Rules of Inference

Why This Matters

Rules of inference are the engine of formal proof—they're what allow you to move from premises to conclusions with absolute certainty. In Formal Logic II, you're being tested on your ability to recognize which rule applies in a given situation, construct valid proof sequences, and understand why each inference preserves truth. These aren't just abstract patterns; they're the building blocks you'll use in every proof you write, whether you're working through a simple argument or tackling a complex derivation.

Don't just memorize the symbolic forms—know what logical work each rule performs. Some rules let you break apart compound statements, others let you build them up, and still others let you chain reasoning across multiple conditionals. Understanding these functional categories will help you spot the right tool for each step in a proof and avoid common errors on exams.

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Conditional Elimination Rules

These rules extract conclusions from conditional (if-then) statements. The key insight: conditionals make promises about what follows from what, and these rules let you cash in on those promises.

Modus Ponens

• Affirms the antecedent to derive the consequent—the most fundamental conditional rule: $P \rightarrow Q$, $P$, therefore $Q$
• Direct forward reasoning that moves from cause to effect; if you know the "if" part is true, the "then" part must follow
• Most commonly used rule in proofs—when you see a conditional and its antecedent as separate premises, apply this immediately

Modus Tollens

• Denies the consequent to reject the antecedent—the contrapositive in action: $P \rightarrow Q$, $\neg Q$, therefore $\neg P$
• Backward reasoning that works from failed predictions; if the effect didn't happen, the cause didn't either
• Essential for indirect proofs and disproving hypotheses—look for this when you have a negated statement matching a consequent

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Conditional Chaining Rules

These rules connect multiple conditionals to extend your reasoning across a chain of implications. Think of them as building bridges between statements that don't directly connect.

Hypothetical Syllogism

• Chains two conditionals into one—if $P \rightarrow Q$ and $Q \rightarrow R$, then $P \rightarrow R$
• Transitivity of implication that lets you skip the middle term; the consequent of one becomes the antecedent of the next
• Look for shared middle terms in your premises—when Q appears as both consequent and antecedent, you can link the chain

Constructive Dilemma

• Combines two conditionals with a disjunction: $(P \rightarrow Q) \land (R \rightarrow S)$, $P \lor R$, therefore $Q \lor S$
• Parallel conditional reasoning—whichever disjunct is true triggers its corresponding conditional
• Useful for case analysis when you know one of two things must be true but not which; preserves the disjunctive structure in the conclusion

Destructive Dilemma

• Dual of Constructive Dilemma using negations: $(P \rightarrow Q) \land (R \rightarrow S)$, $\neg Q \land \neg S$, therefore $\neg P \land \neg R$
• Eliminates both antecedents when both consequents fail; applies Modus Tollens to parallel conditionals simultaneously
• Powerful for ruling out options—when multiple predictions fail, reject all the hypotheses that made them

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Disjunction Rules

These rules work with "or" statements, either building them or breaking them down. Disjunctions assert that at least one option holds—these rules help you work with that structure.

Disjunctive Syllogism

• Eliminates one disjunct to affirm the other: $P \lor Q$, $\neg P$, therefore $Q$
• Process of elimination—if you know "P or Q" and P is false, Q must be the true one
• High-frequency exam rule; whenever you can negate one side of a disjunction, immediately apply this to simplify

Addition

• Introduces a disjunction from a single statement: $P$, therefore $P \lor Q$
• Weakens information by adding possibilities; seems counterintuitive but logically valid since "or" only requires one true disjunct
• Strategic proof tool for setting up Disjunctive Syllogism or Constructive Dilemma—add what you'll need later

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Conjunction Rules

These rules handle "and" statements, combining or separating conjuncts. Conjunctions assert that both parts hold—these rules let you assemble or disassemble them.

Conjunction

• Combines two statements into one compound: $P$, $Q$, therefore $P \land Q$
• Builds complex statements from simpler ones; both premises must be independently established first
• Often the final step in proofs requiring compound conclusions—gather your pieces, then conjoin them

Simplification

• Extracts one conjunct from a conjunction: $P \land Q$, therefore $P$ (or therefore $Q$)
• Breaks apart compound information to isolate what you need for the next step
• Apply early in proofs when premises contain conjunctions—extract the components you'll use separately

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Resolution

This rule is a generalized form of Disjunctive Syllogism used extensively in automated theorem proving.

Resolution

• Resolves complementary literals across disjunctions: $P \lor Q$, $\neg P \lor R$, therefore $Q \lor R$
• Cancels matching positive/negative pairs and combines the remaining disjuncts; the $P$ and $\neg P$ "resolve away"
• Foundation of resolution refutation in AI and logic programming—converts everything to clausal form and repeatedly resolves

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Quick Reference Table

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Self-Check Questions

1. Which two rules both work by denying or eliminating to reach a conclusion, and what's the key structural difference between them?

2. You have the premises $A \rightarrow B$, $B \rightarrow C$, and $A$. Which two rules would you apply, and in what order, to derive $C$?

3. Compare and contrast Addition and Simplification: why does one "weaken" information while the other "strengthens" it, and when would you strategically use each?

4. Given $(P \rightarrow Q) \land (R \rightarrow S)$ as a premise, what additional premise would you need to apply Constructive Dilemma versus Destructive Dilemma?

5. A proof requires you to derive $\neg A \land \neg B$ from conditional premises. Which rule of inference should you look to apply, and what form must your premises take?