Logic Basics: Essential Principles and Common Fallacies

Logic Basics for Students: From Propositions to ProofsLogic is the foundation of clear thinking, rigorous argumentation, and mathematical proof. For students across disciplines — mathematics, computer science, philosophy, or any field that values precise reasoning — learning the basics of logic sharpens your ability to form correct arguments, spot errors in reasoning, and construct proofs that others can verify. This article introduces core concepts in propositional and predicate logic, common logical tools and techniques, proof strategies, and examples to practice.

What is logic?

Logic is the study of valid reasoning. It asks: given some statements (premises), which conclusions follow necessarily? Logic separates good arguments (where the conclusion must be true if the premises are true) from bad ones (where the conclusion might be false despite true premises). There are formal systems for logic that define precise syntax (how formulas are written) and semantics (what formulas mean).

1. Propositional logic (sentential logic)

Propositional logic deals with whole statements (propositions) that are either true or false. It uses logical connectives to build complex statements from simpler ones.

Basic elements

Proposition: a declarative sentence that is either true or false (e.g., “It is raining.”).
Propositional variables: symbols like P, Q, R that stand for propositions.
Logical connectives:
- Negation: ¬P (not P)
- Conjunction: P ∧ Q (P and Q)
- Disjunction: P ∨ Q (P or Q; inclusive unless specified)
- Implication: P → Q (if P then Q)
- Biconditional: P ↔ Q (P if and only if Q)

Truth tables

Each connective has a truth table describing the truth value of a compound proposition given the truth values of its components. For example, implication P → Q is false only when P is true and Q is false.

Logical equivalence and laws

Two propositions are logically equivalent if they have identical truth values in all possible cases. Important equivalences include:

De Morgan’s laws: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q; ¬(P ∨ Q) ≡ ¬P ∧ ¬Q
Distribution: P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
Implication rewrite: P → Q ≡ ¬P ∨ Q

Recognizing equivalences lets you simplify expressions and transform statements into standard forms.

Normal forms

Conjunctive Normal Form (CNF): an AND of OR-clauses (useful in automated theorem proving).
Disjunctive Normal Form (DNF): an OR of AND-clauses.

Conversion to normal forms uses distribution and De Morgan’s laws.

2. Predicate logic (first-order logic)

Propositional logic treats whole statements as atomic. Predicate logic refines this by analyzing internal structure — predicates applied to objects and quantifiers.

Components

Domain (universe): the set of objects under discussion.
Predicates: properties or relations, e.g., P(x) = “x is prime”, R(x,y) = “x < y”.
Terms: variables, constants, and function symbols that refer to objects.
Quantifiers:
- Universal: ∀x P(x) — “for all x, P(x)”
- Existential: ∃x P(x) — “there exists an x such that P(x)”

Examples

∀x (Human(x) → Mortal(x)) means “All humans are mortal.”
∃y (Prime(y) ∧ y > 100) means “There exists a prime greater than 100.”

Scope and binding

A quantifier binds variables in its scope. Free variables make a formula open (not a sentence); sentences have no free variables and have definite truth values relative to a structure.

Translating English to logic

Practice translating carefully: phrase like “Everyone loves someone” can be ambiguous.

∀x ∃y Loves(x,y) — each person loves possibly different someone.
∃y ∀x Loves(x,y) — there is one someone loved by everyone.

3. Logical consequence, validity, and satisfiability

Logical consequence: A formula ψ is a logical consequence of premises Γ (Γ ⊨ ψ) if every interpretation/models that makes all formulas in Γ true also makes ψ true.
Validity: A formula is valid if it is true under every interpretation (tautology in propositional logic).
Satisfiability: A formula is satisfiable if there is some interpretation under which it is true. Unsatisfiable means it is false in all interpretations (a contradiction).

4. Proof systems and methods

Logic has formal proof systems that let you derive conclusions syntactically: natural deduction, sequent calculus, Hilbert-style systems, and resolution for automated reasoning.

Natural deduction (informal sketch)

Natural deduction uses introduction and elimination rules for connectives:

∧-introduction: from P and Q infer P ∧ Q.
∧-elimination: from P ∧ Q infer P (or Q).
→-introduction: assume P, derive Q; then infer P → Q (discharge the assumption).
→-elimination (modus ponens): from P and P → Q infer Q.
¬-introduction/elimination and rules for quantifiers also exist.

Natural deduction mirrors how mathematicians reason informally and is good for constructing human-readable proofs.

Proof strategies

Direct proof: assume premises and use rules to reach the conclusion.
Proof by contrapositive: prove ¬Q → ¬P instead of P → Q.
Proof by contradiction: assume the negation of the conclusion and derive a contradiction, concluding the original must be true.
Proof by induction: for statements about natural numbers — base case + inductive step.

Example — direct proof (simple): Claim: If n is even then n^2 is even. Proof: n = 2k for some k. Then n^2 = 4k^2 = 2(2k^2), which is even.

5. Common logical fallacies and pitfalls

Knowing mistakes helps avoid them.

Affirming the consequent: From P → Q and Q, wrongly infer P.
Denying the antecedent: From P → Q and ¬P, wrongly infer ¬Q.
Equivocation: Using an ambiguous term in different senses.
False cause (post hoc): Inferring causation from mere correlation.

6. Proof examples: from propositions to predicate proofs

Example 1 — Propositional: Premises: (P → Q), (Q → R). Show P → R. Proof: Assume P. From P and P → Q get Q (modus ponens). From Q and Q → R get R. Thus P → R by →-introduction, discharging the assumption.

Example 2 — Predicate: Premises: ∀x (Student(x) → Enrolled(x)), Student(Alice). Show Enrolled(Alice). Proof: From ∀x (Student(x) → Enrolled(x)) instantiate with Alice to get Student(Alice) → Enrolled(Alice). From Student(Alice) and modus ponens infer Enrolled(Alice).

Example 3 — Proof by contradiction: Claim: √2 is irrational. Sketch: Assume √2 = p/q in lowest terms. Square both sides: 2q^2 = p^2. So p^2 is even ⇒ p is even; write p = 2k. Substitute back gives q^2 even ⇒ q even. But then p and q share factor 2, contradicting lowest terms. Thus √2 is irrational.

7. Exercises (with brief answers)

Determine truth table and see if (P → Q) is equivalent to (¬P ∨ Q). Answer: Yes; truth tables match.
Convert (P ∧ (Q ∨ R)) to CNF. Answer: (P ∧ (Q ∨ R)) is already in CNF if treated as a single conjunction with clause (Q ∨ R) combined with P — to distribute: (P ∧ (Q ∨ R)) ≡ (P ∧ Q) ∨ (P ∧ R) which is DNF; CNF would be (P) ∧ (Q ∨ R).
Translate: “There is a person who loves everyone.” Answer: ∃x ∀y Loves(x,y).
Show that from P ∨ Q and ¬P you can infer Q. Answer: Use disjunctive syllogism.
Prove: If every student passed and Alice is a student, then Alice passed. Answer: universal instantiation + modus ponens.

8. Tips for students

Practice translating English statements into logical formulas and back.
Work through truth tables to build intuition about connectives.
Learn common equivalences and normal forms — they’re tools you’ll reuse.
Write proofs in clear steps, stating when you use assumptions or rules.
Study both formal rules (natural deduction) and informal proof-writing (mathematical style).

9. Resources for further study

Introductory textbooks: “Language, Proof and Logic” (Barwise & Etchemendy), “How to Prove It” (Velleman).
Online: lecture notes on propositional and predicate logic, practice problems, automated theorem provers to test formulas.
Programming: implement truth-table checkers, SAT solvers, or simple natural deduction proofs to deepen understanding.

Logic turns vague reasoning into precise structure. For students, mastering the concepts from propositions to proofs opens doors to rigorous mathematics, correct algorithms, and clearer arguments in any discipline.