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In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge.

Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent of new logic, remaining dominant until the advent of predicate logic in the late nineteenth century.

However, even if eclipsed by newer logical systems, term logic still plays a significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand it.

Aristotle's system

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Aristotle's logical work is collected in the six texts that are collectively known as the Organon. Two of these texts in particular, namely the Prior Analytics and On Interpretation, contain the heart of Aristotle's treatment of judgements and formal inference, and it is principally this part of Aristotle's works that is about term logic. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm.[1] Lukasiewicz's approach was reinvigorated in the early 1970s by John Corcoran and Timothy Smiley – which informs modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009.[2]

The Prior Analytics represents the first formal study of logic, where logic is understood as the study of arguments. An argument is a series of true or false statements which lead to a true or false conclusion.[3] In the Prior Analytics, Aristotle identifies valid and invalid forms of arguments called syllogisms. A syllogism is an argument that consists of at least three sentences: at least two premises and a conclusion. Although Aristotle does not call them "categorical sentences", tradition does; he deals with them briefly in the Analytics and more extensively in On Interpretation.[4] Each proposition (statement that is a thought of the kind expressible by a declarative sentence)[5] of a syllogism is a categorical sentence which has a subject and a predicate connected by a verb. The usual way of connecting the subject and predicate of a categorical sentence as Aristotle does in On Interpretation is by using a linking verb e.g. P is S. However, in the Prior Analytics Aristotle rejects the usual form in favour of three of his inventions:

  • P belongs to S
  • P is predicated of S
  • P is said of S

Aristotle does not explain why he introduces these innovative expressions but scholars conjecture that the reason may have been that it facilitates the use of letters instead of terms avoiding the ambiguity that results in Greek when letters are used with the linking verb.[6] In his formulation of syllogistic propositions, instead of the copula ("All/some... are/are not..."), Aristotle uses the expression, "... belongs to/does not belong to all/some..." or "... is said/is not said of all/some..."[7] There are four different types of categorical sentences: universal affirmative (A), universal negative (E), particular affirmative (I) and particular negative (O).

  • A - A belongs to every B
  • E - A belongs to no B
  • I - A belongs to some B
  • O - A does not belong to some B

A method of symbolization that originated and was used in the Middle Ages greatly simplifies the study of the Prior Analytics. Following this tradition then, let:

a = belongs to every
e = belongs to no
i = belongs to some
o = does not belong to some

Categorical sentences may then be abbreviated as follows:

AaB = A belongs to every B (Every B is A)
AeB = A belongs to no B (No B is A)
AiB = A belongs to some B (Some B is A)
AoB = A does not belong to some B (Some B is not A)

From the viewpoint of modern logic, only a few types of sentences can be represented in this way.[8]

Basics

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The fundamental assumption behind the theory is that the formal model of propositions are composed of two logical symbols called terms – hence the name "two-term theory" or "term logic" – and that the reasoning process is in turn built from propositions:

  • The term is a part of speech representing something, but which is not true or false in its own right, such as "man" or "mortal". As originally conceived, all terms would be drawn from one of ten categories enumerated by Aristotle in his Organon, classifying all objects and qualities within the domain of logical discourse.
  • The formal model of proposition consists of two terms, one of which, the "predicate", is "affirmed" or "denied" of the other, the "subject", and which is capable of truth or falsity.
  • The syllogism is an inference in which one proposition (the "conclusion") follows of necessity from two other propositions (the "premises").

A proposition may be universal or particular, and it may be affirmative or negative. Traditionally, the four kinds of propositions are:

  • A-type: Universal and affirmative ("All philosophers are mortal")
  • E-type: Universal and negative ("All philosophers are not mortal")
  • I-type: Particular and affirmative ("Some philosophers are mortal")
  • O-type: Particular and negative ("Some philosophers are not mortal")

This was called the fourfold scheme of propositions (see types of syllogism for an explanation of the letters A, I, E, and O in the traditional square). Aristotle's original square of opposition, however, does not lack existential import.

Term

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A term (Greek áœ…ÏÎżÏ‚ horos) is the basic component of the proposition. The original meaning of the horos (and also of the Latin terminus) is "extreme" or "boundary". The two terms lie on the outside of the proposition, joined by the act of affirmation or denial.

For early modern logicians like Arnauld (whose Port-Royal Logic was the best-known text of his day), it is a psychological entity like an "idea" or "concept". Mill considers it a word. To assert "all Greeks are men" is not to say that the concept of Greeks is the concept of men, or that word "Greeks" is the word "men". A proposition cannot be built from real things or ideas, but it is not just meaningless words either.

Proposition

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In term logic, a "proposition" is simply a form of language: a particular kind of sentence, in which the subject and predicate are combined, so as to assert something true or false. It is not a thought, nor an abstract entity. The word "propositio" is from the Latin, meaning the first premise of a syllogism. Aristotle uses the word premise (protasis) as a sentence affirming or denying one thing or another (Posterior Analytics 1. 1 24a 16), so a premise is also a form of words.

However, as in modern philosophical logic, it means that which is asserted by the sentence. Writers before Frege and Russell, such as Bradley, sometimes spoke of the "judgment" as something distinct from a sentence, but this is not quite the same. As a further confusion the word "sentence" derives from the Latin, meaning an opinion or judgment, and so is equivalent to "proposition".

The logical quality of a proposition is whether it is affirmative (the predicate is affirmed of the subject) or negative (the predicate is denied of the subject). Thus every philosopher is mortal is affirmative, since the mortality of philosophers is affirmed universally, whereas no philosopher is mortal is negative by denying such mortality in particular.

The quantity of a proposition is whether it is universal (the predicate is affirmed or denied of all subjects or of "the whole") or particular (the predicate is affirmed or denied of some subject or a "part" thereof). In case where existential import is assumed, quantification implies the existence of at least one subject, unless disclaimed.

Singular terms

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For Aristotle, the distinction between singular[citation needed] and universal is a fundamental metaphysical one, and not merely grammatical. A singular term for Aristotle is primary substance, which can only be predicated of itself: (this) "Callias" or (this) "Socrates" are not predicable of any other thing, thus one does not say every Socrates one says every human (De Int. 7; Meta. D9, 1018a4). It may feature as a grammatical predicate, as in the sentence "the person coming this way is Callias". But it is still a logical subject.

He contrasts universal (katholou)[9] secondary substance, genera, with primary substance, particular (kath' hekaston)[9][10] specimens. The formal nature of universals, in so far as they can be generalized "always, or for the most part", is the subject matter of both scientific study and formal logic.[11]

The essential feature of the syllogism is that, of the four terms in the two premises, one must occur twice. Thus

All Greeks are men
All men are mortal.

The subject of one premise, must be the predicate of the other, and so it is necessary to eliminate from the logic any terms which cannot function both as subject and predicate, namely singular terms.

However, in a popular 17th-century version of the syllogism, Port-Royal Logic, singular terms were treated as universals:[12]

All men are mortals
All Socrates are men
All Socrates are mortals

This is clearly awkward, a weakness exploited by Frege in his devastating attack on the system.

The famous syllogism "Socrates is a man ...", is frequently quoted as though from Aristotle,[13] but in fact, it is nowhere in the Organon. Sextus Empiricus in his Hyp. Pyrrh (Outlines of Pyrronism) ii. 164 first mentions the related syllogism "Socrates is a human being, Every human being is an animal, Therefore, Socrates is an animal."

The three figures

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Depending on the position of the middle term, Aristotle divides the syllogism into three kinds: syllogism in the first, second, and third figure.[14] If the Middle Term is subject of one premise and predicate of the other, the premises are in the First Figure. If the Middle Term is predicate of both premises, the premises are in the Second Figure. If the Middle Term is subject of both premises, the premises are in the Third Figure.[15]

Symbolically, the Three Figures may be represented as follows:[16]

First figure Second figure Third figure
Predicate â€” Subject Predicate â€” Subject Predicate â€” Subject
Major premise A ------------ B B ------------ A A ------------ B
Minor premise B ------------ C B ------------ C C ------------ B
Conclusion A ********** C A ********** C A ********** C

The fourth figure

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In Aristotelian syllogistic (Prior Analytics, Bk I Caps 4-7), syllogisms are divided into three figures according to the position of the middle term in the two premises. The fourth figure, in which the middle term is the predicate in the major premise and the subject in the minor, was added by Aristotle's pupil Theophrastus and does not occur in Aristotle's work, although there is evidence that Aristotle knew of fourth-figure syllogisms.[17]

Syllogism in the first figure

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In the Prior Analytics translated by A. J. Jenkins as it appears in volume 8 of the Great Books of the Western World, Aristotle says of the First Figure: "... If A is predicated of all B, and B of all C, A must be predicated of all C."[18] In the Prior Analytics translated by Robin Smith, Aristotle says of the first figure: "... For if A is predicated of every B and B of every C, it is necessary for A to be predicated of every C."[19]

Taking a = is predicated of all = is predicated of every, and using the symbolical method used in the Middle Ages, then the first figure is simplified to:[20]

If AaB
and BaC
then AaC.

Or what amounts to the same thing:

AaB, BaC; therefore AaC

When the four syllogistic propositions, a, e, i, o are placed in the first figure, Aristotle comes up with the following valid forms of deduction for the first figure:

AaB, BaC; therefore, AaC
AeB, BaC; therefore, AeC
AaB, BiC; therefore, AiC
AeB, BiC; therefore, AoC

In the Middle Ages, for mnemonic reasons they were called "Barbara", "Celarent", "Darii" and "Ferio" respectively.[21]

The difference between the first figure and the other two figures is that the syllogism of the first figure is complete while that of the second and third is not. This is important in Aristotle's theory of the syllogism for the first figure is axiomatic while the second and third require proof. The proof of the second and third figure always leads back to the first figure.[22]

Syllogism in the second figure

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This is what Robin Smith says in English that Aristotle said in Ancient Greek: "... If M belongs to every N but to no X, then neither will N belong to any X. For if M belongs to no X, neither does X belong to any M; but M belonged to every N; therefore, X will belong to no N (for the first figure has again come about)."[23]

The above statement can be simplified by using the symbolical method used in the Middle Ages:

If MaN
but MeX
then NeX.
For if MeX
then XeM
but MaN
therefore XeN.

When the four syllogistic propositions, a, e, i, o are placed in the second figure, Aristotle comes up with the following valid forms of deduction for the second figure:

MaN, MeX; therefore NeX
MeN, MaX; therefore NeX
MeN, MiX; therefore NoX
MaN, MoX; therefore NoX

In the Middle Ages, for mnemonic reasons they were called respectively "Camestres", "Cesare", "Festino" and "Baroco".[24]

Syllogism in the third figure

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Aristotle says in the Prior Analytics, "... If one term belongs to all and another to none of the same thing, or if they both belong to all or none of it, I call such figure the third." Referring to universal terms, "... then when both P and R belongs to every S, it results of necessity that P will belong to some R."[25]

Simplifying:

If PaS
and RaS
then PiR.

When the four syllogistic propositions, a, e, i, o are placed in the third figure, Aristotle develops six more valid forms of deduction:

PaS, RaS; therefore PiR
PeS, RaS; therefore PoR
PiS, RaS; therefore PiR
PaS, RiS; therefore PiR
PoS, RaS; therefore PoR
PeS, RiS; therefore PoR

In the Middle Ages, for mnemonic reasons, these six forms were called respectively: "Darapti", "Felapton", "Disamis", "Datisi", "Bocardo" and "Ferison".[26]

Table of syllogisms

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Table of syllogisms[27]
Figure Major premise Minor premise Conclusion Mnemonic name
First Figure AaB BaC AaC Barbara
AeB BaC AeC Celarent
AaB BiC AiC Darii
AeB BiC AoC Ferio
Second Figure MaN MeX NeX Camestres
MeN MaX NeX Cesare
MeN MiX NoX Festino
MaN MoX NoX Baroco
Third Figure PaS RaS PiR Darapti
PeS RaS PoR Felapton
PiS RaS PiR Disamis
PaS RiS PiR Datisi
PoS RaS PoR Bocardo
PeS RiS PoR Ferison

Decline of term logic

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Term logic began to decline in Europe during the Renaissance, when logicians like Rodolphus Agricola Phrisius (1444–1485) and Ramus (1515–1572) began to promote place logics. The logical tradition called Port-Royal Logic, or sometimes "traditional logic", saw propositions as combinations of ideas rather than of terms, but otherwise followed many of the conventions of term logic. It remained influential, especially in England, until the 19th century. Leibniz created a distinctive logical calculus, but nearly all of his work on logic remained unpublished and unremarked until Louis Couturat went through the Leibniz Nachlass around 1900, publishing his pioneering studies in logic.

19th-century attempts to algebraize logic, such as the work of Boole (1815–1864) and Venn (1834–1923), typically yielded systems highly influenced by the term-logic tradition. The first predicate logic was that of Frege's landmark Begriffsschrift (1879), little read before 1950, in part because of its eccentric notation. Modern predicate logic as we know it began in the 1880s with the writings of Charles Sanders Peirce, who influenced Peano (1858–1932) and even more, Ernst Schröder (1841–1902). It reached fruition in the hands of Bertrand Russell and A. N. Whitehead, whose Principia Mathematica (1910–13) made use of a variant of Peano's predicate logic.

Term logic also survived to some extent in traditional Roman Catholic education, especially in seminaries. Medieval Catholic theology, especially the writings of Thomas Aquinas, had a powerfully Aristotelean cast, and thus term logic became a part of Catholic theological reasoning. For example, Joyce's Principles of Logic (1908; 3rd edition 1949), written for use in Catholic seminaries, made no mention of Frege or of Bertrand Russell.[28][page needed][need quotation to verify]

Revival

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Some philosophers have complained that predicate logic:

Even academic philosophers entirely in the mainstream, such as Gareth Evans, have written as follows:

"I come to semantic investigations with a preference for homophonic theories; theories which try to take serious account of the syntactic and semantic devices which actually exist in the language ...I would prefer [such] a theory ... over a theory which is only able to deal with [sentences of the form "all A's are B's"] by "discovering" hidden logical constants ... The objection would not be that such [Fregean] truth conditions are not correct, but that, in a sense which we would all dearly love to have more exactly explained, the syntactic shape of the sentence is treated as so much misleading surface structure" (Evans 1977)

Boole’s acceptance of Aristotle

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Commentaria in Analytica priora Aristotelis, 1549

George Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought[29] Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought.[30] According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were “to go under, over, and beyond” Aristotle's logic by:

  1. providing it with mathematical foundations involving equations;
  2. extending the class of problems it could treat– from assessing validity to solving equations; and
  3. expanding the range of applications it could handle– e.g. from propositions having only two terms to those having arbitrarily many.

More specifically, Boole agreed with what Aristotle said; Boole's ‘disagreements’, if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced the four propositional forms of Aristotle's logic to formulas in the form of equations– by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic– another revolutionary idea –involved Boole's doctrine that Aristotle's rules of inference (the “perfect syllogisms”) must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce “No quadrangle that is a square is a rectangle that is a rhombus” from “No square that is a quadrangle is a rhombus that is a rectangle” or from “No rhombus that is a rectangle is a square that is a quadrangle”.

See also

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Notes

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References

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Term logic

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Term logic, also known as syllogistic or categorical logic, is a deductive system of reasoning developed by the ancient Greek philosopher Aristotle in the 4th century BCE, focusing on the logical relations between terms—such as classes or categories—within categorical propositions and syllogisms.[1] It forms the core of Aristotle's logical framework as presented in his Organon, a collection of six treatises including the Categories, On Interpretation, and Prior Analytics, where he defines a syllogism as "an argument in which, certain things having been laid down, something different follows of necessity."[2] Central to term logic are categorical propositions, classified into four types: universal affirmative (A: "All S are P"), universal negative (E: "No S are P"), particular affirmative (I: "Some S are P"), and particular negative (O: "Some S are not P"), each linking a subject term (S) to a predicate term (P) via a copula.[1] Syllogisms, the primary inferential tool, consist of two premises and a conclusion involving three terms (major, minor, and middle), yielding 192 possible moods across three figures in Aristotle's system, of which he identified 14 valid non-modal forms; the syllogistic was later expanded to four figures, increasing possibilities to 256 moods and 24 valid forms overall, using methods like conversion and reductio ad impossibile.[2] Aristotle's term logic emphasized existential assumptions in propositions, presupposing the existence of the subjects discussed, and extended to modal variants incorporating notions of necessity and possibility, as explored in Prior Analytics 1.8–22.[1] Historically, it was refined by Aristotle's successors, the Peripatetics like Theophrastus and Eudemus, who expanded the syllogistic to include more figures and modal combinations, and later synthesized by figures such as Galen in the 2nd century CE before being preserved through medieval commentators like Boethius.[1] Dominant in Western philosophy from antiquity through the Middle Ages and into the early modern period, term logic provided the foundational model for evaluating deductive arguments until the 19th century, when it was largely supplanted by modern predicate and propositional logics developed by thinkers like George Boole and Gottlob Frege.[2] Despite its limitations—such as difficulties handling relational statements, non-existent entities (e.g., "All unicorns are magical"), or complex quantifiers—term logic remains influential in understanding classical reasoning structures and continues to inform philosophical and linguistic analyses.[1]

Historical Development

Aristotle's System

Aristotle developed the foundational system of term logic primarily in his Prior Analytics, composed around 350 BCE, where he formalized deductive reasoning through categorical syllogisms as a method for deriving necessary conclusions from given premises.[3] In this work, Aristotle presented logic as a tool for scientific demonstration, emphasizing the structure of arguments that connect general categories of things to yield valid inferences.[2] At the core of Aristotle's system is the idea of reasoning from premises that classify subjects and predicates into categorical statements, such as "all S are P" or "some S are not P," leading to conclusions that follow necessarily from these relations.[3] This approach treats terms as the basic units representing classes or categories, allowing deductions to proceed without reference to individual instances beyond their categorical membership.[2] Propositions serve as the building blocks in these syllogisms, expressing relations between such terms.[3] Aristotle also outlined immediate inferences, which derive new propositions directly from a single premise without additional terms, serving as essential precursors to more complex syllogistic deductions.[2] For instance, conversion involves swapping the subject and predicate terms, as in transforming "no S is P" to "no P is S," while obversion changes the quality and quantity by negating the predicate, such as converting "all S are P" to "no S are non-P."[2] These operations, discussed in the Prior Analytics, enable the manipulation of categorical statements to reveal equivalences and support broader argumentative structures.[3] This system forms a key part of Aristotle's broader Organon, a collection of six logical treatises that together provide the instruments for rational inquiry, distinguishing the deductive focus of the Prior Analytics from the topical and interpretive logics in works like the Topics and Sophistical Refutations.[2] The Organon reflects Aristotle's view of logic not as an end in itself but as preparatory for philosophical and scientific investigation, marking the first systematic treatment of inference in Western thought.[3]

Decline of Term Logic

The decline of term logic began in the 16th and 17th centuries, coinciding with the rise of empirical science that prioritized inductive methods over deductive syllogistics. Francis Bacon, in his Novum Organum (1620), mounted a significant critique of Aristotelian logic, arguing that the syllogism, rooted in deduction from presumed universals, hindered scientific progress by failing to generate new knowledge from observation; instead, he advocated induction as the path to reliable discoveries about nature.[4] This shift reflected broader humanist and scientific movements that viewed traditional term logic as insufficient for empirical inquiry, favoring approaches that built generalizations from particulars rather than assuming them a priori.[4] Contributing to this erosion were reformist logics like Ramism and the Port-Royal Logic, which critiqued Aristotelian syllogistics for its excessive formality and detachment from everyday language. Petrus Ramus (1515–1572) simplified logic by reducing Aristotle's complex Organon to dichotomous diagrams and emphasizing dialectic over intricate categorical rules, portraying traditional syllogisms as overly scholastic and impractical for natural reasoning.[5] Similarly, the Logic or the Art of Thinking (1662) by Antoine Arnauld and Pierre Nicole, associated with Port-Royal, streamlined syllogistic theory by reinterpreting propositions in terms of idea extensions and dismissing much of the traditional apparatus as superfluous or erroneous, thereby aligning logic more closely with Cartesian clarity and vernacular use.[6] These innovations rendered term logic appear antiquated and rigid, accelerating its marginalization in philosophical and educational discourse. By the 19th century, term logic had largely faded from university curricula, supplanted by algebraic logics that treated reasoning mathematically. George Boole's The Mathematical Analysis of Logic (1847) initially critiqued Aristotelian syllogistics as limited in scope and incapable of handling quantitative relations or probabilities, proposing instead an algebra of classes that quantified logical operations; though Boole later incorporated some syllogistic elements, his framework marked a pivotal departure toward symbolic and extensional methods.[7] This replacement culminated in the late 19th century with the mathematization of logic, as seen in works by Gottlob Frege, which definitively sidelined categorical syllogisms in favor of predicate calculus.[8] Cultural factors during the Enlightenment further diminished term logic's prominence, as intellectual emphasis shifted toward mathematics and probability theory to model uncertainty and rationality. Thinkers like Blaise Pascal and Pierre-Simon Laplace developed probability as a calculus of belief, viewing it as a superior tool for decision-making under incomplete information compared to the absolute categoricals of Aristotelian deduction; this mathematical orientation, encapsulated in Lorraine Daston's analysis of classical probability, prioritized empirical quantification over qualitative syllogistic forms, reshaping philosophical inquiry away from medieval traditions.[9]

Revival in the Modern Era

In the 19th century, interest in term logic experienced a notable resurgence in Britain, driven by efforts to reform and defend Aristotelian syllogistic against the rise of symbolic and algebraic alternatives. Sir William Hamilton, a Scottish philosopher, advanced this revival by introducing the "quantification of the predicate," which extended traditional categorical propositions (e.g., reformulating "All A is B" as "All A is some B") to address limitations in Aristotelian forms while preserving their conceptual core.[10] Similarly, Augustus De Morgan contributed through his 1847 work Formal Logic, where he dissected syllogistic components into generalized relations, incorporating symbolic notation to enhance rather than replace term-based reasoning, thereby bridging traditional logic with emerging mathematical approaches.[10] These initiatives reflected a broader motivation to maintain the intuitive, category-centered structure of Aristotelian logic amid criticisms from figures like George Boole. In the early 20th century, Polish logician Jan Ɓukasiewicz provided a rigorous formalization of Aristotelian syllogistics, treating it as an axiomatic system within modern formal logic. He employed propositional variables to represent moods (A, E, I, O) and developed axioms such as ⊱Aaa\vdash Aaa (universal affirmative reflexivity) and rules for inference, demonstrating the system's completeness for assertoric syllogisms while identifying its limitations relative to predicate calculus.[11] This work, building on his earlier contributions to propositional logic, aimed to clarify Aristotle's original intent through precise mathematical reconstruction. Phenomenology and analytic philosophy further influenced the revival by appreciating term logic's emphasis on category-based structures. Edmund Husserl, in his Logical Investigations (1900–1901), developed a formal ontology of meanings as species, distinguishing categorial forms (e.g., object, relation, unity) that underpin logical judgments and align with term logic's subject-predicate framework, viewing them as ideal, non-psychological necessities for valid inference.[12] This categorial approach reinforced term logic's role in analyzing conceptual relations, influencing later analytic thinkers who valued its focus on intuitive categories over purely extensional models. By mid-century, term logic found applications in linguistics and cognitive science, where syllogisms served as models for natural reasoning processes. In linguistics, P.F. Strawson's 1952 Introduction to Logical Theory defended traditional syllogistics against formalist critiques, arguing that its forms better capture everyday language inferences involving quantifiers and categories.[13] In cognitive psychology, emerging studies from the 1950s onward, such as those examining errors in syllogistic tasks, treated Aristotelian forms as proxies for mental simulation of relations, laying groundwork for theories like mental models that explain human deductive performance beyond strict logical validity.[14] Ɓukasiewicz's 1951 monograph Aristotle's Syllogistic from the Standpoint of Modern Formal Logic encapsulated this era's synthesis, axiomatizing the system to support interdisciplinary analyses of reasoning.[15]

Core Components

Terms

In term logic, a term is defined as a categorematic expression—a word or phrase that independently signifies a concept or denotes a class of things, serving as the fundamental unit for constructing categorical propositions.[16] For example, the term "man" refers to the class encompassing all human males, while "animal" denotes a broader category of living beings.[16] These terms provide the semantic content that allows for assertions about relationships between classes. Within propositions, terms are distinguished by their roles: the subject term is the class about which a predication is made, and the predicate term is the class attributed to or denied of the subject.[16] In the proposition "All men are mortal," "men" functions as the subject term, identifying the class under consideration, and "mortal" serves as the predicate term, describing a property applied to that class. This distinction is essential for analyzing how classes interact in logical judgments. Terms are categorized into categorematic and syncategorematic types based on their semantic independence. Categorematic terms, such as "human" or "planet," possess standalone meaning and can directly function as subjects or predicates in propositions.[17] In contrast, syncategorematic terms, including quantifiers like "all," "some," or "no" and modifiers like "not," lack independent signification and instead modify or connect categorematic terms to form complete expressions.[17] Additionally, terms are used in universal or particular scopes: a universal term, as in "all humans," applies to the entire class, implying full distribution, whereas a particular term, as in "some animals," applies to an indefinite subset, indicating partial distribution.[16] The square of opposition elucidates relations among terms by demonstrating logical incompatibilities and implications in the propositions they form. Contradictory terms, such as a class and its direct negation (e.g., "humans" and "non-humans"), generate propositions that cannot both be true or both false, as seen in the contradictory opposition between universal affirmatives and particular negatives sharing the same subject and predicate terms.[18] This framework highlights how term combinations yield exhaustive and exclusive relations, underpinning the validity of inferences in term logic.

Propositions

In term logic, propositions are declarative statements composed of a subject term, a copula (linking verb such as "is" or "is not"), and a predicate term, asserting a relationship between classes denoted by the terms.[3] These categorical propositions form the premises and conclusions of syllogisms, focusing on inclusion or exclusion between categories without reference to time, modality, or conditionals.[2] The four basic forms of categorical propositions, as outlined by Aristotle in the Prior Analytics, are classified along two axes: quantity (universal or particular) and quality (affirmative or negative).[3] The universal affirmative (A) states "All S are P," indicating that every member of the subject class S belongs to the predicate class P; for example, "All humans are mortal."[2] The universal negative (E) asserts "No S are P," meaning no member of S belongs to P, such as "No humans are immortal."[2] The particular affirmative (I) claims "Some S are P," where at least one member of S belongs to P, like "Some humans are philosophers."[2] Finally, the particular negative (O) declares "Some S are not P," indicating that at least one member of S does not belong to P, for instance, "Some humans are not philosophers."[2] Quantity determines the scope of the subject term: universal propositions (A and E) apply to the entire class, while particular ones (I and O) apply to part of it.[19] Quality distinguishes affirmative propositions (A and I), which assert inclusion, from negative ones (E and O), which assert exclusion.[19] This classification enables systematic analysis of logical relations among propositions. A key concept in evaluating propositions is the distribution of terms, which identifies whether a term refers to all (distributed) or only some (undistributed) members of its class.[19] In A propositions, the subject is distributed, but the predicate is not; in E, both are distributed; in I, neither is; and in O, the subject is not, but the predicate is.[19] This distribution affects validity in inferences, as undistributed terms cannot be assumed to apply universally.[19] Conversion rules allow interchanging the subject and predicate terms while preserving truth, providing a method to derive equivalent propositions.[3] E propositions convert directly to E (e.g., "No S are P" becomes "No P are S"); I to I (e.g., "Some S are P" to "Some P are S"); and A to I (e.g., "All S are P" to "Some P are S," known as simple or accidental conversion).[3] Direct conversion is invalid for O propositions, as "Some S are not P" does not logically yield "Some P are not S."[2]
FormQuantityQualityExampleSubject DistributedPredicate Distributed
AUniversalAffirmativeAll S are PYesNo
EUniversalNegativeNo S are PYesYes
IParticularAffirmativeSome S are PNoNo
OParticularNegativeSome S are not PNoYes

Singular Terms

In term logic, singular terms are expressions that refer to unique individuals, such as proper names like "Socrates" or demonstrative phrases like "this man," distinguishing them from general terms that apply to classes or multiples.[3] These terms primarily serve as subjects in propositions, where they combine with a copula and a universal predicate to form statements about the individual.[2] Aristotle treats singular propositions—affirmations or denials of a predicate of a singular subject—as analogous to universal categorical propositions in terms of logical structure and rules.[3] For instance, a singular affirmative proposition like "Socrates is wise" functions like a universal affirmative (A-form), while "Socrates is not wise" resembles a universal negative (E-form).[2] In this adaptation, the singular subject is considered fully distributed, meaning the predicate is affirmed or denied of the entire, indivisible referent, much as "all" distributes over a universal subject.[3] There are no genuine particular forms (I or O) for singulars, as the subject comprises only one entity, rendering "some" or "some not" inapplicable without altering the proposition's meaning.[2] This treatment allows singular propositions to participate in syllogistic inferences by substituting for universal premises. For example, from the singular affirmative "Socrates is a man" (treated as universally distributing the subject) and the universal affirmative "All men are mortal," one validly infers the singular affirmative "Socrates is mortal" via the first figure syllogism (Barbara).[3] However, singular terms and propositions face limitations in Aristotelian logic, particularly in demonstrative contexts. In the Posterior Analytics, Aristotle argues that scientific knowledge and demonstration require necessary, universal truths about essences, excluding singular propositions about contingent individuals, which cannot yield eternal or general explanations.[3] Thus, while singulars fit within assertoric syllogistics for everyday reasoning, they do not support the apodeictic demonstrations central to Aristotle's epistemology.[3]

Syllogistic Structure

Figures of the Syllogism

In term logic, a syllogism constitutes a deductive argument comprising two premises and a conclusion, wherein the premises share a common term known as the middle term, which facilitates the inference of the conclusion through necessary consequence.[20] This structure, as articulated by Aristotle, ensures that the conclusion follows inescapably from the premises without requiring additional assumptions.[3] Aristotle delineates three primary figures of the syllogism in his Prior Analytics, distinguished by the positional arrangement of the middle term relative to the other terms in the premises.[20] The first figure positions the middle term as the subject in the major premise (which concerns the predicate of the conclusion) and as the predicate in the minor premise (which concerns the subject of the conclusion).[20] In this configuration, the middle term directly connects the extremes, forming a chain-like relation that yields the most straightforward deductions.[3] The second figure arranges the middle term as the predicate in both premises, placing it outside the extremes and requiring conversion of one premise to establish the connection.[20] Here, the middle term serves to contrast or exclude relations between the major and minor terms, often highlighting incompatibilities.[3] In the third figure, the middle term functions as the subject in both premises, again positioned outside the extremes but concluding with a particular relation between them.[20] This arrangement typically results in conclusions about partial inclusions or exclusions, emphasizing the middle term's role in aggregating properties.[3] Throughout these figures, the middle term acts as the pivotal link between the major term (the predicate of the conclusion) and the minor term (the subject of the conclusion), enabling the transfer of attributes across the premises to necessitate the conclusion.[20] To denote these elements abstractly, logicians employ the notation M for the middle term, S for the minor term, and P for the major term, facilitating analysis of the syllogistic forms.[3]

Syllogisms in the First Figure

In the first figure of the syllogism, the middle term functions as the subject of the major premise and the predicate of the minor premise, creating a direct progression from a general statement about the middle term to a more specific one involving the minor term. This configuration, as described by Aristotle in Prior Analytics Book I, Chapter 4, facilitates immediate and intuitive inference, where the major premise establishes a universal relation between the middle and major terms, and the minor premise links the subject term to the middle.[20] Aristotle identified four valid moods in this figure, which he deemed "perfect" because their conclusions follow evidently from the premises without requiring reduction to other forms.[21] The valid moods, later assigned medieval mnemonic names to aid memorization, are Barbara (AAA), Celarent (EAE), Darii (AII), and Ferio (EIO). In Barbara, both premises and the conclusion are universal affirmatives: "All humans are mortal" (major: All M are P); "All Greeks are humans" (minor: All S are M); therefore, "All Greeks are mortal" (All S are P). Celarent features a universal negative major, universal affirmative minor, and universal negative conclusion: "No reptiles are warm-blooded" (No M are P); "All snakes are reptiles" (All S are M); therefore, "No snakes are warm-blooded" (No S are P). Darii involves a universal affirmative major, particular affirmative minor, and particular affirmative conclusion: "All birds are animals" (All M are P); "Some sparrows are birds" (Some S are M); therefore, "Some sparrows are animals" (Some S are P). Ferio consists of a universal negative major, particular affirmative minor, and particular negative conclusion: "No reptiles are birds" (No M are P); "Some lizards are reptiles" (Some S are M); therefore, "Some lizards are not birds" (Some S are not P). These moods exhaust the valid combinations in the first figure, as confirmed in Aristotelian analysis.[22][23] For syllogisms in the first figure to yield valid conclusions, specific rules must hold: the major premise must be universal to ensure the major term is distributed, preventing undistributed major term fallacies; the minor premise must be affirmative to maintain positive linkage through the middle term; and the middle term must be distributed in at least one premise to avoid undistributed middle errors. These conditions stem directly from Aristotle's exposition, ensuring the syllogism's deductive force without additional assumptions.[24] Violations, such as a particular major premise, render the mood invalid, as the inference cannot guarantee the conclusion's scope. The first figure's "perfect" status lies in its alignment with natural reasoning patterns, where universals precede particulars, making it foundational for deriving conclusions in term logic.[21]

Syllogisms in the Second Figure

In the second figure of the syllogism, the middle term serves as the predicate in both premises, linking the major term (predicate of the conclusion) in the first premise and the minor term (subject of the conclusion) in the second premise.[3] The valid moods in this figure are Cesare (EAE), Camestres (AEE), Festino (EIO), and Baroco (AOO).[3][25] Cesare takes the form: No M are P; All S are M; therefore, No S are P. For example, No reptiles are mammals; All snakes are reptiles; therefore, No snakes are mammals.[25][3] Camestres takes the form: All M are P; No S are M; therefore, No S are P. For example, All dogs are mammals; No fish are dogs; therefore, No fish are mammals.[25][3] Festino takes the form: No M are P; Some S are M; therefore, Some S are not P. For example, No birds are mammals; Some penguins are birds; therefore, Some penguins are not mammals.[25][3] Baroco takes the form: All M are P; Some S are not M; therefore, Some S are not P. For example, All wise men are learned; Some men are not wise; therefore, Some men are not learned.[25][3] For validity in the second figure, at least one premise must be negative, and the middle term must be distributed in the negative premise (appearing as the predicate of a universal negative or particular negative proposition).[3][25] These moods yield only negative conclusions and are considered imperfect by Aristotle, requiring reduction to the first figure through operations such as conversion (reversing subject and predicate) or obversion (changing quality while altering the predicate to its complement) to demonstrate their validity.[3][25]

Syllogisms in the Third Figure

In the third figure of the syllogism, the middle term serves as the subject in both premises, with the major premise linking the middle term to the major term (as predicate) and the minor premise linking the minor term (as subject) to the middle term (as predicate). This arrangement, as described in Aristotle's Prior Analytics, produces conclusions relating the minor and major terms, typically in particular form.[20][23] The valid moods of the third figure, using traditional medieval mnemonic names and vowel notations (where A denotes universal affirmative, E universal negative, I particular affirmative, and O particular negative), are as follows:
MoodMajor PremiseMinor PremiseConclusionExample
Darapti (AAI)All M are PAll S are MSome S are PAll metals are elements; All gold is metal; therefore, some gold is elements.[23]
Disamis (IAI)Some M are PAll S are MSome S are PSome metals are conductors; All copper is metal; therefore, some copper is conductors.[23]
Datisi (AII)All M are PSome S are MSome S are PAll metals are elements; Some gold is metal; therefore, some gold is elements.[23]
Felapton (EAO)No M are PAll S are MSome S are not PNo fish are mammals; All tuna are fish; therefore, some tuna are not mammals.[23]
Bocardo (OAO)Some M are not PAll S are MSome S are not PSome animals are not rational; All humans are animals; therefore, some humans are not rational.[23]
Ferison (EIO)No M are PSome S are MSome S are not PNo reptiles are warm-blooded; Some lizards are reptiles; therefore, some lizards are not warm-blooded.[23]
These moods encompass the six valid forms recognized by Aristotle.[20][23] Key rules for validity in the third figure include requiring the major premise to be universal in moods yielding stronger existential inferences, such as Darapti or Felapton, and ensuring at least one premise is particular to avoid invalid universal conclusions, as the structure inherently limits outcomes to particulars.[23] The middle term must also be distributed in at least one premise to connect the extremes properly.[20] These syllogisms characteristically yield particular conclusions, either affirmative or negative, making them particularly suited for establishing existential claims about the distribution of properties among classes rather than universal generalizations.[23]

The Fourth Figure

The fourth figure of the syllogism features the middle term functioning as the subject of the major premise and the predicate of the minor premise, resulting in a structure where the major premise connects the middle term to the major term (M–P) and the minor premise connects the minor term to the middle term (S–M), yielding a conclusion relating the minor to the major term (S–P).[26] This arrangement differs from the first three figures recognized by Aristotle, as it reverses the typical flow of predication in a way that often requires indirect reasoning.[27] Historically, the fourth figure was not part of Aristotle's original system, which explicitly limited syllogisms to three figures in the Prior Analytics (I.23, 41a13–18), viewing such forms as reducible to the first figure through conversion or transposition of premises rather than as a distinct category.[27] Its formal introduction is attributed to Theophrastus, Aristotle's successor at the Lyceum, who expanded the syllogistic framework, though some sources credit Galen with its development in the second century AD; however, Galen himself rejected the fourth figure, insisting that valid syllogisms could only be constructed in the three figures outlined by Aristotle (Institutio Logica, p. 43).[26][26] Medieval logicians later incorporated it fully, assigning mnemonic names to its moods, but it remained controversial and was frequently reduced to the first figure for validation.[28] The valid moods of the fourth figure, as established in traditional syllogistic analysis, are five in number: Bramantip (AAI), Camenes (AEE), Dimaris (IAI), Fesapo (EAO), and Fresison (EIO). These moods adhere to the rules of syllogistic validity, including the requirement that at least one premise be affirmative and the proper distribution of terms, but they do not produce universal affirmative conclusions.[26] For instance, in Fesapo (EAO), the major premise is negative universal ("No M are P"), the minor premise is affirmative universal ("All S are M"), and the conclusion is negative particular ("Some S are not P"), as exemplified by: No fish are mammals; All tuna are fish; therefore, some tuna are not mammals.[29] This figure is considered weaker than the first because its conclusions are invariably particular or negative, lacking the strength of universal affirmatives derivable in the primary figures, and it relies on existential import in the minor premise to avoid fallacies of illicit processes.[27] Consequently, fourth-figure syllogisms were often dismissed or subordinated in classical treatments, serving more as pedagogical tools than foundational forms.[28]

Valid Forms and Analysis

Table of Valid Syllogisms

In term logic, the valid syllogisms are traditionally enumerated as 24 moods distributed across the four figures under the Aristotelian interpretation, which assumes existential import for universal propositions (A and E types).[30] These moods combine the four categorical proposition types—A (universal affirmative: All S is P), E (universal negative: No S is P), I (particular affirmative: Some S is P), and O (particular negative: Some S is not P)—into sequences of three letters denoting the major premise, minor premise, and conclusion, respectively.[3] Medieval logicians developed a mnemonic system to aid memorization, where each valid mood receives a name with vowels corresponding to the proposition types (A, E, I, O) in order, and consonants indicating the reduction method to a first-figure syllogism (e.g., "b" for Barbara suggests no reduction, while "c" may denote conversion). For instance, "Barbara" names the AAA mood in the first figure.[3] The following table lists all 24 valid moods for quick reference, including the figure, mnemonic, mood designation, example premises and conclusion using standard term positions (M for middle term, P for major/predicate term, S for minor/subject term), and brief validity notes. Premises follow the convention of major premise first.[30]
FigureMnemonicMoodPremises and ConclusionValidity Notes
1BarbaraAAAMajor: All M are P (A)
Minor: All S are M (A)
Conclusion: All S are P (A)		Perfect (direct, no reduction)
1CelarentEAEMajor: No M are P (E)
Minor: All S are M (A)
Conclusion: No S are P (E)		Perfect (direct, no reduction)
1DariiAIIMajor: All M are P (A)
Minor: Some S are M (I)
Conclusion: Some S are P (I)		Perfect (direct, no reduction)
1FerioEIOMajor: No M are P (E)
Minor: Some S are M (I)
Conclusion: Some S are not P (O)		Perfect (direct, no reduction)
1BarbariAAIMajor: All M are P (A)
Minor: All S are M (A)
Conclusion: Some S are P (I)		Subaltern of Barbara
1CelarontEAOMajor: No M are P (E)
Minor: All S are M (A)
Conclusion: Some S are not P (O)		Subaltern of Celarent
2CesareEAEMajor: No P are M (E)
Minor: All S are M (A)
Conclusion: No S are P (E)		Reduced via conversion to Celarent
2CamestresAEEMajor: All P are M (A)
Minor: No S are M (E)
Conclusion: No S are P (E)		Reduced via conversion to Celarent
2FestinoEIOMajor: No P are M (E)
Minor: Some S are M (I)
Conclusion: Some S are not P (O)		Reduced via conversion to Ferio
2BarocoAOOMajor: All P are M (A)
Minor: Some S are not M (O)
Conclusion: Some S are not P (O)		Reduced via obversion and conversion to Barbara
2CamestrosAEOMajor: All P are M (A)
Minor: No S are M (E)
Conclusion: Some S are not P (O)		Subaltern of Camestres
2CesaroEAOMajor: No P are M (E)
Minor: All S are M (A)
Conclusion: Some S are not P (O)		Subaltern of Cesare
3DaraptiAAIMajor: All M are P (A)
Minor: All M are S (A)
Conclusion: Some S are P (I)		Reduced via existential import to Darii
3FelaptonEAOMajor: No M are P (E)
Minor: All M are S (A)
Conclusion: Some S are not P (O)		Reduced via conversion to Celarent
3DisamisIAIMajor: Some M are P (I)
Minor: All M are S (A)
Conclusion: Some S are P (I)		Reduced via conversion to Darii
3DatisiAIIMajor: All M are P (A)
Minor: Some M are S (I)
Conclusion: Some S are P (I)		Reduced via conversion to Darii
3BocardoOAOMajor: Some M are not P (O)
Minor: All M are S (A)
Conclusion: Some S are not P (O)		Reduced via obversion to Ferio
3FerisonEIOMajor: No M are P (E)
Minor: Some M are S (I)
Conclusion: Some S are not P (O)		Reduced via conversion to Ferio
4BamalipAAIMajor: All P are M (A)
Minor: All M are S (A)
Conclusion: Some S are P (I)		Reduced via conversion to Darapti
4CamenesAEEMajor: All P are M (A)
Minor: No M are S (E)
Conclusion: No S are P (E)		Reduced via conversion to Camestres
4DimarisIAIMajor: Some P are M (I)
Minor: All M are S (A)
Conclusion: Some S are P (I)		Reduced via conversion to Disamis
4FesapoEAOMajor: No P are M (E)
Minor: All M are S (A)
Conclusion: Some S are not P (O)		Reduced via conversion to Felapton
4FresisonEIOMajor: No P are M (E)
Minor: Some M are S (I)
Conclusion: Some S are not P (O)		Reduced via conversion to Ferison
4CalemosAEOMajor: All P are M (A)
Minor: No M are S (E)
Conclusion: Some S are not P (O)		Subaltern of Camenes
Syllogisms in the second, third, and fourth figures are imperfect and validated by reduction to the perfect moods of the first figure (Barbara and Celarent) through logical operations such as conversion (interchanging subject and predicate while preserving truth for A, E, and I propositions), obversion (altering affirmative to negative or vice versa while changing the predicate to its complement), or reductio ad impossibile (assuming the opposite conclusion to derive a contradiction).[3] This reduction process confirms their equivalence to first-figure forms without altering the underlying validity.[30]

Boole's Acceptance of Aristotle

George Boole, in his 1847 pamphlet The Mathematical Analysis of Logic, initially critiqued Aristotelian syllogistic logic for its narrow focus on enumerating valid forms without a deeper mathematical foundation, yet he ultimately endorsed its core principles by integrating them into his emerging algebraic system of classes.[31][32] Boole viewed Aristotle's logic as a valid but incomplete framework, capable of extension through symbolic algebra, stating that it represented "a collection of scientific truths" worthy of mathematical elaboration.[33] This partial acceptance marked a pivotal shift, as Boole demonstrated that syllogisms could be rigorously analyzed and generalized within his class algebra, bridging classical term logic with modern formal methods.[31] Central to Boole's reinterpretation was the translation of Aristotelian terms into algebraic representations of classes or sets, where the universe is denoted by 1 and the empty class by 0.[32] A universal affirmative proposition like "All S are P" is expressed as the equation $ x(1 - y) = 0 $, where $ x $ symbolizes the class S and $ y $ the class P, indicating that no element of S falls outside P (i.e., $ S \subseteq P $).[32] Particular propositions, which carry existential import in Aristotelian logic, are handled via an auxiliary symbol $ v $ representing a non-zero portion of the universe; for instance, "Some S are P" becomes $ v = xy $, affirming the existence of overlap between the classes.[32] Boole retained Aristotle's assumption of existential import for universals, allowing particulars to follow without modern Boolean restrictions that deny such implications.[33] Boole's algebraic laws, such as the commutative property $ xy = yx $ for class intersections and the expansion of products to resolve combinations, were directly applied to syllogistic derivations, revealing their consistency with deductive reasoning.[32] For contradictions, where classes have no overlap, the law $ xy = 0 $ enforces mutual exclusion, enabling the elimination of intermediate terms in proofs.[32] This approach unified syllogistic forms under a single calculus, as seen in the Barbara syllogism (first figure, AAA): from premises "All M are P" as $ m(1 - p) = 0 $ and "All S are M" as $ s(1 - m) = 0 $, elimination yields "All S are P" as $ s(1 - p) = 0 $, or in set notation, $ (S \cap M) \subseteq P $ from the chained inclusions.[32] Boole's framework demonstrated the formal consistency of term logic with Boolean algebra, influencing subsequent developments by providing a mathematical basis that facilitated extensions into predicate logic and symbolic reasoning.[33][31] His work highlighted how Aristotelian syllogisms, when algebraized, could handle broader propositional structures, laying groundwork for modern logic's evolution beyond categorical forms.[33]

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  21. The essential and the derivative moods of Aristotelian syllogism
  22. [PDF] Aristotle's Syllogistic
  23. [PDF] Methodology of Syllogistics.
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  25. [PDF] The Fourth Figure in Aristotle* - DergiPark
  26. [PDF] Aristotle's Theory of the Assertoric Syllogism - PhilArchive
  27. ON THE FOURTH FIGURE OF THE SYLLOGISM
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  29. Classical Syllogisms - 1000-Word Philosophy: An Introductory ...
  30. George Boole - Stanford Encyclopedia of Philosophy
  31. [PDF] The Mathematical Analysis of Logic - Project Gutenberg
  32. [PDF] Aristotle's Prior Analytics and Boole's Laws of Thought - PhilArchive
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