Logic & Critical Thinking Series Part 2: Deductive Reasoning

graph TD
    MAJOR["Major Premise\n(All humans are mortal)"] --> CONCLUSION["Conclusion\n(Therefore, Socrates\nis mortal)"]
    MINOR["Minor Premise\n(Socrates is a human)"] --> CONCLUSION
    
    MAJOR -.-> MIDDLE["Middle Term\n(human)"]
    MINOR -.-> MIDDLE
    
    CONCLUSION --> VALID{"Valid?"}
    VALID -->|"Form correct\n+ premises true"| SOUND["Sound\nArgument ✓"]
    VALID -->|"Form correct\n+ premises false"| UNSOUND["Valid but\nUnsound"]
                        

Syllogisms: The Classic Form

A syllogism is a deductive argument with exactly two premises and one conclusion, each containing categorical propositions. Invented by Aristotle, syllogisms dominated logical analysis for over two thousand years.

Categorical Propositions

Syllogisms use four types of categorical propositions:

The Structure of a Syllogism

Every syllogism has three terms:

• Major term (P): The predicate of the conclusion
• Minor term (S): The subject of the conclusion
• Middle term (M): Appears in both premises but not the conclusion—it's the "link"

Major premise: All mammals (M) are warm-blooded (P).

Minor premise: All dogs (S) are mammals (M).

Conclusion: Therefore, all dogs (S) are warm-blooded (P).

The middle term "mammals" connects dogs to warm-blooded things, then disappears from the conclusion.

Figures & Moods

The figure of a syllogism describes the position of the middle term:

The mood is the combination of proposition types (A, E, I, O). For example, "AAA" means both premises and the conclusion are universal affirmatives.

Valid mood-figure combinations (there are only 15 unconditionally valid forms):

• Figure 1: AAA (Barbara), EAE (Celarent), AII (Darii), EIO (Ferio)
• Figure 2: EAE (Cesare), AEE (Camestres), EIO (Festino), AOO (Baroco)
• Figure 3: IAI (Disamis), AII (Datisi), OAO (Bocardo), EIO (Ferison)
• Figure 4: AEE (Camenes), IAI (Dimaris), EIO (Fresison)

The medieval names (Barbara, Celarent, etc.) encode the mood in their vowels—a mnemonic device from scholastic logic.

Testing Validity with Venn Diagrams

Venn diagrams provide a visual method for testing syllogism validity:

Common Invalid Syllogisms

Watch out for these frequent errors:

Validity & Soundness: The Critical Distinction

These two concepts are the most important in evaluating deductive arguments. Master them, and you'll never be confused about what makes an argument good or bad.

The Four Combinations

Every deductive argument falls into one of four categories:

The Counterexample Method

To prove an argument form is invalid, find a counterexample: an argument with the same form where the premises are obviously true and the conclusion obviously false.

Example: Is this valid?

If it's raining, the ground is wet.

The ground is wet.

? It's raining.

Counterexample with same form:

If I'm in Paris, I'm in France. (TRUE)

I'm in France. (TRUE)

? I'm in Paris. (FALSE—I could be in Lyon)

The counterexample proves the argument form is invalid. This fallacy is called "affirming the consequent."

Propositional Logic

Propositional logic studies how simple propositions combine with logical connectives to form complex statements. It's the foundation of modern symbolic logic and computer science.

Logical Connectives

Propositions are combined using five basic connectives:

Truth Tables

Truth tables systematically show how compound propositions relate to their components:

Essential Rules of Inference

These are the valid argument forms you can use to construct airtight deductions:

Common Invalid Forms (Fallacies)

These look similar to valid forms but fail:

Next Steps in the Series

Now that you understand deductive reasoning, you're ready to explore its complement—inductive reasoning.

• Part 3: Inductive Reasoning - Probability, generalizations, the scientific method
• Part 4: Logical Fallacies - Formal and informal fallacies, cognitive biases
• Part 5: Argument Analysis - Identifying premises, evaluating evidence
• Part 6: Critical Thinking Applications - Real-world uses, media literacy