Logic Mastery Academy

Logic → Valid Reasoning

Understanding the systematic study of valid reasoning and inference

Why Learn Logic?

Logic Skills → Academic & Professional Success

Academic benefits and practical applications of formal logical reasoning

Essential Terms

Premise₁, Premise₂, ... ⊢ Conclusion

Key vocabulary: propositions, premises, conclusions, and inference rules

Learning Goals

Study + Practice → Mastery

Master inference rules, develop critical thinking, and apply logical principles

Tools of the Trade

Tools + Practice → Understanding

Interactive introduction to logical tools: truth tables, proofs, calculators, and more

Symbols

Learn what each logical symbol means and how to read logical statements.

Variables (P, Q, R)

P, Q, R represent propositions

Propositional variables that represent statements or propositions

P could be "It is raining"

Q could be "I have an umbrella"

Any letter can represent any proposition

Negation (¬)

If P is true, then ¬P is false

The "not" operator - reverses the truth value of a statement

¬P means "not P"

If P is false, then ¬P is true

Conjunction (∧)

True only when both P and Q are true

The "and" operator - true only when both parts are true

P∧Q means "P and Q"

False if either P or Q (or both) is false

Disjunction (∨)

True when P is true, Q is true, or both are true

The "or" operator - true when at least one part is true

P∨Q means "P or Q"

False only when both P and Q are false

Implication (→)

P→Q means "if P then Q"

The "if...then" operator - false only when antecedent is true and consequent is false

False only when P is true and Q is false

True in all other cases

P is the antecedent, Q is the consequent

Biconditional (↔)

P↔Q means "P if and only if Q"

The "if and only if" operator - true when both sides have the same truth value

True when P and Q are both true or both false

False when P and Q have different truth values

Equivalent to (P→Q)∧(Q→P)

True (⊤)

⊤ means "true" or "tautology"

The logical constant representing truth - always true

Always has the truth value true

P∨⊤ is always true (regardless of P)

P∧⊤ simplifies to just P

False (⊥)

⊥ means "false" or "contradiction"

The logical constant representing falsehood - always false

Always has the truth value false

P∧⊥ is always false (regardless of P)

P∨⊥ simplifies to just P

Entailment (⊢)

P, Q ⊢ R means "from P and Q, we can derive R"

Shows that a conclusion logically follows from premises

Left side: premises (what we know)

Right side: conclusion (what follows)

The turnstile symbol means "therefore" or "proves"

Parentheses ( )

Operations inside parentheses are evaluated first

Grouping symbols that control the order of operations

(P∧Q)∨R vs P∧(Q∨R)

Change the meaning: (P∧Q)∨R ≠ P∧(Q∨R)

Use to make complex statements clear

Equivalence (≡)

P≡Q means "P is equivalent to Q"

Concept Only

Shows that two formulas are logically equivalent📚 This concept is taught here but not yet supported in the interactive calculator.

Both formulas have the same truth value in all cases

Can substitute one for the other in any context

Different from biconditional (↔) which is a connective

Universal Quantifier (∀)

∀x P(x) means "for all x, P(x) is true"

Concept Only

Means "for all" or "for every" - applies to all objects📚 This concept is taught here but not yet supported in the interactive calculator.

Used in predicate logic

∀x (Human(x) → Mortal(x)) means "all humans are mortal"

Must be true for every possible value of x

Existential Quantifier (∃)

∃x P(x) means "there exists an x such that P(x) is true"

Concept Only

Means "there exists" or "for some" - applies to at least one object📚 This concept is taught here but not yet supported in the interactive calculator.

Used in predicate logic

∃x (Cat(x) ∧ Orange(x)) means "there exists an orange cat"

Only needs to be true for at least one value of x

Basic Rules

Master the fundamental building blocks of logical reasoning.

Modus Ponens

If P implies Q, and P is true, then Q must be true

P→Q, P ⊢ Q

Modus Tollens

If P implies Q, and Q is false, then P must be false

P→Q, ¬Q ⊢ ¬P

Hypothetical Syllogism

If P implies Q and Q implies R, then P implies R

P→Q, Q→R ⊢ P→R

Disjunctive Syllogism

If P or Q is true, and P is false, then Q must be true

If P is true and Q is true, then P and Q is true

P, Q ⊢ P∧Q

Simplification

If P and Q is true, then P is true (and Q is true)

If P is true, then P or Q is true (for any Q)

P ⊢ P∨Q

Commutation

¬(P∧¬P) (negation of a contradiction)

Reorder operands in conjunctions and disjunctions

Remove or add double negations

Simplify repeated disjunctions or conjunctions

Intermediate Rules

Build on the basics with these more sophisticated reasoning patterns.

Constructive Dilemma

From (P→Q)∧(R→S) and P∨R, infer Q∨S

(P→Q)∧(R→S), P∨R ⊢ Q∨S

Absorption

From P→Q, infer P→(P∧Q)

P→Q ⊢ P→(P∧Q)

Association

Regroup operands in conjunctions and disjunctions

Distribute conjunctions over disjunctions and vice versa

P∧(Q∨R) ⊢ (P∧Q)∨(P∧R)

(P∧Q)∨(P∧R) ⊢ P∧(Q∨R)

P∨(Q∧R) ⊢ (P∨Q)∧(P∨R)

(P∨Q)∧(P∨R) ⊢ P∨(Q∧R)

De Morgan's Laws