INTRODUCTION
There is no extended treatment of complex terms in the Organon. The dispersed comments on the genus and the differentia in the Categories and Topics, while they imply that a species-term is equivalent to a conjunction of differentiae and genus, where the conjuncts belong essentially to the conjunction, do not amount to a general logic of conjunctive terms since many conjunctive terms are not the result of combining differentiae with a genus.
A passage in the Prior Analytics discusses a complex term that is thrown up incidentally in a context of a discussion about pure contingency-syllogisms in the first Figure. It argues that the premises It’s contingent that every C is B and It’s contingent that some B is A do not yield any syllogistic conclusion:
For nothing prevents B from extending beyond A and not being predicated of an equal number of things. Let C be taken as the part by which B extends beyond A; then it is not possible for A to belong either to all of C or to none, or to some, or not to some, given that the possible premisses convert, and it is possible for B to belong to more things than A.Footnote 1
Just below the surface here is the postulation of a conjunctive term B and not-A with which C is identified. So Aristotle is here theorizing with a conjunctive term.
In other texts he theorizes about such terms. For example, in Chapter 3 of the Categories, when he tells us that one species of knowledge does not differ from another by being two-footed, he theorizes about the conjunctive term Two-footed knowledge, which expresses a conceptual impossibility.
The impression one gets from this material is that Aristotle did not believe a formal logic was applicable to all complex terms, and that he did not consider working out such a logic for the limited range of complex terms that are susceptible to formal treatment.
Medieval logicians in the Arabic tradition, however, did offer theoretical treatments of conjunctive terms. In discussing these medieval theories I will represent a conjunctive term composed from the terms J and B by enclosing these two terms, juxtaposed, within brackets thus (JB). Categorical propositions (a-, e-, i- and o-propositions) with predicate B and subject J will be represented by BJa, BJe, BJi, BJo. Inferences from a sequence Π of n categorical premises (n ≥ 0) to a categorical conclusion p will be represented by Π/p.Footnote 2
The theories to be discussed focus on a logical problem that results from the use of conjunctive terms in syllogistic. It seems that such terms ought to obey the Laws of Partial Identity: B(JB)a, J(JB)a. But, if every (JB) is B and every (JB) is J then, by the third-figure syllogism Darapti, it follows that some B is J. It seems that the premises are logical truths. So it seems to follow that ‘Some B is J’ is a logical truth. And yet there are false statements of the form ‘Some B is J’: for example, it is false that some man is stone. It seems that any solution to this puzzle (the puzzle of Partial Identity and Darapti) will need to find a way of avoiding the assertion of all i-propositions, or else will have to explain why syllogistic logic cannot accommodate conjunctive terms.
ABHARĪ1
In addressing the puzzle, Abharī distinguishes on the one hand a reading of BJa according to which the subject-term is taken ‘according to extramental existence’Footnote 3 and on the other hand several readings where (in the words of Avicenna as quoted by Ṭūsī) ‘we have propositions whose subjects are matters to the existence of which no adversion is made’.Footnote 4 Abharī’s claim is that the proposition BJa has different meanings on different occasions.
On some occasions it has a meaning of the first type, and on some a meaning of the second type.
On the first reading he understands the subject J to stand for every J in extramental existence ‘whether the J is in the present or before it or after’.Footnote 5 This is not an ut nunc reading in the sense understood by the Latin logicians, since it embraces more than the presently existing Js. It reads the universal affirmative proposition as making a statement about all actual Js, past, present and future.
A syllogism such as Darapti, which derives a particular conclusion from two universal premises, can be valid on a semantics such as this, provided that all affirmations (both universal and particular) imply that their subject-terms are true of something (past, present or future).
What applies to the ordinary terms of categorical propositions also applies to conjunctive terms when they appear in such propositions. In order for an affirmation with a conjunctive term (JB) as subject to be true, that conjunctive term must be true of something past, present or future. In other words, for such an affirmation to be true on Abharī’s first reading, it is required that some B be J, whether in the past or in the present or in the future.
Thus the first reading implies that if no B has ever been J, and none is J, and none ever will be J, all affirmations with (JB) as subject will be false. Now, there are indeed conjunctive terms whose conjuncts have never been predicated of one another, and are not so predicated, and never will be – terms like ‘stone man’. Consequently, there are propositions of the form (JB)(JB)a that are false on Abharī’s first reading; and therefore there are propositions of the form JJa (the Law of Identity) that are false on that reading. By the same token, there are propositions of the form J(JB)a that are false on Abharī’s first reading. The Law of Partial Identity does not hold. So, on this reading of the meaning of affirmative categorical propositions, the puzzle of Partial Identity and Darapti is solved because even though the reasoning is valid its premises are false.
Nevertheless, Ṭūsī reports Avicenna as describing this type of analysis as ‘foolish and vulnerable to criticism’.Footnote 6 He quotes Avicenna as saying that such an analysis gives us a statement only about some of the things falling under the subject-term ‘and yet our utterance “Every B” is more general than that’. The thought is that the meaning of some of our utterances is not exhausted by an analysis that confines itself to what is in the external world. In his defence Abharī could argue that he is proposing the first reading only as covering some of our utterances, not all. In any case, he goes on to consider a second reading that allows the subjects of categorical propositions to extend beyond the external world.
ABHARĪ2
Abharī suggests that BJa may be read as stating that ‘everything that would be a J were it to exist, would be B in so far as it were to exist’.Footnote 7 He provides the following gloss: ‘This means that everything that is the implicant of J is the implicant of B’. So, taking this notion of implicant as an undefined relation between terms, and representing it by an arrow ⇒, the second analysis of BJa can be represented as stating that any D that is an implicant of J is an implicant of B, i.e.
![$$\forall D\left[ {D \Rightarrow J \supset D \Rightarrow B} \right].$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:5163:20160420050100346-0754:S0957423909990105_eqnU1.gif?pub-status=live){kind=small}
BJe would then mean
where B is a metathetic term signifying the logical complement of B.
BJi would mean
![$$\exists D\left[ {D \Rightarrow J \wedge D \Rightarrow B } \right].$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:43970:20160420050100346-0754:S0957423909990105_eqnU3.gif?pub-status=live){kind=small}
This is implied by the universal affirmative, because there is always a D such that D is an implicant of J, namely J itself. Given there is such a D, and that any implicant of J is an implicant of B, it follows that there is a D that is an implicant both of J and of B.
BJo would mean
The Law of Identity JJa, which was not valid on the first reading, becomes valid on this second reading, because it is always true that ∀D[D ⇒ J ⊃ D ⇒ J]. Whatever is an implicant of J is an implicant of J.
The Law of Partial Identity J(JB)a is also valid, because it is always true that ∀D[D ⇒ (JB) ⊃ D ⇒ J]. Whatever is an implicant of (JB) is an implicant of J. This principle is not formulated in the abstract by Abharī, but he does assume particular instances of it, in the form of the examples ‘Everything that is a man and a stone is a man’, ‘Everything that is a man and a stone is a stone’, ‘Everything that is a man and a non-animal is a man’ and ‘Everything that is a man and a non-animal is a non-animal’. These examples arise because in the above formulas it is understood that D ‘extends to the impossible’.Footnote 8
Darapti is valid on this second reading: if ∀D[D ⇒ J ⊃ D ⇒ B] and ∀D[D ⇒ J ⊃ D ⇒ A] then ∃D[D ⇒ B ∧ D ⇒ A]. Given that any implicant of J is an implicant of both B and A (and given that there is always an implicant of J), there must be a term that is an implicant of both B and A, namely J itself.
Since Darapti is valid, and B(JB)a and J(JB)a are valid, it follows that BJi is valid. This result could not be obtained on the first reading. But Abharī states it for the second reading: ‘From this it is clear that the i-proposition is always true’. He also notes that the e-proposition is always false:
The e-proposition on this account would never be true, because when you say no man is a stone its contradictory is true. This is because everything that is a man and a stone is a man, and everything that is a man and a stone is a stone, so some man is a stone.Footnote 9
The assertion of all i-propositions and the rejection of all e-propositions have consequences regarding the doctrine of Obversion. There are two principles of Obversion to consider. One goes from an affirmative proposition to a negative:
The other goes from negative to affirmative:
On Abharī’s preliminary statement of this second semantics, affirmative-to-negative Obversion is invalid. This follows from the fact that the e-proposition is always false, given that some a-propositions are true. Abharī thinks that some a-propositions are true; but he adds that when BJa is true BJi is also true. He gives an example:
So if you say every man is an animal, it is also true that some men are non-animals. This is because everything that is a man and a non-animal is a man, and everything that is a man and a non-animal is a non-animal, so some man is a non-animal.Footnote 10
At the same time we can observe that negative-to-affirmative Obversion is valid because its premise is always false. So the Laws of Obversion turn out exactly contrary to the Aristotelian doctrine according to which affirmatives obvert to negatives but negatives do not obvert to affirmatives.Footnote 11
Ṭūsī raises an objection against this doctrine of Obversion:
What is strange is that he judged that the e-proposition on this interpretation is not true because it cannot be true with its contradictory, as he explained; yet he did not judge that the a-proposition is also not true, in spite of the fact that he explained that it can be true with its opposite (I mean the metathetic proposition whose predicate is opposed to its predicate). Rather here he judged that two opposites can be true together, yet he did not judge there that two contradictories can be true together. But it’s necessary that he judge either the impossibility of the two universals being true together, or the possibility of opposites in universals being true together.Footnote 12
Strange indeed are the consequences of Abharī’s second semantics. In this semantics, unlike that underlying the Aristotelian doctrine of Obversion, BJa does not entail BJe, and so BJa and BJi are not incompatible. At the same time, BJe and BJi are incompatible, as they are according to Aristotle. However, the strangeness of these results when compared to the standard Aristotelian logic should not be thought to imply that they do not genuinely follow on Abharī’s second semantics.
Coming to the heart of the second semantics, Ṭūsī criticizes Abharī for his statement that the a-proposition means that every implicant of J is an implicant of B:
This explanation is misleading, because the thing need not be said of its implicant. What is meant here by every J is every thing of which J is said, not every thing such that, were it to exist, J would exist. For many things are such that were they to exist, J would exist, yet J is not said of them: the perfect causes of J; the things of which J is a part; the things from which J is not separated in existence.Footnote 13
Ṭūsī has uncovered an important ambiguity in the notion of an implicant. We can best see the ambiguity by treating the implicant-relation as resting on a more basic relation of entailment between propositions (which we can designate by the sign →). A statement of the form J ⇒ B might be a conditional proposition stating that the proposition Something is J entails the proposition Something is B:
![$$\exists x\left[ {Jx} \right] \to \exists x\left[ {Bx} \right].$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:62558:20160420050100346-0754:S0957423909990105_eqnU7.gif?pub-status=live){kind=small}
(There being a J entails there being a B.) Or it might be a categorical proposition stating that any proposition x is J entails the corresponding proposition x is B:
![$$\forall x\left[ {Jx \to Bx} \right].$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:52839:20160420050100346-0754:S0957423909990105_eqnU8.gif?pub-status=live){kind=small}
(Being J entails being B.) The conditional proposition states a necessary but not a sufficient condition for the truth of BJa. This can be shown by an example. In the inference ‘The proposition Theft exists entails the proposition Property exists; the proposition Property exists entails the proposition Property-owners exist; so the proposition Theft exists entails the proposition Property-owners exist’, it would be absurd to re-word the conclusion as the categorical proposition Theft is a property-owner.Footnote 14 Ṭūsī argues that the original proposition BJa must be analysed as a categorical not a conditional proposition because ‘the thing need not be said of its implicant’.Footnote 15
This is a fair objection; but it is one that points clearly to a reply. Clearly, Abharī’s response can be to say that he intended to speak of implicants in the second sense, so that to say that any D that is an implicant of J is an implicant of B is to say the following:
![$$\forall D\left[ {\forall x\left\{ {Dx \to Jx} \right\} \supset \forall x\left\{ {Dx \to Bx} \right\}} \right].$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:85546:20160420050100346-0754:S0957423909990105_eqnU9.gif?pub-status=live){kind=small}
Any D such that being D entails being J is such that being D entails being B.
Ṭūsī sees Abharī’s second semantics in the context of a well-known objection to Alfarabi’s opinion that the subject of BJa stands for whatever is potentially J. (The objection points out that propositions predicating B of the Js don’t predicate B of entities that in future may become J.)
Yet this is something no one held, because the later validating scholars [in saying these things] refuted Alfarabi (in spite of his eminence in the verification of the philosophical sciences such that he was known as the Second Teacher) in his claim that when we say every J it includes what may be said to be J. [According to this view], one must understand from “black” all that is not black among those things which may become black. So [given the position of Alfarabi they meant to refute], how could they permit the entry of what is impossible to be black into it?Footnote 16
He thinks that Abharī, having opted for an analysis in terms of implicants, must have inferred from the fact the antecedent of a conditional may be impossible that the subject of BJa may also be impossible.
And Abharī, having imagined the implicational relation [to be present], took the connective conditional to explain the quantification of the categorical; he compounded universality of predication (I mean, generalization for individuals) with the universality of the conditional (I mean, generalization for circumstances). So he fell into absurdities which stuck to him because it is permissible that the antecedent in a conditional be absurd. This is the judgment that the subject of the sentence covers impossible things, such that if it is said every man, that which is a non-animal man is included by way of it.Footnote 17
His thought seems to be that Alfarabi’s analysis is not improved by extending the subject to the impossible; on the contrary, this makes matters worse.
Ṭūsī goes on to cast doubt on ‘the necessity of the implication from the whole to its part’, i.e. the particular case of the Law of Partial Identity, namely the principle according to which a contradictory conjunction implies both of its contradictory components:
This is due to the possibility that the truth of true propositions won’t remain on a given assumption supposed for the antecedent, if it’s possible that it be absurd. […]
His subject – “non-animal man” – has the sense “the thing that is rational animal and non-animal”. “Animal which is non-animal”: how is it possible that it have as predicate “animal” for which it is impossible to be non-animal? Indeed it would be [just as] possible to predicate of it “man non-man”.Footnote 18
Here he is pointing to (what some see as) the strangeness of assuming a contradictory conjunction and then inferring one of the conjuncts, when that inference is contradicted by the original assumption.
As an improvement on the preliminary formulation of the second reading, Abharī proposes to restrict the subject to ‘what is not impossible’:
So if we say every J is B in accordance with the essence, we mean everything that is not impossible in itself or through another that is the implicant of J is the implicant of B.Footnote 19
As Ṭūsī puts it,
[…] he fled from these repugnant consequences, going some way back towards the truth by restricting the subject to what is not impossible in itself or through another.Footnote 20
Assuming that a proposition is impossible if and only if it is the implicant of a contradiction, we may consider the following formulation of Abharī’s suggestion that a proposition BJa be read as stating that J is not the implicant of a contradiction, and any implicant of J is an implicant of B:
![$$\neg \exists D\left[ {J \Rightarrow D \wedge \neg D} \right] \wedge \forall D\left[ {D \Rightarrow J \supset D \Rightarrow B} \right]$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:99830:20160420050100346-0754:S0957423909990105_eqnU10.gif?pub-status=live){kind=small}
(where J is an implicant of D is understood as stating that being J entails being B). The first conjunct here is not trivial, since some terms (e.g. terms like stone man) are impossible in the relevant sense. The i-proposition should be read
![$$\neg \exists D\left[ {J \Rightarrow D \wedge \neg D} \right] \wedge \exists D\left[ {D \Rightarrow J \wedge D \Rightarrow B} \right].$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:4946:20160420050100346-0754:S0957423909990105_eqnU11.gif?pub-status=live){kind=small}
The o- and e-propositions are the contradictories of these.
Abharī doesn’t tell us anything further about this restriction to his second semantics; however, some consequences of his assumptions are evident. The Law of Identity is invalid because its subject might be impossible, and the Law of Partial Identity is invalid for the same reason: for example, it is not true on the restricted version of his second semantics that every stone man is a stone man, or that every stone man is a man. Darapti is still valid: if J is possible and every implicant of J is an implicant of both B and A, then B is possible (since whatever is implied by the possible is possible), and some implicant of B is an implicant of A (namely any implicant of J). But Darapti cannot (as with the first reading) be used to derive the truth of all i-propositions. And in fact some i-propositions (such as ‘Some stone man is a man’) are false on this semantics. Equally, some e-propositions are true.
Obversion, on this semantics, behaves according to Aristotelian principles, and quite contrary to its behaviour on the unrestricted version of his second semantics. Affirmative-to-negative Obversion is valid. This can be proved by trying to construct a counter-example, i.e. a case in which BJa and BJi are both true (the type of case that Ṭūsī objected to). If there is such a case then J is possible and every implicant of J is an implicant of B, and at the same time there is a D which is possible and is an implicant of both J and B. So let this D be D 1. Now, since D 1 ⇒ J and every implicant of J is an implicant of B, it follows that D 1 ⇒ B. But D 1 ⇒ B. So D 1 is impossible, contrary to our assumption. Thus, no counter-example to affirmative-to-negative can be consistently supposed, and this form of Obversion is valid.
But negative-to-affirmative Obversion is invalid. It’s true that no stone man is a man; but it’s false that every stone man is a non-man, because any term that entails both stone and man has to be impossible.
Abharī’s first semantics also makes Obversion conform to Aristotelian principles. If all of the Js (past, present and future) are B then none is non-B. But if none is non-B it doesn’t follow that there have been, or are, or will be, any actual Js; so it doesn’t follow that all past, present and future Js are B. Thus, the restricted version of the second semantics is formally similar to the first semantics. They will also be materially identical if the possible is identical with what is at some time actual.
Ṭūsī accuses Abharī of falling into a contradiction on this semantics. He argues that if all (or no) Js may be B (i.e. if Every J is B and No J is B hold as special possibilities – i.e. two-way possibilities), then Abharī’s restricted semantics will make it true that every BJ is B and that every BJ is J. So we can conclude that some J is B. Similarly, if all (or no) Js may be B (i.e. if Every J is B̅ and No J is B̅ hold as special possibilities), then it will be true that every BJ is B and that every BJ is J. So we can conclude that some J is B:
[…] if we say every J is B by special possibility or no J is B by special possibility (and B is something whose joining and not joining to J is neither essentially nor extrinsically impossible, but [one state or] the other is only true according to two [different] times, then it will be true that all that is J and B is J necessarily and all that is J and B is B necessarily, and they produce some of what is J is B necessarily. Similarly it is true that all that is J and not B is J necessarily and all that is J and not B is not B necessarily, and they produce some of what is J is not B necessarily. So it is true that some J is necessarily B and some J is necessarily not B, along with the truth of every J is B by a special possibility and no J is B by a special possibility. From this it follows that two contradictories and two opposites have been conjoined truly. This only follows due to Abharī’s considering implication [a way to understand] categoricals.Footnote 21
But it’s not clear that this charge is justified. The two sets of assumptions lead to the conclusions that some J is B and that some J is B; but these do not contradict one other. Nor does Ṭūsī explain why he sees the problem as springing from Abharī’s conditional analysis of BJa. The restricted version of Abharī’s second semantics can meet Ṭūsī’s objections. In addition it stays close to Aristotelian principles in accepting Darapti and affirmative-to-negative Obversion, while not holding that every i-proposition is true. But, for anyone who takes the possible to coincide with the sometime actual, it is indistinguishable from the first semantics.
Abharī proposes a third semantics, which takes on a mentalist tinge:
And sometimes [“every J”] is used according to mental existence, so what is meant by every J is B is that everything that is J in the mind is B in the mind.Footnote 22
On this reading, it appears that the premises of the Darapti syllogism are true assuming that someone thinks they are true. It cannot, however, be assumed that Darapti is valid in the sense that if its premises are true then its conclusion is true – where truth is relativized to a particular thinker – because someone may think the premises are true without thinking the conclusion is true. The conclusion of the reasoning is not necessarily true, for similar reasons.
Ṭūsī comments that this is ‘something no one could hold … because many absurdities are found in the mind’.Footnote 23 The subject ‘Every man’ would have to include flying men, since it is possible to imagine a man flying – ‘and the aforementioned absurdities return’. And there is no reason to think that Abharī would disagree with this evaluation.
MODERN TREATMENTS
Some of Abharī’s and Ṭūsī’s discussions have counterparts in modern solutions to the puzzle of Partial Identity and Darapti.
J.N. Keynes drew attention to the puzzle in his Formal Logic:
“Whatever P and Q may stand for, we may show à priori that some P is Q. For All PQ is Q by the law of identity, and similarly All PQ is P; therefore, by a syllogism in Darapti, Some P is Q.”
How would you deal with this paradox?Footnote 24
Keynes refers to section 342 for a solution, saying that “this example seems to shew that the enquiry – how far assumptions with regard to existence are involved in syllogistic processes – is not irrelevant or unnecessary”. What section 342 shows is that the puzzle is capable of four solutions. If all propositions have existential import with regard to their subject, then the Laws of Partial Identity fail but Darapti is valid. The same result emerges if affirmative proposition have existential import but negatives do not. If no proposition has existential import, the Laws of Partial Identity are valid but Darapti is not. The same result emerges if universal propositions do not have existential import but particulars do.Footnote 25 In any case, the derivation of Some B is J is blocked.
It will be seen from this that both the semantics that Abharī believes to apply to ordinary language – his first semantics, and the restricted version of his second semantics – give the same results as the assumption that all propositions have existential import with regard to their subject, or the assumption that affirmatives but not negatives have such existential import.
Taking up the puzzle from Keynes, A.J. Baker considers a “logic of referential or non-empty terms” where all propositions assume that the subject is not empty and that the predicate is not universal.Footnote 26 He states:
Putative propositions containing terms that apply to everything, as well as those containing vacuous terms, will be excluded from our system.Footnote 27
On Baker’s account, an expression (JB) is a well-formed conjunctive term only if it’s true that some J is B and it’s true that something is either not J or not B. This reading is designed to validate both forms of Obversion – affirmative-to-negative we well as negative-to-affirmative.Footnote 28
A.N. Prior pointed out that on Baker’s assumptions the propositional forms J(JB)a and B(JB)a come out as true whenever they are well-formed.Footnote 29 In this sense, these two propositional form express ‘laws’ in Baker’s logic. However, even though Darapti is valid by Baker’s account, BJi does not express a law: there are well-formed false propositions of this form, e.g. Some man is a stone. Prior correctly draws the inference:
[…] thus Bakerian validity cannot be guaranteed to take us from laws to laws.Footnote 30
The explanation for this situation is as follows. According to Baker, a proposition’s well-formedness depends on the world’s being in a particular state. For example, in order for All brown cows are cows to be well-formed, the world has to contain brown cows as well as non-cows. The existence of brown cows ensures that the subject is non-empty, and also that the predicate is non empty; the existence of non-cows ensures that both subject and predicate are non-universal. Similarly, the well-formedness of All brown cows are brown requires that the world contains brown cows and also contains non-brown things. Now, the propositions All brown cows are cows and All brown cows are brown entail Some cows are brown by Darapti. But the well-formedness of this proposition requires only that the world contain cows and brown things, and non-cows and non-brown things; it does not require that the world contains brown cows (as required by the premises of our Darapti). Thus, the world could be such that the conclusion of our Darapti was well-formed but its premises were not well-formed. This is why the fact that the premises express laws does not entail that the conclusion expresses a law. The truth (and thus the well-formedness) of the premises doesn’t entail anything about a situation in which the premises are ill-formed; therefore it doesn’t entail anything about a situation in which the premises are ill-formed but the conclusion is well-formed; but there are circumstances in which the premises are ill-formed but the conclusion is well-formed; therefore the fact that the premises are true whenever they are well-formed doesn’t entail that the conclusion is true in all circumstances in which it is well-formed.
CONCLUSION
Abharī’s explorations of the puzzle of Partial Identity and Darapti are summarized in the Table below.
He argues that in our ordinary utterances, universal affirmative propositions sometimes have the first meaning, and sometimes the second as restricted. He convincingly shows that the other two readings lead to serious difficulties. Ṭūsī’s objections to the first reading and to the second in its preliminary form are well taken: the first reading cannot account for all of our utterances, and Abharī’s use of the notion of an implication requires clarification. The clarification, once provided, should lead Abharī to revise both the preliminary and the restricted versions of the second reading. Some of Ṭūsī’s other objections are less damaging.Footnote 31
The Abharī/Ṭūsī debate about conjunctive terms remains of continued interest because of its high level of logical sophistication. Abharī, like Keynes, considers a range of possible solutions to the paradox of Partial Identity and Darapti; but he is more ambitious than Keynes in specifying which solutions match ordinary linguistic usage.
