Rules and Fallacies for Categorical Syllogisms

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Chapter Outline

   • Introduction
   • Learning Objectives
   • Rules of Standard-form of Categorical Syllogisms
   • Fallacies of Violating the Rules
   • Summary
   • Practice Test
   • Reference


Since the validity of a categorical syllogism depends solely upon its logical form, it is relatively simple to state the conditions under which the premises of syllogisms succeed in guaranteeing the truth of their conclusions. Relying heavily upon the medieval tradition, Copi & Cohen provide a list of six rules, each of which states a necessary condition for the validity of any categorical syllogism. Violating any of these rules involves committing one of the formal fallacies, errors in reasoning that result from reliance on an invalid logical form.In this lesson we will concentrate on the rules required for a standard-form of categorical syllogism and the fallacies created for violating these rules.

Learning Objectives
After reading this chapter, you are expected to learn about:

• Know the necessary condition for the validity of any categorical syllogism

• Explain how violating any of these rules involves committing fallacies.

Rules and Fallacies for Categorical Syllogisms

The following rules must be observed in order to form a valid categorical syllogism:

Rule-1. A valid categorical syllogism will have three and only three unambiguous categorical terms.

The use of exactly three categorical terms is part of the definition of a categorical syllogism, and we saw earlier that the use of an ambiguous term in more than one of its senses amounts to the use of two distinct terms. In categorical syllogisms, using more than three terms commits the fallacy of four terms.

 Fallacy: Four terms


	Power tends to corrupt
       Knowledge is power
       Knowledge tends to corrupt

Justification: This syllogism appears to have only three terms, but there are really four since one of them, the middle term “power” is used in different senses in the two premises. To reveal the argument’s invalidity we need only note that the word “power” in the first premise means “ the possession of control or command over people,” whereas the word “power” in the second premise means “the ability to control things.

Rule-2. In a valid categorical syllogism the middle term must be distributed in at least one of the premises.

In order to effectively establish the presence of a genuine connection between the major and minor terms, the premises of a syllogism must provide some information about the entire class designated by the middle term. If the middle term were undistributed in both premises, then the two portions of the designated class of which they speak might be completely unrelated to each other. Syllogisms that violate this rule are said to commit the fallacy of the undistributed middle.

Fallacy: Undistributed middle


	All sharks are fish
       All salmon are fish
       All salmon are sharks

Justification: The middle term is what connects the major and the minor term. If the middle term is never distributed, then the major and minor terms might be related to different parts of the M class, thus giving no common ground to relate S and P.

Rule-3. In a valid categorical syllogism if a term is distributed in the conclusion, it must be distributed in the premises.

A premise that refers only to some members of the class designated by the major or minor term of a syllogism cannot be used to support a conclusion that claims to tell us about every menber of that class. Depending which of the terms is misused in this way, syllogisms in violation commit either the fallacy of the illicit major or the fallacy of the illicit minor.

Fallacy: Illicit major; illicit minor


	All horses are animals
       Some dogs are not horses
       Some dogs are not animals


       All tigers are mammals
       All mammals are animals
       All animals are tigers

Justification: When a term is distributed in the conclusion, let’s say that P is distributed, then that term is saying something about every member of the P class. If that same term is NOT distributed in the major premise, then the major premise is saying something about only some members of the P class. Remember that the minor premise says nothing about the P class. Therefore, the conclusion contains information that is not contained in the premises, making the argument invalid.

Rule-4. A valid categorical syllogism may not have two negative premises.

The purpose of the middle term in an argument is to tie the major and minor terms together in such a way that an inference can be drawn, but negative propositions state that the terms of the propositions are exclusive of one another. In an argument consisting of two negative propositions the middle term is excluded from both the major term and the minor term, and thus there is no connection between the two and no inference can be drawn. A violation of this rule is called the fallacy of exclusive premises.

Fallacy: Exclusive premises


	No fish are mammals
       Some dogs are not fish
       Some dogs are not mammals

Justification: If the premises are both negative, then the relationship between S and P is denied. The conclusion cannot, therefore, say anything in a positive fashion. That information goes beyond what is contained in the premises.

Rule-5. If either premise of a valid categorical syllogism is negative, the conclusion must be negative.

An affirmative proposition asserts that one class is included in some way in another class, but a negative proposition that asserts exclusion cannot imply anything about inclusion. For this reason an argument with a negative proposition cannot have an affirmative conclusion. An argument that violates this rule is said to commit the fallacy of drawing an affirmative conclusion from a negative premise.

Fallacy: Drawing an affirmative conclusion from a negative premise, or drawing a negative conclusion from an affirmative premise.


	All crows are birds
       Some wolves are not crows
       Some wolves are birds

Justification: Two directions, here. Take a positive conclusion from one negative premise. The conclusion states that the S class is either wholly or partially contained in the P class. The only way that this can happen is if the S class is either partially or fully contained in the M class (remember, the middle term relates the two) and the M class fully contained in the P class. Negative statements cannot establish this relationship, so a valid conclusion cannot follow. Take a negative conclusion. It asserts that the S class is separated in whole or in part from the P class. If both premises are affirmative, no separation can be established, only connections. Thus, a negative conclusion cannot follow from positive premises. Note: These first four rules working together indicate that any syllogism with two particular premises is invalid.

Rule-6. In valid categorical syllogisms particular propositions cannot be drawn properly from universal premises.

Because we do not assume the existential import of universal propositions, they cannot be used as premises to establish the existential import that is part of any particular proposition. The existential fallacy violates this rule. Although it is possible to identify additional features shared by all valid categorical syllogisms (none of them, for example, have two particular premises), these six rules are jointly sufficient to distinguish between valid and invalid syllogisms.

Fallacy: Existential fallacy


	All mammals are animals
       All tigers are mammals
       Some tigers are animals

Justification: On the Boolean model, Universal statements make no claims about existence while particular ones do. Thus, if the syllogism has universal premises, they necessarily say nothing about existence. Yet if the conclusion is particular, then it does say something about existence. In which case, the conclusion contains more information than the premises do, thereby making it invalid.

|||| Summary

Rule 1: There must be exactly three unambiguous categorical terms

Fallacy = Four terms

Rule 2: Middle term must be distributed at least once.

Fallacy = Undistributed Middle

Rule 3: All terms distributed in the conclusion must be distributed in one of the premises.

Fallacy = Illicit major; Illicit minor

HINT: Mark all distributed terms first Remember from Chapter 1 that a deductive argument may not contain more information in the conclusion than is contained in the premises. Thus, arguments that commit the fallacies of illicit major and illicit minor commit this error.

Rule 4: Two negative premises are not allowed.

Fallacy = Exclusive premises The key is that "nothing is said about the relation between the S class and the P class."

Rule 5: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.

Fallacy = Drawing an affirmative conclusion from a negative premise. OR

Drawing a negative conclusion from affirmative premises. OR Any syllogism having exactly one negative statement is invalid.

Note the following sub-rule: No valid syllogism can have two particular premises. The last rule is dependent on quantity.

Rule 6: If both premises are universal, the conclusion cannot be particular.

Fallacy =Existential Fallacy}}

Key Points

The key points of this chapter are as follows:


Practice Test

Log Answer Type Questions

1. Name the fallacies committed and the rules broken by invalid syllogisms of the following forms.

            (a) AAA-2
            (b) EAA-1
            (c) OEO-4

2. Name the fallacies committed and the rules broken by the following syllogisms that are invalid.

  (a)   All criminals actions are wicked deeds.
        All prosecutions for murder are criminal actions.
        All prosecutions for murder are wicked deeds.
  (b)   Some parrots are not pests.
        All parrots are pets.
        No pets are pests.

Self-Assessment Questions (SAQs)

Put tick mark(√) on the correct answer:

  1. How many rules does a categorical syllogism contain?
      (a)2 (b)4 (c)6 (d)8
  2. Absence of three terms in a categorical syllogism commits fallacy of:
      (a)four terms (b) undistributed middle (c) exclusive premises (d) existential
  3. What type of fallacy occurs if there is no distribution of middle term in the premises?
      (a)four terms (b) undistributed middle (c) exclusive premises (d) existential
  4. The violation of the rule of having at least one affirmative premise in a valid categorical syllogism commits:
      (a)four terms (b) undistributed middle (c) exclusive premises (d) existential

Answers to SAQs

        1. (c)
        2. (a)
        3. (b) 
        4. (c) 

References and Further Readings

Copi, I.M and Cohen, C (1996). Introduction to Logic. Prentice -Hall of India Private Limited, New Delhi.Ninth Edition.

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