Propositional Logic Rules

Why This Matters

Propositional logic rules are the engine of formal reasoning—they're how you move from premises to conclusions in a valid, airtight way. In Formal Logic I, you're being tested on your ability to recognize, apply, and justify each inference step. These rules fall into patterns: some let you break apart compound statements, others let you build them up, and still others let you chain implications together. Understanding these categories means you won't just memorize ten disconnected rules—you'll see the logical architecture underneath.

Here's what separates students who ace proofs from those who struggle: knowing which rule to reach for and why. When you see a conditional, do you need Modus Ponens or Hypothetical Syllogism? When you hit a negated compound, is it time for De Morgan's Laws? Don't just memorize the formulas—know what logical work each rule performs and when it's your best tool.

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Conditional Elimination Rules

These rules let you extract conclusions from conditional statements. When you have an "if-then" relationship, these are your go-to tools for drawing inferences.

Modus Ponens

• Affirms the consequent by affirming the antecedent—given $P \rightarrow Q$ and $P$, conclude $Q$
• The most fundamental inference rule in propositional logic; appears in nearly every proof you'll write
• Direction matters: you must have the antecedent $P$ true, not the consequent—affirming $Q$ is a formal fallacy

Modus Tollens

• Denies the antecedent by denying the consequent—given $P \rightarrow Q$ and $\neg Q$, conclude $\neg P$
• Exploits the contrapositive relationship: if the consequent fails, the antecedent must also fail
• Key for indirect proofs where you're working backward from a negation to establish what can't be true

Hypothetical Syllogism

• Chains conditionals together—given $P \rightarrow Q$ and $Q \rightarrow R$, conclude $P \rightarrow R$
• Creates transitivity of implication: the "middle term" $Q$ drops out, linking $P$ directly to $R$
• Essential for multi-step proofs where you need to connect premises that don't share variables directly

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Disjunction Rules

These rules govern "or" statements—how to use them and how to create them. Remember that logical disjunction is inclusive (both can be true).

Disjunctive Syllogism

• Eliminates one disjunct to affirm the other—given $P \lor Q$ and $\neg P$, conclude $Q$
• Works by process of elimination: if one option is ruled out, the other must hold
• Symmetric rule: you can also use $\neg Q$ to derive $P$—either disjunct can be eliminated

Addition

• Introduces a disjunction from a single true statement—given $P$, conclude $P \lor Q$ for any $Q$
• Seems counterintuitive but is logically valid: adding possibilities can't make a true statement false
• Strategic use in proofs: often needed to set up Constructive Dilemma or match a target conclusion's form

Constructive Dilemma

• Combines disjunction with two conditionals—given $P \lor Q$, $P \rightarrow R$, and $Q \rightarrow S$, conclude $R \lor S$
• Handles "either way" reasoning: whichever disjunct is true, you get a corresponding conclusion
• Powerful but premise-heavy: requires three premises, so look for it when you have multiple conditionals available

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Conjunction Rules

These rules handle "and" statements—building them up and breaking them down. They're simpler than conditional rules but essential for managing compound premises.

Conjunction

• Combines two true statements into one—given $P$ and $Q$ separately, conclude $P \land Q$
• Order doesn't matter logically: $P \land Q$ is equivalent to $Q \land P$
• Often the final step in proofs where the conclusion is a compound statement

Simplification

• Extracts individual components from a conjunction—given $P \land Q$, conclude $P$ (or conclude $Q$)
• Reverses conjunction: what was built up can be broken down
• Apply it immediately when you receive a conjunction as a premise—extract both parts for maximum flexibility

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Negation and Equivalence Rules

These rules transform statements by manipulating negations. They're essential for simplifying expressions and converting between logically equivalent forms.

Double Negation

• Eliminates or introduces paired negations—$\neg\neg P$ is equivalent to $P$
• Works both directions: you can remove $\neg\neg$ or add it strategically
• Often necessary after Modus Tollens or other rules that introduce negations you need to clean up

De Morgan's Laws

• Transforms negated compounds into compound negations—$\neg(P \land Q) \equiv \neg P \lor \neg Q$ and $\neg(P \lor Q) \equiv \neg P \land \neg Q$
• The negation "flips" the connective: AND becomes OR, OR becomes AND
• Memorization tip: negation distributes over the compound but changes the operator—think of it as "breaking through" the parentheses

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Quick Reference Table

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Self-Check Questions

1. You have $P \rightarrow Q$, $Q \rightarrow R$, and $P$ as premises. Which two rules would you use to derive $R$, and in what order?

2. Compare Modus Ponens and Modus Tollens: what do they share structurally, and what distinguishes the direction of their reasoning?

3. Given the premise $\neg(A \lor B)$, what can you derive using De Morgan's Laws? How does this differ from applying De Morgan's to $\neg(A \land B)$?

4. A proof requires you to derive $P \lor R$ from the premise $P$. Which rule applies, and why might this feel counterintuitive at first?

5. You're stuck in a proof with premises $A \land B$ and $A \rightarrow C$. Outline the two-step strategy using Simplification and another rule to derive $C$.