### 1.

In Aristotle’s construction of the Categorical Syllogistic, he draws a distinction between perfect and imperfect syllogisms.1placeholder Aristotle suggests that all the first figure assertoric syllogisms, as well as some first figure modal syllogisms, are perfect. This article offers an epistemological explanation for Aristotle’s distinction between perfect and imperfect syllogisms. I argue that in Aristotle’s view, the perfect syllogisms are perfect because it is immediately evident that the middle terms in these syllogisms are responsible for the predicative relationship between the major and minor terms in the conclusion. The distinction between perfect and imperfect syllogisms reflects Aristotle’s concerns for conducting scientific investigations through demonstrative argumentations. Aristotle makes the first figure moods perfect because they work better in aiding an investigator to uncover the proper explanatory middle term of a given conclusion.

### 2.

In *Prior Analytics* 24b25, Aristotle defines a perfect syllogism as:

“Now I call a syllogism perfect if it requires nothing beyond the things posited for the necessity to be evident; I call a syllogism imperfect if it requires one or more things that are indeed necessary because of the terms laid down, but that have not been taken among the premises.” (24b23-27)

To truly understand what Aristotle means by a perfect syllogism, it is crucial to decipher the meaning of ‘for the necessity to be evident’. Following Patzig’s influential work, it is now widely accepted that what Aristotle means by perfect syllogisms is that the validity of these syllogisms is immediately evident.2placeholder Patzig argued that the validity perfect syllogisms is immediately evident due to their transitivity. The middle term of a perfect syllogism mediates between the two extremes, transiting the predicative relationship in the premises to the conclusion. For instance, if a cup is in the cupboard, and the cupboard is in the room, then it is immediately evident that the cup is also in the room.3placeholder Even though I agree with Patzig’s view that the perfect syllogisms are perfect by virtue of their transitivity, I disagree with his view on why the transitivity of perfect syllogisms makes them perfect. Following the traditional interpretation that the validity of perfect moods is immediately evident, the validity of first figure moods is immediately evident through the principle of *dictum de omni et de nullo*. What this means is that we can determine that the first figure moods are valid by virtue of the definition of ‘belong to all’ and ‘belonging to none’.4placeholder Therefore, the perfect moods require no further deductions to show that their conclusions necessarily follow from their premises. However, Striker pointed out that the validity of some second figure moods like *Cesare* and *Camestres* is also immediately evident. Take the second figure mood *Cesare* (MeN, MaX à NeX) as an example. Since no member of N is contained in M, we know that N cannot be predicated of any member of M. Since X is contained in M, it is immediately evident that N cannot belong to X. The validity of this mood is immediately evident through the principle of *dictum de omni et de nullo*. Therefore, as Striker pointed out, it is doubtful as to whether there is truly a difference in degree of obviousness between some of the second/third figure moods and the first figure moods.5placeholder

I think the idea that the first figure moods are perfect because their validity is immediately evident is misleading. In *Prior Analytics* 28a1-7, Aristotle says:

“But it is evident also that all the deductions in these figure are imperfect: for all are made perfect by certain supplementary assumption, which either are contained in the terms of necessity or are assumed as hypotheses, i.e., when we prove per impossibile.”

Aristotle suggests that imperfect moods are imperfect because they are made perfect by “supplementary assumptions”. If the traditional interpretation of imperfect syllogisms is correct, then what this means is that the imperfect moods can be proved through the perfect ones. In this sense, the imperfect moods can be reduced to perfect ones because they can be shown valid through the perfect moods.6placeholder Therefore, we can eliminate the redundant moods to produce a more compact set of deductive rules.7placeholder However, if this is what Aristotle means, then his words about how imperfect moods require supplementary assumption to be made perfect is quite strange. When the validity of the imperfect moods is proved through direct deduction, the validity of the moods is proved through the original assumptions. We prove a given syllogism by converting the premises using the conversion rules. Take the proof for MaN, MeX à NeX as an example. We convert the second premise MeX to XeM using the e-e conversion rule. Then, we can write down a first figure syllogism XeM, MaN à XeN (*Celarent*). We can then convert the conclusion XeN into NeX using e-e conversion rule. Therefore, MaN, NeX à NeX. Aristotle must think that NeX counts as a supplementary assumption, it is an assumption that is “contained in terms of necessity” by the e-e conversion rule.8placeholder However, why does Aristotle think that NeX count as a supplementary assumption when it is directly entailed by the original assumption XeN?

I believe there is an extra-logical reason for why Aristotle thinks that a syllogism provable through simple conversion requires supplementary assumptions. This view, I believe, is the result of Aristotle carrying his concerns for scientific investigations to his construction of his logical system. In the following sections, I will explore how Aristotle’s theory of demonstrative science necessitates a conclusion-to-premise order of construction of arguments. Aristotle’s concern for constructing syllogisms that could lead to some known conclusion makes the first figure moods the most scientific ones in conducting scientific investigations. In this sense, Aristotle does not believe that imperfect moods require supplementary assumptions to be proved through the perfect moods, but rather the imperfect moods require additional assumptions to be rewritten into the perfect ones.

### 3.

In my view, it is impossible to fully appreciate Aristotle’s categorical syllogistic and its unique features without taking his epistemological project into consideration. Therefore, it is important to understand Aristotle’s demonstrative argumentation and how it leads to scientific discoveries. Corcoran argued that demonstrative arguments are deductions in which the premises are known to be true while the conclusion are not known to be true. Investigators demonstrate the conclusion by deducing it from the known premises.9placeholder In this way, the knower gain knowledge of the conclusion through true statements instead of mere beliefs. However, I believe this interpretation misrepresents Aristotle’s view on how to construct demonstrative arguments. It overlooks Aristotle’s epistemological commitments for the sake of pertaining to a consistent interpretation of his logical project.

Many scholars have correctly argued that the goal of scientific demonstrations is to search for a middle term that is explanatory of a given conclusion.10placeholder This view suggests that demonstrative arguments have conclusions that are known to be true in search of the premises that could reveal the proper explanatory middle term of a conclusion. The middle term is the common predicative term that necessitates and explains (or preliminarily explains) the predicative relationship between the predicate and the subject of the known conclusion. Following this interpretation, I aim to explore Aristotle’s requirements for constructing demonstrative argumentations that is consistent with the goal of securing the correct explanatory middle terms.

In the *Prior Analytics*, Aristotle mentions the process of upward and downward predications:

“We shall explain in another place that there is an upward limit also to the process of predicating; for the present we must assume this.” (An. Pr. 1. 27 43a36-38).

We can find Aristotle’s detailed discussion of upward and downward predications in *Posterior Analytics* 1.20:

“Now it is clear that it is not possible for the terms in between to be indefinitely many if the predications come to a stop downwards and upwards – I mean by upwards, towards the more universal; and by downward, towards the particular.” (82a22-25)

Aristotle suggests that in searching for a middle term for a given conclusion, we could uncover predications that are either more universal or more particular. In search for an explanation why, we need to construct demonstrations in an upward direction to identify the more universal predications:

“Again, if demonstration is a probative deduction of an explanation and the reason why, and the universal is more explanatory (for that to which something belongs in itself, is itself explanatory for itself; and the universal is primitive: therefore the universal is explanatory); hence the universal demonstration is better; for it is more a demonstration of the explanation and the reason why it is the case.” (An. Post. 1.24 85b25-27)

When it comes to explaining what a certain state of affair is, we can uncover multiple prior predicate terms until we have reached the immediate explanatory principle. As Aristotle says in *Posterior Analytics* 1. 23 84b30-35:

“When you have to prove something, you should assume what is predicated primitively of B. Let it be C; and let D be predicated similarly of this (C). And if you always proceeds in this way no proposition and nothing belonging outside A will ever be assumed in the proof, but the middle term will always be thickened, until they become indivisible and single. It is single when it becomes immediate; and a single proposition

simpliciteris an immediate one.”

Aristotle shows that the process of upward predication allows us to uncover the most universal predicate (A) of the object of investigation (B) where A has all the qualities that can be predicated of B. Constructing syllogisms allows us to uncover prior predicate terms until we have reached the immediate explanatory principle. This process is what Aristotle refers to as the “thickening” of the middle terms.

Uncovering upward predications requires an investigator to uncover the proper explanatory middle term of an observed phenomenon through identifying the premises that could lead to a given conclusion:

“For the middle term is the explanation, and in all cases that is sought. Is it eclipsed? Is there some explanation or not? After that, aware that there is one, we seek what it is. For the explanation of a substance being not this or that but

simpliciter, or if its being notsimpliciterbut one of the things which belong to it in itself or accidentally – that is the middle term.” (An. Post.2.1 90a7-15)

Aristotle says that through constructing scientific demonstrations, an investigator seeks the explanatory middle term of either why something is what it is or why it possesses some non-essential qualities. This process of uncovering the explanatory middle necessitates a conclusion-to-premise order of constructing arguments, where an investigator demonstrates a given conclusion by finding the premises that could derive this conclusion. If Aristotle is concerned with constructing demonstrative arguments in developing a system of logic, then it is natural for him to categorize a group of syllogisms that is more suitable for this purpose.

### 4.

To better understand Aristotle’s view, we need to look at how Aristotle makes imperfect moods perfect. I will take Aristotle’s proof of *Festino* as an example. In showing how to make *Festino* perfect, Aristotle says:

“For if M belongs to no N, but to some O, it is necessary that N does not belong to some O. For since the negative is convertible, N will belong to no M; but M was admitted to belong to some O; therefore, N will not belong to some O; for a deduction is found by means of the first figure.”

In this passage, Aristotle shows that the arrangement of premises MeN, MiX leads to the conclusion NoX. It is not hard to observe that it requires no additional deduction to show that MeN and MiX leads to the conclusion NoX. Given that no N is M, and some X is M; it is obvious that some X cannot be N. Those X that are M cannot be N, as no N is M. We can observe the validity of this syllogism immediately through a Euler diagram. If this is the case, why does Aristotle say that this syllogism requires supplementary assumptions? This, I believe, is the result of Aristotle’s prioritizing the argument forms in which the middle terms are positioned in the middle of the major and the minor terms. Following this interpretation, the second and third figure moods require additional assumptions to be rewritten into the first figure moods where the middle term lies between the two extremes.11placeholder Aristotle prioritizes the first figure moods because when the middle term sits in the middle, it is evident that the middle term is responsible for the predicative relationship between the predicate and the subject of the conclusion.

I think Hintikka is correct in pointing out that the distinction between perfect and imperfect syllogisms is mostly psychological. He argued that any arbitrary instantiation of the general terms of a first figure syllogism will allow people to immediately perceive the validity of the syllogism. This view implies that when Aristotle is constructing his system of logic, he is not purely focusing on logic. He is also concerned with the arguers’ perception or their grasping of a syllogism. Hintikka attributes Aristotle’s distinction between perfect and imperfect syllogisms to his concern for practicality.12placeholder However, I believe there is an epistemological explanation to it.

When the middle term is positioned in the middle of the two extremes, it is the subject of the major and the predicate of the minor term. The transitivity of first figure moods would make it evident that the predicative relationship between the predicate and subject term in a conclusion is established through the middle term. If we have a syllogism in the form of AB, BC à AC, then it is evident to us that the middle term transits the predicative relationships between AB and BC to AC. However, in the second and third figures, since the middle term lies at the end of the schema, it is not clear that the middle term transits the predicative relationships in the premises to the conclusion. Let us consider a set of numerical examples for comparison. Given that x<2 and 2<y, it is immediately evident that x<y. This is akin to the transitivity we see in the first figure syllogisms, where the middle term clearly attributes the predicate of the conclusion to the subject. However, if we consider the conditions x<3 and y<3, the relationship between x and y is not explicitly revealed by the number Three. This sort of relationship is similar to the second and third figure moods. Based on the original forms of these moods, it is not immediately clear that the predicative relationship between the major and minor terms is so by virtue of the common term.

Take *Celarent* (AeB, BaC à AeC) for example, it is immediately clear that the middle term is responsible for why the major term is negatively universally predicated of the minor term in the conclusion, given that the minor term is the subject of the conclusion and major term is the predicate of the conclusion. The middle term carries the universally negative relationship, illustrated in the major premise, to the minor term. However, the role of the middle term is not immediately clear in the second and third figures. Take *Cesare* (MaN, MeX à NeX) for example. Based on the form of the syllogism (MNX), it is not immediately clear that the predicative relationship between N and X is so by virtue of the middle term, as the middle term is predicated of N and X independently. However, we could work this relationship out, that no X is N because N possesses a quality (M) that does not belong to X.13placeholder

Following this interpretation, I believe Aristotle prioritizes the first figure because the form of this figure is more apt in allowing us to uncover the explanatory principle of a given conclusion. In the *Posterior Analytics*, Aristotle says that:

“Again, in cases in which the middle is positioned outside – for in these too the demonstration is of the fact and not of the reason why; for the explanation is not mentioned. E.g. why does the wall not breathe? Because it is not an animal. For if this were explanatory of breathing – i.e., if the denial is explanatory of something’s not belonging, the affirmation is explanatory of its belonging (e.g. if imbalance in the hot and cold elements is explanatory of not being healthy, their balance is explanatory of being healthy), and similarly too if the affirmation is explanatory of something’s belonging, the denial is of its not belonging. But when things are set out in this way what we have said does not result; for not every animal breathes. The deduction of such an explanation comes about in the middle figure. E.g. let A be animal, B breathing, C wall: then A belongs to every B (for everything breathe is an animal), but to no C, so that B too belongs to no C – therefore the wall does not breathe.” (

An. Post.78b15-30)

In this passage, Aristotle states that when the middle term is positioned outside the extremes in a demonstrative argument, the demonstration does not reveal the explanatory principle of the conclusion. I believe that Aristotle’s concerns about the positioning of the middle term in demonstrative science is what motivates him to distinguish perfect syllogisms and imperfect syllogisms. What this means is that when we construct a syllogism in the form of the second or the third figure moods for demonstrative science, we run the risk of misidentifying the explanatory middle term.

Barnes pointed out that the middle figure can produce a demonstration that reveals the explanatory principle of the conclusion. For instance:

Having lungs holds of all breathing things.

Having lungs holds of no walls.

∴ Breathing holds of no walls.

Therefore, it has been suggested that “the middle term is positioned outside” does not refer to the structure of syllogism, but to the remoteness of the explanation it offers. 14placeholder This interpretation implies that when Aristotle gives the example of the middle term being positioned outside, he is not considering the second and third figure mood structure. He is only concerned with whether the middle term offers an immediate explanation for a proposition like “no walls breathe”. However, this interpretation does not align well with Posterior Analytics 1.14, a chapter that immediately follows this passage, where Aristotle establishes that the first figure is the most scientific. It makes better sense that when Aristotle suggests that “the middle term is positioned outside”, he has the second and third figure moods in mind.

From the perspective of a conclusion-to-premise construction of syllogisms, we can reliably find the proper instances of the middle term (since the major and the minor terms are given) that could lead to the given conclusion in the first figure. The first figure is most reliable in allowing us to uncover the proper explanatory middle term of the conclusion, as the middle term is both contained in the major term and contains the minor term. In the imperfect figures, on the other hand, since both the major and the minor term are in relation to the middle term independently, these figures are less reliable in allowing us to uncover the explanatory middle term of the conclusion. This makes the discovery of the proper middle term in the second and third figures somewhat arbitrary. For example, if we write down the conclusion “no walls breathe” to search for the explanatory principle for the conclusion through the first figure, we would be searching for a set of items all of which possess the quality of breathing, but do not hold of walls: all *x* breathes, and no wall is *x*. We would have excluded “animal” in the first place, as we know not all animals breathe. However, if we look for the middle term in the second or the third figure, we might construct a demonstration that does not reveal the proper explanatory middle term: animal belongs to all breathing things (since everything that breathe is an animal), animal belongs to no walls; therefore, no walls breathe. We are looking for a common feature true of breathing and walls independently. On this note, the search for a middle term under the first figure is more constricted (as we are looking for a term that sits between the extremes) than under the second and third figures.

While the validity of the moods of the second and the third figures might be equally immediately evident as the first figure moods, they are less efficient in constructing a syllogism that allows us to locate the proper middle term that attributes the predicate of a given conclusion to the subject.15placeholder From the perspective of constructing syllogisms out of the concern of locating the explanatory middle terms, it is easier and more intuitive to search for a set of premises that could lead us to the desired conclusion through the first figure moods.

In light of this interpretation, we can explain why Aristotle says that additional assumptions need to be made to make imperfect syllogisms perfect. I believe that by requiring additional assumptions, Aristotle means that new information needs to be supplied to write an imperfect syllogism into a perfect one. In *Posterior Analytics* I.22, Aristotle emphasizes how reversing the subject and predicate of proposition changes the underlying subject, a significant alteration in inquiries:

“For when I say that the white thing is a log, then I say that which is accidentally white is a log; and not that the white thing is the underlying subject for the log; for it is not the case that, being white or just what is some white, it came to be a log, so that it is not a log except accidentally. But when I say that the log is white, I do not say that something else is white and that that is accidentally a log, as when I say that the musical thing is white; but the log is the underlying subject which did come to be white without being something other than just what is a log or a particular log.” (An. Post. 83a5-14)

In this sense, changing the order of the terms means that we are changing the underlying subject of the sentence. Suppose we have a second figure syllogism: All men are rational, no raven is rational, therefore, no man is a raven (*Cesare*). To write the syllogism in the first figure mood, we need to convert the second premise into ‘no rational thing is a raven’. Even though the proposition ‘no raven is rational’ and ‘no rational thing is a raven’ are logically equivalent, they are two different assumptions. The first proposition is an assumption about ravens, while the second proposition is an assumption about rational things. The process of finding out whether no raven is rational requires an investigator to find all the ravens in the world and find out if any of these ravens is rational. Finding out whether no rational thing is a raven requires finding out all the rational things in the world to see if any of these things is a raven. Even though finding out one of these facts implies that the investigator immediately knows the other fact, they are essentially different assumptions about the world. Therefore, rewriting the imperfect syllogisms into the perfect forms requires supplementing new assumptions (as a direct conversion changes the underlying subject of the original premise). It is clear then that when Aristotle shows how imperfect moods are reducible to the perfect ones, he has the investigative purposes of syllogisms in mind.

### 5. Conclusion

Aristotle’s distinction between a perfect and imperfect syllogism can be sufficiently explained by his interest in conducting demonstrative arguments which necessitate a conclusion-to-premise order of construction of syllogisms. His interest in demonstrative science compels him to prioritize a form of valid syllogistic structure in which the middle term sits in the middle of the two extremes. It is also for this reason that Aristotle attempts to show how all syllogisms can be reduced to the first figure. While the validity of some syllogisms of the second and third figure are immediately evident, these syllogisms, by their original form, do not immediately reveal how the middle term transits the predicative relationship from the premises to the conclusion. Therefore, they are less obvious in demonstrating the explanatory role of the middle term. When we are looking for the middle term through constructing a syllogism that leads to a known conclusion, these syllogisms have a greater chance of leading us astray from the correct explanatory middle term. We might end up focusing on propositions or terms that are not explanatory of a given conclusion. Therefore, Aristotle makes these syllogistic moods imperfect.

Works Cited

Barnes, Jonathan. “Posterior Analytics.” Chap. 39-113 In *The Complete Works of Aristotle*, edited by Jonathan Barnes. Princeton: Princeton University Press, 1995.

Bochenski, I.M. *A History of Formal Logic.* Notre Dame, 1960.

Boger, George. *Aristotle’s Syllogistic Underlying Logic: His Model with His Proofs of Soundness and Completeness.* Studies in Logic. Edited by Dov Gabbay. Vol. 92: College Publications, 2022.

———. “Completion, Reduction and Analysis: Three Proof-Theoretic Processes in Aristotle’s Prior Analytics.” *History and Philosophy of Logic * (1998).

Bronstein, David. *Aristotle on Knowledge and Learning: The Posterior Analytics.* Oxford: Oxford University Press, 2016.

Corcoran, John. “Aristotle’s Demonstrative Logic.” *History and Philosophy of Logic *30 (2009): 1-20.

———. “Aristotle’s Natural Deduction System.” In *Ancient Logic and Its Modern Interpretations*, edited by John Corcoran, 85-132. Boston: D. Reidel Publishing Company, 1974.

———. “Deductions and Reductions Decoding Syllogistic Mnemonic.” *Entelekya Logico-Metaphysical Review * (2018): 5-39.

Ebert, Theodor. “What Is a Perfect Syllogism in Aristotelian Syllogistic?”. *Ancient Philosophy * (2015): 351-74.

Ferejohn, Michael T. *Formal Causes: Definition, Explanation, and Primacy in Socratic and Aristotelian Thought.* Oxford: Oxford University Press, 2013.

Hintikka, Jaakko. “Aristotle’s Incontinent Logician.” *Analyses of Aristotle * (2004): 139-52.

Kevin L. Flannery. “A Rationale for Aristotle’s Notion of Perfect Syllogisms.” *Notre Dame Journal of Formal Logic *3: 455-71.

Kneale, William, and Martha Kneale. *The Development of Logic.* Oxford, 1975.

Łukasiewicz, Jan. *Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic.* Oxford: Clarendon Press, 1957.

Patterson, Richard. “Aristotle’s Perfect Syllogisms, Predication, and the “Dictum De Omni”.” *Synthese *96, no. 3 (1993): 359-78.

Patzig, Günther. *Aristotle’s Theory of the Syllogism: A Logico-Philosophical Study of Book a of the Prior Analytics.* Dordrecht: D. Reidel, 1968.

Peterson, James B. “The Forms of the Syllogism.” *The Philosophical Review *8 (1899): 371-85.

Striker, Gisela. *Aristotle: Prior Analytics Book I.* Clarendon Aristotle Series. Oxford Oxford University Press, 2009.

———. “Perfection and Reduction in Arsitotle’s Prior Analytics.” In *Rationality in Greek Thought*, edited by Gisela Striker Michael Frede, 203-21. Oxford: Clarendon Press Oxford, 2002.

In this paper, I will be adopting the medieval mnemonic of syllogisms. The medieval mnemonic:

First figure moods: AxB, BxC à AxC (ABC)

Second figure moods: MxN, MxX à NxX (MNX)

Third figure moods: PxR, ZxR à PxZ (PZR)

The medieval mnemonic of the moods represents Aristotle’s move of putting the predicate to the left of the subject term in a proposition using words such as “belongs to”. They capture the forms of the syllogistic moods by the positioning of the middle terms: In the first figure, the middle term (B) lies between the extremes; in the second figure, the middle term (M) is to the left of both extremes; in the third figure, the middle term (R) is to the right of both extremes. For a detailed introduction to the medieval mnemonic, see John Corcoran, “Deductions and Reductions Decoding Syllogistic Mnemonic,” *Entelekya Logico-Metaphysical Review* (2018).

In the Prior Analytics, Aristotle introduces a system of logic that has predicative sentences as premises. Aristotle points out that there are four predicative relationships: predicated of all, predicated of none, predicated of some, and not predicated of some. These predicative relationships are traditionally denoted as a, e, i, o. Aristotle also introduces three conversion rules: a-i conversion, e-e conversion, and i-i conversion. If A is predicated of B, then B is predicated of some A. If A is predicated of no B, then B is predicated of no A. If A is predicated of some B, then B is predicated of some A. In systematizing the syllogistic, Aristotle develops three figures: the first, the middle, and the last. They are also referred to as the first, the second, and the third figure. Aristotle calls the terms in a syllogism the major, the middle, and the minor terms by their relative position to the middle term. In the first figure, the middle term sits in the middle of the two extremes. This means that in the premises, the major term is predicated of the middle term, while the middle term is predicated of the minor term. The first figure contains four moods: Barbara, Celarent, Darii, and Ferio. In the second figure, the middle term is positioned to the left of both extremes in the premises, that is, it is predicated of both the major and the minor term. The major term is the term that is closer to the middle term, while the minor term is the term that is further away from the middle term. This figure contains four moods: Cesare, Camestres, Festino, and Baroco. In the third figure, the middle term is positioned to the right of both extremes in the premises, that is, the major and the minor term are both predicated of the middle term. This figure contains six moods: Darapti, Felapton, Disamis, Datisi, Bocardo, Ferison. In Aristotle’s syllogistic, the major term is always the predicate of the conclusion, while the minor term is the subject.

Gisela Striker, *Aristotle: Prior Analytics Book I*, Clarendon Aristotle Series, (Oxford Oxford University Press, 2009); Gisela Striker, “Perfection and Reduction in Arsitotle’s Prior Analytics,” in *Rationality in Greek Thought*, ed. Gisela Striker Michael Frede (Oxford: Clarendon Press Oxford, 2002); Theodor Ebert, “What is a Perfect Syllogism in Aristotelian Syllogistic?,” *Ancient Philosophy* (2015); William Kneale and Martha Kneale, *The Development of Logic* (Oxford, 1975). Richard Patterson, “Aristotle’s Perfect Syllogisms, Predication, and the “Dictum de Omni”,” *Synthese* 96, no. 3 (1993).

Günther Patzig, *Aristotle’s Theory of the Syllogism: A Logico-Philosophical Study of Book A of the Prior Analytics* (Dordrecht: D. Reidel, 1968). Also, Ebert, “What is a Perfect Syllogism in Aristotelian Syllogistic?.”

Patterson, “Aristotle’s Perfect Syllogisms, Predication, and the “Dictum de Omni”.”

Cf. p.83 Striker, *Aristotle: Prior Analytics Book I*.

In this view, Aristotle intends to show that the imperfect moods are reducible to perfect moods because he treats the perfect moods as inference rules or as logical theorems. Scholars who adopt a natural deductionist interpretation of the categorical syllogistic believes that the first figure moods are inferential rules, while those who adopt an axiomatic interpretation takes them to be logical theorems or axioms. For the purpose of this article, I will not dive too deep into this debate. I believe scholars of both camp fail to provide a sufficient logical justification for Aristotle’s treatment of the first figure moods, either as inferential rules or axioms. For the natural deductionist interpretation, see George Boger, “Completion, reduction and analysis: three proof-theoretic processes in aristotle’s prior analytics,” *History and Philosophy of Logic* (1998). For the axiomatic interpretation, see p.134 Patzig, *Aristotle’s Theory of the Syllogism: A Logico-Philosophical Study of Book A of the Prior Analytics*.

Jan Łukasiewicz, *Aristotle’s Syllogistic From the Standpoint of Modern Formal Logic* (Oxford: Clarendon Press, 1957). I.M. Bochenski, *A History of Formal Logic* (Notre Dame, 1960). P. 7, 175 George Boger, *Aristotle’s Syllogistic Underlying Logic: His Model with his Proofs of Soundness and Completeness*, ed. Dov Gabbay, vol. 92, Studies in Logic, (College Publications, 2022).

For proofs of these syllogisms through simple conversion, see p.111 John Corcoran, “Aristotle’s Natural Deduction System,” in *Ancient Logic and its Modern Interpretations*, ed. John Corcoran (Boston: D. Reidel Publishing Company, 1974).

John Corcoran, “Aristotle’s Demonstrative Logic,” *History and Philosophy of Logic* 30 (2009).

Bronstein argued that Aristotle distinguishes scientific knowledge from non-scientific knowledge. Scientific knowledge is acquired through constructing demonstrations that uncover the explanation of an observed phenomenon without deducing it as the conclusion of the demonstration. Therefore, Bronstein posited that an expert scientist could acquire new scientific knowledge by uncovering the explanatory principle of an explanandum, which is signified by the middle term. Esp. p. 64 David Bronstein, *Aristotle on Knowledge and Learning: The Posterior Analytics* (Oxford: Oxford University Press, 2016). Ferejohn also argued that Aristotle recognizes that knowledge comes in degrees, with the highest type of knowledge being “episteme” or “unqualified knowledge”. Hence, He argued that the technical notion of Aristotelian demonstration refers to a sequence of discursive reasoning that “ties down” its conclusion, aiming to uncover the explanation of the truth of the fact. According to Ferejohn, demonstrations aim to illustrate the aitia of some observed phenomenon that is signified by the middle term. Esp. p.68, p. 99-102 Michael T. Ferejohn, *Formal Causes: Definition, Explanation, and Primacy in Socratic and Aristotelian Thought* (Oxford: Oxford University Press, 2013).

Peterson correctly pointed out that the first figure moods, as inference rules, are not applicable to the second figure as to the third. Peterson showed that by reducing certain imperfect moods into the perfect ones, we do not reach the conclusion of the original syllogisms, but its converse. P.380-381 James B. Peterson, “The Forms of the Syllogism,” *The Philosophical Review* 8 (1899).

Jaakko Hintikka, “Aristotle’s Incontinent Logician,” *Analyses of Aristotle* (2004).

Flannery argued for a somewhat similar view. He suggested that “when Aristotle says that a syllogism is perfect, he means that during the process of constructing a mental picture of what a syllogism says, once a person has grasped the premises, he sees the conclusion. The operation is simple and smooth. One reads through the syllogism without a hitch: i.e., without having to make any suppositions about different steps in the process.” Flannery did not note the epistemological significance of this requirement, that the smooth grasping of the deduction immediately reveals the role the middle term plays in a syllogism. However, his reading does correctly reveal that the distinction between perfect and imperfect syllogisms lies in the distinction between their forms. The form of perfect syllogisms allows us to grasp the deductive process immediately. P.461 Kevin L. Flannery, “A Rationale For Aristotle’s Notion of Perfect Syllogisms,” *Notre Dame Journal of Formal Logic* 3.

P.157 Jonathan Barnes, “Posterior Analytics,” in *The Complete Works of Aristotle*, ed. Jonathan Barnes (Princeton: Princeton University Press, 1995).

Striker pointed out that reduction to the first figure is not required to establish validity of some of the syllogistic moods since they seem to be immediately valid through the Euler diagrams. The author pointed out that Calerant, Cesare, and Cametres seem to share the same Euler diagram. On this note, even if the first figure moods are meant to be the deductive rules, we still need to explain why Aristotle chooses these moods. Striker suggested that “One might suspect, indeed, that he found it rather convenient to have all perfect syllogisms assembled in one and the same figure, and did not bother to look further.” P.217 Striker, “Perfection and Reduction in Arsitotle’s Prior Analytics.”