Future Contingents, Bivalence, and the Excluded Middle in Aristotle

1 Introduction

In the last four decades, the problem of future contingents in Aristotle has received comparatively little attention. In the decades before, by contrast, this problem gave rise to an enormous amount of literature. The reason why the topic has receded into the background seems to be that there is a communis opinio among scholars, an authoritative solution to the problem of future contingents in Aristotle. This solution is usually called ‘the standard interpretation of Int. 9.’ Int. 9 is the famous chapter in which Aristotle discusses the problem of future contingents. According to the standard interpretation of Int. 9, the principle of bivalence (PB), which says that every sentence^[1] is either true or false, does not apply to future contingents, for Aristotle holds that future contingents lack a truth value at the time they are produced. Some of those who endorse the standard interpretation – standardists, as I call them – further maintain that the principle of excluded middle (PEM), which says that in every pair of contradictory sentences, one member must be true,^[2] does not apply to pairs of contradictory future contingents. Other standardists aim to rescue PEM by adducing the method of supervaluations, according to which a tautology such as ‘There will or will not be a sea battle tomorrow,’ an instance of the law of excluded middle (p ∨ ¬p), is true, although neither disjunct is true.

This paper asks whether PB applies to future contingents and PEM to pairs of contradictory future contingents. I aim to show three things. First, I argue that standardists have misunderstood what is really at stake in Int. 9: Aristotle does not reject PB tout court, but rather a specific interpretation of it, according to which every sentence is either true or false at present. He does endorse another interpretation, according to which every sentence is either true or false at some point. Second, I argue against the standardists who claim that PEM fails for pairs of contradictory future contingents. Again, I show that Aristotle rejects a specific interpretation of it, according to which one contradictory must be true at present. He endorses another interpretation, according to which one contradictory must be true at some point. Third, I argue against another group of standardists who attempt to rescue PEM by working with the supervaluational method. I show that this method is unsuitable for explaining how PEM does not fail. In light of my criticisms against the standardists, I introduce a new interpretation of Int. 9, according to which neither PB nor PEM fails for (pairs of contradictory) future contingents. I call this ‘the universalist interpretation of Int. 9.’ All in all, I pursue two goals: I want to reopen the debate on future contingents in Aristotle by casting doubt on some tenets of the standard interpretation, and I present a new, modest proposal which I hope will inform new debates on future contingents in Aristotle.

Sections 2 and 3, which are expository in nature, set out the dialectic and show that the standard interpretation fails to overcome problems that my universalist interpretation is intended to cope with. In sections 4 to 7, I discuss pieces of evidence that cast doubt on the adequacy of the standard interpretation, and I explain whether, and if so how, my universalist interpretation fares any better. In section 8, I address the consequences of my interpretation for other semantic principles in Aristotle, his syllogistic logic, and theory of demonstrative science.

2 Three Problems: Future Contingents, Bivalence, and the Excluded Middle

Three problems are at the heart of this paper. These problems are distinct, even though (ii) below may initially appear like a paraphrase of (i).

1. The problem of future contingents is about whether or not future contingents are either true or false at present.

2. The problem of bivalence is about whether or not PB applies to future contingents.

3. The problem of the excluded middle is about whether or not PEM applies to pairs of contradictory future contingents.

Here are the relevant semantic principles:

1. Every sentence is either true or false.^[3]

2. It is necessary that in every contradictory pair, one^[4] member is true.^[5]

A brief note on terminology: I use the adverbs ‘now’ and ‘at present’ interchangeably throughout this paper. The contrast they are supposed to bring out is between the time future contingents refer to (the future) and the time at which future contingents are produced (now or at present). I use the verb ‘to produce [a sentence]’ to express the process or result of a sentence’s phonetic or mental manifestation. A sentence produced is thus uttered as a declaration or held as a belief. I ignore written manifestations of sentences because Aristotle has nearly nothing to say about them. Future contingents are singular future-tense sentences about contingent events, such as ‘There will be a sea battle tomorrow’ and its contradictory, ‘There will not be a sea battle tomorrow.’^[6]

Let me go through each of the problems in turn. (i) is what Int. 9 is about. Int. 9 can be divided into three parts.^[7] In 18a28–33, Aristotle summarizes some of the upshots of Int. 7–8 and states the results of the discussion that is to follow. In 18a34–19a22, he conducts a reductio ad absurdum.^[8] The assumption for reductio is the affirmative answer to the question of whether future contingents are at present either true or false: if they are, the future is predetermined, everything is necessary, and there is no contingency. In 19a23–b4, Aristotle presents his solution based on the negative answer to the question just mentioned, that is, that future contingents are not either true or false at present. Hence, the problem of future contingents takes center stage in Int. 9.

(ii) and (iii) are closely related to (i). If future contingents are neither true nor false at present, PB and PEM are non-universal principles. I use the phrases ‘universal’ and ‘non-universal principles’ to indicate that the principles do or do not apply universally to the members of their domain: if PB and PEM are universal, they apply to all (pairs of contradictory) sentences, and if they are non-universal, they do not apply to all (pairs of contradictory) sentences. If future contingents are sentences that are neither true nor false at present, PB does not apply to all sentences but apparently only to sentences that are not future contingents. Similarly, if no future contingents are either true or false at present, contradictory future contingents, too, will now lack truth values. But if PEM requires that in all pairs of contradictory sentences, one member be true, and no member of a pair of contradictory future contingents is true at present, PEM does not apply to all pairs of contradictories but apparently only to those that are not future contingents. It thus seems as if Aristotle’s solution to the problem of future contingents entails that PB and PEM are non-universal principles. However, if that is really the case, the upshot of Int. 9 comes at a considerable cost. Here are four drawbacks.

1. Aristotle states PB as a universal principle and PEM as a necessary and universal principle.^[9] There are no indications beyond Int. 9 that Aristotle acknowledges any exceptions to the universality of PB and the necessity and universality of PEM. Aristotle would contradict himself if he treated these principles as permitting exceptions.

2. The view that PB and PEM fail for (pairs of contradictory) future contingents is in tension with the views of the ancient commentators. For instance, Alexander of Aphrodisias (in APr. 400.34–35 Wallies, in Top. 183.24–25 Wallies), Philoponus (in APr. 436.5–9 Wallies), and Simplicius (in Cat. 195.21–23 Kalbfleisch) emphasize that a contradiction always divides the true and the false. This is a statement of the so-called rule of contradictory pairs:

1. In every contradictory pair, one member is true and the other false.^[10]

As can easily be shown, RCP is derivable from PB and PEM.^[11] Since every sentence is either true or false (PB), and one member of any pair of contradictory sentences must be true (PEM), either the affirmative contradictory is true and the negative false, or the affirmative contradictory is false and the negative true. This is precisely what RCP amounts to. Hence, if PB or PEM is suspended for (pairs of contradictory) future contingents, RCP too will be suspended, and this contradicts the commentators’ views.^[12]

3. One would expect PB and PEM to be universal principles precisely because of their principality.^[13] If they are non-universal principles, it is hard to see why we should treat them as principles. It would also be difficult to see why Aristotle treats PEM as a syllogistic (APo. I.2, 72a14–17, Met. Γ.3, 1005b7) and demonstrative principle (Met. B.2, 996b26–997a15) when it in fact is neither a necessary nor a universal principle. What else should make PEM a syllogistic and demonstrative principle if not its universality and necessity?

4. There would be an asymmetry between PEM and the principle of non-contradiction:

1. It is impossible that both members of a contradictory pair are simultaneously true.^[14]

While PEM may appear to fail for pairs of contradictory future contingents, PNC does not: it is not the case that both members of a pair of contradictory future contingents are simultaneously true because they are not either true or false at present. That is, the standard interpretation must make sense of PEM and PNC coming apart with regard to universality.

In sum, the upshot of PB and PEM failing to apply to (pairs of contradictory) future contingents is undesirable from an exegetical, systematic, and historical point of view. Interpreters of Int. 9 must take a stance on whether Int. 9 contradicts views Aristotle holds elsewhere in his writings. I take it that, on balance, a reading on which he does not contradict himself is preferable. Accordingly, the bulk of this paper lays out a reading on which Aristotle’s views are consistent. I argue that there is a way of reconciling PB’s and PEM’s universality and principality as well as PEM’s necessity with the fact that future contingents now lack truth values. This is why I dub my interpretation ‘universalist.’

3 The Standard and Universalist Interpretations Compared

Before I lay out my reading, I should note that the standard interpretation of Int. 9 is very widely held.^[15] It addresses all argumentative steps throughout Int. 9. My universalist interpretation, by contrast, does not present a new exegesis of the chapter as a whole. The points of departure concern the interpretation of Aristotle’s semantic principles and adjacent topics, such as his conception of truth. I pursue a primarily systematic goal, namely, to make sense of Aristotle’s semantic principles and theory given the difficulties raised by Int. 9. I focus on the following three core claims of the standard interpretation, and I will turn to a fourth claim further below:^[16]

The Core of the Standard Interpretation

1. Future contingents are not either true or false now.

2. Future contingents will be either true or false when the event they assert has or has not occurred.

3. PB is suspended for future contingents.

(i) and (ii) relate to the semantic status of future contingents. (iii) states something (i) alone or (i) and (ii) together presumably entail, namely, that PB is suspended for future contingents. PB thus fails for future contingents and is restricted to all other sentences.^[17] Consequently, it is not the case that every sentence is either true or false. Due to (ii), however, future contingents will receive one out of two truth values, which is why Dorothea Frede states that “PB is just suspended, not discarded.”^[18] Still, the standardist picture suggests that Aristotle’s PB is non-universal.

It is crucial to be clear about what exactly future contingents are. In Int. 9, Aristotle appears to focus on phonetically manifested sentences (declarations).^[19] This entails that mentally manifested sentences (beliefs), too, are at issue because “things in speech follow things in discursive thought (τὰ μὲν ἐν τῇ φωνῇ ἀκολουθεῖ τοῖς ἐν τῇ διανοίᾳ)” (Int. 14, 23a32–33). Hence, future contingents are declarations or beliefs. This is in line with the flow of De Interpretatione, which treats spoken sounds and thoughts with combination or division (declarations and beliefs) as bearers of truth and falsehood (see especially Int. 1, 16a10–18). Aristotle contrasts declarations with non-declarative sentences, such as prayers, by stating that truth and falsehood only occur in declarations (Int. 4, 17a1–5). Declarations (ἀποφάνσεις) come in two qualities – as affirmative declarations (affirmations, καταφάσεις) or negative declarations (denials, ἀποφάσεις) – and they are a kind of sentence. Declarative or non-declarative sentences (λόγοι), in turn, are a kind of significant spoken sound (φωνὴ σημαντική, Int. 4, 16b26). However, Aristotle is also ready to distinguish between internal and external λόγοι (APo. I.10, 76b26–27), the former arguably corresponding to beliefs and the latter to declarations. Beliefs (ὑπολήψεις) are mental truth-value bearers,^[20] and Aristotle makes it clear that opinions (δόξαι), a subclass of beliefs, are mental counterparts of affirmations and denials, which are phonetically realized (Int. 14, 23a32–39 and 24b1–6). I distinguish Aristotle’s non-shareable truth-value bearers (sentences that are either phonetically manifested as declarations or mentally manifested as beliefs) from propositions in the contemporary sense, which are shareable.^[21]

Next, future contingents are sentence tokens rather than sentence types. According to (ii) above, a future contingent like ‘There will be a sea battle tomorrow’ will acquire a truth value when the event it asserts has or has not occurred. This seems to rule out sentence types since it is questionable whether the sentence type ‘There will be a sea battle tomorrow’ could ever acquire a truth value. Due to the indexical ‘tomorrow,’ this sentence type changes its truth conditions every day and therefore appears to escape the possibility of ever being semantically evaluated. Aristotle appears to countenance sentence types (see especially Cat. 5, 4a21–b19),^[22] but as far as I can see, there is nothing in Int. 9 that speaks in favor of sentence types rather than sentence tokens being at issue. Accordingly, Gerhard Seel refers to future contingents as “speech- and thought-event[s].”^[23] Future contingents are thus tokens of sentences that lack a truth value at the time they are produced but will acquire one at some point in the future.^[24]

Sentences in Aristotle are token-reflexive: a sentence such as ‘Socrates is wise’ may have different truth values when produced at different times, for example, false when Socrates is a child (t₁) and true when he is 50 (t₂).^[25] This means that sentence tokens are implicitly temporally indexed – not in the sense that a temporal index occurs in a sentence (‘Socrates is wise at t₁’) but in the sense that temporal information belongs to a sentence’s truth conditions (‘Socrates is wise’ is false when produced at t₁ but true when produced at t₂). In contrast to its corresponding sentence type, then, the sentence token ‘There will be a sea battle tomorrow’ may be semantically evaluated the day after the sentence is produced. What acquires a truth value is the declaration (token) uttered or the belief (token) held the day before.

I will now turn to the fourth claim of the standard interpretation. Upon closer examination, it turns out that ‘the standard interpretation’ is an umbrella term for a family of interpretations of which two strands are incompatible. These strands differ in how they answer the problem of the excluded middle. The A-brand of the standard interpretation, as I call it, has it that PEM is suspended for pairs of contradictory future contingents, while according to the B-brand, PEM is not suspended.

The Standard Interpretation A

1. PEM is suspended for pairs of contradictory future contingents.

The Standard Interpretation B

1. PEM is not suspended for pairs of contradictory future contingents.

(iv.a) states something (i) alone or (i) and (ii) together presumably entail, namely, that PEM is suspended for pairs of contradictory future contingents because future contingents lack a truth value at the time they are produced. By contrast, (iv.b) purports to leave the universality of PEM unaffected.

I do not find (i) and (ii) troubling, but I do worry about (iii) and (iv) (in both of its brands) and their consequences. My own approach to Aristotle’s solution is to some extent similar to the standardists’:

The Universalist Interpretation

1. Future contingents are not either true or false now.

2. Future contingents will be either true or false when the event they assert has or has not occurred.

3. PB is not suspended for future contingents.

4. PEM is not suspended for pairs of contradictory future contingents.

Whereas the standard interpretation has it that PB is suspended for future contingents, I maintain that PB is not suspended. I claim that PB is unrestricted.^[26] Moreover, I maintain that PEM is not suspended for pairs of contradictory future contingents. As we shall see in section 6, though, my (iv*) has little to do with (iv.b): the standardist view that purports to retain PEM’s universality apparently does not apply to PEM, but rather to the law of excluded middle from modern logic, which is distinct from Aristotle’s PEM. In sum, the standard interpretation and my universalist interpretation are incompatible: it cannot be the case that PB both is and is not suspended for future contingents. Considering also that the standard interpretation is incoherent with regard to its stand on PEM, we can say that it relies on a rather different conception of Aristotle’s semantic principles than my universalist interpretation.

Some scholars identify the standard with the traditional interpretation. The originators of the traditional interpretation are the ancient commentators Ammonius and Boethius.^[27] I believe it is wrong to identify these interpretations.^[28] To name but one point where the traditional interpretation diverges from the standard and the universalist interpretations: Ammonius and Boethius never mention PB. PB was first coined and introduced into the debate on future contingents in Aristotle by Jan Łukasiewicz in the first half of the twentieth century.^[29] Note that one of the problems in this paper is which exact formulation of PB to attribute to Aristotle. Accordingly, it is not obvious either which exact formulation of PB to attribute to Ammonius and Boethius. It is important to distinguish between the standard and the traditional interpretations because otherwise one uncritically ascribes modern notions to ancient authors and is likely to adapt ancient texts to our expectations. For instance, while I do agree with modern scholars in crediting Aristotle with some formulation of PB, I disagree with almost all of them in its exact interpretation. Besides, by identifying the standard with the traditional interpretation, standardists can illegitimately claim authority for their views.

To sum up, all standardists believe PB fails for future contingents. A-standardists additionally believe that PEM fails for pairs of contradictory future contingents. Universalists, by contrast, believe that PB and PEM do not fail for (pairs of contradictory) future contingents. However, universalists disagree with B-standardists on how PEM does not fail.

4 The Principle of Bivalence Temporalized^[30]

Standardists hold that PB in Aristotle is suspended for future contingents. Future contingents are sentences that now lack a truth value but will acquire one once the event they assert has or has not occurred. Hence, PB does not apply to them at the time they are produced but will apply to them when the event they assert has or has not occurred. This means that standardists presuppose that PB in Aristotle is temporally indexed so as to state that every sentence is now either true or false. Because PB is temporally indexed this way, it does not apply to future contingents. I do not believe this is the right way of conceiving of PB in Aristotle. I maintain that it is better conceived of as stating that every sentence is either true or false at some point. Let us juxtapose PB’s two rival interpretations:^[31]

1. Every sentence is now either true or false.^[32]

2. Every sentence is either true or false at some point.^[33]

To illustrate how PB_now and PB_point apply to sentences, let us examine the present contingent ‘Socrates is a philosopher’ and the future contingent ‘There will be a sea battle tomorrow.’ What makes ‘Socrates is a philosopher’ either true or false is the corresponding state of affairs that now either obtains or fails to obtain. Hence, ‘Socrates is a philosopher’ is now either true or false, which entails that both PB_now and PB_point apply to it (‘at some point’ includes the present moment). By contrast, what makes ‘There will be a sea battle tomorrow’ either true or false is the corresponding state of affairs that will either obtain or fail to obtain the day after the sentence is produced. Hence, ‘There will be a sea battle tomorrow’ is either true or false tomorrow, which entails that PB_point, but not PB_now, applies to it. PB_point does not entail that some sentences, like ‘Socrates is a philosopher,’ will be either true or false at some point in the future, even though their truth values can now be ascertained. Instead, PB_point states that every sentence is such that it is either true or false, whether now or later.

‘Either–or’ indicates exclusive disjunction because the same sentence cannot simultaneously be true and false.^[34] PB_now underlies the standard interpretation and PB_point the universalist interpretation. PB_now implies PB_point, but not vice versa: whatever has a truth value now necessarily has a truth value at some point, but whatever has a truth value at some point need not have a truth value now, namely, if a sentence acquires a truth value at some point later than now. Hence, PB_point is weaker than PB_now. However, PB_now is and PB_point is not suspended for future contingents.

These two interpretations of PB are temporalized insofar as they involve temporal terminology. One might object to temporalized interpretations by requiring that PB be construed as a timeless principle.^[35] I doubt that Aristotle’s PB is timeless for the following four reasons. 1. There is no direct evidence in Aristotle for PB being either timeless or temporalized. Still, the fact that Aristotle’s statements of PB appear timeless is insufficient to prove its alleged timelessness. PB, in any of its formulations, involves truth, and we will see later that truth in Aristotle is inherently temporalized. 2. There is reason to suspect an anachronism in the expectation that Aristotle’s PB is timeless. In contrast to modern non-tense-logical systems, time does play a role in Aristotle’s analysis of sentences in the first few chapters of De Interpretatione (see especially Int. 1, 16a16–18, 2, 16a19–21 and b1–5, 3, 16b6–7 and b16–18, 5, 17a9–12 and a23–24, 6, 17a26–31; see also the next section). It should therefore not come as a surprise if Aristotle’s PB turns out to be temporalized. 3. PNC and the ontological version of the principle of non-contradiction are temporalized, too: they involve the notion of simultaneity.^[36] It would be strange to assume that only PNC and its ontological cognate are temporalized, whereas PB (and perhaps other semantic principles in Aristotle) are timeless. 4. There is a temporalized statement of the ontological version of the principle of excluded middle: “so that everything necessarily is or is not, and will be or will not be (εἶναι μὲν <δὴ> ἢ μὴ εἶναι ἅπαν ἀνάγκη καὶ ἔσεσθαί γε ἢ μή)” (Int. 9, 19a28–29; Ackrill’s translation, modified). It appears questionable to require that Aristotle’s semantic principles be timeless while their ontological counterparts are not.

Given these four reasons, I shall assume that Aristotle’s PB is temporalized. The question is: which temporalized interpretation should we attribute to Aristotle, PB_now or PB_point? Let us see what the texts say. Consider the following two passages:

T1: Cat. 4, 2a7–8 (my translation)

ἅπασα γὰρ δοκεῖ κατάφασις ἤτοι ἀληθὴς ἢ ψευδὴς εἶναι

For every assertion seems to be either true or false

 

T2: An. III.6, 430b26–27 (Shields’s translation, modified)

ἔστι δ’ ἡ μὲν φάσις τι κατά τινος, ὥσπερ καὶ ἡ ἀπόφασις, καὶ ἀληθὴς ἢ ψευδὴς πᾶσα·

It is also the case that every affirmation, just as every denial, is something of something and is true or false.

In T1, the word ‘assertion (κατάφασις)’ functions as a general term for both affirmations (καταφάσεις) and denials (ἀποφάσεις).^[37] In T2, the word ‘φάσις’ is a shorthand for ‘κατάφασις (affirmation),’ which in this context does not mean ‘assertion.’^[38] T2 additionally states that affirmations and denials say something of something (τι κατά τινος). This phrase expresses the predicative structure of sentences (APr. I.1, 24a16–17 and a26–28).^[39] In T1, Aristotle appears hesitant to state tout court that every sentence is either true or false. Instead, he states that it seems so (δοκεῖ). I will account for this in a moment. T2, on the other hand, is more straightforward and therefore appears to be in tension with T1. Now contrast these two passages with the beginning of Int. 9:

T3: Int. 9, 18a28–35 (Ackrill’s translation, modified)

[i] Ἐπὶ μὲν οὖν τῶν ὄντων καὶ γενομένων ἀνάγκη τὴν κατάφασιν ἢ τὴν ἀπόφασιν ἀληθῆ ἢ ψευδῆ εἶναι […]. [ii] ἐπὶ δὲ τῶν καθ’ ἕκαστα καὶ μελλόντων οὐχ ὁμοίως. [iii] Εἰ γὰρ πᾶσα κατάφασις καὶ ἀπόφασις ἀληθὴς ἢ ψευδής, καὶ ἅπαν ἀνάγκη ὑπάρχειν ἢ μὴ ὑπάρχειν.

[i] With regard to what is and what has been, it is necessary for the affirmation or the denial to be true or false. […] [ii] But with individuals that are going to be, it is different. [iii] For if every affirmation and denial is true or false, it is necessary for everything either to be the case or not to be the case.

In [i], Aristotle refers to pairs of present-tense contradictories and past-tense contradictories for which the following rule holds: one member of each contradictory pair is true and the other false.^[40] In [ii], he says that this rule does not hold in the same way for (pairs of contradictory) future contingents (in fact, Aristotle refers to the things future contingents are about, and that these things are contingent, in contrast to necessary, is indicated by the word ‘μελλόντων;’ GC II.11, 337b3–7). In [iii], Aristotle lays down PB as an assumption for the reductio that occupies the middle part of the chapter.^[41]

Temporality has center stage in T3. T3 contrasts past- and present-tense contradictories with future contingent contradictories. In the part that follows T3, Aristotle spells out the absurdities resulting from the view that future contingents are now either true or false. Hence, PB in this passage seems adequately interpreted as PB_now, which underlies the standard interpretation.^[42] This, however, is different for T2. Recall that one of the upshots of Int. 9 is that some sentences – future contingents – are not either true or false now. Accordingly, if PB in T2 were interpreted in terms of PB_now, we would have to charge Aristotle with self-contradiction because of the absurd consequence that every (T2) and not every (Int. 9) sentence is now either true or false. It is therefore charitable to interpret PB in T2 in terms of PB_point, which underlies the universalist interpretation. T1, in turn, is readily interpreted as adopting a middle position between T2 and T3: Aristotle seems to be aware of the problems that an interpretation of PB as PB_now entails and therefore expresses himself cautiously. Another reason, which I consider to be decisive, is that Int. 9 is intended to rule out an interpretation of PB in terms of PB_now: Int. 9 demonstrates that PB is better interpreted as the weaker PB_point. This interpretation has to underlie T2 unless we want to accuse Aristotle of self-contradiction. Thus, while the stronger PB_now underlies Aristotle’s discussion in Int. 9 (only to be reduced ad absurdum), the weaker PB_point underlies Aristotle’s discussions elsewhere. I consider this strong evidence for the view that Aristotle endorses PB_point and rejects PB_now.

But why does Aristotle discuss PB_now, to begin with? I believe PB_now is a plausible thesis to inquire into and defend. Chrysippus, for instance, famously defended PB (here: ‘Every assertible (ἀξίωμα) is either true or false’), claiming that future contingents are now either true or false (Cicero, De Fato 20–21). He would therefore champion PB_now (‘Every assertible is now either true or false’) instead of PB_point (‘Every assertible is either true or false at some point’). Aristotle and Chrysippus are thus exponents of incompatible solutions to the problem of bivalence: Chrysippus would think that PB_now applies to future contingents, which Aristotle denies. Moreover, recall that T1 and T2 are opaque as to whether the principle in question is equivalent to PB_now or PB_point. PB_now seems like a natural way of interpreting PB in T1 and T2. The reflex to interpret PB as PB_now is, in fact, attested right at the beginning of Int. 9 (in T3-[iii] and the part that follows it). Only Aristotle’s discussion of future contingents in Int. 9 prompts the idea that PB_point is likely at issue in T1 and T2. Otherwise, T1, T2, and Int. 9 are inconsistent, an option envisaged by standardists but not by universalists.

Standardists might retort by questioning the universality of PB_point. For instance, they may find that a sentence such as ‘There will be a sea battle at some point’ may fail to be either true or false at any point in time. For if a sea battle never occurs, there is nothing that will either verify or falsify the sentence. This entails that PB_point is not a universal principle after all. Here are two replies. Firstly, I do not believe that ‘at some point’ is a term. One reason for this is that the contradictory of the sentence above would be ‘There will not be a sea battle at any point,’ and the opposition between ‘at some point’ and ‘(… not …) at any point’ is reminiscent of the opposition between the quantifiers ‘some’ and ‘no,’ and quantifiers are not terms (Int. 7, 17b11–12, 10, 20a9–10). Something similar applies to the contradictory pair ‘There will never be a sea battle’ and ‘There will not never be a sea battle,’ the former’s sense being identical to the above ‘There will not be a sea battle at any point’ and the latter’s to the above ‘There will be a sea battle at some point.’ Secondly, terms should not depend on the quality of the sentences in which they occur. However, in the examples in question, ‘at some point’ is tied to affirmative quality (that is, it would be weird to say ‘There will not be a sea battle at some point’), and ‘at any point’ is tied to negative quality (that is, it would be weird to say ‘There will be a sea battle at any point’). Considering both replies together, it seems that the contradictories above are grammatically but not logically well-formed, and one cannot require logically ill-formed sentences to fall under (any interpretation of) PB (for a comparable case, see Int. 7, 17a12–16).^[43]

As I mentioned above, PB_point is a principle weaker than PB_now, and its weakness possibly accounts for the fact that it has never been attributed to Aristotle, at least to the best of my knowledge. In section 8, I argue that PB_point is not too weak a principle. Hence, a lot is gained if we dismiss the idea that Aristotle’s PB is interpreted as PB_now, or for that matter, as a timeless principle.

5 Aristotle’s Account of Declarations

In her influential monograph on future contingents in Aristotle, Frede contends that Aristotle contradicts himself within the same treatise: in Int. 4, Aristotle claims that all declarations are either true or false, whereas in Int. 9, he claims that future contingents, which are declarative, are not either true or false.^[44] She explains this inconsistency by surmising that Aristotle composed the various parts of De Interpretatione at different stages of his intellectual development.^[45] I find her conjecture unconvincing. We know nothing about the composition of De Interpretatione. Frede’s story is therefore bound to be speculative. I believe there is a more straightforward story to tell, so let us take a look at Aristotle’s account of declarations (in what follows until the end of this section, ‘sentence’ does not only stand for ‘declarative sentence’ but encompasses both declarative and non-declarative sentences):

T4: Int. 4, 17a1–5 (my translation)

Ἔστι δὲ λόγος ἅπας μὲν σημαντικός, […]. ἀποφαντικὸς δὲ οὐ πᾶς, ἀλλ’ ἐν ᾧ τὸ ἀληθεύειν ἢ ψεύδεσθαι ὑπάρχει. οὐκ ἐν ἅπασι δὲ ὑπάρχει, οἷον ἡ εὐχὴ λόγος μέν, ἀλλ’ οὔτε ἀληθὴς οὔτε ψευδής.

Every sentence is significant […], but not every sentence is declarative, but only those in which there is truth or falsehood. These do not belong to all sentences, though; for example, a prayer is a sentence but is neither true nor false.

Aristotle is focusing on declarations, that is, phonetically manifested declarative sentences, because he is referring to λόγοι, which he characterizes as spoken sounds (Int. 4, 16b26). However, his account also applies to beliefs, that is, mentally manifested declarative sentences (APo. I.10, 76b24–27; see section 3). Accordingly, T4 applies to all kinds of declarative sentences, not only phonetically manifested ones, so whenever I talk about declarations, I effectively mean both declarations and beliefs. T4 invokes some formulation of PB because every declaration is said to be either true or false.^[46] The question is whether we should interpret PB in terms of PB_now or PB_point. Is Aristotle’s account of declarations in conflict with his discussion of future contingents? Does Aristotle contradict himself?

There is no need to see a conflict here, let alone a contradiction. T4 says that truth or falsehood belongs to declarations. Future contingents clearly meet this condition. As Francesco Ademollo puts it, future contingents are “fit for being true or false, even though they are not actually true or false.”^[47] Declarations are distinguished from non-declarative sentences, such as prayers, in that the latter are never true or false: neither truth nor falsehood belongs to non-declarative sentences.^[48] If PB in T4 is interpreted in the spirit of my universalist interpretation (PB_point; here: ‘Every declaration is either true or false at some point’), there is no conflict between Aristotle’s account of declarations and future contingents: future contingents meet the condition of being either true or false at some point. If, however, PB in T4 is interpreted in the spirit of the standard interpretation (PB_now; here: ‘Every declaration is now either true or false’), there will indeed be tension with Int. 9: future contingents do not meet the condition of now being either true or false. But Aristotle’s account of declarations does not require truth values now assigned to declarations. Hence, it is preferable to interpret PB in T4 in the spirit of my universalist interpretation. Standardists, on the other hand, will have to charge Aristotle of contradicting himself again.

It turns out that the standardists’ view that future contingents violate Aristotle’s account of declarations entails two further problems: (i) standardists fail to explain the sense in which future contingents are both truth-value bearers and truth-value gaps at the time they are produced, a view Aristotle seems to countenance; and (ii) standardists are committed to denying that the disjunctive attribute of being either true or false specifically differentiates declarations from other species of non-declarative sentences. As we proceed, it will become plain that (i) and (ii) are closely related.

Let us start with (i). From the beginning of De Interpretatione onwards, Aristotle prepares us to consider a possible interaction between the grammatical category of tense and the truth values of declarative sentences. For instance, at Int. 1, 16a9–18, he says that truth and falsehood have to do with the combination and division of nouns and verbs and that nouns, such as ‘human,’ are not yet true or false unless finite forms of ‘to be’ or ‘not to be’ are added, either simpliciter (that is, in the present) or in one of their temporally inflected forms (that is, in the past or the future).^[49] It follows that linguistic strings such as ‘Human is,’ ‘Human was not,’ ‘Human will be,’ and ‘Human will not be’ are truth-value bearers because they have to do with the combination (expressed by affirmations, for example, ‘Human will be’) and division (expressed by denials, for example, ‘Human will not be’) of nouns (‘human’) and verbs (‘to be,’ ‘not to be;’ Aristotle calls negated verbs ‘infinite (ἀόριστα),’ Int. 3, 16b12–15). Similarly, he argues at Int. 2, 16a32–b5, that inflected nouns (which for Aristotle include proper names), such as ‘Philo’s,’ do not form truth-value bearers if a verb, such as ‘is,’ ‘was,’ or ‘will be,’ is added, whereas non-inflected nouns, such as ‘Philo,’ do. Hence, ‘Philo will be’ and ‘Philo will not be’ are truth-value bearers. At Int. 3, 16b6–9, Aristotle defines the verb in terms of what co-signifies time, and this feature differentiates the verb from the noun.^[50]

These and other passages indicate that the grammatical category of tense, which signifies the present, past, or future, plays a significant role in Aristotle’s syntactic analysis of declarative sentences and that future-tense sentences – including future contingents – are truth-value bearers. In fact, we may identify sentences such as ‘Philo will be’ and ‘Philo will not be’ as future contingents, provided, for instance, that Philo does not exist at the time these sentences are produced, for (some) Philo may or may not come into being at some point, just like a sea battle that may or may not happen tomorrow. Hence, we may commit Aristotle to treating future contingents as truth-value bearers right from the beginning of De Interpretatione. The following passage confirms this analysis: “it is surely not only true or false that Cleon is pale, but also that he was or will be (ἔστι γε οὐ μόνον τὸ ψεῦδος ἢ ἀληθὲς ὅτι λευκὸς Κλέων ἐστίν, ἀλλὰ καὶ ὅτι ἦν ἢ ἔσται)” (An. III.6, 430b4–5; Shields’s translation, modified). The sentence ‘Cleon will be pale’ is a future contingent, and Aristotle expressly treats it as a truth-value bearer.

Now, recall that both standardists and universalists treat future contingents as declarative sentences that now lack a truth value. But how can something that lacks a truth value be a truth-value bearer? The standard interpretation seems to entail that future contingents are not truth-value bearers at the time they are produced but will become truth-value bearers when the event they assert has or has not occurred. This is problematic since future contingents are declarative, and being declarative is the mark of truth-value bearers, as we saw in T4 and will see in more detail below. The universalist interpretation, by contrast, is apt to provide a concrete sense in which future contingents are truth-value bearers already at the time they are produced: future contingents are either true or false at some point, and having a truth value at some point is necessary for something to qualify as a sentential truth-value bearer in Aristotle.^[51] In other terms, something that lacks a truth value may nevertheless be a truth-value bearer. The universalist interpretation thus seems to have another advantage over the standard interpretation, for it supplies an explanation for how a declarative sentence that now lacks a truth value can nonetheless be a truth-value bearer.

Consider next (ii), according to which standardists deny that the attribute of being either true or false specifically differentiates declarations from non-declarative sentences. The universalist interpretation treats the disjunctive attribute of being either true or false at some point as the differentia of declarative sentences. Accordingly, universalists think that T4 presents a definition of declarations. Standardists, by contrast, claim that T4 does not state a definition of declarations because the account in T4 does not accommodate future contingents, which are truth-value gaps at the time they are produced. However, standardists presuppose that it only accommodates declarations that are now either true or false, and there is no need to accept this.

According to Ammonius (in Int. 66.10–19 Busse), T4 states a definition per genus proximum et differentiam specificam of declarations (ἀποφάνσεις or λόγοι ἀποφαντικοί). He believes that the genus of simple sentences (ἁπλοῖ λόγοι) includes the species of declarations and the various species of non-declarative sentences, such as vocative, interrogative, or imperative sentences (in Int. 64.26–65.2 Busse). Both declarations and non-declarative sentences are significant (σημαντικοί), but what specifically differentiates declarations from all species of non-declarative sentences is the fact that truth or falsehood only occurs in declarations. It is true that Aristotle does not explicitly state that T4 is a definition of declarations. However, in the programmatic opening of De Interpretatione, Aristotle announces the following:

T5: Int. 1, 16a1–2 (my translation)

Πρῶτον δεῖ θέσθαι τί ὄνομα καὶ τί ῥῆμα, ἔπειτα τί ἐστιν ἀπόφασις καὶ κατάφασις καὶ ἀπόφανσις καὶ λόγος.

First, we must establish what a noun is and what a verb is, next what a denial, an affirmation, a declaration, and a sentence are.

This sounds as if Aristotle is setting out to define all these items, although he only says that he will establish them in whichever sense of the word. Indeed, Ammonius (in Int. 9.4–27 Busse) argues at length that of the various senses of the word ‘establish (θέσθαι),’ Aristotle is using it here in the sense of ‘define (ὁρίσασθαι)’ (likewise Boethius, in Int. II 13.29–14.2 Meiser) because the ‘what is (τί ἐστιν)’ phrase that follows it indicates a definition. This exegesis squares well with how Aristotle explicates these items in the following chapters, for his accounts of the noun (Int. 2, 16a19–20), verb (Int. 3, 16b6–8), denial and affirmation (Int. 6, 17a25–26), declaration (T4), and sentence (Int. 4, 16b26–28) sound very much like definitions per genus proximum et differentiam specificam. In fact, Ammonius (in Int. 47.18–20, 80.31–35 Busse) holds that Aristotle defines the noun, verb, affirmation, and denial, and we have just seen that he also thinks Aristotle defines the declaration.^[52]

Standardists, by contrast, are bound to deny that Aristotle is defining declarations in T4 and that the attribute of being either true or false specifically differentiates declarations from non-declarative sentences.^[53] This, in turn, entails that standardists are likely to believe that Aristotle does not define the other items either. Otherwise, they would have to explain why Aristotle defines some of them but not others, such as the declaration. Hence, standardists will likely interpret Aristotle’s program in T5 differently than universalists and commentators such as Ammonius and Boethius. Universalists can provide a solution to the problem owing to which standardists deny that T4 states a definition of declarations: if it were a definition, then all declarations, including future contingents, would have to be either true or false because this is supposed to differentiate declarations from non-declarative sentences specifically. Unlike universalists, standardists preclude the possibility that the differentia is such that declarations are either true or false at some point rather than now.

6 The Principle of Excluded Middle Temporalized

So far, I argued that, on the universalist interpretation, Int. 9 does not introduce a borderline case for Aristotle’s PB, and his account of declarations in Int. 4 is not in tension with his rationale in Int. 9. Standardists may argue, however, that I have not yet addressed how my interpretation bears on PEM. If it were to turn out that PEM is best understood in ways that invoke the standardist ‘now temporality’ rather than the ‘point temporality’ of my reading, this would weaken my case. Consider, then, the A-brand of the standard interpretation and its take on PEM (I turn to the B-brand below).

As I mentioned in section 3, the A-brand has it that PEM fails for pairs of contradictory future contingents.^[54] PEM, just like PB, is a principle that governs the distribution of truth values to sentences. Specifically, PEM is a general semantic thesis about pairs of contradictory sentences and their truth values, stating that one member must be true in every pair of contradictories. The standardist rationale that aims to show that PEM fails for pairs of contradictory future contingents runs as follows: if future contingents now lack a truth value, then so do both members of a pair of contradictory future contingents, and this entails that it is not necessary that in a pair of contradictory future contingents, one member is true. Therefore, PEM fails for pairs of contradictory future contingents. Some A-standardists do not seem to find this particularly problematic. For instance, Dorothea Frede claims that “Aristotle treats PB and [P]EM on a par, that is, both are suspended in the same way [for (pairs of contradictory) future contingents] but while [P]EM is harmless PB is not.”^[55] This should strike us as surprising. Recall that Aristotle treats PEM as a necessary principle. By contrast, it is not entirely clear whether he also treats PB as a necessary principle. It thus appears that PEM’s failure is indeed more harmful than PB’s. Moreover, Frede claims that Aristotle’s defense of PEM in Met. Γ is not evidence against the view that he restricts PEM’s scope so as not to apply to pairs of contradictory future contingents.^[56] She fails to acknowledge, however, that Aristotle treats PEM as a universal principle. Only Int. 9, if at all, gives rise to the idea that PEM is a non-universal principle, though this is by no means plain. Her view thus comes at the cost of attributing incoherence and inconsistency to Aristotle.

It is worth highlighting that there is not a single passage in Int. 9 that unequivocally expresses a failure of PEM for pairs of contradictory future contingents.^[57] I assume the reason why A-standardists endorse this view is due to how they interpret PEM (and, perhaps, PB). Sure enough, the standardist rationale above relies on a temporalized interpretation of PEM in terms of one of two contradictories having to be true at present. It is because neither member of a pair of contradictory future contingents is now true that PEM is supposed to fail for them. I do not believe that this is how PEM is properly interpreted. Instead, it is sufficient if one of two contradictories is true at some point, whether now or later. Here are the two rival interpretations of PEM:

1. It is necessary that in every contradictory pair, one member is now true.

2. It is necessary that in every contradictory pair, one member is true at some point.

A-standardists presuppose an interpretation of PEM as PEM_now, which fails for pairs of contradictory future contingents. PEM_point, by contrast, does not: since one member of a pair of contradictory future contingents will turn out true when the event it asserts has or has not occurred, and PEM_point requires that one of two contradictories be true at some point, PEM_point applies to pairs of contradictory future contingents. Hence, PEM is a universal principle, provided that it is interpreted as PEM_point. PEM_point is a principle weaker than PEM_now: the latter implies the former but not vice versa. This analysis aligns with how I interpret Aristotle’s PB so that PB and PEM yield a coherent, albeit fragmentary, picture of Aristotle’s semantic principles. One remaining task will be to show why it makes sense to attribute weaker principles to Aristotle than has yet been assumed (see section 8). Another task will be to show that these temporalized interpretations sit well with other semantic principles, namely, PNC and RCP (see section 8).

Consider next the B-brand of the standard interpretation. B-standardists adduce the method of supervaluations in accounting for how PB may fail for future contingents, while PEM does not fail for pairs of contradictory future contingents. This method was introduced by Bas van Fraassen (1966) in order to make a case for how arguments may retain their validity and – more importantly for current purposes – how tautologies may retain their truth when the constituent propositions of these arguments and tautologies are truth-value gaps. He distinguishes between classical (that is, bivalent) valuations and supervaluations. For convenience, let us consider the language of propositional logic, PL. A classical valuation is a function that assigns either of the values true or false to each proposition of PL. A supervaluation is a function that assigns the value true exactly to those propositions assigned the value true by all classical valuations, and likewise for the value false.^[58] To see the difference, consider an arbitrary proposition p, such as ‘It is raining.’ Suppose that the classical valuation function v₁ assigns it the value true and another one, v₂, the value false. These are trivially all the values p can take. The supervaluation function s does not assign p to either of these values because it is not the case that p is made true or false by all classical valuations. By contrast, the tautology p ∨ ¬p (‘It is raining or it is not raining’) is assigned the value true by v₁ and v₂. Hence, s, too, assigns it the value true. B-standardists argue in a similar vein to show how future contingents may lack a truth value, while disjunctions of contradictory future contingents are true: even if p stands for a future contingent (a truth-value gap), p ∨ ¬p is nevertheless true. p ∨ ¬p is a classical tautology (an instance of the law of excluded middle), and this entails that the supervaluation function assigns it the value true.^[59] At first blush, then, there seems to be a way to plausibly claim that PB fails for future contingents, while PEM does not fail for pairs of contradictory future contingents.^[60]

The problem with this proposal is that it does not meet Aristotle on his own terms. The supervaluational approach may be apt to explain the universality of the law of excluded middle (and other classical tautologies). However, the law of excluded middle and Aristotle’s PEM have little in common apart from the fact that they are named after some excluded middle. Besides, the supervaluational approach tacitly attributes a proto-account of truth-functional connectives to Aristotle. However, he does not discuss connectives, truth-functionality, or the notion of tautology.^[61] In effect, the proposal makes it seem as if Aristotle was groping towards a solution suited for modern logical systems.

The law of excluded middle is syntactic (logical) and not semantic (metalogical), whereas PEM is semantic (metalogical) and not syntactic (logical). The reason for this is that PEM involves the truth predicate, and the law of excluded middle does not. This entails that PEM is not translatable into PL, and the law of excluded middle and PEM come apart because they are operative on different linguistic levels. Hence, the supervaluational approach leaves open how it applies to PEM. Next, it is unlikely that PEM applies to disjunctions rather than to individual sentences arranged in contradictory pairs that are not disjunctive. According to PEM, one member of any contradictory pair must be true, and this seems to rule out that PEM concerns disjunctions of contradictories rather than simple sentences. If one were to insist that PEM concerns disjunctions of contradictories, one would have to explain what kind of disjunction PEM involves, for Aristotle did not countenance disjunctions in the modern truth-functional sense. By contrast, saying that PEM concerns simple sentences is more straightforward because it sits well with Aristotle’s logic of simple sentences. Hence, it is questionable to impute the crucial step in the B-standardists’ supervaluational rationale to Aristotle: the step from truth-valueless propositions, p and ¬p, to true disjunctions consisting of truth-valueless propositions, p ∨ ¬p.

The supervaluational approach also struggles to make sense of Aristotle’s solution in the final part of Int. 9 (19a23–b4). To get clear about this difficulty, a close look at the passage, which I divide into two parts (19a23–32 and 19a32–b4), is needed.

T6: Int. 9, 19a23–32 (Ackrill’s translation, modified)

[i] Τὸ μὲν οὖν εἶναι τὸ ὄν, ὅταν ᾖ, καὶ τὸ μὴ ὂν μὴ εἶναι, ὅταν μὴ ᾖ, ἀνάγκη· οὐ μέντοι οὔτε τὸ ὂν ἅπαν ἀνάγκη εἶναι οὔτε τὸ μὴ ὂν μὴ εἶναι. οὐ γὰρ ταὐτόν ἐστι τὸ ὂν ἅπαν εἶναι ἐξ ἀνάγκης, ὅτε ἔστιν, καὶ τὸ ἁπλῶς εἶναι ἐξ ἀνάγκης· ὁμοίως δὲ καὶ ἐπὶ τοῦ μὴ ὄντος. [ii] καὶ ἐπὶ τῆς ἀντιφάσεως ὁ αὐτὸς λόγος. εἶναι μὲν <δὴ> ἢ μὴ εἶναι ἅπαν ἀνάγκη καὶ ἔσεσθαί γε ἢ μή· οὐ μέντοι διελόντα γε εἰπεῖν θάτερον ἀναγκαῖον. λέγω δέ, οἷον ἀνάγκη μὲν ἢ ἔσεσθαι ναυμαχίαν αὔριον ἢ μὴ ἔσεσθαι, οὐ μέντοι γενέσθαι ναυμαχίαν αὔριον ἀναγκαῖον οὐδὲ μὴ γενέσθαι· γενέσθαι μέντοι ἢ μὴ γενέσθαι ἀναγκαῖον.

[i] What is, necessarily is, when it is; and what is not, necessarily is not, when it is not. But not everything that is, necessarily is; and not everything that is not, necessarily is not. For to say that everything that is, is of necessity, when it is, is not the same as saying unconditionally that it is of necessity. Similarly with what is not. [ii] And the same account holds for a contradiction: everything necessarily is or is not, and will be or will not be; but one cannot divide and say that one or the other is necessary. I mean, for example: it is necessary for there either to be or not to be a sea battle tomorrow; but it is not necessary for a sea battle to take place tomorrow, nor for one not to take place – though it is necessary for one to take place or not to take place.

In [i], Aristotle differentiates between things whose being or non-being is conditionally or unconditionally necessary. [ii] applies this distinction to contradictions and is thus of special importance to my topic. One might be tempted to say in modern parlance that Aristotle distinguishes between the necessity – presumably the necessary truth – of the tautological disjunction p ∨ ¬p and the non-necessity – presumably the non-necessary truth – of either of the disjuncts, p or ¬p. One may thus take Aristotle to say that we cannot infer from the necessary truth of p ∨ ¬p (for instance, ‘There will or will not be a sea battle tomorrow’) that either disjunct is necessarily true (that either ‘There will be a sea battle tomorrow’ or ‘There will not be a sea battle tomorrow’ is necessarily true). However, such an interpretation is at odds with the views of most commentators, who think that this part of Aristotle’s solution to the problem of future contingents is not about contradictory sentences but rather about contradictory states of affairs or the like.^[62] Consequently, it would be wrong to interpret the necessity in [ii] in terms of necessary truth.

That said, the passage makes it clear that, according to Aristotle, “everything necessarily is or is not, and will be or will not be (εἶναι μὲν <δὴ> ἢ μὴ εἶναι ἅπαν ἀνάγκη καὶ ἔσεσθαί γε ἢ μή)” (19a28–29) and “it is necessary for there either to be or not to be a sea battle tomorrow (ἀνάγκη μὲν ἢ ἔσεσθαι ναυμαχίαν αὔριον ἢ μὴ ἔσεσθαι)” (19a30). This passage can seem to speak in favor of the supervaluational approach. Aristotle seems aware of the tautological character of these disjunctions. The B-standardists’ supervaluational approach appears to answer why, for instance, Aristotle considers it necessary that there either will or will not be a sea battle tomorrow. Note, however, that Aristotle is not talking about the truth of disjunctions whose disjuncts are truth-value gaps. T6 does not even mention truth. Accordingly, the passage does not invoke the schema or any instances of PEM, for T6 is not about pairs of contradictory sentences and their truth values. It seems, instead, that the passages quoted at the beginning of this paragraph involve the schema (one in the present tense, one in the future tense) and an instance of the ontological version of the principle of excluded middle.^[63] This is the reason why I doubt that the supervaluational approach is apt to provide a solution to the problem of the excluded middle (‘Does PEM apply to pairs of contradictory future contingents, or doesn’t it?’), which concerns the principle’s semantic version.

Let us now turn to the second half of the finale of Int. 9:

T7: Int. 9, 19a32–b4 (Ackrill’s translation, modified, emphases not added)

Ὥστε, ἐπεὶ ὁμοίως οἱ λόγοι ἀληθεῖς ὥσπερ τὰ πράγματα, δῆλον ὅτι ὅσα οὕτως ἔχει ὥστε ὁπότερ’ ἔτυχε καὶ τὰ ἐναντία ἐνδέχεσθαι, ἀνάγκη ὁμοίως ἔχειν καὶ τὴν ἀντίφασιν. ὅπερ συμβαίνει ἐπὶ τοῖς μὴ ἀεὶ οὖσιν ἢ μὴ ἀεὶ μὴ οὖσιν. τούτων γὰρ ἀνάγκη μὲν θάτερον μόριον τῆς ἀντιφάσεως ἀληθὲς εἶναι ἢ ψεῦδος, οὐ μέντοι τόδε ἢ τόδε, ἀλλ’ ὁπότερ’ ἔτυχεν, καὶ μᾶλλον μὲν ἀληθῆ τὴν ἑτέραν, οὐ μέντοι ἤδη ἀληθῆ ἢ ψευδῆ. Ὥστε δῆλον ὅτι οὐκ ἀνάγκη πάσης καταφάσεως καὶ ἀποφάσεως τῶν ἀντικειμένων τὴν μὲν ἀληθῆ τὴν δὲ ψευδῆ εἶναι· οὐ γὰρ ὥσπερ ἐπὶ τῶν ὄντων, οὕτως ἔχει καὶ ἐπὶ τῶν μὴ ὄντων, δυνατῶν δὲ εἶναι ἢ μὴ εἶναι, ἀλλ’ ὥσπερ εἴρηται.

So, since sentences are true according to how the things are, it is clear that wherever these are such as to allow of contraries as chance has it, the same necessarily holds for a contradiction also. This happens with things that are not always so or are not always not so. With these it is necessary for one or the other part of a contradiction to be true or false – not, however, this one or that one, but as chance has it, or for one to be true rather than the other, yet not already true or false. Clearly, then, it is not necessary that of every affirmation and opposite denial one should be true and the other false. For what holds for things that are does not hold for things that are not but may possibly be or not be; with these it is as we have said.

Aristotle now turns to the truth values of future contingents. Sentences about necessary things (‘things that are always so’) or impossible things (‘things that are always not so’) do not receive their truth values as chance has it. Hence, one can always tell in advance which truth value a sentence about something necessary or impossible receives. For instance, every token of ‘Every human is a bird’ is false, and every token of ‘Every human is a bipedal land animal’ is true, no matter when these sentences are produced. Accordingly, ‘Every human will be a bird’ is now false, and ‘Every human will be a bipedal land animal’ is now true. By contrast, sentences about contingent things (“things that are not always so or are not always not so (τοῖς μὴ ἀεὶ οὖσιν ἢ μὴ ἀεὶ μὴ οὖσιν)”; 19a36) do receive their truth values as chance has it, and one cannot tell in advance which truth value a sentence about something contingent receives. For instance, ‘It is raining’ is sometimes true (say, at time t₁ and place p) and sometimes false (say, at time t₂ and place p), depending on the states of affairs that obtain at the time and the place the sentence is produced.

Future contingents behave differently, for just as contingent future events are not yet settled, neither are the truth values of sentences about contingent future events. Accordingly, both members of a pair of contradictory future contingents lack a truth value at present. This entails that “it is not necessary that of every affirmation and opposite denial one should be true and the other false (οὐκ ἀνάγκη πάσης καταφάσεως καὶ ἀποφάσεως τῶν ἀντικειμένων τὴν μὲν ἀληθῆ τὴν δὲ ψευδῆ εἶναι)” (19b1–2). Here, Aristotle negates RCP, according to which in every contradictory pair, one member is true and the other false. Strangely, though, Aristotle appears to say the opposite just one sentence before: “With these [namely, pairs of contradictory future contingents] it is necessary for one or the other part of a contradiction to be true or false (τούτων γὰρ ἀνάγκη μὲν θάτερον μόριον τῆς ἀντιφάσεως ἀληθὲς εἶναι ἢ ψεῦδος)” (19a36–37). This sounds like an affirmation of RCP for pairs of contradictory future contingents, but Aristotle adds important qualifications: “not, however, this one or that one, but as chance has it, or for one to be true rather than the other, yet not already true or false (οὐ μέντοι τόδε ἢ τόδε, ἀλλ’ ὁπότερ’ ἔτυχεν, καὶ μᾶλλον μὲν ἀληθῆ τὴν ἑτέραν, οὐ μέντοι ἤδη ἀληθῆ ἢ ψευδῆ)” (19a37–39).^[64] What should we do about this apparent contradiction?^[65]

On my universalist interpretation, these passages do not contradict each other: RCP fails only if it is supposed to apply to contradictories at present (19b1–2), but it does not fail if it is interpreted as applying to contradictories at some point (19a36–39).^[66] But if RCP fails, then so does PEM because RCP is derived from PEM, as I showed in section 2. On my interpretation, however, PEM only fails if interpreted as PEM_now, but it does not fail if interpreted as PEM_point. The supervaluational approach, by contrast, does not seem to make sense of T7. Aristotle does not discuss the truth of disjunctions (such as ‘There will or will not be a sea battle tomorrow’). Instead, he focuses on the truth values of simple sentences (such as ‘There will be a sea battle tomorrow’ and ‘There will not be a sea battle tomorrow’). In contrast to T6, then, PEM is really at issue in T7 because of the danger of RCP possibly failing.

To put it in a nutshell, the B-standardists’ supervaluational approach does not actually account for PEM holding of pairs of contradictory future contingents, and it comes with technical and conceptual machinery without sufficient parallels in Aristotle.

7 Temporalized Truth and the Definitions of Truth and Falsehood in Met. Γ.7

So far, I defended PB_point and PEM_point. I have yet to address a premise that I introduced early on, namely, that Aristotle’s notion of truth is temporalized. To elucidate and support this premise, a closer look at the notions of ‘either true or false at some point’ (occurring in PB_point), ‘true at some point’ (occurring in PEM_point), and ‘false at some point’ (which has not occurred up until now but may be added by analogy) is needed. I refer to these notions comprehensively as ‘temporalized truth’ to indicate that the phrase ‘at some point,’ combined with a semantic value, is currently at issue. I aim to show that the temporalization of Aristotle’s semantic principles is not due to the principles themselves but rather to his notion of truth, which is inherently temporalized.^[67] As a rule (allowing for exceptions, as we will see shortly), sentences presuppose some correspondence to what they assert in order to be either true or false.^[68] However, future contingents lack such correspondence, although they are declarative.

Aristotle presents the following definitions of truth and falsehood:

T9: Met. Γ.7, 1011b26–27 (my translation)

τὸ μὲν γὰρ λέγειν τὸ ὂν μὴ εἶναι ἢ τὸ μὴ ὂν εἶναι ψεῦδος, τὸ δὲ τὸ ὂν εἶναι καὶ τὸ μὴ ὂν μὴ εἶναι ἀληθές

For to assert of what is that it is not or of what is not that it is, is false; and <to assert> of what is that it is and of what is not that it is not, is true;

We may restate these definitions as follows:

1. To assert of what is that it is or to assert of what is not that it is not, is true.

2. To assert of what is that it is not or to assert of what is not that it is, is false.

According to one interpretation, the definitions are elliptic and properly supplemented as follows:^[69]

1. To assert of what is F that it is F or to assert of what is not F that it is not F, is true.

2. To assert of what is F that it is not F or to assert of what is not F that it is F, is false.

In other terms, an affirmative predicative sentence is true iff it asserts of what is F that it is F, and false otherwise, and a negative predicative sentence is true iff it asserts of what is not F that it is not F, and false otherwise. Hence, a sentence of the form ‘x is F’ is true iff x is F, and false otherwise; and a sentence of the form ‘x is not F’ is true iff x is not F, and false otherwise.^[70] Let us instantiate these schemata using our sample future contingents:

1. ‘There will be a sea battle tomorrow’ is true iff there will be a sea battle tomorrow, and false otherwise.

2. ‘There will not be a sea battle tomorrow’ is true iff there will not be a sea battle tomorrow, and false otherwise.

These truth conditions clearly indicate that future contingents are either true or false, albeit not now.^[71] Future contingents are declarative and hence truth-value bearers, though the fact that they signify a contingent event in the future bars them from now having a truth value. For them to be able to correspond to what they assert and thus be truth-evaluable, some time span has to elapse. In the case at hand, one member of the pair of contradictory future contingents will become true and the other false the day after these sentences are produced. Until then, they are neither true nor false because they do not fulfill the presupposition of corresponding to what they assert. Other future-tense sentences – those that do not signify a contingent event in the future – behave differently, as we can see in the following contradictories:

1. ‘Socrates will be mortal’ is true iff Socrates will be mortal, and false otherwise.

2. ‘Socrates will not be mortal’ is true iff Socrates will not be mortal, and false otherwise.

Assuming for the sake of argument that Socrates exists, (iii) is a singular future-tense sentence about something necessary, or a future necessity, and (iv) is a singular future-tense sentence about something impossible, or a future impossibility. The difference between (i) and (ii), on the one hand, and (iii) and (iv), on the other, is that (iii) and (iv) now have a truth value. I maintain that the reason for this is that there is something in the present that makes (iii) true and (iv) false: the sentences’ truth values are causally dependent on the relation between the things signified by their subject and predicate terms.^[72] Socrates is a human being, and humans qua bipedal land animals have essential attributes, such as being material substances. Anything material that comes into being, like a human individual, exists at one time and does not exist at another (Cael. I.12, 283a29–b5). Hence, if Socrates now exists, it is necessary that he will die at some point. This entails that he is and will be mortal during his lifetime, making (iii) true and (iv) false. Strictly speaking, then, (iii) and (iv) do not correspond or fail to correspond to what they assert because each asserts a future state of affairs. Instead, they are respectively made true and false by something presently obtaining.^[73] Future contingents, by contrast, now lack a truth value because there cannot be a relation of correspondence between the states of affairs and what future contingents assert, and because they do not indicate any causal relations necessitating their truth or falsehood already at the time they are produced.

To generalize, we may state that temporalized truth has to be at work in the definitions of truth and falsehood in T9. The definitions, as such, must be sufficiently general to accommodate all predicative sentences, including borderline cases. But if future contingents are predicative sentences that are neither true nor false, then the definitions do not accommodate all predicative sentences. Hence, the definitions are not definitions – contradiction.^[74] One way out of this impasse is to assume that Aristotle presents definitions of temporalized truth and falsehood. Assuming that the definitions should be sufficiently general to accommodate all predicative sentences, and given that there are predicative sentences that cannot be either true or false at present but will become either true or false sometime in the future, the definitions should be taken to involve temporalized truth in order to accommodate the borderline case of future contingents. In a sense, then, temporalized truth is parasitic on the definitions of truth and falsehood because they seem to accommodate all predicative sentences already at the time they are produced, except future contingents, which the definitions nonetheless accommodate in virtue of being temporalized.

We are now in a position to add another reason why the standard interpretation is questionable: it either contradicts the definitions or leaves it open how they accommodate future contingents. Interpreters of Int. 9 must address the question of how the definitions accommodate future contingents. All standardists maintain that PB fails for future contingents, and some also believe that PEM fails for pairs of contradictory future contingents. This is because they presuppose an interpretation of PB and PEM in terms of PB_now and PEM_now. The question now is: how would standardists interpret the definitions? It is clearly not the case that Aristotle presents definitions of timeless truth and falsehood, as we would expect in modern philosophy of language. The most straightforward answer seems to be that standardists would interpret the definitions similarly to PB and PEM, that is, as now applying to sentences. However, this answer is unattractive because it contradicts the definitions: they would fail to accommodate future contingents and would, therefore, not qualify as definitions. I am not aware of any other standardist answers that address this problem without producing another inconsistency within Aristotle’s semantic theory.^[75] Note that the B-standardists’ supervaluational approach also fails to do justice to the definitions given that it assumes disjunctions to be true, although the definitions do not refer to the truth conditions of complex sentences.^[76]

On my universalist account, by contrast, the fact that future contingents now lack a truth value is not a problem for the definitions provided that we interpret them as involving temporalized truth. Here are the definitions again adapted accordingly: an affirmative predicative sentence is true at some point iff it asserts of what is F at that point that it is F at that point, and false otherwise; a negative predicative sentence is true at some point iff it asserts of what is not F at that point that it is not F at that point, and false otherwise.^[77] In this way, the definitions accommodate not only sentences that are either true or false already at present but also future contingents. Thus, universalists can explain how the definitions of truth and falsehood accommodate future contingents using the same argumentative apparatus as in their explanation of how PB and PEM apply to (pairs of contradictory) future contingents.

8 Semantic Principles, Demonstrative Science, and Temporalized Truth

Interpreting the definitions in T9 as definitions of temporalized truth provides a solid basis for claiming that Aristotle’s semantic principles (PB, PEM, RCP, and PNC) are temporalized because his notions of truth and falsehood are temporalized. For if the notions of truth and falsehood occurring in Aristotle’s semantic principles are inherently (that is, per definitionem) temporalized, the principles themselves are temporalized because they involve temporalized notions. Put differently, Aristotle’s semantic principles inherit their temporalized character from the temporalized notions of truth and falsehood occurring in them. Hence, nothing is lost if we interpret PB and PEM as PB_point and PEM_point, as the initial worry went (recall sections 4 and 6). PB_point and PEM_point do not seem to be too weak interpretations of PB and PEM. On the contrary, it seems that the standardists’ PB_now and PEM_now are too strong interpretations of PB and PEM.

Proponents of the standard interpretation may raise the concern of whether temporalized truth also figures in semantic principles other than PB and PEM, namely, RCP and PNC. RCP states that in every contradictory pair, one member is true and the other false. For RCP not to fail for pairs of contradictory future contingents, it has to be interpreted according to temporalized truth to state that in every contradictory pair, one member is true at some point and the other false at that point. RCP is a positive principle in that it attributes truth to one contradictory and falsehood to the other, though it does not state which contradictory has which truth value. If RCP is interpreted as now requiring truth values assigned to contradictories (RCP_now), it fails for pairs of contradictory future contingents, which now lack truth values. Only an interpretation of RCP in terms of temporalized truth (RCP_point) guarantees that it applies to all contradictory pairs. Let us juxtapose RCP_now and the universalist RCP_point:

1. In every contradictory pair, one member is now true and the other now false.

2. In every contradictory pair, one member is true at some point and the other false at that point.

Things stand slightly differently in the case of PNC. PNC states the impossibility of both members of a contradictory pair being simultaneously true. Hence, PNC excludes there being a contradictory pair in which both members have the truth value true. This makes PNC a negative principle, in contrast to PB, PEM, and RCP, which assign truth values to (pairs of contradictory) sentences and are therefore positive. I presume it is because PNC does not distribute any truth values to pairs of contradictories that it does not conflict with pairs of contradictory future contingents, which lack truth values at the time they are produced.^[78] Again, let us juxtapose PNC_now and the universalist PNC_point:

1. It is impossible that both members of a contradictory pair are now simultaneously true.

2. It is impossible that both members of a contradictory pair are ever simultaneously true.

The fact that PNC states an impossibility and is thus negative affects the way temporalized truth figures in PNC: the negation of ‘at some point’ is ‘never,’ such that PNC_point amounts to the thesis that contradictories cannot ever be simultaneously true.

Finally, standardists may object that Aristotle’s syllogistic logic and theory of demonstrative science do not seem to mention the idea of something being either true or false at some point. They may also point out that PB_point does not warrant the inference from a sentence not being true (false) to it being false (true) because future contingents are neither true nor false at present. Likewise, PEM_point and RCP_point do not warrant the inference from one member of a contradictory pair not being true (false) to the other one being false (true) because both members of a pair of contradictory future contingents lack a truth value at present. But shouldn’t these semantic inferences be at the heart of Aristotle’s logical theory? I do think so, but it is crucial to see that future contingents do not play a role in Aristotle’s syllogistic logic and theory of demonstrative science. Temporalized (that is, past-tense and future-tense) sentences do not seem to be categorical. Otherwise, we would expect Aristotle to discuss their syntax or inferential relations in the opening chapters of the Analytics or to examine sample deductions involving temporalized sentences. For example, Aristotle does not say that a necessary sentence, such as ‘Every human is necessarily bipedal,’ entails the correlative past-tense sentence, ‘Every human was bipedal,’ or future-tense sentence, ‘Every human will be bipedal’ (though it is likely that he would consider these inferences valid). This means that in Aristotle’s logical theory, the universalist PB_point, PEM_point, and RCP_point are equivalent to PB_now, PEM_now, and RCP_now, respectively. This is so, not because the standard interpretation conceives of Aristotle’s semantic principles correctly, but because his logical theory leaves no room for the borderline case of future contingents, and future contingents are the only cases that disrupt the semantic inferences mentioned at the beginning of this paragraph.

In a word, the scope of Aristotle’s syllogistic logic and theory of demonstrative science is, in a certain sense, narrower than his philosophy of language: the former deals with fewer kinds of sentences than the latter. For instance, De Interpretatione discusses future contingents, past-tense sentences, and existential sentences, and these kinds of sentences do not play a role in his syllogistic logic (existential sentences do play a role in his theory of demonstrative science, but they do not appear to serve as inputs for the syllogistic machinery upon which his theory of demonstrative science is erected). Aristotle nonetheless treats such sentences as truth-value bearers, so we need to explain how PB and related principles apply to them.

In sum, Int. 9 prompts us to rethink PB, PEM, RCP, and PNC. I claim that due to the problem of future contingents, the first three principles do not hold without qualification. In contrast to most scholars, however, I do not believe that this qualification amounts to a failure to apply to (pairs of contradictory) future contingents. Instead, the qualification concerns the truth values semantic principles assign to (pairs of contradictory) sentences. Aristotle conceives truth as something temporalized, and this feature is inherited by his semantic principles, including PNC, even though it is not affected by the problem of future contingents. If we interpret Aristotle’s PB, PEM, and RCP in terms of temporalized truth, they do not fail for (pairs of contradictory) future contingents.

9 Conclusion

According to Aristotle, future contingents lack a truth value at the time they are produced. Does this entail that PB (‘Every sentence is either true or false’) is suspended for future contingents and that PEM (‘It is necessary that in every contradictory pair, one member is true’) is suspended for pairs of contradictory future contingents? Adherents of the standard interpretation of Int. 9 think that PB fails. Moreover, some standardists think that PEM, too, fails, whereas others try to rescue this principle by adducing the method of supervaluations. Against this, I argued that the supervaluational method is unsuitable for showing how PEM should not be suspended. I explained that PB and PEM only fail if they require truth values assigned to sentences already at the time they are produced. PB and PEM do not fail if they require truth values assigned to sentences either at the time they are produced or at some point in the future, and this is the bedrock of my universalist interpretation of Int. 9. I also showed that the standard interpretation conflicts with Aristotle’s account of declarations in Int. 4 and appears to be at odds with the definitions of truth and falsehood in Met. Γ.7. There is an intimate connection between time and Aristotle’s semantic foundations (his conception of truth and falsehood as well as his semantic principles, such as PB and PEM), and I hope this paper is a step towards more clarity in this regard.