Did Aristotle have a Doctrine of Existential Import?

2.1 Categorical Propositions

In De Interpretatione 5, Aristotle divides “single statement-making” sentences or propositions into two kinds: (a) the “simple” propositions, which consist of “affirming or denying of something,” and (b) the “composite” propositions, which are “compounded of simple statements” and “single in virtue of a connective” (DI 5, 17a16–22).^[27] It is usual to call the former categorical propositions.^[28] The characterization of simple or categorical propositions^[29] divides them into affirmations (or affirmatives) and negations (or negatives), as in De Interpretatione 6 (17a25). In De Interpretatione 7 (17a38–17b3), he presents a division based on the division of “actual things” into two kinds: the “particular” (e.g., Callias) and the “universal” (e.g., man) (DI 7, 17a38–b3). With reference to this division, he divides categoricals into two kinds:

1. The singular categoricals: those “about particulars, e.g. Socrates is white–Socrates is not white” (DI 7, 17a28–29).

2. The general categoricals: those “about a universal” (DI 7, 17a29–30).

Now, singular categoricals are instances of the two forms:

Instances of the affirmative and negative forms resulting from replacing “ s ” and “ A ” in them with singular terms (e.g., “Socrates”) and general terms (e.g., “just”), respectively, are singular categoricals: singular affirmatives (e.g., “Socrates is just”) and negatives (e.g., “Socrates is not just”). The terms replacing “ s ” and “ A ” are called their subject and predicate terms, respectively.

In the Topics (Top.) II.1, Aristotle divides general categoricals into universal and particular categoricals:

Of problems some are universal, others particular. Universal problems are such as “Every pleasure is good” and “No pleasure is good;” particular problems are such as “Some pleasure is good” and “Some pleasure is not good.”^[30]

And he elaborates on divisions of general categoricals in the Prior Analytics I.2:

Every [general] proposition states that something either belongs or must belong or may belong; of these some are affirmative, others negative, in respect to each of the three modes; again some affirmative and negative propositions are universal, others particular, others indefinite.^[31]

Thus there are four kinds of (definite) general categoricals:^[32]

Replacing “A” and “B” in the forms with general terms yields general categoricals. For example, replacing “A” and “B” in “Every A is B ” with “pleasure” or “human” and “good” or “just,” respectively, yields universal affirmatives: “Every pleasure is good” and “Every human is just.”^[34] The representative examples of the four kinds of categoricals given in the Prior Analytics I.2 (25a5–13) are as follows:

The general terms replacing “A” and “B” are called the subject and predicate terms, respectively.^[35]

Now, whether a categorical proposition is affirmative or negative determines its quality, and whether it is singular, universal, or particular its quantity.

Universal and particular categoricals are definite (or quantified) general categoricals, which involve quantifiers modifying subject terms: “every,” “some,” and “no.” By contrast, indefinite (or unquantified) general categoricals do not involve quantifiers; they are instances of “ A is B ” and “ A is not B ” (e.g., “Man is an animal,” “Pleasure is not good”). It is useful for the present purpose to focus on definite general categoricals, and I will henceforth use “general categorical (proposition)” for definite general categoricals unless noted otherwise.

In both the Topics II.1 and the Prior Analytics I.2, we have seen, Aristotle gives instances of “Some A is not B ” (e.g., “Some pleasure is not good,” “Some animal is not man”) as examples of particular negatives. In the discussion of opposite (i.e., contrary or contradictory) general categoricals in De Interpretatione (DI 7, 17b16–26), however, he does not discuss instances of “Some A is not B ” and instead uses an instance of “Not every A is B ” (e.g., “Not every man is white”) to give a contradictory of a universal affirmative (e.g., “Every man is white”). And he uses instances of “Not every A is B ” interchangeably with “Some A is not B ” in the Prior Analytics. In the Prior Analytics I.1, he says:

By universal I mean a statement that something belongs to all or none of something; by particular that it belongs to some or not to some or not to all.^[36]

And in the Posterior Analytics, he holds that “Some A is not B ” is the same (in meaning) as “Not every A is B ”: “not to belong to some and to belong not to every are the same” (APo. II.11, 62a11–13).

So it is usual in traditional logic to assume that “Some A is not B ” and “Not every A is B ” are logically equivalent. As noted in Section 1, however, some traditional logicians (e.g., Abelard) draw a logical distinction between them. So it is useful to distinguish them in discussing various traditional systems of logic. To do so, I call instances of “Some A is not B ” O -propositions, and those of “Not every A is B ” A ¯ -propositions. (I consider the former particular negatives of the primary kind, and the latter those of the secondary kind.)

2.2 Square of Opposition

At the core of systems of traditional logic are theses about logical relations between general categoricals with the same subject and predicate terms. Say that categoricals (e.g., “Every human is just” and “Some human is just”), singular or general, match, if they have the same subject and predicate terms. Then most traditional systems include four theses about matching general categoricals:

1. Particular affirmatives do not imply matching universal affirmatives ( I α β ⇏ A α β ).^[37]

2. Subalternation: Universal affirmatives imply matching particular affirmatives ( A α β ⇒ I α β ).

3. EI -contradiction: Universal negatives and matching particular affirmatives are contradictories ( I α β ≁ E α β ).

4. AO -contradiction: Universal affirmatives and matching particular negatives are contradictories ( O α β ≁ A α β , A ¯ α β ≁ A α β ).^[38]

Now, say that two propositions are contraries, if they cannot both be true but can both be false;^[39] and that they are subcontraries, if their contradictories are contraries. Then Tr0–Tr3 imply both Tr1-a and Tr1-b:

Assuming Tr2–Tr3, both Tr1-a and Tr1-b are equivalent to the conjunction of Tr0–Tr1. Call the doctrine of general categoricals consisting of Tr0–Tr3 the orthodox traditional doctrine.

Some proponents of traditional logic^[41] hold a weaker doctrine, the one resulting from replacing Tr3 in the orthodox traditional doctrine with its weak cousin:

Call this doctrine the moderate traditional doctrine. The doctrine includes Tr1-a but not Tr1-b,^[42] and it is equivalent to the theory of opposition Aristotle presents in De Interpretatione (DI 5, 17b16–26), where he does not discuss O -propositions (e.g., “Some human is not white”) while discussing A ¯ -propositions (e.g., “Not every human is white”).^[43] In the Prior Analytics, however, he gives O -propositions as primary examples of particular negatives while taking A ¯ -propositions to be logically equivalent to them (Section 2.1). Adding the assumption of logical equivalence between the two kinds of particular negatives to the theory of opposition yields the orthodox traditional doctrine.^[44] This doctrine is encapsulated in the diagram called the (traditional) square of opposition that dates back to Apuleius^[45] and Boethius^[46] (Figure 1):

Figure 1

The traditional square of opposition.

So the doctrine is often called the square of opposition (doctrine).^[47]

2.3 Traditional Doctrines of Existential Import

It is usual for modern logicians to hold that traditional logic makes a serious error in accepting the subalternation thesis (Tr1). Tr1 is incompatible, they argue, with the existence of categoricals with empty subject terms. The objection dates back to Brentano, who presents the modern doctrine of (general) categoricals, which consists of Tr0 and Tr2–Tr3.^[48] The doctrine, which results from removing Tr1 from the orthodox traditional doctrine (Tr0–Tr3), assumes an analysis of general categoricals in terms of general existential propositions, i.e., instances of the form “There is an A ” (in symbols, ∃ ! α ), where “ A ” (or “ α ”) is for general terms (e.g., “There is a human who is just”).^[49] To formulate the modern objection to subalternation, however, it is not necessary to assume an analysis of general categoricals. Russell states an objection as follows:

[…] general propositions [viz., universal categoricals] are to be interpreted as not involving existence. When I say, for instance, “All Greeks are men,” .... It is to be considered ... as not implying that [i.e., existence].... If it happens that there were no Greeks, … the proposition that “All Greeks are men” ... would be true.... because the contradictory of any general [viz., universal] statement does assert existence and is therefore false in this case.^[50]

In this passage, he argues that the universal “All Greeks are men” does not imply the existence of a Greek because its contradictory, “Some Greeks are not men,” does. Assuming that “Some Greeks are men” also implies the existence of a Greek, the argument yields an objection to subalternation.

We can give a general formulation of this objection that lays out its basic assumptions. To do so, it is useful to use the notion of existential import that an early proponent of the modern doctrine of general categoricals, Venn, uses to compare and contrast the doctrine with traditional ones (e.g., the orthodox traditional doctrine).^[51]

Venn uses “existential import” as a short for “the import [i.e., semantic content] of propositions, as regards the actual existence of their subjects and predicates.”^[52] And he says that a general categorical (e.g., “Every Greek is a human”) has existential import (with regard to its subject term), if the proposition implies the existence of its subjects, i.e., denotations of its subject term (e.g., Greeks).^[53] To capture this notion, it is useful to consider logical relations between categorical and existential propositions. Say that an existential proposition (e.g., “There is a goat-stag”) matches a categorical proposition (and vice versa), if the term featured in the existential proposition (e.g., “goat-stag”) is the same as the subject term of the categorical proposition. Then we can say that a categorical proposition (e.g., “Every goat-stag lives in Athens”) has existential import, if it implies the matching existential proposition (e.g., “There is a goat-stag”).^[54]

Now, we can formulate the basic assumptions of the modern objection to subalternation:

These theses are incompatible with the subalternation thesis (Tr1):

To prove this, it is useful to note that Tr1 and I -EI imply the existential import thesis on universal affirmatives:

1. Existential import thesis on universal affirmatives: Universal affirmatives have existential import ( A α β ⇒ ∃ ! α ).

We can now prove Theorem 1:

Now, the modern analyses of general categoricals presuppose G-ET and are designed to yield I - and O -EI while implementing Tr2–Tr3. And proponents of the modern doctrine take Tr3, I -EI, O -EI, and G-ET for granted. So they might use Theorem 1 to reject the subalternation thesis (Tr1). Call the modern objection to Tr1 that rests on the theorem the existential import objection. Then we can see that the objection to Tr1 that Russell presents in the passage quoted above^[55] boils down to the existential import objection.^[56] And call the existential import doctrine that results from removing Tr1 from the orthodox traditional doctrine (Tr0–Tr3) while adding I -EI, O -EI, and G-ET the modern doctrine of existential import of (general) categoricals.

Those who accept traditional systems of logic might reply to the existential import objection by rejecting one (or more) of the four theses assumed in the objection: Tr3, I -EI, O -EI, and G-ET. And some traditional logicians hold traditional doctrines of existential import by adding all but one of these theses to the moderate or orthodox traditional doctrine. Ockham, who holds the orthodox traditional doctrine (Tr0–Tr3), rejects O -EI. Abelard, who accepts the moderate traditional doctrine (Tr0–Tr2), rejects Tr3. And De Morgan, an early pioneer of modern logic who, like Boole,^[57] accepted traditional logic (including the orthodox traditional doctrine), would reject G-ET by requiring terms of general categoricals in traditional systems to be non-empty;^[58] in formulating a traditional system of syllogisms, he says “The affirmative [general categorical] proposition requires the existence of both terms: the negative proposition, of one,” and proposes to “make the existence of terms ... preceding conditions [i.e., presuppositions] of the propositions.”^[59]

Note that all the existential import doctrines mentioned above (the modern and the three traditional ones) accept I -EI. This might suggest that it is usually considered implausible to reject this thesis. And it is usual to take Aristotle to implicitly assume the thesis. But I think there is a good reason to take him not to hold or assume it,^[60] which means that he would not accept any of the three traditional doctrines of existential import. To present the grounds for this interpretation, it is useful to examine problems of the interpretation that takes him to hold the existential import doctrine Ockham holds.

3.1 General Theses

The affirmative–negative doctrine assumes a division of all categoricals into affirmatives (e.g., I ) and negatives (e.g., E ) that yields the following thesis:

Then the doctrine states whether affirmative and negative categoricals are true or false when their subject terms (e.g., “Socrates,” “pleasure”) are empty (i.e., they do not denote anything). Say that categoricals (“Pegasus flies,” “Every centaur is an extraterrestrial”) are vacuous, if they have empty subject terms. Then we can formulate two main theses of the doctrine as follows:

On these theses, “Every centaur is an extraterrestrial” is false (AN1), while “Some centaur is not an extraterrestrial” is true (AN2). Note that assuming AN0, AN1 and AN2 are equivalent.

Ockham holds AN0–AN2 while assuming that subject terms of categoricals might be empty. In Part I of Summa Logicae, where he discusses how negative existential propositions (e.g., “A white man does not exist”) can be true,^[64] he states both theses:

In affirmative propositions a term [viz., the subject term] is always asserted to supposit for something. Thus, if it supposits for nothing the proposition is false. However, in negative propositions the assertion is either that the term does not supposit for something or that it supposits for something of which the predicate is truly denied.^[65]

Affirmatives are false and negatives true, on this account, if they are vacuous, i.e., their subject terms denote (or “supposit for”) nothing.^[66]

Proponents of the AN-doctrine, like Ockham, usually formulate its main theses as vacuous truth or falsity theses (e.g., AN1–AN2). In modern discussion of traditional logic, however, it is usual to take the doctrine to involve existential import theses. For vacuous falsity theses are equivalent to existential import theses. To formulate the existential import thesis equivalent to AN2, we need to use a generalization of Venn’s notion of existential import that applies to singular categoricals as well. Call instances of “ s exists (in symbols, ∃ ! s ),”^[67] where “ s ” is for singular terms, singular existential propositions. Then say that singular or general existential propositions (e.g., “Socrates exists,” “There is a human”) match categorical propositions (e.g., “Socrates is just,” “Some human is just”), if the terms featured in the former (e.g., “Socrates,” “human”) are the same as the subject terms of the latter. Then we can formulate the existential import thesis equivalent to AN1 as follows:

1. The affirmative thesis (EI-formulation): Affirmative categoricals have existential import.^[68]

1. The negative thesis (EI-formulation): Negative categoricals do not have existential import.

This thesis does not follow from AN0–AN2.^[70] But it follows from AN2 and a generalization of G-ET:

1. Existential thesis: Existential propositions (singular or plural) are not logical truths.

AN3, which is assumed by those who hold AN1–AN2, codifies the assumption that terms of categoricals might be empty, for if some terms (e.g., “human”) might have no denotation, the existential propositions featuring them (e.g., “There is a human”) cannot be logical truths. To see that AN2–AN3 imply AN2*, note that AN2 is equivalent to AN2 ′ :

1. Contradictories of existential propositions imply matching negative categoricals.

Note also that AN2* implies AN3, for “Some human is just,” for example, must imply “There is a human” if this proposition is a logical truth.

AN2–AN3 imply AN2*, we have seen, while AN2* implies AN3. And AN1 and AN1* are equivalent, while AN0–AN1 imply AN2 (for assuming AN0, as noted above, AN1 and AN2 are equivalent). So we can take the basic theses of the affirmative–negative doctrine to be: AN0, AN1 (or AN1*), and AN3 (or AN2*).

3.2 Specific Theses on Singular and General Categoricals

The affirmative–negative doctrine, as noted above, assumes a division of all categoricals into affirmatives and negatives that yields AN0. Affirmatives in the usual division include singular, universal, and particular affirmatives, and negatives singular, universal, and particular negatives. This division yields specific theses of the doctrine on singular and general categoricals:

1. Singular affirmative–negative doctrine ( S -AN)

   • S-AN0. Singular contradictories thesis: S + s α and S − s α are contradictories ( S − s α ≁ S + s α ).

   • S-AN1. Singular affirmative thesis ( S + -VF): Singular affirmatives must be false if vacuous.

   • S-AN1*. Singular affirmative thesis ( S + -EI): Singular affirmatives have existential import ( S + s α ⇒ ∃ ! s ).

   • S-AN3. Singular existential thesis: Singular existential propositions are not logical truths.

2. General affirmative–negative doctrine ( G -AN)

   • G-AN0. General contradictories thesis: Universal and particular negatives are contradictories of particular and universal affirmatives, respectively (Tr2, Tr3 (or Tr3 ′ )).

   • G-AN1. General affirmative thesis ( G + -VF): General affirmatives must be false if vacuous ( A -VF, I -VF).

   • G-AN1*. General affirmative thesis ( G + -EI): General affirmatives have existential import ( A -EI, I -EI).

   • G-AN3. General existential thesis: General existential propositions are not logical truths.

3.3 General Negatives and O -Propositions

As AN0 and AN1 (or AN1*) imply AN2, so do G -AN0 and G -AN1 (or G -AN1*) imply the negative thesis on general categoricals:

1. Universal and particular negatives must be true if vacuous:

   1. Universal negatives must be true if vacuous ( E -VT).

   2. Particular negatives must be true if vacuous.

1. Vacuous truth thesis on A ¯ : A ¯ -propositions must be true if vacuous.

2. Vacuous truth thesis on O : O -propositions must be true if vacuous.

So we can formulate two versions of the (general) AN-doctrine. The strong version, the orthodox (general) AN-doctrine, assumes the orthodox traditional doctrine (Tr0–Tr3) and takes O -propositions to be negatives to include O -VT. The weak version, the moderate (general) AN-doctrine, does not include O -VT, for it assumes only the moderate traditional doctrine (Tr0–Tr2). And those who hold the moderate (general) AN-doctrine might take O -propositions to be affirmatives to hold O -VF:

1. Vacuous falsity thesis on O : O -propositions must be false if vacuous.

1. Existential import thesis on O : O -propositions have existential import.

[...] [consider particular categoricals, such as] “Some man runs” and ... “Some animal is not a man” .... it is sufficient for the truth of such propositions that the subject term and the predicate refer to [or supposit for] some same thing, if the proposition is affirmative.... But if such a proposition is negative, it is required ... either that the subject refers to nothing or that it refers to something to which the predicate does not refer .... Thus .... it suffices for the truth of a negative ... particular proposition that the subject refers to nothing.^[71]

On this account, the O -proposition “Some animal is not a man,” for example, is true if the subject term “animal” is empty.^[72] We can take Ockham to draw this conclusion by assuming Tr3 to accept O -VT.

By contrast, Abelard argues against Tr3 by rejecting O -VT:

[...] Similarly for categorical propositions, where the only real truth-value inverting contradiction .... of any arbitrary positive [or affirmative] proposition appears to be the one that has the negation preposed to it .... For example, the contradictory of Every human is human is Not every human is human, and not Some human is not human, since there are situations where the first and the third are simultaneously false. For when not a single human exists, both of these two propositions are false: Every human is human and Some human is not human .... [So] it must be incorrect to call them contradictory opposites.^[73]

In this argument, he assumes O -VF to argue that “Some human is not a human,” for example, is not a contradictory of “Every human is a human” because both must be false if there is no human. While Abelard disagrees with Ockham on O -propositions, we might take him to agree with Ockham on other general propositions to hold the restriction of the moderate AN-doctrine to general categoricals (the moderate G -AN), which includes A ¯ -VT but not O -VT.^[74] In his commentary on Aristotle’s Categories, Kitāb al-Maqūlāt (in short, al-Maqūlāt) or Book of the Categories,^[75] Al-Fārābī (872–950 CE) has a systematic discussion of truth or falsity of vacuous categoricals in elaborating on Aristotle’s theory of opposition. In the discussion, which assumes the singular and general existential theses ( S -AN3, G -AN3), he does not state overarching theses applying to all affirmatives or negatives (e.g., AN1, AN2), but formulates vacuous truth or falsity theses on specific kinds of affirmatives and negatives:

• Affirmative theses: S + -VF, A -VF, I -VF

• Negative theses: S − -VT, E -VT, A ¯ -VT

• Singular AN-doctrine ( S -AN): S + -VF, S − -VT.

• Moderate general AN-doctrine (moderate G -AN): A -VF, I -VF, E -VT, A ¯ -VT.

4.1 The Affirmative–Negative Interpretation: Singular Categoricals

Proponents of the AN-interpretation take Aristotle to assume the AN-doctrine as a generalization of his discussion of singular contradictories. In the Categories 10 (13b12–37),^[83] he holds the singular contradictories thesis ( S -AN0): matching singular affirmatives and negatives (e.g., “Socrates is sick” and “Socrates is not sick”) are contradictories, that is, “one [of them] will always be false and the other true” (13b26–27).^[84] To do so, he contrasts matching singular affirmatives and negatives with singular affirmatives with the same subject term and contrary predicates (e.g., “Socrates is sick” and “Socrates is well”). And he argues that the former are contradictories while the latter are not:

[...] [Consider] contraries said with combination, [such as] ‘Socrates is well’ ... [and] ‘Socrates is sick’.... not even with these is it necessary always for one to be true and the other false.... neither ‘Socrates is sick’ nor ‘Socrates is well’ will be true if Socrates himself does not exist at all.... But with an affirmation and negation one will always be false and the other true whether he exists or not. For take ‘Socrates is sick’ and ‘Socrates is not sick’: if he exists it is clear that one or the other of them will be true or false, and equally if he does not; if he does not exist ‘he is sick’ is false but ‘he is not sick’ is true.^[85]

While holding that “Socrates is sick” and “Socrates is not sick,” for example, have different truth values in all logically possible situations, including when Socrates does not exist, he holds that “Socrates is sick” and “Socrates is well,” for example, are both false if Socrates does not exist. Proponents of the AN-interpretation takes Aristotle to hold the same for all singular affirmatives to attribute to him the singular affirmative thesis ( S -AN1): singular affirmatives must be false if vacuous.

And we might take the passage to assume that singular existential propositions (e.g., “Socrates exists”) are not logical truths, for the discussion in the passage assumes that it is not logically false that Socrates does not exist. If it is, whether “Socrates is sick” and “Socrates is well” have different truth values if Socrates does not exist is irrelevant to the issue of whether or not they are contradictories. Moreover, Aristotle argues in the passage that “Socrates is sick” and “Socrates is not sick” are not contradictories because both are false if Socrates does not exist. If this is logically false, however, the former does not follow from the latter.^[86]

4.2 The Affirmative–Negative Interpretation: General Categoricals

On the AN-interpretation, Aristotle presupposes or assumes the general affirmative–negative doctrine ( G -AN). This doctrine includes theses about general categoricals that draw parallels with key theses of the singular AN-doctrine:

1. General contradictories thesis: Universal and particular negatives are contradictories of particular and universal affirmatives, respectively (Tr2, Tr3 (or Tr3 ′ )).

2. General affirmative thesis ( G + -VF): General affirmatives must be false if vacuous ( A -VF, I -VF).

3. General affirmative thesis ( G + -EI): General affirmatives have existential import ( A -EI, I -EI).

4. General existential thesis: General existential propositions are not logical truths.

G -AN1* follows from Tr1 (i.e., subalternation) and I -EI (i.e., the existential import thesis on particular affirmatives) (see Section 3.2). And those who take Aristotle to accept G -AN hold that he assumes I -EI in his treatments of general categoricals (e.g., the theory of opposition). For example, Parsons says:

[ I -EI] is rarely made explicit, rather it is taken for granted by most logicians, ancient, medieval, and modern alike. Aristotle presupposes it in his proof of simple conversion at Prior Analytics 25a14–16.^[87]

In the passage of the Prior Analytics Parsons refers to, Aristotle proves the E -conversion thesis, the thesis that universal negatives imply their converses ( E β α ⇒ E β α ):

First then take a universal negative with the terms A and B . Now if A belongs to no B , B will not belong to any A , for if it does belong to some ... (say to C ), it will not be true that A belongs to no B — for C is one of the B s .^[88]

In this passage, he presents an indirect proof of E -conversion. To prove that E β α (or “ A belongs to no B ”) implies E α β (or “ B belongs to no A ”), he assumes a contradictory of the latter, “it does belong to some”: I α β (or “ B belongs to some A ”). Then he draws a contradiction from E β α and I α β . To do so, it seems, he invokes I α β to assume that A belongs to something (e.g., C ) that is a B (or is “one of the B s ”). In making this assumption, Parsons takes Aristotle to presuppose that the particular affirmative I α β implies that there is something that is an A , i.e., a denotation of the term “ A .”

4.3 Particular Negatives

The orthodox general AN-doctrine (i.e., the general AN-doctrine that includes Tr0–Tr3) includes two theses on O -propositions:

1. Vacuous truth thesis on O : O -propositions must be true if vacuous.

2. O -propositions do not have existential import.

But it is usual to consider this an implausible response, for O -propositions are usually taken to have existential import.^[89] This gives rise to a problem with the AN-interpretation: The interpretation, it seems, attributes to Aristotle an implausible thesis he has never stated.^[90] So some Aristotle scholars reject it. For example, Spade takes the AN-doctrine not to originate from Aristotle but to result from later attempts to extend his logic to deal with vacuous categoricals:

Aristotle has nothing to say about [truth or falsity of vacuous general categoricals] ... when later authors ... sought to extend this logical machinery [i.e., the square of opposition] to accommodate non-denoting terms, it became plain that something was going to give. Generally, it was the O-form that was compromised. Despite its ordinary sense in Greek and Latin (and in English, too), the O-form “Some S is not P” was taken as not having existential import.^[91]

While O -propositions have existential import in Greek and Latin as well as in English, Spade holds, most of those who inherit the square of opposition (i.e., the orthodox traditional doctrine) came to deny it to apply the doctrine to vacuous general categoricals.

Some proponents of the AN-interpretation respond to this objection by arguing that Aristotle does not hold Tr3; propositions he considers particular negatives do not include O -propositions, they hold, for they are A ¯ -propositions. For example, Thompson says “A literal rendering of his examples of O gives ‘Not every S is P,’ which is to be taken simply the denial that P is truly affirmed universally of S” and “the usual rendering of O as ‘Some S exists and is not P’ certainly does not express what Aristotle meant.”^[92] Wedin says “Aristotle never writes the o statement as ‘Some A is not B’,”^[93] and Parsons “Ackrill’s translation [of De Interpretatione] contains something a bit unexpected: Aristotle’s articulation of the O form is not the familiar ‘Some S is not P’... it is rather ‘Not every S is P.”’^[94] While using “ O ” for particular negatives, they hold that Aristotle limited them to A ¯ -propositions (i.e., instances of “Not every S is P ”). This response is based on Aristotle’s formulation of the theory of opposition in De Interpretatione 7 (17b16ff), as is made clear in Parsons’s statement. In this formulation of the theory, as noted in Section 2.2, he indeed does not discuss O -propositions and holds only Tr3 ′ : A ¯ -propositions are contradictories of matching A -propositions. But this does not mean that he did not include O -propositions among particular negatives, nor that he made no commitment to Tr3.

In the Topics and the Prior Analytics (but not in De Interpretatione), we have seen (Section 2.1), Aristotle divides (determined) general categoricals into affirmative and negatives and into universals and particulars. In both works, his primary examples of particular negatives are O -propositions: “Some pleasure is not good” (Top. II.1, 108b35–109a3), and “Some animal is not a man (Man does not belong to some animal)” and “Some man is not an animal (Animal does not belong to some man)” (APr. I.2, 25a10–13). He gives the latter two examples to prove that “the particular negative need not convert” (i.e., O α β does not imply O β α ), for one of them is true while the other false. Then, as we have seen (Section 2.1), he includes A ¯ -propositions, too, among particular negatives (APr. I.1, 24a17–19), and holds that O -propositions are “the same” (in meaning) as matching A ¯ -propositions (APo, II.11, 62a11–13). And we can see that he takes them to be logically equivalent. In elaborating on the thesis mentioned above (i.e., O α β does not imply O β α ), he assumes that “Not every man is an animal” implies “Some animal is not man” while “Every man is an animal” implies the contradictory of “Some man is not an animal” (APr. I.2, 24b22–25).

While including both A ¯ - and O -propositions among particular negatives, we have seen, Aristotle takes the latter to be the same in meaning as the former. We can take this view to include the thesis of logical equivalence between A ¯ and O :

Tr4. A ¯ α β is logically equivalent to O α β .

And Tr4 and Tr3 ′ imply Tr3. So adding Tr4 to the De Interpretatione formulation of the theory of opposition yields the usual formulation of the theory encapsulated in Boethius’s square.^[95] It is this theory, which is equivalent to the orthodox traditional doctrine (Tr0–Tr3), that Aristotle assumes in presenting the theory of conversion in the Prior Analytics I.2 (25a1–25).

So Aristotle would not agree with Abelard but with Ockham to accept Tr3–Tr4. This means that those who hold the AN-interpretation would have to take him to hold the orthodox AN-doctrine (Ockham’s). He cannot be taken to fall a step short of the doctrine to hold just its moderate cousin (Al-Fārābī’s). Some (e.g., Spade) might take this to cast doubt on the AN-interpretation. I think there are more convincing reasons to reject the interpretation altogether. Aristotle holds theses about singular and general categoricals independent of Tr3–Tr4 that conflict with both the singular and general affirmative theses.

5.1 Singular Obversion

On the AN-doctrine, negative categoricals cannot imply affirmative categoricals with the same subject term. For example, “Socrates is not wise” cannot imply “Socrates is unwise,” an affirmative with a privative predicate term. For if Socrates does not exist so that they are both vacuous, the former is true but the latter is false.^[97] But Aristotle holds that singular negatives imply some singular affirmatives with the same subject terms. In De Interpretatione 10, he says:

It is clear too that, with regard to particulars [i.e., singular subjects], if it is true, when asked something, to deny it, it is true also to affirm something. For instance: “Is Socrates wise? No. Then Socrates is not-wise.”^[98]

In this passage, he considers two propositions related to, e.g., the question “Is Socrates wise?”:

He holds that (7b) follows from (7a), the proposition capturing the negative answer to the question. While (7a), where the negative particle “not” is added to “is,” is a negative, however, (7b) is an affirmative. It has the indefinite predicate term “not-wise,” which results from adding “not” to the general term “wise,”^[99] and is taken to affirm something (viz., not-wise) of Socrates.

Let “not- A ” (in symbols, α ¯ ) be an indefinite general term resulting from adding “not” to the general term “ A ” (in symbols, α ).^[100] Then say that “s is not- A ” (in symbols, S + s α ¯ ) is the obverse of “s is not A ” (in symbols, S − s α ). Then we can formulate the thesis Aristotle holds in the passage quoted above as an obversion thesis:

1. Singular obversion: “s is not A ” implies “s is not- A ” ( S − s α ⇒ S + s α ¯ ).

Defenders of the AN-interpretation might hold that Aristotle might consider “Socrates is not-wise” a negative proposition of a special kind. But he must consider it an affirmative. Otherwise it would not help to support the thesis that the negative answer to a question involving a singular subject term implies an affirmative proposition. Moreover, he says that “it is not-white,” for example, differs from “it is not white” in being an “affirmation” (APr. I.46, 52a25).

Some who hold the AN-interpretation deny that Aristotle holds S -Obv in the passage. To do so, they argue that the clause “with regard to particulars” indicates that he restricts the questions and propositions in question (e.g., (7a)–(7b)) to those with subjects.^[101] But the discussion of singular categoricals in the passage is directly followed by a discussion of general categoricals:

With universals, on the other hand, the corresponding affirmation is not true, but the negation is true. For instance: ‘Is every man wise? No. Then every man is not-wise.’ This is false, but ‘then not every man is wise’ is true; this is the opposite statement, the other is the contrary.^[102]

Those who hold the AN-interpretation cannot take this discussion to assume that the questions and propositions on “universals” in question are supposed to have subjects, for they must take him to consider “Not every man is wise” the “opposite” (viz., contradictory) of “Every man is wise” whether or not their subjects exist. If so, the previous discussion on propositions on “particulars” cannot be taken to be restricted to those whose subjects exist.^[103]

Now, some proponents of the AN-interpretation take Aristotle to deny obversion theses (including S -Obv) in the Prior Analytics I.46. But I do not think the discussion in the chapter yields a rejection of S -Obv.

In the Prior Analytics I.46, Aristotle says:

In establishing or refuting, it makes some difference whether we suppose the expressions “not to be this” and “to be not this” are identical or different in meaning, e.g., “not to be white” and “to be not white.” ... They do not mean the same thing, nor is “to be not white” the negation of “to be white,” but rather “not to be white.”^[104]

Some scholars take him to assume the AN-doctrine to reject obversion theses (e.g., S -Obv) in this passage, for “s is not A ” is true while “s is not- A ” is false, on the doctrine, if the subject does not exist.^[105] But he does not hold this (or consider vacuous categoricals) to argue that “not to be this” (or “not to be A ”) and “to be not this” (or “to be not- A ”) “do not mean the same thing” (51b7–8). To do so, he compares singular propositions that might seem to involve “is not white” and “is not-white”:

He argues that (8a) does not imply (8b) because if their subject (i.e., the referent of “it”) is something that is not a log (e.g., a stone), (8a) is true but (8b) is false. He takes this to suffice to show that “it is not-white” and “it is not white” “mean different things” (52a24), for he probably assumes that otherwise, (8a) and (8b) must have the same meaning.

But (8b) does not involve an instance of “to be not this” (or “to be not- A ”), for the complement “is” in the sentence is not the indefinite “not-white” but the compound “not-white log” (or “a log that is not-white”). Nor do “is not white” and “is not-white” figure as constituents of (8a) and (8b). Thus the failure of logical equivalence between (8a) and (8b) does not yield a logical difference between “not to be A ” (e.g., “is not white”) and “to be not- A ” (e.g., “is not-white”). Now, we might take Aristotle to hold that “not to be a B that is A ” (e.g., “is not a log that is white (or a white log)”) and “to be a B that is not- A ” (e.g., “is a log that is not-white (or a not-white log)”) do not have the same meaning. But this thesis, which he shows to be correct with (8a)–(8b), is compatible with S -Obv.^[107]

By holding S -Obv in De Interpretatione 10, Aristotle must reject both S -AN1 ( S + -VF) and S -AN2 ( S − -VT), for the two theses are equivalent assuming S -AN0 (the singular contradictories thesis), which he holds in the Categories 10 (13b12–37). In De Interpretatione 11, too, Aristotle rejects S -AN1.^[108]

5.2 Copula and Existence

The singular AN-doctrine implies the existential import thesis on singular affirmatives ( S + -EI). This thesis is equivalent to the singular affirmative thesis ( S -AN1 or S + -VF) and conflicts with the singular obversion thesis ( S -Obv). In De Interpretatione 10, as we have seen, Aristotle holds S -Obv. Moreover, he directly denies S + -EI in De Interpretatione 11.

In a well-known passage in De Interpretatione 11, Aristotle discusses whether singular affirmatives have existential import:

It is [sometimes] true to speak of the particular case even without qualification; e.g., to say that some particular man is a man or some particular white man is white. Not always, though.... [Consider] For example, Homer is something (e.g., [is] a poet). Does it follow that he is? No, for the “is” is accidentally predicated of Homer. Thus, [only] where predicates ... are predicated in its own right and not accidentally, ... it will be true to speak of the particular thing even without qualification.^[109]

Here, Aristotle raises a question about the logical relation between two propositions involving “is”:

1. Homer is ̲ a poet.

2. Homer is ̲ .

The question is whether (9), for example, implies (10) just as “Socrates is a white ̲ man” implies “Socrates is white ̲ ,” where “white” is used without being, so to speak, qualified by “man.” We can take the “is” in (10) to be for existence. If so, the question can be considered one on existential import: whether (9) has existential import. And he answers the question in the negative: (9) does not imply (10). This means that he rejects S + -EI.^[110]

Note that this does not mean that Aristotle holds that all singular affirmatives can be true while the matching existential propositions are false. S + -EI is a general thesis applying to all singular affirmatives: No instance of the form “s is A ( S + s α )” can be true while the matching instance of “s exists ( ∃ ! s )” is false. (The same holds mutatis mutandis for S + -VF, which is equivalent to S + -EI.) To reject the general thesis, it is not necessary to hold that “Homer is alive,” “Homer is existent,” etc. can all be true while “Homer is (i.e., exists)” is false. It is sufficient to give a single counterexample, a singular affirmative that can be true while the matching existential is false. And Aristotle takes (9) to be such a proposition, for Homer is a poet while he is dead and thus does not exist.^[111]

In the passage, Aristotle raises the question of existential import while considering whether something predicated with qualification of an individual subject can be predicated “even without qualification” (21a18–19). Suppose that Cato is a white man and a good cobbler. Then white man and good cobbler are predicated of him, and we might say that white and good are predicated of him as qualified with man and cobbler, respectively. Does this mean that they can also be predicated of him without the qualifications? That is, must Cato be white (simpliciter)? Must he be good (simpliciter)? Aristotle gives the affirmative answer to the first of the latter two questions. But he would not take the same to hold for the second, for Cato, who is a good cobbler, might still not be good (simpliciter) and even be a bad person. And he would take the relation between (9) and (10), where “is” figures without qualification (i.e., “Homer is (simpliciter)”), to be like that between “Cato is a good cobbler” and “Cato is good (simpliciter),” not that between “Cato is a white man” and “Cato is white (simpliciter).” “Cato is a white man” implies “Cato is white (simpliciter),” because the former, like the latter, indicates white being predicated of Cato “in its own right,” viz., independently of man (21a28). By contrast, “Cato is a good cobbler” does not imply “Cato is good (simpliciter),” for the former, unlike the latter, indicates good being predicated of Cato only by piggybacking on cobbler. Aristotle takes the same to hold for (9). (9), unlike (10), indicates is being predicated of Homer “accidentally,” i.e., by accompanying, or piggybacking on, poet (21a28).

In discussing the existential import of singular affirmatives, we have seen, Aristotle draws on a distinction between two uses of the verb “be” (or “is”):

1. The existential use (“est” secundi adiecti), where “is” figures as the second element of a proposition, as in “a man is” and “a man is not” (DI 10, 19b10–18)

2. The copulative use (“est” tertii adiecti), where “is” is “predicated additionally as a third thing” in addition to the subject and predicate terms, as in “a man is just” and “a man is not just” (DI 10, 19b19–20)^[112]

Whitaker, who attributes the AN-doctrine (or the singular AN-doctrine) to Aristotle,^[114] gives the same analysis of Aristotle’s reason for holding that (9) does not imply (10).^[115] He does not point out that this view conflicts with the singular AN-doctrine.^[116] But others who give the same analysis of the passage do. Ackrill, who takes Aristotle to hold the (singular) AN-doctrine, asks “How ... is this view to be reconciled with the contention ... that ‘Homer is a poet’ does not entail ‘Homer is’?”^[117]

Wedin argues that Aristotle’s thesis on (9) does not conflict with the singular AN-doctrine. He gives the same reading of Aristotle’s talk of accidental predication, which he calls “the non-decomposition interpretation” of the thesis.^[118] Then he holds that the interpretation yields a defense of the AN-interpretation. He says:

[...] when Aristotle blocks the inference from ‘Homer is a poet’ to ‘Homer is,’ he is concerned not with T1 [i.e., S + -EI] at all but only with the fact that the predicate in the first sentence is not subject to decomposition principle. To say this is just to say that “is” has no significance independent of its occurrence in the original phrase “is a poet.”^[119]

In this defense of the AN-interpretation, he holds that Aristotle’s only concern in the discussion of (9) is whether “is a poet” in the proposition is decomposable into two independent parts (“is” and “a poet”) just as “white man” in “Smith is a white man” is so. But what is his only or main concern in the discussion is beside the point. In order to hold that “is a poet,” unlike “white man” (but like “good cobbler”), is not so decomposable, Aristotle does hold, as Wedin agrees, that (9) does not imply (10), taking the former to be true while the latter is false. And this thesis contradicts S + -EI. If so, the discussion of decomposition gives rise to a problem for the AN-interpretation, which takes Aristotle to hold S + -VF and thus S + -EI.^[120]

While agreeing that Aristotle denies that (9) implies (10), Mignucci^[121] holds that this does not yield a counterexample to S + -EI. The existential proposition in question, which is false because Homer is now dead, pertains to temporal existence (viz., existence at present), not “the kind of existence which matters for the existential import of propositions” that he calls “logical existence,” the existence “not necessarily linked to the distinction between now and having existed in the past.”^[122] Because Homer existed in the past, which means that he exists in the “logical” or temporally unrestricted sense, one might hold that (9) is not vacuous and implies the temporally unrestricted existential “Homer is” that is not interchangeable with “Homer is now.”

I agree with Mignucci about Aristotle’s discussion of (9). But Aristotle immediately adds a discussion directly relating to existence in the temporally unrestricted sense:

[...] Thus, [only] where predicates ... are predicated in their own right and not accidentally, ... it will be true to speak of the particular thing even without qualification. [This is why] It is not true to say that what is not, since it is thought about, is something that is....^[123]

In this passage, Aristotle discusses two propositions:

(11a) does not imply (11b) for the same reason that (9) does not imply (10): (11b) involves an accidental (viz., indirect) predication of “is.” And this yields a direct counterexample to S + -EI. For (11b) cannot be taken just to mean that what does not exist exists now. Aristotle denies even a weaker thesis: What does not exist at all in the temporally unrestricted (or atemporal) sense exists in the same sense, existing at present or in the past or future (or in the atemporal sense). So (11b) must be taken to mean that what is not exists in this weak sense. And Aristotle takes even (11b) on this reading to be false while (11a) is true.

By distinguishing the existential and copulative uses of the verb “be,” we have seen, Aristotle rejects S + -EI: “ s is A ” implies “ s is (i.e. exists).” In the Sophistical Refutations (De Sophisticis Elenchis, SE), he makes the same point:

[...] it is not the case that what is not, even if it is something, is without qualification.^[124]

And he elaborates on what he considers fallacies resulting from ignoring the distinction between the two uses:

Those [i.e., fallacies] that depend on whether an expression is used without qualification or in a certain respect and not strictly, occur whenever an expression used in a particular sense is taken as though it were used without qualification, e.g. “If what is not is an object of opinion, then what is not is;” for it is not the same thing to be something and to be without qualification. Or again, “What is, is not, if it is not a particular kind of being, e.g. if it is not a man.” For it is not the same thing not to be something and not to be without qualification .... ^[125]

While agreeing that Aristotle denies S + -EI in the discussion of (9) and (11a) in DI 11, Kahn holds that the discussion shows that Aristotle does not have “the concept of copula,” for “It is here, if anywhere, that Aristotle could have used the concept of copula if he had one. Instead he is obliged to rely on the ambiguous notion of ‘accidental predication.’”^[126] Aristotle does not use the word “copula” (or its Greek cousin) in the discussion, but this does not mean that he does not have the concept of copula. He draws a clear distinction between two uses of “be” in DI and SE, gives accounts of their syntactic and semantic differences,^[127] and appeals to the differences to make an important point in the history of philosophy: While what is not can be thought about, this does not mean that it is, where “is” is not used as a copula but without qualification by a complement, implicit, or explicit. And this point, we have seen, involves rejection of S + -EI.^[128]

5.3 Singular Contradictories

It is usual to take Aristotle to hold the singular AN-doctrine ( S -AN) in the Categories 10 (13b12–37). But he holds theses contradicting the doctrine, as we have seen, in De Interpretatione 10–11. Does this mean that he holds incompatible theses in the two works? I think not.

To hold the singular contradictories thesis ( S -AN0), on the usual interpretation, Aristotle assumes the singular affirmative and negative theses. In presenting S -AN0 in the Categories 10, he discusses two singular categoricals:

(12b) is the matching negative of (12a). And Aristotle holds that they have different truth values whether or not Socrates exists, for “if he [= Socrates] does not exist ‘he is sick’ is false but ‘he is not sick’ is true” (Cat. 10 13b33–34). The usual interpretation takes him to hold the same on all matching singular affirmatives and negatives, which yields both the singular affirmative and negative theses:

1. Vacuous falsity thesis on singular affirmatives: Singular affirmatives must be false if vacuous ( S -AN1).

2. Vacuous truth thesis on singular negatives: Singular negatives must be true if vacuous ( S -AN2).

If so, why does hold that (12a) and the like (e.g., Socrates has sight) are false if Socrates does not exist? Like “Homer is a poet” and “What is not is conceivable,” (12a) involves the predicative use or the so-called accidental predication of ‘is’ and is an instance of the form “s is A ( S + s α ).” So he might not take the proposition to (logically) imply the matching existential proposition. Still, he might take (12a) to differ semantically from “Homer is a poet” and “What is not is conceivable” for special semantic features of the term “sick” that it does not share with “poet” and “conceivable.” To see this, it would be useful to consider (13a)–(13b):

(13a)–(13b) are also singular affirmatives, i.e., instances of the same form. But (13a) cannot be true unless Socrates exists, for the predicate term “existent” (unlike “conceivable”) requires the subject term “Socrates” to denote something that exists for the proposition to be true; nor can (13b) be true unless Socrates exists, because “alive” requires the subject term to denote something alive for its truth while anything alive must exist. One might hold that it is essentially the same with (12a): its predicate term “sick” requires the subject term to denote something that is sick while anything sick must be alive, which means that it must exist. Unlike “sick,” however, “poet” in “Homer is a poet” does not require Homer to exist at present, nor does “conceivable” in “What is not is conceivable” require the existence at any time of what is not. So I think Aristotle’s discussion of singular contradiction in the Categories 10 invokes semantic features of the likes of (12a) not shared by all singular affirmatives to take them to be false if vacuous.^[129]

If so, the reason he illustrates the singular contradictories thesis ( S -AN0) with (12a) and (12b) is not that he takes them to be representatives of all singular affirmatives and negatives in being false and true, respectively, if their subject does not exist. To argue for the thesis, I think he uses the specific examples to respond to a potential objection: the affirmative and negative pair, like their affirmative cousins with contrary predicate terms (e.g., “Socrates is sick” and “Socrates is well”), are not genuine contradictories because both are false if their subject does not exist. With this objection in mind, he holds that if a singular affirmative (e.g., (12b)) is false because its subject does not exist, its matching negative (e.g., (12a)) is still its contradictory because the negative is true, not false, in that case. This response does not commit him to S + -VF or S − -VT and is compatible with the discussions of singular obversion and existential import in De Interpretatione 10–11, where he holds S -Obv and denies S + -VF (Section 5.1–2).^[130]

6.1 Empty General Terms

Many modern studies of Aristotle’s syllogistic take the system to presuppose or require that no empty terms figure in general categoricals dealt with in the system. In the Prior Analytics I.38, however, Aristotle discusses application of syllogistic rules to a categorical with the empty subject term “goat-stag”:

This shows that the interpretation that takes him to ban vacuous categoricals from his syllogistic is not correct, as some proponents of the AN-interpretation, among others, point out.^[131] Moreover, I think, the discussion shows that the AN-interpretation is also incorrect.

(14) is assumed to be true in the discussion, where Aristotle explains how to use syllogistic rules (e.g., Barbara) to prove a group of special categoricals that include (14):

A term which is repeated in the propositions ought to be joined in the first extreme .... for example ... if a deduction should be made proving that there is knowledge of justice, that it is good, the expression “that it is good” (or ‘qua good’) should be joined to the first term. Let A stand for knowledge that it is good, B for good, C for justice. It is true to predicate A of B . For of the good there is knowledge that it is good. Also it is true to predicate B of C . For justice is ... a good.... Similarly if it should be proved that the healthy is an object of knowledge quâ good, of goat-stag an object of knowledge quâ not existing or man perishable qŭa an object of sense.^[132]

In this passage, he explains that we can prove “of justice, there is knowledge that it is good (or justice can be known to be good)” with a syllogism:

• Of the good there is knowledge that it is good.

• Justice is a good.

• ∴ Of justice there is knowledge that it is good.

• A nonexistent is an object of knowledge quâ not existing.

• A goat-stag is a nonexistent.

• ∴ A goat-stag is an object of knowledge quâ not existing.^[133]

This means that Aristotle must reject I -VF (and I -EI), for G + -VF (which consists of A -VF and I -VF) follows from I -VF and Tr1 (subalternation). It might be useful to illustrate this with (14). We might take (14) to have the universal affirmative reading. If so, on Tr1, it implies the matching particular affirmative:

1. Some goat-stag is an object of knowledge quâ not existing.

1. There is a goat-stag.

Some might propose to read (14) as a singular categorical involving “the goat-stag”:

1. The goat-stag is an object of knowledge quâ not existing.^[135]

1. Every nonexistent is an object of knowledge quâ not existing.

1. There is (i.e., exists) a nonexistent.

6.2 Ecthesis and Self-predication

Proponents of the AN-interpretation hold that Aristotle assumes or presupposes the existential import thesis on particular affirmatives ( I -EI) in presenting the theory of opposition. Parsons, we have seen (Section 3.2), takes him to invoke this assumption in the Prior Analytics I.2 (25a14–16), where he proves the E -conversion thesis:

1. E -conversion: Universal negatives imply their converses ( E α β ⇒ E β α ).

I think this argument for attributing G -AN to Aristotle has serious problems. According to G -AN, particular negatives do not have existential import. In proving the validity of some syllogisms (e.g., Baroco, Bocardo), however, Aristotle applies to particular negatives the same method he applies to particular affirmatives to prove E -conversion.

The method, called ecthesis (or exposition), is similar to the existential quantifier elimination rule in introducing a new term by invoking certain available propositions (e.g., particular affirmatives, existential quantifications).^[136] In the proof of E -conversion, we have seen, Aristotle invokes “ B belongs to some A ” to introduce a new term “ C ” to assume both “ A belongs to C ” and “ B belongs to C .” And he explains how to use the method to prove Darapti:

• Darapti. Every S is P . Every S is R . ∴ Some R is P .

[...] whenever both P and R belong to every S , it follows that P will ... belong to some R .... It is possible to demonstrate this ... by exposition [ecthesis]. For if both P and R belong to every S , should one of the S s , e.g., N , be taken, both P and R will belong to this [e.g., N ], and thus P will belong to some R . (APr. I.6, 28a18–26)

In this proof, where he calls the method “ecthesis,” he invokes one of the two universal affirmative premisses to introduce a new term “ N ” for “one of the S s ” (i.e., something denoted by “ S ”). Now, consider Bocardo and Baroco:

• Bocardo. Every S is R . Some S is not P . ∴ Some R is not P .

• Baroco. Every N is M . Some O is not M . ∴ Some O is not N .

Aristotle, we have seen, applies ecthesis to particular negatives as well. So proponents of the AN-interpretation (e.g., Parsons) would have to take him to assume that particular negatives, too, have existential import ( O -EI). But this conflicts with the (general) AN-doctrine.^[137]

Now, some studies of Aristotle’s syllogistic takes ecthesis to introduce new general terms, instead of new singular terms. I think he might accept two versions of the method: the singular and general versions. The talk of, e.g., “one of the S s ” in the Bocardo ecthesis (APr. I.6, 28b21) suggests that the new term is singular (the singular version), and the talk of “a part of the subject, to which in each case the predicate does not belong” in relation to modal cousins of Bocardo and Baroco (APr. I.8, 30a9–10) seems to suggest that the new term is general (the general version). But this does not help to defend the AN-interpretation, for the interpretation conflicts with the general term version of ecthesis as well. The proof of Bocardo or Baroco that uses this method invokes the negative “Some S is not P ” to introduce a general term, “ E ” (which amounts to the compound term “ S that is not P ”), to add two universal categoricals:

1. Every E is S .

2. No E is P .

Wedin defends the AN-interpretation by holding that the ecthesis proofs (or most of them) are not essential to Aristotle’s syllogistic, for he gives two other methods of proof (i.e., indirect proof and proof by conversion) in the Prior Analytics, and says that one can use them to prove, e.g., Bocardo and Baroco.^[139] But this is beside the point. The question is not (i) whether Aristotle must have used ecthesis on particular negatives, but (ii) whether he considers it legitimate to apply the method to them. Clearly, the answer to (ii) is yes. If so, one cannot take him to assume the general AN-doctrine. Moreover, Aristotle says that it is necessary to use ecthesis to prove modal cousins of Bocardo and Baroco: “the demonstration [of the modal Bocardo and Baroco] will not take the same form [as that involving indirect proof or proof by conversion], but it is necessary [to use proof] by the exposition of a part of the subject, to which in each case the predicate does not belong” (APr. I.8, 30a9–10). So Wedin concedes that “here, finally, we may have some positive evidence for the existential reading” of particular negatives, i.e., O -EI.^[140]

Some might attempt to defend the AN-interpretation by rejecting Parsons’s assumption that Aristotle’s use of ecthesis on I -propositions is sufficient to show that he assumes I -EI. In that case, they can hold that his use of ecthesis on O -propositions does not mean that he assumes O -EI, either. I think they are right to reject the usual assumption on presupposition of ecthesis. But rejecting the assumption would not help to defend the interpretation, for those who reject it would have no good reason to attribute I -EI to Aristotle.

Moreover, the ecthesis on O -propositions (in short, O -ecthesis) yield self-predication theses directly conflicting with the AN-doctrine:

1. A-self-predication: “Every A is A ” is a logical truth ( ⊧ A α α ).

2. O-self-predication: “Some A is not A ” is a logical falsity ( ⊧ ¬O α α ).

So I think Łukasiewicz is right to take Aristotle to assume in the Prior Analytics II.15 (64a20–31) that “sentences like ‘Some science is not science’ cannot be true” (Łukasiewicz, Aristotle’s Syllogistic, 9f). Considering only the discussion in the chapter, he makes the reservation that “It is not very probable that Aristotle knew” the generalization that all instances of ‘Some A is not A ’ must be false. But the proof of O -SF shows that he is committed to the thesis by accepting the validity of O -ecthesis.^[142]

Note that the proof, which uses O -ecthesis, does not assume the singular affirmative or negative thesis ( S -AN1 or S -AN2). It uses only the singular contradictories thesis ( S -AN0): S + c α and S − c α are contradictories. And we can see that Aristotle’s use of ecthesis on particular affirmatives or negatives does not presuppose that the propositions have existential import. The ecthesis on, e.g., “Some A is B ” assumes that the proposition cannot be true unless there is a singular term “ s ” such that both “ s is an A ” and “ s is a B ” are true. But this does not mean that “Some A is B ” has existential import, unless the singular propositions (e.g., “ s is an A ”) have existential import. And despite the usual interpretation of his discussion of S -AN0 in the Categories 10, as we have seen (Section 5), Aristotle does not take all singular affirmatives to have existential import, which means that “ s is an A ” can be true while both “ s exists” and “There is (i.e., exists) an A ” are false. He takes, e.g., “What is not is (an object that is) thought about” (=(11a)) to be true while taking “What is not exists” (=(11b)) and “There is (i.e., exists) an object that is thought about” to be false.

Moreover, we can use the inference rules used in ecthesis proofs to give a general reason to reject I -EI: to deny that singular categoricals have existential import ( S -EI), Aristotle must also deny that particular affirmatives have existential import ( I -EI). Consider the following:

Aristotle takes (20a) and (20b) to be true but (20d) to be false (DI 11, 21a18–33). Assuming S -EI, however, (20a)–(20b) imply (20d), for they imply (20c) and (20c), on S -EI, implies (20d).

Some might object that Aristotle would reject (20a) while accepting its obverse:

1. (a) What is not is not existent.

But (21a) implies (20a) on the singular obversion thesis, which he holds in DI 10 (see Section 5.1). Moreover, even on the AN-doctrine, (21a) together with the affirmative (20b) implies (20a). Now, some might object that Aristotle need not take (20a)–(20b) to imply (20c). But he is committed to the validity of the inference from “ s is A ” and “s is B ” to “Some A is B ” in his ecthesis proof of E -conversion in the Prior Analytics I.2 (see Section 4.2). In the proof, he holds that “No A is a B ,” “ C is an A ,” and “ C is a B,” where “ C ” is a singular term, lead to a contradiction. If so, “ C is an A ” and “ C is a B ” must imply the contradictory of “No A is a B ”: “Some A is a B .”^[143]

6.3 Obversion and Contraposition

While suggesting that the orthodox AN-doctrine is objectionable for including O -VT (Section 4.3), Church holds that the doctrine “does preserve all the properly Aristotelian inferences” (“History of the Question of Existential Import,” 423, my italics). In making this assessment, he assumes that Aristotle does not accept obversion and contraposition:

1. E -obversion: “No A is B ” and “Every A is not- B ” are logically equivalent ( E α β ⇔ A α β ¯ ).^[144]

2. A -contraposition: “Every A is B ” and “Every not-B is not- A ” are logically equivalent‘ ( A α β ⇔ A β ¯ α ¯ ).^[145]

8.1 On Existential Import of Singular Propositions

Consider two instances of the singular affirmative form “ s is A ”:

In the Categories 10 and De Interpretatione 11, we have seen (Sections 4.2 and 5.2), Aristotle holds apparently incompatible theses on (31a)–(31b):

1. (31a) must be false, if Socrates does not exist. (Cat. 10, 13b25–35)

2. (31b) is true, but “What is not is (or exists)” is false. (DI 11, 21a32–33)

1. Singular affirmatives must be false if vacuous.

Now, S ⁺-VF contradicts T2. If (31b) is true, on S + -VF, its subject (what is not) must exist, which means that “What is not is (or exists)” must also be true. But I do not think there is a genuine conflict between T1 and T2, for it is not necessary to take T1 to be meant to hold for all singular categoricals, including (31b). Moreover, one cannot deny that Aristotle holds T2 about one specific singular categorical (viz., (31b)).^[169] He also holds S -obversion, which conflicts with S + -VF (Section 5.2). If so, he must be taken to reject S + -VF by denying that (31b) implies “What is not is (or exists).”

Aristotle, we have seen (Section 5.2), rejects S + -EI by holding T2. If so, he must take (31a), too, to lack existential import, that is, not to (logically) imply “Socrates is (i.e., exists).” For the former and the latter have the same logical forms, in his view, as (31b) and “What is not is (or exists),” respectively: (31a) and (31b) are singular affirmatives while “Socrates is (i.e., exists)” and “What is not is (or exists)” are matching singular existential propositions. So I think he must hold the opposite of S + -EI:

1. No-import thesis on singular affirmatives: Singular affirmatives do not have existential import (i.e., S + s α does not imply ∃ ! s ).

(To facilitate the comparison, it would be useful to replace “what is not” with a co-referential proper name, “Notie”, as in (32b).) Aristotle denies that (32b) is a logical truth, which means that (31b) has existential import. Similarly, he would deny that (32a) is a logical truth, for it has the same logical form as (32b). But he can still hold that (32a) differs from (32b) in an important respect. While (32b) can be (and in fact is) false, (32a) cannot be false because it is an analytic truth for semantic connections between “sick” and “exist” mediated by “alive”: (a) Anything sick must be alive, and (b) Anything alive must exist. In holding T1, then, I think Aristotle invokes semantic features of “sick” that some general terms (e.g., “conceivable,” “thought about”) do not share. And it is not necessary, to be sure, to take all general terms to be like “sick” to hold the singular contradictories thesis: S + s α and S − s α are contradictories. For example, (31b) is true while Notie (i.e., what is not) does not exist, but (31b) is still a contradictory of the negative “What is not is not thought about,” which is false.

Now, consider a singular affirmative with the special predicate term “existent”:

1. Socrates is an existent.

So we can divide singular affirmatives into three kinds:

1. Instances of S + s η

2. The likes of (31a)

3. The likes of (31b)

1. Singular affirmatives that imply matching existentials (e.g., (33))

2. Singular affirmatives that do not imply but semantically entail matching existentials (e.g., (31a)).

3. Singular affirmatives that do not semantically entail matching existentials (e.g., (31b))

8.2 On Existential Import of General Propositions

While rejecting S + -EI, as noted in Section 3.3, Abelard assumes I -EI.^[170] While agreeing with him on singular categoricals, Aristotle would disagree on general categoricals. He holds many theses conflicting with I -EI (Section 6). Moreover, he must deny I -EI, as we have seen (Section 6), for he rejects S + -EI and assumes the I -introduction rule:

1. “ s is A ” and “s is B ” imply “Some A is B .”

And I think he must hold I -NEI:

1. No-import thesis on particular affirmatives: Particular affirmatives do not have existential import (i.e., I α β does not logically imply ∃ ! α ).

Now, note that (34a)–(34b) involve general terms figuring in singular affirmatives belonging to SA2 (i.e., “(one who is) sick,” “(thing) that lives in Athens”). Singular affirmatives involving such terms do not have existential import but semantically entail matching existentials. The same holds for particular affirmatives involving them. For example, (34a) and (34b) do not imply but semantically entail matching existentials, for “If someone is sick, it exists” and “If something lives in Athens, it exists” are not logical but analytic truths that rest on semantic connections that “sick” and “living in Athens” have to “exist.”^[172] But (34c)–(34d) do not even semantically entail matching existentials, Aristotle would argue, for “If something is conceivable, it exists” and “If something is nonexistent, it exists” are not even analytic truths (he would consider them false). So I think Aristotle might divide particular affirmatives, too, into three kinds:

1. Particular affirmatives that imply matching existentials (e.g., “Some existent is a centaur”)

2. Particular affirmatives that do not imply but semantically entail matching existentials (e.g., (34a), (34b)).

3. Particular affirmatives that do not semantically entail matching existentials (e.g., (34c), (34d))

8.3 The No-Import Doctrine

Aristotle holds theses that lead to rejections of S + -EI and I -EI, we have seen, and we can take him to hold theses on singular and particular affirmatives opposite to them: S + -NEI and I -NEI. If so, we can take the same to hold for all categorical propositions, singular or general. On this interpretation, Aristotle holds a thoroughly negative doctrine of existential import, the no-import doctrine, which holds that categorical propositions, singular or general, do not have existential import (unless they involve special logical terms, e.g., “existent”). We can combine this doctrine with other logical theses Aristotle holds: the singular contradictories thesis, the orthodox traditional doctrine (or the square of opposition), A -contraposition, and S - and E -obversion, etc. Then we can get a traditional system of logic that includes the no-import doctrine of existential import. I think the system implements the no-import interpretation of Aristotle’s assumptions on existential import and vacuous categoricals.

On this interpretation, Aristotle has a clear-cut, if radical, response to the modern, existential import objection to traditional logic: while the objection assumes I -EI and O -EI, both assumptions are false. On the interpretation, then, Aristotle holds a doctrine on existential import diametrically opposite to the modern doctrine as far as particular categoricals are concerned. By contrast, the affirmative–negative doctrine can be taken to result from making a compromise.

By adding S + -EI and I -EI to the singular contradictories thesis and the theory of opposition, the AN-doctrine yields a simple formula on truth values of vacuous singular and general categoricals. The no-import doctrine does not. To see whether singular affirmatives (e.g., “Socrates is sick,” “Homer is a poet”) are true, on the doctrine, it is necessary to consider the meaning (and reference) of their predicate terms whether they are vacuous or not. We can see that “Socrates is sick,” for example, must be false if vacuous, but we can see even this not simply by noting that it is an instance of “ s is A ,” but by considering the meaning of “sick” to see that the proposition semantically entails “Socrates exists.” And the doctrine of existential import and of the logic of vacuous categoricals does not determine which general terms yield, e.g., singular affirmatives that cannot be true unless their subjects exist. On the no-import doctrine, this is not a question to be addressed by a doctrine of existential import. It is a question belonging to semantics, not logic proper, while a doctrine of existential import addresses only logical issues on vacuous categoricals. The same holds for the question whether “Socrates is sick” semantically entails “Socrates exists.” And the thesis that it does, which I take Aristotle to hold, is not a part of his doctrine of existential import, for issues on semantic entailment does not belong to logic proper. But it is useful, I think, to clarify the nature of the semantic thesis to resolve the apparent conflict between the existential import theses propounded in, e.g., De Interpretatione 10–11 and the semantic thesis suggested in the Categories 10.