# Syllogism

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A syllogism (Greek : συλλογισμός, syllogismos, 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.

## Contents

In its earliest form, defined by Aristotle, from the combination of a general statement (the major premise) and a specific statement (the minor premise), a conclusion is deduced. For example, knowing that all men are mortal (major premise) and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal. [1]

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. [2] From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably. This article is concerned only with this historical use. The syllogism was at the core of historical deductive reasoning, whereby facts are determined by combining existing statements, in contrast to inductive reasoning in which facts are determined by repeated observations.

Within an academic context, the syllogism was superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift (Concept Script; 1879). However, syllogisms remain useful in some circumstances, and for general-audience introductions to logic. [3] [4]

## Early history

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. [2]

### Aristotle

Aristotle defines the syllogism as "a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so." [5] Despite this very general definition, in Prior Analytics , Aristotle limits himself to categorical syllogisms that consist of three categorical propositions, including categorical modal syllogisms. [6]

The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle. Prior to the mid 12th century, medieval logicians were only familiar with a portion of Aristotle's works, including such titles as Categories and On Interpretation , works that contributed heavily to the prevailing Old Logic, or logica vetus . The onset of a New Logic, or logica nova , arose alongside the reappearance of Prior Analytics , the work in which Aristotle developed his theory of the syllogism.

Prior Analytics, upon rediscovery, was instantly regarded by logicians as "a closed and complete body of doctrine," leaving very little for thinkers of the day to debate and reorganize. Aristotle's theory on the syllogism for assertoric sentences was considered especially remarkable, with only small systematic changes occurring to the concept over time. This theory of the syllogism would not enter the context of the more comprehensive logic of consequence until logic began to be reworked in general in the mid 14th century by the likes of John Buridan.

Aristotle's Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise, that is, a premise containing the modal words 'necessarily', 'possibly', or 'contingently'. Aristotle's terminology, in this aspect of his theory, was deemed vague and in many cases unclear, even contradicting some of his statements from On Interpretation. His original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use and would be replaced by new distinctions and new theories altogether.

### Medieval syllogism

#### Boethius

Boethius (c. 475 – 526) contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the 12th century, his textbooks on the categorical syllogism were central to expanding the syllogistic discussion. Rather than in any additions that he personally made to the field, Boethius's logical legacy lies in his effective transmission of prior theories to later logicians, as well as his clear and primarily accurate presentations of Aristotle's contributions.

#### Peter Abelard

Another of medieval logic's first contributors from the Latin West, Peter Abelard (1079–1142), gave his own thorough evaluation of the syllogism concept and accompanying theory in the Dialectica—a discussion of logic based on Boethius's commentaries and monographs. His perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelard's distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a more coherent concept of Aristotle's modal syllogism model.

#### John Buridan

John Buridan (c. 1300 – 1361), whom some consider the foremost logician of the later Middle Ages, contributed two significant works: Treatise on Consequence and Summulae de Dialectica, in which he discussed the concept of the syllogism, its components and distinctions, and ways to use the tool to expand its logical capability. For 200 years after Buridan's discussions, little was said about syllogistic logic. Historians of logic have assessed that the primary changes in the post-Middle Age era were changes in respect to the public's awareness of original sources, a lessening of appreciation for the logic's sophistication and complexity, and an increase in logical ignorance—so that logicians of the early 20th century came to view the whole system as ridiculous. [7]

## Modern history

The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself is about drawing valid conclusions from assumptions (axioms), rather than about verifying the assumptions. However, people over time focused on the logic aspect, forgetting the importance of verifying the assumptions.

In the 17th century, Francis Bacon emphasized that experimental verification of axioms must be carried out rigorously, and cannot take syllogism itself as the best way to draw conclusions in nature. [8] Bacon proposed a more inductive approach to the observation of nature, which involves experimentation and leads to discovering and building on axioms to create a more general conclusion. [8] Yet, a full method of drawing conclusions in nature is not the scope of logic or syllogism.

In the 19th century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Immanuel Kant famously claimed, in Logic (1800), that logic was the one completed science, and that Aristotelian logic more or less included everything about logic that there was to know. (This work is not necessarily representative of Kant's mature philosophy, which is often regarded as an innovation to logic itself.) Although there were alternative systems of logic elsewhere, such as Avicennian logic or Indian logic, Kant's opinion stood unchallenged in the West until 1879, when Gottlob Frege published his Begriffsschrift (Concept Script). This introduced a calculus, a method of representing categorical statements (and statements that are not provided for in syllogism as well) by the use of quantifiers and variables.

A noteworthy exception is the logic developed in Bernard Bolzano's work Wissenschaftslehre (Theory of Science, 1837), the principles of which were applied as a direct critique of Kant, in the posthumously published work New Anti-Kant (1850). The work of Bolzano had been largely overlooked until the late 20th century, among other reasons, due to the intellectual environment at the time in Bohemia, which was then part of the Austrian Empire. In the last 20 years, Bolzano's work has resurfaced and become subject of both translation and contemporary study.

This led to the rapid development of sentential logic and first-order predicate logic, subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many.[ original research? ] The Aristotelian system is explicated in modern fora of academia primarily in introductory material and historical study.

One notable exception, to this modern relegation, is the continued application of Aristotelian logic by officials of the Congregation for the Doctrine of the Faith, and the Apostolic Tribunal of the Roman Rota, which still requires that any arguments crafted by Advocates be presented in syllogistic format.

### Boole's acceptance of Aristotle

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 . [9] [10] Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought . [11] 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: [11]

1. providing it with mathematical foundations involving equations;
2. extending the class of problems it could treat, as solving equations was added to assessing validity; and
3. expanding the range of applications it could handle, such as expanding propositions of 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 Aristotle's four propositional forms to one form, the form of equations, which by itself was 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."

## Basic structure

A categorical syllogism consists of three parts:

1. Major premise
2. Minor premise
3. Conclusion

Each part is a categorical proposition, and each categorical proposition contains two categorical terms. [12] In Aristotle, each of the premises is in the form "All A are B," "Some A are B", "No A are B" or "Some A are not B", where "A" is one term and "B" is another:

More modern logicians allow some variation. Each of the premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate of the conclusion); in a minor premise, this is the minor term (i.e., the subject of the conclusion). For example:

Major premise: All humans are mortal.
Minor premise: All Greeks are humans.
Conclusion: All Greeks are mortal.

Each of the three distinct terms represents a category. From the example above, humans, mortal, and Greeks: mortal is the major term, and Greeks the minor term. The premises also have one term in common with each other, which is known as the middle term; in this example, humans. Both of the premises are universal, as is the conclusion.

Major premise: All mortals die.
Minor premise: All men are mortals.
Conclusion: All men die.

Here, the major term is die, the minor term is men, and the middle term is mortals. Again, both premises are universal, hence so is the conclusion.

### Polysyllogism

A polysyllogism, or a sorites, is a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion. For example, one might argue that all lions are big cats, all big cats are predators, and all predators are carnivores. To conclude that therefore all lions are carnivores is to construct a sorites argument.

## Types

There are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes the form (note: M – Middle, S – subject, P – predicate.):

Major premise: All M are P.
Minor premise: All S are M.
Conclusion: All S are P.

The premises and conclusion of a syllogism can be any of four types, which are labeled by letters [13] as follows. The meaning of the letters is given by the table:

codequantifiersubjectcopulapredicatetypeexample
AAllSarePuniversal affirmativeAll humans are mortal.
ENoSarePuniversal negativeNo humans are perfect.
ISomeSarePparticular affirmativeSome humans are healthy.
OSomeSare notPparticular negativeSome humans are not clever.

In Prior Analytics , Aristotle uses mostly the letters A, B, and C (Greek letters alpha , beta , and gamma ) as term place holders, rather than giving concrete examples. It is traditional to use is rather than are as the copula, hence All A is B rather than All As are Bs. It is traditional and convenient practice to use a, e, i, o as infix operators so the categorical statements can be written succinctly. The following table shows the longer form, the succinct shorthand, and equivalent expressions in predicate logic:

FormShorthandPredicate logic
All A is BAaB${\displaystyle \forall x(A(x)\rightarrow B(x))}$ or ${\displaystyle \neg \exists x(A(x)\land \neg B(x))}$
No A is BAeB${\displaystyle \neg \exists x(A(x)\land B(x))}$ or ${\displaystyle \forall x(A(x)\rightarrow \neg B(x))}$
Some A is BAiB${\displaystyle \exists x(A(x)\land B(x))}$
Some A is not BAoB${\displaystyle \exists x(A(x)\land \neg B(x))}$

The convention here is that the letter S is the subject of the conclusion, P is the predicate of the conclusion, and M is the middle term. The major premise links M with P and the minor premise links M with S. However, the middle term can be either the subject or the predicate of each premise where it appears. The differing positions of the major, minor, and middle terms gives rise to another classification of syllogisms known as the figure. Given that in each case the conclusion is S-P, the four figures are:

Figure 1 Figure 2 Figure 3 Figure 4 M–P P–M M–P P–M S–M S–M M–S M–S

(Note, however, that, following Aristotle's treatment of the figures, some logicians—e.g., Peter Abelard and John Buridan—reject the fourth figure as a figure distinct from the first.)

Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, though this makes no difference logically). Each premise and the conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogism BARBARA below is AAA-1, or "A-A-A in the first figure".

The vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms. Even some of these are sometimes considered to commit the existential fallacy, meaning they are invalid if they mention an empty category. These controversial patterns are marked in italics. All but four of the patterns in italics (felapton, darapti, fesapo and bamalip) are weakened moods, i.e. it is possible to draw a stronger conclusion from the premises.

Figure 1Figure 2Figure 3Figure 4
BarbaraCesareDatisiCalemes
CelarentCamestresDisamisDimatis
DariiFestinoFerisonFresison
FerioBarocoBocardoCalemos
BarbariCesaroFelaptonFesapo
CelarontCamestrosDaraptiBamalip

The letters A, E, I, and O have been used since the medieval Schools to form mnemonic names for the forms as follows: 'Barbara' stands for AAA, 'Celarent' for EAE, etc.

Next to each premise and conclusion is a shorthand description of the sentence. So in AAI-3, the premise "All squares are rectangles" becomes "MaP"; the symbols mean that the first term ("square") is the middle term, the second term ("rectangle") is the predicate of the conclusion, and the relationship between the two terms is labeled "a" (All M are P).

The following table shows all syllogisms that are essentially different. The similar syllogisms share the same premises, just written in a different way. For example "Some pets are kittens" (SiM in Darii) could also be written as "Some kittens are pets" (MiS in Datisi).

In the Venn diagrams, the black areas indicate no elements, and the red areas indicate at least one element. In the predicate logic expressions, a horizontal bar over an expression means to negate ("logical not") the result of that expression.

It is also possible to use graphs (consisting of vertices and edges) to evaluate syllogisms. [14]

### Examples

 M: menS: Greeks      P: mortal

#### Barbara (AAA-1)

All men are mortal. (MaP)
All Greeks are men. (SaM)
All Greeks are mortal. (SaP)
 M: reptileS: snake      P: fur

#### Celarent (EAE-1)

Similar: Cesare (EAE-2)

No reptile has fur. (MeP)
All snakes are reptiles. (SaM)
No snake has fur. (SeP)

 M: rabbitS: pet      P: fur

#### Darii (AII-1)

Similar: Datisi (AII-3)

All rabbits have fur. (MaP)
Some pets are rabbits. (SiM)
Some pets have fur. (SiP)

#### Ferio (EIO-1)

Similar: Festino (EIO-2), Ferison (EIO-3), Fresison (EIO-4)

No homework is fun. (MeP)
Some reading is not fun. (SoP)
 M: mammalS: pet      P: cat

#### Baroco (AOO-2)

All cats are mammals. (PaM)
Some pets are not mammals. (SoM)
Some pets are not cats. (SoP)
 M: catS: mammal      P: pet

#### Bocardo (OAO-3)

Some cats are not pets. (MoP)
All cats are mammals. (MaS)
Some mammals are not pets. (SoP)

 M: manS: Greek      P: mortal

#### Barbari (AAI-1)

All men are mortal. (MaP)
All Greeks are men. (SaM)
Some Greeks are mortal. (SiP)

 M: reptileS: snake      P: fur

#### Celaront (EAO-1)

Similar: Cesaro (EAO-2)

No reptiles have fur. (MeP)
All snakes are reptiles. (SaM)
Some snakes have no fur. (SoP)
 M: hoovesS: human      P: horse

#### Camestros (AEO-2)

Similar: Calemos (AEO-4)

All horses have hooves. (PaM)
No humans have hooves. (SeM)
Some humans are not horses. (SoP)
 M: flowerS: plant      P: animal

#### Felapton (EAO-3)

Similar: Fesapo (EAO-4)

No flowers are animals. (MeP)
All flowers are plants. (MaS)
Some plants are not animals. (SoP)
 M: squareS: rhomb      P: rectangle

#### Darapti (AAI-3)

All squares are rectangles. (MaP)
All squares are rhombuses. (MaS)
Some rhombuses are rectangles. (SiP)

### Table of all syllogisms

This table shows all 24 valid syllogisms, represented by Venn diagrams. Columns indicate similarity, and are grouped by combinations of premises. Borders correspond to conclusions. Those with an existential assumption are dashed.

figureA ∧ AA ∧ EA ∧ IA ∧ OE ∧ I
1
2
3
4

## Terms in syllogism

With Aristotle, we may distinguish singular terms, such as Socrates, and general terms, such as Greeks. Aristotle further distinguished types (a) and (b):

1. terms that could be the subject of predication; and
2. terms that could be predicated of others by the use of the copula ("is a").

Such a predication is known as a distributive, as opposed to non-distributive as in Greeks are numerous. It is clear that Aristotle's syllogism works only for distributive predication, since we cannot reason All Greeks are animals, animals are numerous, therefore all Greeks are numerous. In Aristotle's view singular terms were of type (a), and general terms of type (b). Thus, Men can be predicated of Socrates but Socrates cannot be predicated of anything. Therefore, for a term to be interchangeable—to be either in the subject or predicate position of a proposition in a syllogism—the terms must be general terms, or categorical terms as they came to be called. Consequently, the propositions of a syllogism should be categorical propositions (both terms general) and syllogisms that employ only categorical terms came to be called categorical syllogisms.

It is clear that nothing would prevent a singular term occurring in a syllogism—so long as it was always in the subject position—however, such a syllogism, even if valid, is not a categorical syllogism. An example is Socrates is a man, all men are mortal, therefore Socrates is mortal. Intuitively this is as valid as All Greeks are men, all men are mortal therefore all Greeks are mortals. To argue that its validity can be explained by the theory of syllogism would require that we show that Socrates is a man is the equivalent of a categorical proposition. It can be argued Socrates is a man is equivalent to All that are identical to Socrates are men, so our non-categorical syllogism can be justified by use of the equivalence above and then citing BARBARA.

## Existential import

If a statement includes a term such that the statement is false if the term has no instances, then the statement is said to have existential import with respect to that term. It is ambiguous whether or not a universal statement of the form All A is B is to be considered as true, false, or even meaningless if there are no As. If it is considered as false in such cases, then the statement All A is B has existential import with respect to A.

It is claimed Aristotle's logic system does not cover cases where there are no instances. Aristotle's goal was to develop "a companion-logic for science. He relegates fictions, such as mermaids and unicorns, to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is 'a phrase signifying a thing's essence.'... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." [15] However, many logic systems developed since do consider the case where there may be no instances.

However, medieval logicians were aware of the problem of existential import and maintained that negative propositions do not carry existential import, and that positive propositions with subjects that do not supposit are false.

The following problems arise:

1. (a) In natural language and normal use, which statements of the forms All A is B, No A is B, Some A is B and Some A is not B have existential import and with respect to which terms?
2. In the four forms of categorical statements used in syllogism, which statements of the form AaB, AeB, AiB and AoB have existential import and with respect to which terms?
3. What existential imports must the forms AaB, AeB, AiB and AoB have for the square of opposition to be valid?
4. What existential imports must the forms AaB, AeB, AiB and AoB have to preserve the validity of the traditionally valid forms of syllogisms?
5. Are the existential imports required to satisfy (d) above such that the normal uses in natural languages of the forms All A is B, No A is B, Some A is B and Some A is not B are intuitively and fairly reflected by the categorical statements of forms AaB, AeB, AiB and AoB?

For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:

"All flying horses are mythical" is false if there are no flying horses.
If "No men are fire-eating rabbits" is true, then "There are fire-eating rabbits" is true; and so on.

If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid (e.g. BaC,AaB->AiC).

These problems and paradoxes arise in both natural language statements and statements in syllogism form because of ambiguity, in particular ambiguity with respect to All. If "Fred claims all his books were Pulitzer Prize winners", is Fred claiming that he wrote any books? If not, then is what he claims true? Suppose Jane says none of her friends are poor; is that true if she has no friends?

The first-order predicate calculus avoids such ambiguity by using formulae that carry no existential import with respect to universal statements. Existential claims must be explicitly stated. Thus, natural language statements—of the forms All A is B, No A is B, Some A is B, and Some A is not B—can be represented in first order predicate calculus in which any existential import with respect to terms A and/or B is either explicit or not made at all. Consequently, the four forms AaB, AeB, AiB, and AoB can be represented in first order predicate in every combination of existential import—so it can establish which construal, if any, preserves the square of opposition and the validity of the traditionally valid syllogism. Strawson claims such a construal is possible, but the results are such that, in his view, the answer to question (e) above is no.

On the other hand, in modern mathematical logic, however, statements containing words "all", "some" and "no", can be stated in terms of set theory. If the set of all A's is labeled as s(A) and the set of all B's as s(B), then:

• "All A is B" (AaB) is equivalent to "s(A) is a subset of s(B)", or s(A) ⊆ s(B)
• "No A is B" (AeB) is equivalent to "The intersection of s(A) and s(B) is empty", or ${\displaystyle s(A)\cap s(B)=\emptyset }$
• "Some A is B" (AiB) is equivalent to "the intersection of s(A) and s(B) is not empty", or ${\displaystyle s(A)\cap s(B)\neq \emptyset }$
• "Some A is not B" (AoB) is equivalent to "s(A) is not a subset of s(B)"

By definition, the empty set is a subset of all sets. From this it follows that, according to this mathematical convention, if there are no A's, then the statements "All A is B" and "No A is B" are always true whereas the statements "Some A is B" and "Some A is not B" are always false. This, however, implies that AaB does not entail AiB, and some of the syllogisms mentioned above are not valid when there are no A's.

## Syllogistic fallacies

People often make mistakes when reasoning syllogistically. [16]

For instance, from the premises some A are B, some B are C, people tend to come to a definitive conclusion that therefore some A are C. [17] [18] However, this does not follow according to the rules of classical logic. For instance, while some cats (A) are black things (B), and some black things (B) are televisions (C), it does not follow from the parameters that some cats (A) are televisions (C). This is because in the structure of the syllogism invoked (i.e. III-1) the middle term is not distributed in either the major premise or in the minor premise, a pattern called the "fallacy of the undistributed middle".

Determining the validity of a syllogism involves determining the distribution of each term in each statement, meaning whether all members of that term are accounted for.

In simple syllogistic patterns, the fallacies of invalid patterns are:

## Related Research Articles

The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.

Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion.

In term logic, the square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle making the distinction between two oppositions: contradiction and contrariety. However, Aristotle did not draw any diagram. This was done several centuries later by Apuleius and Boethius.

Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word infer means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, contradistinguishing abduction from induction.

In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to logic that began with Aristotle and was developed further in ancient history mostly by his followers, the peripatetics, but largely fell into decline by the third century CE. 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, and remained dominant until the advent of modern predicate logic in the late nineteenth century. This entry is an introduction to the term logic needed to understand philosophy texts written before it was replaced as a formal logic system by predicate logic. Readers lacking a grasp of the basic terminology and ideas of term logic can have difficulty understanding such texts, because their authors typically assumed an acquaintance with term logic.

The fallacy of the undistributed middle is a formal fallacy that is committed when the middle term in a categorical syllogism is not distributed in either the minor premise or the major premise. It is thus a syllogistic fallacy.

The Prior Analytics is a work by Aristotle on deductive reasoning, known as his syllogistic, composed around 350 BCE. Being one of the six extant Aristotelian writings on logic and scientific method, it is part of what later Peripatetics called the Organon. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Jan Łukasiewicz of a revolutionary paradigm. His approach was replaced in the early 1970s in a series of papers by John Corcoran and Timothy Smiley—which inform modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009.

The fallacy of four terms is the formal fallacy that occurs when a syllogism has four terms rather than the requisite three. This form of argument is thus invalid.

In logic, a middle term is a term that appears in both premises but not in the conclusion of a categorical syllogism. Example:

The fallacy of exclusive premises is a syllogistic fallacy committed in a categorical syllogism that is invalid because both of its premises are negative.

Early Islamic law placed importance on formulating standards of argument, which gave rise to a "novel approach to logic" in Kalam . However, with the rise of the Mu'tazili philosophers, who highly valued Aristotle's Organon, this approach was displaced by the older ideas from Hellenistic philosophy. The works of al-Farabi, Avicenna, al-Ghazali and other Persian Muslim logicians who often criticized and corrected Aristotelian logic and introduced their own forms of logic, also played a central role in the subsequent development of European logic during the Renaissance.

In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category are included in another. The study of arguments using categorical statements forms an important branch of deductive reasoning that began with the Ancient Greeks.

In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that:

In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.

In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.

The False Subtlety of the Four Syllogistic Figures Proved is an essay published by Immanuel Kant in 1762.

A premise or premiss is a statement that an argument claims will induce or justify a conclusion. It is an assumption that something is true.

In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formulas. The validity of an argument—its being valid—can be tested, proved or disproved, and depends on its logical form.

Logic is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.

## References

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13. According to Copi, p. 127: 'The letter names are presumed to come from the Latin words "AffIrmo" and "nEgO," which mean "I affirm" and "I deny," respectively; the first capitalized letter of each word is for universal, the second for particular'
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