show that the semantics of $\mathsf{L}$ vindicates a formal version of the principles P1–P3 outlined in Section 3.

Let us consider the following schemas, which are the most obvious formal counterparts of P1–P3 in $\mathsf{L}$:

P1f 

$T(⌜\alpha ⌝,t)\vee T(⌜\neg \alpha ⌝),t)$

P2f 

$T(⌜T(⌜\alpha ⌝,t)⌝,t)\equiv T(⌜\alpha ⌝,t)$

P3f 

$T(⌜\neg T(⌜\alpha ⌝,t)⌝,t)\equiv T(⌜\neg \alpha ⌝,t)$

The following holds about P1f–P3f for any model of $\mathsf{L}$ as a direct consequence of the definitions in Section 5.

Proposition 1.

For any assignment σ and any moment–history pair $m/h$, if ${\left[t\right]}_{\sigma ,h}\in M$, σ satisfies $T(⌜\alpha ⌝,t)\vee T(⌜\neg \alpha ⌝),t)$ at $m/h$.

Proof. 

Either $\sigma$ satisfies $\alpha$ at ${\left[t\right]}_{\sigma ,h}/h$ or it does not. By Clause 2 of Definition 8 it follows that either $\sigma$ satisfies $\alpha$ at ${\left[t\right]}_{\sigma ,h}/h$ or $\sigma$ satisfies $\neg \alpha$ at ${\left[t\right]}_{\sigma ,h}/h$. By Clause 7 of Definition 8, this means that either $\sigma$ satisfies $T(⌜\alpha ⌝,t)$ at $m/h$ or $\sigma$ satisfies $T(⌜\neg \alpha ⌝,t)$ at $m/h$. Therefore, by Clause 4 of Definition 8, $\sigma$ satisfies $T(⌜\alpha ⌝,t)\vee T(⌜\neg \alpha ⌝,t)$ at $m/h$. □

Proposition 2.

For any assignment σ and any moment–history pair $m/h$, if ${\left[t\right]}_{\sigma ,h}\in M$, σ satisfies $T(⌜T(⌜\alpha ⌝,t)⌝,t)\equiv T(⌜\alpha ⌝,t)$ at $m/h$.

Proof. 

We suppose that $\sigma$ satisfies $T(⌜T(⌜\alpha ⌝,t)⌝,t)$ at $m/h$. By Clause 7 of Definition 8, this means that $\sigma$ satisfies $T(⌜\alpha ⌝,t)$ at ${\left[t\right]}_{\sigma ,h}/h$, so that $\sigma$ satisfies $\alpha$ at ${\left[t\right]}_{\sigma ,h}/h$. By the same clause, we get that $\sigma$ satisfies $T(⌜\alpha ⌝,t)$ at $m/h$. The proof of the converse is similar, so we get that $\sigma$ satisfies $T\left(T.\right(⌜\alpha ⌝,t),t)\equiv T(⌜\alpha ⌝,t)$ at $m/h$. □

Proposition 3.

For any assignment σ and any moment–history pair $m/h$, if ${\left[t\right]}_{\sigma ,h}\in M$, σ satisfies $T(⌜\neg T(⌜\alpha ⌝,t)⌝,t)\equiv T(⌜\neg \alpha ⌝,t)$ at $m/h$.

Proof. 

We suppose that $\sigma$ satisfies $T(⌜\neg T(⌜\alpha ⌝,t)⌝,t)$ at $m/h$. By Clause 7 of Definition 8, this means that $\sigma$ satisfies $\neg T(⌜\alpha ⌝,t)$ at ${\left[t\right]}_{\sigma ,h}/h$. By Clause 2 of Definition 8, it follows that $\sigma$ does not satisfy $T(⌜\alpha ⌝,t)$ at ${\left[t\right]}_{\sigma ,h}/h$; hence, by Clause 7 of Definition 8, that $\sigma$ does not satisfy $\alpha$ at ${\left[t\right]}_{\sigma ,h}/h$. Again by Clause 2 of Definition 8, $\sigma$ satisfies $\neg \alpha$ at ${\left[t\right]}_{\sigma ,h}/h$. From this and Clause 7 of Definition 8, we get that $\sigma$ satisfies $T(⌜\neg \alpha ⌝,t)$ at $m/h$. The proof of the converse is similar, so we get that $\sigma$ satisfies $T(⌜\neg T(⌜\alpha ⌝,t)⌝,t)\equiv T(⌜\neg \alpha ⌝,t)$ at $m/h$. □

Propositions 1–3 show that P1f–P3f are always satisfied provided that t denotes a moment, which means that they are always satisfied when the second term in the scope of T has its intended interpretation. This is not to say, however, that P1f–P3f are valid in the sense of Definition 11. If t does not denote a moment, a formula that instantiates one of P1f–P3f can fail to be satisfied by an assignment at a moment–history pair.

In order to provide a general formulation of P1–P3 that takes into account the possibility that t does not denote a moment, we need to express at the syntactic level the property of being a moment. Let us assume that S is a monadic predicate whose extension is M in any model. Then, P1f–P3f can be generalized as follows:

P1fu 

$\forall x(Sx\supset (T(⌜\alpha ⌝,x)\vee T(⌜\neg \alpha ⌝,x))$

P2fu 

$\forall x(Sx\supset (T(⌜\left(T\right(⌜\alpha ⌝,x)⌝,x)\equiv T(\alpha ,x)\left)\right)$

P3fu 

$\forall x(Sx\supset (T(⌜\neg T(⌜\neg \alpha ⌝,x)⌝,x)\equiv T(⌜\neg \alpha ⌝,x)\left)\right)$

P1fu–P3fu are universally quantified formulas, so by Clause 5 of Definition 8, each of them is satisfied by an assignment $\sigma$ just in case every x-variant ${\sigma }^{\prime }$ of $\sigma$ satisfies the open formula in the scope of the universal quantifier. It is easy to see that, when ${\sigma }^{\prime }$ satisfies the antecedent, that is, when x denotes a moment according to ${\sigma }^{\prime }$, the consequent is also satisfied in virtue of Propositions 1–3. Therefore, P1fu–P3fu turn out to be valid according to Definition 11.

The facts just outlined show that T behaves in the way suggested in Section 3. More specifically, P1fu implies that bivalence holds for T, hence it captures the Ockhamist idea that every future contingent is either true or false. P2fu and P3fu imply that T preserves transparency, a fundamental property of truth which makes perfect sense in an Ockhamism perspective.

A final remark about the formal principles considered is that they could feature in an axiomatic theory of truth. Although the proofs of Propositions 1–3 provided above rely on the assumption that $\alpha$ is grounded, P1f–P3f can be shown to hold for any $\alpha$, for they can be shown to hold in a fixed-point semantics of the kind suggested in Section 6. The same goes for P1fu–P3fu. So it is reasonable to expect that a coherent Ockhamist theory of truth can be developed by combining some formal version of P1–P3 with further axioms that concern the other connectives of $\mathsf{L}$.8

8. Back to the Initial Question

What has been said so far suggests that the two objections against Ockhamism considered in Section 2 can be countered. Recall the scenario represented in Figure 1, where $m_0$ is today, $m_1$ is a sunny tomorrow, and $m_2$ is a rainy tomorrow. Let $\mathcal{F}$ be a frame $\langle M,<\rangle$ such that

• $m_0,m_1,m_2\in M$,
• $m_0<m_1$ and $m_0<m_2$,
• $m_0,m_1\in h_1$ and $m_0,m_2\in h_2$.

Let $\mathcal{M}$ be a model $\langle \mathcal{F},D,I\rangle$ where the following holds for three individual constants $t_0,t_1,t_2$ of $\mathsf{L}$.

• $I\left(t_0\right)=m_0$;
• $I\left(t_1\right)=m_1$;
• $I\left(t_2\right)=m_2$.

Let $\alpha$ be a closed formula of $\mathsf{L}$ such that ${\left[\alpha \right]}_{m_0/h_1}=1$ and ${\left[\alpha \right]}_{m_0/h_2}=0$ in $\mathcal{M}$. We can assume that $\alpha$ represents (1) as evaluated in the situation described in Section 2, that is, as true today relative to a history that leads to a sunny tomorrow and as false today relative to a history that leads to a rainy tomorrow.

Now consider the formula $T(⌜\alpha ⌝,t_0)$ in which truth is ascribed to $\alpha$ relative to $m_0$. This formula represents (2). In $\mathcal{M}$, ${\left[T(⌜\alpha ⌝,t_0)\right]}_{m/h_1}=1$ for any $m/h_1$, given that ${\left[\alpha \right]}_{m_0/h_1}=1$. This means that (2) is true as uttered today relative to $h_1$, that is, today one correctly says that (1) is true. By contrast, the same formula $T(⌜\alpha ⌝,t_0)$ turns out to be false relative to $m_0/h_2$, given that ${\left[\alpha \right]}_{m_0/h_2}=0$. That is, today it is false to say that (1) is true.

Finally, consider the formulas $T(⌜\alpha ⌝,p\left(t_1\right))$ and $T(⌜\alpha ⌝,p\left(t_2\right))$, which represent (3) as uttered by two assessors located, respectively, at $m_1$ and $m_2$. In $\mathcal{M}$, we obtain ${\left[T(⌜\alpha ⌝,p\left(t_1\right))\right]}_{m_1/h_1}=1$. For any assignment $\sigma$, ${\left[p\left(t_1\right)\right]}_{\sigma ,h_1}={\left[t_0\right]}_{\sigma ,h_1}$, so ${\left[T(⌜\alpha ⌝,p\left(t_1\right))\right]}_{m/h_1}={\left[T(⌜\alpha ⌝,t_0)\right]}_{m/h_1}$ for any $m\in h_1$. Similarly, we get that ${\left[T(⌜\alpha ⌝,p\left(t_2\right))\right]}_{m_2/h_2}=0$. For any assignment $\sigma$, ${\left[p\left(t_2\right)\right]}_{\sigma ,h_2}={\left[t_0\right]}_{\sigma ,h_2}$, so ${\left[T(⌜\alpha ⌝,p\left(t_2\right))\right]}_{m/h_2}={\left[T(⌜\alpha ⌝,t_0)\right]}_{m/h_2}$ for any $m\in h_2$.

Note that $\mathcal{M}$ does not provide an answer to the question whether (1)–(3) are actually true, or true simpliciter, for $\mathcal{M}$ does not say which of the two histories is the actual history. If $h_1$ is the actual history, (1)–(3) are true. Instead, if $h_2$ is the actual history, (1)–(3) are false. In any case, the truth value of (1)–(3) depends on what happens in the actual history.

When one imagines that a future contingent $\alpha$ is assessed at a context c, one reasons as if c were the actual context. Since a model of $\mathsf{L}$ simply provides a set of possible contexts—that is, a set of moment–history pairs—without specifying what is actual and what is not, to imagine that a given context is actual is to make a hypothesis about the model. Thus, when one imagines that (1) is evaluated at $m_2$, one imagines that $h_2$ is the actual history, and so that the relevant moment–history pair that includes $m_0$ is $m_0/h_2$. According to our semantics, (3) turns out to be false in that case, for the same reason for which (1) is false, namely that it is rainy at $m_2$. In other words, Ockhamism does not predict that an assessor located at $m_2/h_2$ should evaluate (1) as true as uttered the day before. Similar considerations apply to (4).