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. The following steps will help you understand syllogisms.

Steps

1. Know the basic structure of syllogisms. A syllogism has three parts: major premise, 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 the conclusion: the major term in the major premise, which forms the predicate of the conclusion, and the minor term in the minor premise, which forms the subject of the conclusion. 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 and predicate of the conclusion, "parrot" is the minor term and subject of the conclusion, and "bird" is the middle term.
   
2. Think of each term as representing a category. For example, "animal" is a category composed of everything that can be described as an animal.
   
3. Understand each part is expressed as "Some/all/no A is/are [not] B," with four 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) is expressed as "no A is/are B," abbreviated as AEB. The particular affirmative (symbolized as 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.
   
4. Determine the figure of the syllogism. Depending on whether the middle term serves 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 and predicate 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 are birds"
     Conclusion:......S-P..........e.g., "All parrots are animals".
     
     
   • Second figure: The middle term serves as predicate in the major premise and predicate in the minor premise. Thus, second figure takes the form:
     
     Major premise: P-M..........e.g., "No foxes are birds"
     Minor premise: S-M..........e.g., "All parrots are birds"
     Conclusion:......S-P..........e.g., "No parrots are foxes".
     
     
   • Third figure: The middle term serves as subject in the major premise and subject 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 birds are mortals"
     Conclusion:......S-P..........e.g., "Some mortals are animals".
     
     
   • Fourth figure: The middle term serves as predicate in the major premise and subject 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 are animals"
     Conclusion:......S-P..........e.g., "Some animals are not birds".
     
     
5. Determine whether a given syllogism is valid: by checking to see if it fits into one 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, the conclusion 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 four figures, so 4*4*4*4=256) of syllogism, only 19 of them are valid. The valid forms for each figure is given below, with their mnemonic names (each containing three vowels specifying 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.
       
     • 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.
       

Tips

• Note that if either of the premises is negative, the conclusion must also be negative. If both premises are affirmative, the conclusion must also be affirmative.
• In order for a syllogism to be valid, at least one of the two premises must contain a universal form. If both premises are particulars, then no valid conclusion can follow. For example, if "some cats are black" and "some black things are tables", it does not follow that "some cats are tables".
• Drawing out or visualising Venn Diagrams can help in understanding distribution of terms in determining whether a given syllogism is valid or not.
  • The universal affirmative (A) is represented as one circle (the subject) entirely within another circle (the predicate).
  • The universal negative (E) is represented as two mutually exclusive, 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 solving problems of categorical syllogisms: instead of using them in the purely set-theoretical manner described above (also known as "Euler Circles").
    • Draw three intersecting circles and use shading to indicate absence (or impossibility), leave blank to indicate "no knowledge", and a small '+' sign to indicate presence.
    • Now a valid categorical statement will have one of four forms:
      • either a lens fully shaded
      • a lune fully shaded
      • a '+' mark in a lens
      • a '+' mark in a lune
    • The syllogism is valid (in the classical Aristotelian sense)if the circles representing Major and Minor Premises are one of four forms: either a lens or lune fully shaded, or a '+' mark in lens or lune.
    • This method works conveniently only for syllogisms of three categorical statements only: Minor Premise, Major Premise and Conclusion.
• In order for a valid conclusion to be made, the middle term must be distributed in at least one of the premises, to allow the major and minor terms be linked. Avoid the fallacy of the undistributed middle. For example, from "All dogs love food", and "John loves food", it does not follow that "John is a dog".
• Understand the distribution of terms. A categorical term is said to be distributed if all individual members of that category are accounted for, for example, in "all men are mortal", the term "men" is distributed because every member belonging to that category is accounted for, in this case, as mortal. Note how each of the four variations distributes (or not) 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 the predicate 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 premises must be affirmative. If both premises are negative, then no valid conclusion can follow. If both premises are negative, the middle cannot establish any link between the major and minor terms.

Warnings

• Beware of the fallacy of the illicit major, where the major term is undistributed in the major premise but distributed in the conclusion. An example of this is in the form All A are B; no C are A. Therefore, no C are B. For instance, "All cats are animals"; "no dogs are cats"; therefore, "no dogs are animals": this syllogism is invalid because the major term "animals" is undistributed in the major premise, but distributed in the conclusion.
• Beware of the fallacy of the illicit minor, where the minor term is undistributed in the minor premise but distributed in the conclusion. An example of this is in the form All A are B; all A are C. Therefore, all C are B. For instance, "All cats are mammals"; "all cats are animals"; therefore, "all animals are mammals": this syllogism is invalid because the minor term "animals" is undistributed in the minor premise (because not all animals are cats), but distributed in the conclusion.

Related wikiHows

• How to Reason
• How to Logically Organize Your Thoughts
• How to Widen Your Logic and Knowledge

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