how to understand syllogisms_ 5 steps (with pictures) - wikihow

9/10/2014 How to Understand Syllogisms: 5 Steps (with Pictures) - wikiHow

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How to Understand Syllogisms

A syllogism is a logical argument composed of three parts: the major premise,the minor premise, and the conclusion inferred from the premises. Thefollowing steps will help you understand syllogisms.

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Know the basic structure of syllogisms. A syllogism has three parts: majorpremise, minor premise, and conclusion. Each part is composed of two categorical

terms (terms that denote categories; such as birds, animals, etc.), linked in the form"Some/all A is/are [not] B." Each of the premises has one term in common with theconclusion: the major term in the major premise, which forms the predicate of theconclusion, and the minor term in the minor premise, which forms the subject of theconclusion. The categorical term in common in the premises is called the "middle term".For example: Major premise: All birds are animals. Minor premise: All parrots are birds.Conclusion: All parrots are animals. In this example, "animal" is the major term andpredicate of the conclusion, "parrot" is the minor term and subject of the conclusion, and"bird" is the middle term.

Think of each term as representing a category. For example, "animal" is acategory composed of everything that can be described as an animal.

Understand each part is expressed as "Some/all/no A is/are [not] B," withfour possible variation. The universal affirmative (symbolized as A) is expressed

as "all A is/are B," abbreviated as AaB. The universal negative (symbolized as E) isexpressed as "no A is/are B," abbreviated as AeB. The particular affirmative (symbolizedas I) is expressed as "some A is/are B," abbreviated as AiB. The particular negative(symbolized as O) is expressed as "some A is/are not B," abbreviated as AoB.

Determine the figure of the syllogism. Depending on whether the middle termserves as subject or predicate in the premises, a syllogism may be classified as

one of four possible figures:

First figure: The middle term serves as subject in the major premise andpredicate in the minor premise. Thus, first figure takes the form: Major premise:M-P..........e.g., "All birds are animals" Minor premise: S-M..........e.g., "All parrots

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are birds" Conclusion:......S-P..........e.g., "All parrots are animals".Second figure: The middle term serves as predicate in the major premise andpredicate in the minor premise. Thus, second figure takes the form: Majorpremise: P-M..........e.g., "No foxes are birds" Minor premise: S-M..........e.g., "Allparrots are birds" Conclusion:......S-P..........e.g., "No parrots are foxes".Third figure: The middle term serves as subject in the major premise andsubject in the minor premise. Thus, third figure takes the form: Major premise:M-P..........e.g., "All birds are animals" Minor premise: M-S..........e.g., "All birdsare mortals" Conclusion:......S-P..........e.g., "Some mortals are animals".Fourth figure: The middle term serves as predicate in the major premise andsubject in the minor premise. Thus, fourth figure takes the form: Major premise:P-M..........e.g., "No birds are cows" Minor premise: M-S..........e.g., "All cows areanimals" Conclusion:......S-P..........e.g., "Some animals are not birds".

Determine whether a given syllogism is valid: by checking to see if it fits intoone of the valid forms of syllogism for the given figure. A syllogism is valid if and

only if the conclusion necessarily follows the premises, i.e., if the premises are true, theconclusion must be true. Although there are 256 possible forms (4 possible variations (a,e, i, o) for each part, three parts (major premise, minor premise, conclusion), and fourfigures, so 4*4*4*4=256) of syllogism, only 19 of them are valid. The valid forms for eachfigure is given below, with their mnemonic names (each containing three vowelsspecifying the form of the part (a, e, i, o) in order of major premise, minor premise,conclusion):

First figure has 4 valid forms: Barbara, Celarent, Darii, Ferio

Barbara (AAA): for example,All birds are animals.All parrots are birds.All parrots are animals.Celarent (EAE): for example,No birds are foxes.All parrots are birds.No parrots are foxes.Darii (AII): for example,All dogs are animals.Some mammals are dogs.Some mammals are animals.Ferio (EIO): for example,No dogs are birds.Some mammals are dogs.Some mammals are not birds.

Second figure has 4 valid forms: Cesare, Camestres, Festino, Baroco

Cesare (EAE): for example,No foxes are birds.All parrots are birds.No parrots are foxes.Camestres (AEE): for example,All foxes are animals.No trees are animals.No trees are foxes.Festino (EIO): for example,No restaurant food is healthy.Some recipes are healthy.Some recipes are not restaurant foods.



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Baroco (AOO): for example,All liars are evil-doers.Some doctors are not evil-doers.Some doctors are not liars.

Third figure has 6 valid forms: *Darapti, Disamis, Datisi, Felapton, Bocardo,Ferison

Darapti (AAI): for example,All men are fallible.All men are animals.Some animals are fallible.Disamis (IAI): for example,Some books are precious.All books are perishable.Some perishable things are precious.Datisi (AII): for example,All books are imperfect.Some books are informative.Some informative things are imperfect.Felapton (EAO): for example,No snakes are good to eat.All snakes are animals.Some animals are not good to eat.Bocardo (OAO): for example,Some websites are not helpful.All websites are Internet resources.Some internet resources are not helpful.Ferison (EIO): for example,No lepers are allowed to enter the church.All lepers are human.Some humans are not allowed to enter the church.

Fourth figure has 5 valid forms: Bramantip, Camenes, Dimaris, Fesapo,Fresison

Bramantip (AAI): for example,All pigs are unclean.All unclean things are best avoided.Some things that are best avoided are pigs.Camenes (AEE): for example,All trees are plants.No plants are birds.No birds are trees.Dimaris (IAI): for example,Some evil doers are lawyers.All lawyers are human.Some humans are evil doers.Fesapo (EAO): for example,No meals are free.All free things are desirable.Some desirable things are not meals.Fresison (EIO): for example,No dogs are birds.Some birds are pets.Some pets are not dogs.



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Note that if either of the premises is negative, the conclusion must alsobe negative. If both premises are affirmative, the conclusion must alsobe affirmative.In order for a syllogism to be valid, at least one of the two premisesmust contain a universal form. If both premises are particulars, then novalid conclusion can follow. For example, if "some cats are black" and"some black things are tables", it does not follow that "some cats aretables".Drawing out or visualising Venn Diagrams can help in understandingdistribution of terms in determining whether a given syllogism is valid ornot.

The universal affirmative (A) is represented as one circle (thesubject) entirely within another circle (the predicate).The universal negative (E) is represented as two mutuallyexclusive, non-overlapping circles.The particulars (I, O) are represented as two intersecting circles,each with area in common and area not in common with the other.There is another way to mark up Venn Diagrams when solvingproblems of categorical syllogisms: instead of using them in thepurely set-theoretical manner described above (also known as"Euler Circles").

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Draw three intersecting circles and use shading to indicateabsence (or impossibility), leave blank to indicate "noknowledge", and a small '+' sign to indicate presence.Now a valid categorical statement will have one of four forms:

either a lens fully shadeda lune fully shadeda '+' mark in a lensa '+' mark in a lune

The syllogism is valid (in the classical Aristotelian sense)if thecircles representing Major and Minor Premises are one of fourforms: either a lens or lune fully shaded, or a '+' mark in lens orlune.This method works conveniently only for syllogisms of threecategorical statements only: Minor Premise, Major Premise andConclusion.

In order for a valid conclusion to be made, the middle term must bedistributed in at least one of the premises, to allow the major and minorterms be linked. Avoid the fallacy of the undistributed middle. Forexample, from "All dogs love food", and "John loves food", it does notfollow that "John is a dog".Understand the distribution of terms. A categorical term is said to bedistributed if all individual members of that category are accounted for,for example, in "all men are mortal", the term "men" is distributedbecause every member belonging to that category is accounted for, inthis case, as mortal. Note how each of the four variations distributes (ornot) the terms:

In "All A are B" propositions, the subject (A) is distributed.In "No A are B" propositions, both the subject (A) and the predicate(B) are distributed.In "Some A are B" propositions, neither the subject nor thepredicate are distributed.In "Some A are not B" propositions, the predicate (B) is distributed.

In order for a syllogism to be valid, at least one of the two premisesmust be affirmative. If both premises are negative, then no validconclusion can follow. If both premises are negative, the middle cannotestablish any link between the major and minor terms.

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Beware of the fallacy of the illicit major, where the major term isundistributed in the major premise but distributed in the conclusion. Anexample of this is in the form All A are B; no C are A. Therefore, no Care B. For instance, "All cats are animals"; "no dogs are cats";therefore, "no dogs are animals": this syllogism is invalid because themajor term "animals" is undistributed in the major premise, butdistributed in the conclusion.Beware of the fallacy of the illicit minor, where the minor term isundistributed in the minor premise but distributed in the conclusion. Anexample of this is in the form All A are B; all A are C. Therefore, all Care B. For instance, "All cats are mammals"; "all cats are animals";therefore, "all animals are mammals": this syllogism is invalid becausethe minor term "animals" is undistributed in the minor premise (becausenot all animals are cats), but distributed in the conclusion.

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