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if both middle terms are the predicate is it valid

if both middle terms are the predicate is it valid

2 min read 21-01-2025
if both middle terms are the predicate is it valid

Is a Categorical Syllogism Valid if Both Middle Terms are the Predicate?

Determining the validity of a categorical syllogism hinges on the relationship between its three terms: the major term (P), the minor term (S), and the middle term (M). A syllogism's structure dictates whether the conclusion logically follows from the premises. This article explores the validity of syllogisms where the middle term (M) functions as the predicate in both premises. The answer, as we'll see, is a nuanced "sometimes," depending on the specific form of the syllogism.

Understanding Categorical Syllogisms

Before delving into the specific scenario, let's briefly review the structure of a categorical syllogism. It consists of three parts:

  • Major Premise: A statement relating the major term (P) to the middle term (M).
  • Minor Premise: A statement relating the minor term (S) to the middle term (M).
  • Conclusion: A statement relating the major term (P) to the minor term (S).

Each premise and the conclusion are categorical propositions, which can be of four types: A, E, I, or O. The arrangement of these proposition types determines the syllogism's mood. The position of the terms (S, P, M) in each proposition determines its figure.

When the Middle Term is the Predicate in Both Premises

Let's consider syllogisms where the middle term (M) always appears as the predicate. This limits the possible forms significantly. For example:

  • Example 1 (Invalid):

    • Major Premise: All M are P. (A Proposition)
    • Minor Premise: All M are S. (A Proposition)
    • Conclusion: Therefore, All S are P. (Invalid Conclusion)

This syllogism is invalid. While both premises are true, they don't guarantee the truth of the conclusion. The middle term is distributed in both premises, which is a necessary but not sufficient condition for validity. The relationship between S and P remains ambiguous.

  • Example 2 (Potentially Valid):

    • Major Premise: No M are P. (E Proposition)

    • Minor Premise: No M are S. (E Proposition)

    • Conclusion: Therefore, All S are P. (Invalid Conclusion)

    • Alternative Conclusion: Therefore, Some S are not P. (Invalid Conclusion)

This syllogism is also invalid. The premises tell us that M and P are entirely separate, as are M and S. However, this doesn't dictate any necessary relationship between S and P.

  • Example 3 (Potentially Valid):

    • Major Premise: All P are M (A Proposition)
    • Minor Premise: Some S are M. (I Proposition)
    • Conclusion: Therefore, Some S are P (Possibly Valid)

This is a type of syllogism which might be valid. The structure allows for the possibility that some members of S are also members of P, but doesn't guarantee it. Whether it's valid depends on whether the specific conclusion is valid relative to the premises.

The Importance of Figure and Mood

The validity of a syllogism isn't solely determined by whether the middle term is the predicate in both premises. The figure (the arrangement of the terms) and mood (the types of propositions) are equally crucial. There are 256 possible syllogistic forms (4 moods x 4 figures x 4 positions of the terms), but only a fraction are valid.

To determine validity definitively, one needs to use techniques like Venn diagrams or traditional square of opposition methods to map the premises and see if the conclusion is a necessary consequence.

Conclusion

While it's not inherently invalid to have the middle term as the predicate in both premises, such a structure doesn't guarantee validity. Many syllogisms with this characteristic will be invalid. The determination of validity always requires a thorough analysis considering the syllogism's mood and figure using established logical methods. Simply observing that the middle term is the predicate in both premises is insufficient for determining whether the syllogism is valid.

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