Unit 5: Testing Syllogism Validity

Table of Contents

This unit provides two methods for testing whether a syllogism is Valid or Invalid.

Copi's Six Rules for Testing Syllogisms

A categorical syllogism is valid if, and only if, it follows all of these six rules. If it breaks even one rule, it is invalid, and the rule it breaks is the fallacy it commits.

Rules of Distribution

These rules relate to the terms (S, P, M).

Rule 1: The middle term (M) must be distributed at least once.

Fallacy: Fallacy of the Undistributed Middle.
(Example: "All P are M. All S are M." Here M is the predicate in two A-propositions and is never distributed. We can't link S and P.)

Rule 2: If a term (S or P) is distributed in the conclusion, it must also be distributed in its premise.

This rule has two associated fallacies:

Fallacy: Fallacy of Illicit Major.
(The Major Term, P, is distributed in the conclusion but not in the major premise.)

Fallacy: Fallacy of Illicit Minor.
(The Minor Term, S, is distributed in the conclusion but not in the minor premise.)

Rules of Quality

These rules relate to affirmative and negative propositions.

Rule 3: No conclusion can be drawn from two negative premises.

Fallacy: Fallacy of Exclusive Premises.
(If both premises are "No..." or "Some...not...", they just state exclusion. We can't mediate a link.)

Rule 4: If one premise is negative, the conclusion must be negative. (And vice-versa: If the conclusion is negative, one premise must be negative).

Fallacy: Fallacy of Drawing an Affirmative Conclusion from a Negative Premise (or vice-versa).

Rules of Quantity

These rules relate to universal and particular propositions. (Note: These rules are sometimes combined with the others).

Rule 5: If both premises are universal, the conclusion cannot be particular.

(This rule is only for the Boolean (modern) interpretation. In the traditional Aristotelian view, this is not a fallacy. This is the "Existential Fallacy" problem.)

Rule 6: No conclusion can be drawn from two particular premises.

Fallacy: (This often breaks another rule, like Undistributed Middle, but it's a good rule of thumb.)

(Note: Your syllabus likely follows Copi's rules, which are sometimes listed as 4 or 5. Rules 1-4 are the core of Aristotelian logic. The "Fallacy of Four Terms" - having more than 3 terms - is also a rule, but it's usually assumed a syllogism has only 3 terms.)

Summary of Rules and Fallacies

Rule Fallacy if Broken
1. Middle term (M) must be distributed at least once. Undistributed Middle
2. A term (S or P) distributed in conclusion must be distributed in its premise. Illicit Major (for P) or Illicit Minor (for S)
3. Cannot have two negative premises. Exclusive Premises
4. A negative premise requires a negative conclusion (and vice-versa). Affirmative from Negative (or vice-versa)

Venn Diagram Technique for Testing Syllogism

This is a modern, visual method for testing validity. It is based on the Boolean (modern) interpretation, which does not assume existential import.

How to Draw the Diagram

A syllogism has three terms (S, P, M). To test it, we draw three overlapping circles, labeled S, P, and M. This creates 8 distinct regions (including the area outside all circles).

[Diagram Placeholder: Three-Circle Venn Diagram]

Three overlapping circles labeled S, P, and M, showing 8 numbered regions.
1: S only, 2: S and M only, 3: M only, 4: S and P only, 5: S, P, and M, 6: P and M only, 7: P only, 8: Outside all.

How to Test a Syllogism

  1. Draw the three-circle diagram.
  2. Diagram ONLY the premises. Do NOT diagram the conclusion.
  3. How to Diagram:
  4. Check for Validity: After diagramming *only* the premises, look at the diagram. Is the conclusion *already* represented on the diagram?

Example 1: (Valid: AAA-1)

Syllogism:
1. All M are P
2. All S are M
------------------
C. All S are P

  1. Draw the three S, P, M circles.
  2. Diagram Premise 1 ("All M are P"): We shade the part of M that is *outside* P. (This includes the "S and M only" region and the "M only" region).
  3. Diagram Premise 2 ("All S are M"): We shade the part of S that is *outside* M. (This includes the "S only" region and the "S and P only" region).
  4. Check the Conclusion ("All S are P"): We look at the diagram. Is the conclusion "All S are P" (which means the part of S *outside* P is empty) already shown? YES. When we shaded Premise 2, we shaded the "S and P only" region, and when we shaded Premise 1, we shaded the "S and M only" region. The *entire* part of S that is not P is now shaded.
  5. Result: VALID.
[Diagram Placeholder: Venn Diagram Test for AAA-1]

A 3-circle diagram. The area of S outside M is shaded. The area of M outside P is shaded. The resulting diagram clearly shows the area of S outside P is fully shaded, proving the conclusion "All S are P."

Example 2: (Invalid: AOO-1)

Syllogism:
1. All M are P
2. Some S are not M
------------------
C. Some S are not P

  1. Draw the three S, P, M circles.
  2. Diagram Premise 1 (Universal): ("All M are P"): We shade the part of M that is *outside* P.
  3. Diagram Premise 2 (Particular): ("Some S are not M"): We must place an 'X' in the part of S that is *outside* M. This area is split into two regions: "S only" and "S and P only." Since our shading from step 2 did *not* eliminate either region, we must place the 'X' on the line between S-only and S-and-P-only.
  4. Check the Conclusion ("Some S are not P"): We look at the diagram. Does it *necessarily* show "Some S are not P"? This would require an 'X' to be *definitively* in the S-area *outside* the P-circle. Our 'X' is "on the fence." It *could* be in the "S and P" part. Since the conclusion is not *guaranteed*, the argument is...
  5. Result: INVALID.

    6. Aristotelian vs. Boolean Standpoint

    Exam Tip: This is a common point of confusion.

    How to test with Aristotelian view: If (and only if) an argument is invalid on the Boolean test, AND it has two universal premises and a particular conclusion (like AAI-1 or EAO-3), you do *one more step*: Look at the circle of the class that is universally affirmed (e.g., in AAI-1, the S class). If that class's one remaining "unshaded" area is empty, you can add an "X" there (because S *must* exist). If this "X" makes the conclusion true, the argument is Conditionally Valid (valid from the Aristotelian standpoint).