7.3 Venn Diagrams, Moods & Figures, Formal and Informal Fallacies

Venn Diagrams, Moods & Figures, Formal and Informal Fallacies

Quick Answer: The form of a syllogism is determined by its Mood (proposition types) and Figure (position of the Middle term). Of 256 possible forms, only 15 are unconditionally valid in Boolean logic. Syllogisms can be verified using three-circle Venn diagrams. Structural violations produce formal fallacies, while deceptive reasoning in natural language produces informal fallacies categorized into relevance, defective induction, presumption, and ambiguity.

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1. Moods and Figures of Categorical Syllogisms

The logical validity of a categorical syllogism depends exclusively on its structural form, defined by two parameters: Mood and Figure.

1. Mood

The Mood of a syllogism is a three-letter code representing the types of categorical propositions ($A, E, I, O$) that constitute the argument, listed in standard order:

1. Major Premise
2. Minor Premise
3. Conclusion

Example: An EIO mood denotes an $E$ major premise, an $I$ minor premise, and an $O$ conclusion. With 4 proposition types across 3 positions, there are $4^3 = 64$ possible moods.

2. Figures

The Figure identifies the arrangement of the Middle Term ($M$) in the premises. There are four distinct figures:

 Figure 1 (Sub-Prae)     Figure 2 (Prae-Prae)     Figure 3 (Sub-Sub)      Figure 4 (Prae-Sub)
      M ── P                   P ── M                   M ── P                  P ── M
       \                            │                   │                            /
        \                           │                   │                           /
      S ── M                   S ── M                   M ── S                  M ── S
      ──────                   ──────                   ──────                  ──────
   ∴  S ── P                ∴  S ── P                ∴  S ── P               ∴  S ── P

• Figure 1 (Left-Slant): $M$ is the Subject of the major premise and Predicate of the minor premise.
• Figure 2 (Right-Vertical): $M$ is the Predicate of both premises.
• Figure 3 (Left-Vertical): $M$ is the Subject of both premises.
• Figure 4 (Right-Slant): $M$ is the Predicate of the major premise and Subject of the minor premise.

The 15 Unconditionally Valid Syllogisms (Boolean Logic)

Combining 64 moods with 4 figures generates $64 \times 4 = 256$ total syllogistic forms. In modern Boolean logic, exactly 15 forms are unconditionally valid:

[!NOTE] Under classical Aristotelian logic, 9 additional conditionally valid forms (such as AAI-1, EAO-1, AAI-3, EAO-3, AAI-4, EAO-4) are recognized under the condition that the subject class possesses existential import (is non-empty).

2. Testing Syllogisms with Venn Diagrams

To evaluate a three-term categorical syllogism using a Venn diagram:

1. Draw three interlocking circles representing the three terms: Subject ($S$, bottom-left), Predicate ($P$, bottom-right), and Middle ($M$, top-center). This creates 8 distinct regions.
2. Diagram only the two premises. Do NOT diagram the conclusion.
   • Universal propositions ($A, E$): Shade out the entire region that is asserted to be empty.
   • Particular propositions ($I, O$): Place an '$\mathbf{X}$' in the region asserted to contain at least one member. If the '$\mathbf{X}$' could belong to either of two adjacent sub-regions, place it directly on the dividing boundary line.
3. Evaluation Rule: Inspect the resulting diagram for the conclusion. If the conclusion is already fully depicted without adding any further marks, the argument is valid; otherwise, it is invalid.

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3. Formal Fallacies

A formal fallacy is an error in the structural form of a deductive argument, rendering it invalid regardless of content:

• Fallacy of Undistributed Middle: Occurs when the middle term ($M$) is not distributed in at least one premise, preventing it from linking $S$ and $P$.
  • Example: All dogs ($P$) are mammals ($M$); All cats ($S$) are mammals ($M$); Therefore, all cats ($S$) are dogs ($P$).
• Fallacy of Illicit Major: The major term ($P$) is distributed in the conclusion but is undistributed in the major premise.
  • Example: All carnivores ($M$) are animals ($P^u$); No reptiles ($S$) are carnivores ($M$); Therefore, no reptiles are animals ($P^d$).
• Fallacy of Illicit Minor: The minor term ($S$) is distributed in the conclusion but is undistributed in the minor premise.
  • Example: All tigers ($M$) are mammals ($P$); All tigers ($M$) are striped ($S^u$); Therefore, all striped creatures ($S^d$) are mammals ($P$).
• Fallacy of Exclusive Premises: Attempting to deduce a conclusion from two negative premises ($EE, EO, OE, OO$).
• Existential Fallacy: Inferring a particular conclusion ($I$ or $O$) from two universal premises ($AA, AE, EA$) from the Boolean standpoint without establishing existential instantiation.

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4. Comprehensive Taxonomy of Informal Fallacies

Informal fallacies are flaws in reasoning that arise from deceptive content, ambiguous language, or irrelevant context rather than abstract logical syntax. They are categorized into four major families:

1. Fallacies of Relevance

The premises are logically irrelevant to the truth of the conclusion, even if psychologically persuasive:

• Argumentum ad Hominem (Attack against the Person):
  • Abusive: Attacking the advocate's personal character, intelligence, or integrity rather than addressing their argument.
  • Circumstantial: Dismissing a claim by alleging that the advocate is motivated by personal self-interest or circumstance.
  • Tu Quoque (You Too / Appeal to Hypocrisy): Defending oneself by alleging that the accuser is guilty of the same behavior.
• Argumentum ad Populum (Appeal to the Masses / Bandwagon): Claiming a proposition is true merely because the majority believes it, or appealing to group identity, snobbery, or popular sentiment.
• Argumentum ad Baculum (Appeal to Force / Threat): Employing coercion, implicit intimidation, or physical threats to compel assent.
• Argumentum ad Misericordiam (Appeal to Pity): Evoking sympathy, guilt, or pity to win acceptance of an unproven conclusion.
• Straw Man Fallacy: Caricaturing, exaggerating, or misrepresenting an opponent's argument to make it easier to attack and refute.
• Red Herring: Introducing an irrelevant, emotionally charged side-issue to divert attention away from the primary argument.
• Ignoratio Elenchi (Missing the Point / Irrelevant Conclusion): Presenting premises that support one specific conclusion, but drawing a completely different conclusion.

2. Fallacies of Defective Induction

The premises are relevant but too weak or inadequate to justify the conclusion:

• Argumentum ad Ignorantiam (Appeal to Ignorance): Arguing that a claim is true because it has not been proven false, or false because it has not been proven true.
• Argumentum ad Verecundiam (Appeal to Inappropriate Authority): Citing an authority whose expertise lies outside the relevant field of inquiry.
• Hasty Generalization (Secundum Quid): Drawing a universal conclusion from an isolated instance or an unrepresentative, statistically inadequate sample.
• False Cause (Post Hoc Ergo Propter Hoc & Non Causa Pro Causa):
  • Post Hoc: Assuming that because event $Y$ followed event $X$, event $X$ must have caused event $Y$.
  • Non Causa: Treating something as the cause of an effect when it is merely a correlated factor.

3. Fallacies of Presumption

The premises assume what they purport to prove, creating unjustified cognitive leaps:

• Petitio Principii (Begging the Question / Circular Reasoning): Assuming the truth of the conclusion within the premises, often through synonymous phrasing.
• Complex Question (Loaded Question): Asking a question that embeds an unproven, incriminating presupposition (e.g., "Have you stopped mistreating your research assistants?").
• False Dilemma (False Dichotomy / Black-and-White Thinking): Artificially presenting only two extreme alternatives when viable middle options exist.
• Slippery Slope: Claiming without causal evidence that taking a single initial step will trigger an uncontrollable chain reaction leading to disaster.

4. Fallacies of Ambiguity

Arise from linguistic confusion, semantic shifts, or syntactic amphiboly:

• Equivocation: Shifting the meaning of a key word or phrase between premises and conclusion in the same argument.
• Amphiboly: Ambiguity resulting from faulty grammatical structure or punctuation, allowing a sentence to be interpreted in multiple ways.
• Fallacy of Composition: Erroneously reasoning that because every individual part possesses a property, the whole system must possess that property.
• Fallacy of Division: Erroneously reasoning that because the whole entity possesses a property, each individual component part must possess that property.