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Syllogism

Syllogism

Overview
A syllogism is a kind of logical argument in which one proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...

 (the conclusion) is inferred
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...

 from two or more others (the premise
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...

s) of a certain form. In antiquity, there were two rival theories of the syllogism: Aristotelian syllogistic and Stoic syllogistic.
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Aristotle's syllogism as simple as ABC by new Raval's notation
Encyclopedia
A syllogism is a kind of logical argument in which one proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...

 (the conclusion) is inferred
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...

 from two or more others (the premise
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...

s) of a certain form. In antiquity, there were two rival theories of the syllogism: Aristotelian syllogistic and Stoic syllogistic.

In the Prior Analytics
Prior Analytics
The Prior Analytics is Aristotle's work on deductive reasoning, specifically the syllogism. It is also part of his Organon, which is the instrument or manual of logical and scientific methods....

, Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

 defines the syllogism as "a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so." (24b18–20)

Despite this very general definition, in the Prior Analytics, Aristotle limits himself to categorical syllogisms, which consist of three categorical proposition
Categorical proposition
A categorical proposition contains two categorical terms, the subject and the predicate, and affirms or denies the latter of the former. Categorical propositions occur in categorical syllogisms and both are discussed in Aristotle's Prior Analytics....

s. These included categorical modal
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

 syllogisms. From the Middle Ages onwards, "categorical syllogism" and "syllogism" were mostly used interchangeably, and the present article is concerned with this traditional use of "syllogism" only. The syllogism was at the core of traditional deductive reasoning
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

, where facts are determined by combining existing statements, in contrast to inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...

 where facts are determined by repeated observations. The syllogism was superseded by first-order predicate logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

 following the work of Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...

, in particular his Begriffsschrift
Begriffsschrift
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...

 (Concept Script) (1879).

Basic structure


A categorical syllogism consists of three parts: the major premise, the minor premise and the conclusion.

Each part is a categorical proposition
Categorical proposition
A categorical proposition contains two categorical terms, the subject and the predicate, and affirms or denies the latter of the former. Categorical propositions occur in categorical syllogisms and both are discussed in Aristotle's Prior Analytics....

, and each categorical proposition contains two categorical terms. 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. "All A are B," and "No A are B" are termed universal propositions; "Some A are B" and "Some A are not B" are termed particular propositions. 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, it is the minor term (the subject) of the conclusion. For example:
Major premise: All men are mortal.
Minor premise: All Greeks are men.
Conclusion: All Greeks are mortal.


Each of the three distinct terms represents a category. In the above example, "men," "mortal," and "Greeks." "Mortal" is the major term; "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, "man." Both of the premises are universal, as is the conclusion.
Major premise: All mortals die.
Minor premise: Some men are mortals.
Conclusion: Some men die.


Here, the major term is "die", the minor term is "men," and the middle term is "mortals". The major premise is universal; the minor premise and the conclusion are particular.

A sorites
Polysyllogism
A polysyllogism is a string of any number of propositions forming together a sequence of syllogisms such that the conclusion of each syllogism, together with the next proposition, is a premise for the next, and so on...

 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, if one argues that a given number of grains of sand does not make a heap and that an additional grain does not either, then to conclude that no additional amount of sand will make a heap is to construct a sorites argument.

Types of syllogism



Although there are infinitely many possible syllogisms, there are only a finite number of logically distinct types. We shall classify and enumerate them below. Note that the syllogism above has the abstract form:
Major premise: All M are P.
Minor premise: All S are M.
Conclusion: All S are P.


(Note: M – Middle, S – subject, P – predicate. See below for more detailed explanation.)

The premises and conclusion of a syllogism can be any of four types, which are labeled by letters as follows. The meaning of the letters is given by the table:
code quantifier subject copula predicate type example
a All S are P universal affirmatives All humans are mortal.
e No S are P universal negatives No humans are perfect.
i Some S are P particular affirmatives Some humans are healthy.
o Some S are not P particular negatives Some humans are not clever.


In Analytics, Aristotle mostly uses the letters A, B and C (actually, the Greek letters alpha
Alpha
Alpha is the first letter of the Greek alphabet. Alpha or ALPHA may also refer to:-Science:*Alpha , the highest ranking individuals in a community of social animals...

, beta
Beta
Beta is the second letter of the Greek alphabet. Beta or BETA may also refer to:-Biology:*Beta , a genus of flowering plants, mostly referred to as beets*Beta, a rank in a community of social animals...

 and gamma
Gamma
Gamma is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. It was derived from the Phoenician letter Gimel . Letters that arose from Gamma include the Roman C and G and the Cyrillic letters Ge Г and Ghe Ґ.-Greek:In Ancient Greek, gamma represented a...

) as term place holders, rather than giving concrete examples, an innovation at the time. 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 to enable the categorical statements to be written succinctly thus:
Form Shorthand
All A is B AaB
No A is B AeB
Some A is B AiB
Some A is not B AoB


This particular syllogistic form is dubbed BARBARA (see below) and can be written neatly as BaC,AaB -> AaC.

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
Major premise: M–P P–M M–P P–M
Minor premise: S–M S–M M–S M–S


Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, although 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 above 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
Existential fallacy
The existential fallacy, or existential instantiation, is a logical fallacy in Boolean logic while it is not in Aristotelian logic. In an existential fallacy, we presuppose that a class has members even when we are not explicitly told so; that is, we assume that the class has existential import.An...

, meaning they are invalid if they mention an empty category. These controversial patterns are marked in italics.
Figure 1 Figure 2 Figure 3 Figure 4
Barbara Cesare Datisi Calemes
Celarent Camestres Disamis Dimatis
Darii Festino Ferison Fresison
Ferio Baroco Bocardo Calemos
Barbari Cesaro Felapton Fesapo
Celaront Camestros Darapti Bamalip


The letters A, E, I, O have been used since the medieval Schools
Scholasticism
Scholasticism is a method of critical thought which dominated teaching by the academics of medieval universities in Europe from about 1100–1500, and a program of employing that method in articulating and defending orthodoxy in an increasingly pluralistic context...

 to form mnemonic
Mnemonic
A mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...

 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 actually 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 is Datisi).

In the Venn diagrams the black areas tell, that there are no elements, and the red areas tell, that there is at least one element.

Examples



Barbara (AAA-1)

   All men are mortal. (MaP)
   All Greeks are men. (SaM)
∴
Therefore sign
In a mathematical proof, the therefore sign is a symbol that is sometimes placed before a logical consequence, such as the conclusion of a syllogism. The symbol consists of three dots placed in an upright triangle and is read therefore. It is encoded at . While it is not generally used in formal...

 All Greeks are mortal. (SaP)



Celarent (EAE-1)


Similar: Cesare (EAE-2)
   No reptiles have fur. (MeP)
   All snakes are reptiles. (SaM)
∴ No snakes have fur. (SeP)






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 homework. (SiM)
∴ Some reading is not fun. (SoP)



Baroco (AOO-2)

   All informative things are useful. (PaM)
   Some websites are not useful. (SoM)
∴ Some websites are not informative. (SoP)



Bocardo (OAO-3)

   Some cats have no tails. (MoP)
   All cats are mammals. (MaS)
∴ Some mammals have no tails. (SoP)



----

Barbari (AAI-1)

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






Celaront (EAO-1)


Similar: Cesaro (EAO-2)
   No reptiles have fur. (MeP)
   All snakes are reptiles. (SaM)
∴ Some snakes have no fur. (SoP)



Camestros (AEO-2)


Similar: Calemos (AEO-4)
   All horses have hooves. (PaM)
   No humans have hooves. (SeM)
∴ Some humans are not horses. (SoP)



Felapton (EAO-3)


Similar: Fesapo (EAO-4)
   No flowers are animals. (MeP)
   All flowers are plants. (MaS)
∴ Some plants are not animals. (SoP)



Darapti (AAI-3)

   All squares
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

 are rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

s. (MaP)
   All squares are rhombs
Rhombus
In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...

. (MaS)
∴ Some rhombs are rectangles. (SiP)



Table of all syllogisms


This table shows all 24 valid syllogisms, represented by Venn diagram
Venn diagram
Venn diagrams or set diagrams are diagrams that show all possible logical relations between a finite collection of sets . Venn diagrams were conceived around 1880 by John Venn...

s.

(9 of them, on the right side of the table, require that one category must not be empty.)

Syllogisms of the same type are in the same row, and very similar syllogisms are in the same column.
1
Barbara
Celarent
Darii
Ferio
Barbari
Celaront
2
Cesare
Camestres
Festino
Baroco
Cesaro
Camestros
3
Datisi
Disamis
Ferison
Bocardo
Felapton
Darapti
4
Calemes
Dimatis
Fresison
Calemos
Fesapo
Bamalip

Terms in syllogism


We may, with Aristotle, distinguish singular terms such as Socrates and general terms such as Greeks. Aristotle further distinguished (a) terms that could be the subject of predication, and (b) terms that could be predicated of others by the use of the copula (is are). (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 for 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 to enable a term to be interchangeable — that is 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 employing just 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, would not be a categorical syllogism. An example of such would be 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 it would be necessary to 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 so that the statement is false if the term has no instances (is not instantiated) then the statement is said to entail existential import with respect to that term. In particular, a universal statement of the form All A is B has existential import with respect to A if All A is B is false if there are no As.

The following problems arise:
(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?
(b) 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?
(c) What existential imports must the forms AaB, AeB, AiB and AoB have for the square of opposition be valid?
(d) What existential imports must the forms AaB, AeB, AiB and AoB to preserve the validity of the traditionally valid forms of syllogisms?
(e) 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 Ahab, Abe, Ail and Alb?


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 mythological" is false if there are not flying horses.

If "No men are fire-eating rabbits" is true, then "There are fire-eating rabbits" is false.


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 the problems of such ambiguity by using formulae that carry no existential import with respect to universal statements; existential claims have to 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 exactly represented in first order predicate calculus in which any existential import with respect to terms A and/or B is made explicitly 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 that it can establish which construal, if any, preserves the square of opposition and the validly of the traditionally valid syllogism. Strawson claims that such a construal is possible, but the results are such that, in his view, the answer to question (a) above is no.

Syllogism in the history of logic


Aristotelian syllogistic dominated Western philosophical thought from the 3rd Century to the 17th Century. At that time, Sir Francis Bacon rejected the idea of syllogism and deductive reasoning by asserting that it was fallible and illogical. Bacon offered a more inductive approach to logic in which experiments were conducted and axioms were drawn from the observations discovered in them.

In the 19th Century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Kant
Immanuel Kant
Immanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....

 famously claimed, in Logic (1800), that logic was the one completed science, and that Aristotelian logic more or less included everything about logic there was to know. Though it should be kept in mind that this work is not necessarily representative of Kant's mature philosophy, and that this philosophy is generally regarded as an innovation to logic itself. Though there were alternative systems of logic such as Avicennian logic
Logic in Islamic philosophy
Logic played an important role in Islamic philosophy .Islamic Logic or mantiq is similar science to what is called Traditional Logic in Western Sciences.- External links :*Routledge Encyclopedia of Philosophy: , Routledge, 1998...

 or Indian logic
Indian logic
The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...

 elsewhere, Kant's opinion stood unchallenged in the West until 1879 when 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.

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. 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
Congregation for the Doctrine of the Faith
The Congregation for the Doctrine of the Faith , previously known as the Supreme Sacred Congregation of the Roman and Universal Inquisition , and after 1904 called the Supreme...

, and the Apostolic Tribunal of the Roman Rota, which still requires that arguments crafted by Advocates be presented in syllogistic format.

Syllogistic Fallacies



People often make mistakes when reasoning syllogistically.

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. However, this does not follow according to the rules of classical logic. For instance, while some cats (A) are black (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 first, the mood of the syllogism invoked is illicit (III), and second, the supposition of the middle term is variable between that of the middle term in the major premise, and that of the middle term in the minor premise (not all "some" cats are by necessity of logic the same "some black things").

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:
  • Undistributed middle
    Fallacy of the undistributed middle
    The fallacy of the undistributed middle is a logical fallacy, and more specifically a formal fallacy, that is committed when the middle term in a categorical syllogism is not distributed in the major premise...

    : Neither of the premises accounts for all members of the middle term, which consequently fails to link the major and minor term.
  • Illicit treatment of the major term
    Illicit major
    Illicit major is a logical fallacy committed in a categorical syllogism that is invalid because its major term is undistributed in the major premise but distributed in the conclusion.This fallacy has the following argument form:#All A are B...

    : The conclusion implicates all members of the major term (P — meaning the proposition is negative); however, the major premise does not account for them all (i.e., P is either an affirmative predicate or a particular subject there).
  • Illicit treatment of the minor term
    Illicit minor
    Illicit minor is a logical fallacy committed in a categorical syllogism that is invalid because its minor term is undistributed in the minor premise but distributed in the conclusion....

    : Same as above, but for the minor term (S — meaning the proposition is universal) and minor premise (where S is either a particular subject or an affirmative predicate).
  • Exclusive premises
    Fallacy of exclusive premises
    The fallacy of exclusive premises is a syllogistic fallacy committed in a categorical syllogism that is invalid because both of its premises are negative.Example of an EOO-4 invalid proposition:...

    : Both premises are negative, meaning no link is established between the major and minor terms.
  • Affirmative conclusion from a negative premise
    Affirmative conclusion from a negative premise
    Affirmative conclusion from a negative premise is a logical fallacy that is committed when a categorical syllogism has a positive conclusion, but one or two negative premises.For example:...

    : If either premise is negative, the conclusion must also be.
  • Negative conclusion from affirmative premises
    Negative conclusion from affirmative premises
    Negative conclusion from affirmative premises is a syllogistic fallacy committed when a categorical syllogism has a negative conclusion yet both premises are affirmative...

    : If both premises are affirmative, the conclusion must also be.
  • Existential fallacy
    Existential fallacy
    The existential fallacy, or existential instantiation, is a logical fallacy in Boolean logic while it is not in Aristotelian logic. In an existential fallacy, we presuppose that a class has members even when we are not explicitly told so; that is, we assume that the class has existential import.An...

    : This is a more controversial one. If both premises are universal, i.e. "All" or "No" statements, one school of thought says they do not imply the existence of any members of the terms. In this case, the conclusion cannot be existential; i.e. beginning with "Some". Another school of thought says that affirmative statements (universal or particular) do imply the subject's existence, but negatives do not. A third school of thought says that the any type of proposition may or may not involve the subject's existence, and although this may condition the conclusion it does not affect the form of the syllogism.

See also


  • Enthymeme
    Enthymeme
    An enthymeme , in its modern sense, is an informally stated syllogism with an unstated assumption that must be true for the premises to lead to the conclusion. In an enthymeme, part of the argument is missing because it is assumed...

  • Other types of syllogism:
    • Disjunctive syllogism
      Disjunctive syllogism
      A disjunctive syllogism, also known as disjunction-elimination and or-elimination , and historically known as modus tollendo ponens,, is a classically valid, simple argument form:where \vdash represents the logical assertion....

    • Hypothetical syllogism
      Hypothetical syllogism
      In logic, a hypothetical syllogism has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic, it is a short-hand for the theory of consequence.-Propositional logic:...

    • Polysyllogism
      Polysyllogism
      A polysyllogism is a string of any number of propositions forming together a sequence of syllogisms such that the conclusion of each syllogism, together with the next proposition, is a premise for the next, and so on...

    • Prosleptic syllogism
      Prosleptic syllogism
      A prosleptic syllogism is a class of syllogisms that use a prosleptic proposition as one of the premises. The term originated with Theophrastus of Eresus, although Aristotle did briefly mention such syllogisms by a different name in his Prior Analytics....

    • Quasi-syllogism
      Quasi-syllogism
      Quasi-syllogism is a term that is sometimes used to describe what might be otherwise called a categorical syllogism but where one of the premises is singular, and thus not a categorical statement.For example:#All men are mortal#Socrates is a man...

    • Statistical syllogism
      Statistical syllogism
      A statistical syllogism is a non-deductive syllogism. It argues from a generalization true for the most part to a particular case .-Introduction:Statistical syllogisms may use qualifying words like "most", "frequently", "almost never", "rarely",...

  • Syllogistic fallacy
    Syllogistic fallacy
    Syllogistic fallacies are logical fallacies that occur in syllogisms. They include:Any syllogism type :*fallacy of four termsOccurring in categorical syllogisms:*related to affirmative or negative premises:...

  • The False Subtlety of the Four Syllogistic Figures
  • Venn diagram
    Venn diagram
    Venn diagrams or set diagrams are diagrams that show all possible logical relations between a finite collection of sets . Venn diagrams were conceived around 1880 by John Venn...


External links