PHI-DSC-201 (Logic I): Unit 4: Syllogisms and Dilemmas

Contact Hours: 60 | Full Marks: 100 (ESE=70/CCA=30)

Table of Contents

1. Disjunctive Syllogism
2. Hypothetical Syllogism
3. Dilemmas and their Testing

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Disjunctive Syllogism

A **Disjunctive Syllogism** (DS) is a valid deductive argument form whose major premise is a **disjunction** (an 'either-or' statement) and whose minor premise is the negation of one of the disjuncts, and whose conclusion is the affirmation of the other disjunct.

Structure and Formula

• Major Premise (Disjunction): P or Q (P ∨ Q)
• Minor Premise (Negation): Not P (¬P)
• Conclusion (Affirmation): Therefore, Q (∴ Q)

Example: "Either the lights are on or the circuit is broken. The lights are not on. Therefore, the circuit is broken."

Formula: (P ∨ Q), ¬P / ∴ Q

Important Note: The Strong Sense of "Or"

For the Disjunctive Syllogism to be valid, the 'or' in the major premise must be understood in the **inclusive sense** (P or Q or both) OR the **exclusive sense** (P or Q but not both). In modern logic, DS is valid even with the inclusive 'or'. However, arguments based on affirming a disjunct (e.g., P, ∴ ¬Q) are invalid (Fallacy of Affirming a Disjunct).

Hypothetical Syllogism

A **Hypothetical Syllogism** (HS) is a valid deductive argument form composed of at least one **hypothetical** (conditional or 'if-then') proposition, and often links three propositions together. The simplest form contains three conditional statements.

Pure Hypothetical Syllogism (PHS)

Links two conditional statements to infer a third conditional statement.

• Premise 1: If P, then Q (P → Q)
• Premise 2: If Q, then R (Q → R)
• Conclusion: Therefore, if P, then R (∴ P → R)

Example: "If it rains, the grass is wet. If the grass is wet, the sun will dry it. Therefore, if it rains, the sun will dry it."

Formula: (P → Q), (Q → R) / ∴ (P → R)

Dilemmas and their Testing

A **Dilemma** is a powerful and often rhetorically effective argument form that contains a **disjunctive premise** (the 'horns' of the dilemma) and two or more **conditional premises**. The dilemma presents a choice between two (or more) undesirable alternatives.

Types of Dilemmas

Dilemmas are classified based on the nature of their conclusion (simple or complex) and the nature of their minor premises (constructive or destructive).

Testing (Refuting) Dilemmas

A dilemma is usually refuted not by proving it invalid, but by showing that its premises are false or unacceptable. There are three classic ways to rebut a dilemma:

1. Taking it by the Horns (Falsifying the Disjunctive Premise):

   Reject the disjunctive premise (P ∨ R) by proving that there is a third alternative (a 'third horn') that was not considered. Prove that both P and R are false.

2. Escaping between the Horns (Denying a Conditional Premise):

   Reject a conditional premise (P → Q) by showing that the consequent (Q) does not necessarily follow from the antecedent (P). Show a case where P is true but Q is false.

3. Rebuttal by a Counter-Dilemma (Rhetorical Refutation):

   Construct a new dilemma whose conclusion is opposite to the original dilemma. This is a rhetorical device that proves the original argument form *may* be fallacious, but does not prove the original conclusion is false.

Exam Focus: Dilemma Testing

Be prepared to write out an example of a dilemma and then explicitly show how it can be rebutted using all three methods. Rebuttal by a Counter-Dilemma is often a Complex Destructive form with the consequents negated and switched.

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