phil 279 lecture 4 fol rules

Conjunction Introduction (∧ I)

Premise 1: A<br>Premise 2: B<br>Conclusion: A ∧ B

Conjunction Elimination (∧ E)

Premise: A ∧ B<br>Conclusion: A (or B)

Disjunction Introduction (∨ I)

Premise: A<br>Conclusion: A ∨ B (or B ∨ A)

Disjunction Elimination (∨ E)

Premise 1: A ∨ B<br>Subproof 1: Assume A, derive D<br>Subproof 2: Assume B, derive D<br>Conclusion: D

Conditional Introduction (→ I)

Subproof:<br>• Assumption: A<br>• ...<br>• Inference: B<br>Conclusion: A → B

Conditional Elimination (→ E)

Premise 1: A → B<br>Premise 2: A<br>Conclusion: B

Biconditional Introduction (↔ I)

To show A ↔ B, prove both A → B and B → A in separate directions

Biconditional Elimination (↔ E)

From A ↔ B, extract conditional directions or components

Negation Introduction (¬ I) / Indirect Proof

Subproof:<br>• Assumption: A<br>• ...<br>• Inference: ⊥ (Contradiction)<br>Conclusion: ¬A

Contradiction

Premise 1: A
Premise 2: ¬A
Conclusion: ⊥

Explosion / Bottom Elimination (⊥ E)

Premise: ⊥<br>Conclusion: Any formula A

Theorem

If it is provable with no premises (⊢ C)

Logical Equivalence

If they can be proved from each other

Inconsistent Set of Sentences

If ⊥ can be derived from them

Entailment Symbol (⊢)

Read as "proves"; A₁, A₂, …, Aₙ ⊢ C