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 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.

What separates students who ace proofs from those who struggle is 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.

---

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. It appears in nearly every proof you'll write.
• Direction matters: you must have the antecedent $P$ true, not the consequent. Concluding $P$ from $P \rightarrow Q$ and $Q$ is the formal fallacy of affirming the consequent.

Modus Tollens

• Denies the antecedent by denying the consequent: given $P \rightarrow Q$ and $\neg Q$, conclude $\neg P$
• This exploits the contrapositive relationship: $P \rightarrow Q$ is logically equivalent to $\neg Q \rightarrow \neg P$. If the consequent is false, the antecedent must also be false.
• Especially useful in 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 $P \rightarrow R$
• The "middle term" $Q$ drops out, linking $P$ directly to $R$. This is the transitivity of implication.
• Essential for multi-step proofs where you need to connect premises that don't directly share variables.

---

Disjunction Rules

These rules govern "or" statements: how to use them and how to create them. Remember that logical disjunction is inclusive, meaning both disjuncts can be true simultaneously.

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.
• This rule is symmetric: you can also use $\neg Q$ to derive $P$.

Addition

• Introduces a disjunction from a single true statement: given $P$, conclude $P \lor Q$ for any $Q$
• This can feel counterintuitive at first. But it's logically valid because adding possibilities can never make a true statement false. If "it's raining" is true, then "it's raining or pigs can fly" is also true.
• Strategically useful in proofs when you need to set up Constructive Dilemma or match the form of a target conclusion.

Constructive Dilemma

• Combines a 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.
• Premise-heavy (requires three premises), so look for it when you have a disjunction and multiple conditionals available.

---

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 $Q$)
• When you receive a conjunction as a premise, consider extracting both parts right away. This gives you maximum flexibility for later steps.

---

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

• $\neg\neg P$ 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. For example, if Modus Tollens gives you $\neg\neg A$, apply Double Negation to get $A$.

De Morgan's Laws

Two rules that transform negated compounds into compound negations:

• $\neg(P \land Q) \equiv \neg P \lor \neg Q$
• $\neg(P \lor Q) \equiv \neg P \land \neg Q$

The negation "flips" the connective: AND becomes OR, OR becomes AND. Think of the negation as "breaking through" the parentheses, but it changes the operator as it does so. These work in both directions, so you can also go from $\neg P \lor \neg Q$ back to $\neg(P \land Q)$.

---

Quick Reference Table

---

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 $C$.