Valid Argument Forms

Why This Matters

Valid argument forms are the building blocks of logical proof—they're the moves you're allowed to make when constructing an argument or evaluating someone else's reasoning. In Formal Logic I, you're being tested on your ability to recognize these forms, apply them correctly in proofs, and understand why each inference is truth-preserving. These aren't arbitrary rules; each form captures a fundamental relationship between logical connectives that guarantees if your premises are true, your conclusion must be true.

Think of these forms as falling into distinct categories: inference rules that let you derive new statements, equivalence rules that let you swap logically identical expressions, and quantifier rules that bridge general claims and specific instances. Don't just memorize the symbolic patterns—know what logical principle each form exploits and when to reach for it in a proof. The difference between a struggling student and a confident one is understanding the why behind each rule.

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Conditional Inference Rules

These rules govern how we reason with "if-then" statements (conditionals). The conditional $P \rightarrow Q$ creates a one-way logical dependency—and these rules show you exactly how to exploit that dependency to derive new conclusions.

Modus Ponens

• Affirm the antecedent, derive the consequent—if you have $P \rightarrow Q$ and $P$, you can conclude $Q$
• The most fundamental inference rule in propositional logic; you'll use this constantly in proofs
• Pattern recognition tip: look for a conditional and its antecedent appearing separately in your premises

Modus Tollens

• Deny the consequent, derive the negation of the antecedent—if you have $P \rightarrow Q$ and $\neg Q$, conclude $\neg P$
• Essential for indirect reasoning and disproving hypotheses; the contrapositive in action
• Common exam trap: students confuse this with the invalid "denying the antecedent" fallacy

Hypothetical Syllogism

• Chain conditionals together—from $P \rightarrow Q$ and $Q \rightarrow R$, derive $P \rightarrow R$
• Transitivity of implication allows you to build longer inferential chains from shorter ones
• Proof strategy: use this to connect distant premises when you see a "bridge" variable appearing in two conditionals

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Disjunction Rules

These rules handle "or" statements (disjunctions). The disjunction $P \lor Q$ asserts that at least one disjunct is true—and these rules let you either eliminate options or introduce new ones.

Disjunctive Syllogism

• Eliminate one disjunct to conclude the other—from $P \lor Q$ and $\neg P$, derive $Q$
• Process of elimination formalized; if it's not this one, it must be that one
• Proof tip: when you have a disjunction and can derive the negation of one side, this rule closes the gap

Addition

• Introduce a disjunction from any true statement—from $P$, derive $P \lor Q$ for any $Q$
• Seems oddly permissive but is logically valid; adding alternatives never makes a true statement false
• Strategic use: often needed to set up Constructive Dilemma or match a desired conclusion form

Constructive Dilemma

• Combine two conditionals with a disjunction—from $P \rightarrow Q$, $R \rightarrow S$, and $P \lor R$, derive $Q \lor S$
• "Fork in the road" reasoning: either way leads somewhere, so one of those destinations is reached
• Complex but powerful—look for this when you have multiple conditionals and need to preserve optionality

Destructive Dilemma

• The contrapositive version of Constructive Dilemma—from $P \rightarrow Q$, $R \rightarrow S$, and $\neg Q \lor \neg S$, derive $\neg P \lor \neg R$
• Eliminates antecedents when consequents fail; if neither outcome happened, neither cause occurred
• Less common in basic proofs but essential for understanding the symmetry of dilemma reasoning

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Conjunction Rules

These rules govern "and" statements (conjunctions). The conjunction $P \land Q$ requires both conjuncts to be true—so you can build them from separate truths or break them apart.

Conjunction

• Combine two true statements—from $P$ and $Q$ (separately established), derive $P \land Q$
• The only way to introduce $\land$ in a proof; you must have proven both parts independently
• Proof tip: often the final step when your conclusion is a conjunction

Simplification

• Extract either conjunct—from $P \land Q$, derive $P$ (or derive $Q$)
• Break apart compound premises to access the pieces you need for other rules
• Don't overlook this rule—students sometimes forget they can "unpack" conjunctions mid-proof

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Equivalence Rules: Negation and Transformation

Equivalence rules let you replace expressions with logically identical alternatives. Unlike inference rules, equivalences work in both directions and can be applied to subformulas within larger expressions.

Double Negation

• Two negations cancel out—$\neg \neg P$ is equivalent to $P$
• Works both ways: introduce or eliminate double negations as needed
• Frequently necessary to set up Modus Tollens or clean up after other transformations

De Morgan's Laws

• Distribute negation across connectives—$\neg(P \land Q) \equiv (\neg P \lor \neg Q)$ and $\neg(P \lor Q) \equiv (\neg P \land \neg Q)$
• Negation flips the connective: and becomes or, or becomes and
• Exam essential: these appear constantly; practice until the transformation is automatic

Transposition (Contraposition)

• Flip and negate a conditional—$P \rightarrow Q$ is equivalent to $\neg Q \rightarrow \neg P$
• The logical basis for Modus Tollens; both forms say exactly the same thing
• Proof strategy: transpose when you have the negation of a consequent and need to derive something

Material Implication

• Rewrite conditionals as disjunctions—$P \rightarrow Q$ is equivalent to $\neg P \lor Q$
• Reveals the "hidden or" inside every if-then statement
• Use this when you need to apply disjunction rules to a conditional or vice versa

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Equivalence Rules: Structure and Rearrangement

These rules let you reorganize logical expressions without changing their truth value. They're about flexibility in how you write formulas, not deriving new information.

Commutation

• Order doesn't matter for $\land$ and $\lor$—$P \land Q \equiv Q \land P$ and $P \lor Q \equiv Q \lor P$
• Rearrange conjunctions and disjunctions to match required forms or improve readability
• Note: this does NOT apply to conditionals; $P \rightarrow Q$ is not equivalent to $Q \rightarrow P$

Association

• Grouping doesn't matter for repeated connectives—$(P \land Q) \land R \equiv P \land (Q \land R)$
• Regrouping parentheses lets you access different pairs for other rules
• Same principle applies to disjunctions: $(P \lor Q) \lor R \equiv P \lor (Q \lor R)$

Distribution

• Distribute one connective over another—$P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)$
• Works both directions and for both connective combinations; like distributing multiplication over addition
• Key transformation for converting between conjunctive and disjunctive normal forms

Absorption

• A statement absorbs redundant information—$P \rightarrow Q$ is equivalent to $P \rightarrow (P \land Q)$
• If P guarantees Q, then P guarantees both P and Q; the antecedent is already "included"
• Useful for simplification or when you need to introduce a conjunction into a consequent

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Advanced Forms and Concepts

These forms extend beyond basic propositional logic or represent important logical concepts you'll encounter throughout your studies.

Exportation

• Restructure nested implications—$(P \land Q) \rightarrow R$ is equivalent to $P \rightarrow (Q \rightarrow R)$
• "Currying" in logic: a function of two inputs becomes a function returning a function
• Proof flexibility: choose whichever form makes your current inference easier

Universal Instantiation

• Apply general claims to specific cases—from $\forall x \, Px$, derive $Pa$ for any constant $a$
• The bridge from predicate to propositional reasoning; what's true of all is true of each
• Foundation of predicate logic proofs—you'll use this whenever you need to "unpack" a universal statement

Tautology

• A statement true under all interpretations—like $P \lor \neg P$ (law of excluded middle)
• Not an inference rule but a concept; tautologies can be introduced at any point in a proof
• Validity connection: an argument is valid if and only if the conditional linking premises to conclusion is a tautology

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Quick Reference Table

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Self-Check Questions

1. Both Modus Ponens and Modus Tollens use a conditional premise—what's the key difference in what additional premise each requires, and what conclusion each produces?

2. You have the premises $P \rightarrow Q$, $Q \rightarrow R$, and $P$. Which two rules would you use (in order) to derive $R$?

3. Compare De Morgan's Laws and Distribution: both transform expressions with mixed connectives. When would you use each one?

4. If you need to derive $\neg P$ and you have $P \rightarrow Q$ as a premise, what other premise do you need and which rule applies?

5. (FRQ-style) Given premises $A \rightarrow B$, $C \rightarrow D$, and $A \lor C$, construct a proof deriving $B \lor D$. Name each rule you apply.