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Logic Terms and Definitions

About This Glossary

This glossary organizes 28 propositional logic terms into two categories that cover the building blocks of logical reasoning and their meaning.

Syntax defines the structural side of propositional logic across 11 entries: propositions (elementary and compound), well-formed formulas, literals, the five standard logical connectives (negation, conjunction, disjunction, and their roles in building expressions), and the two canonical normal forms -- Disjunctive Normal Form (DNF) and Conjunctive Normal Form (CNF). These terms describe how valid logical expressions are constructed from symbols.

Semantics addresses meaning and truth across 17 entries: the conditional and biconditional connectives, the anatomy of implications (antecedent, consequent, converse, contrapositive, inverse), logical equivalence, the three truth-value classifications (tautology, contradiction, contingency), satisfiability, truth tables, assignments, absorption, and the foundational laws of excluded middle and non-contradiction. These terms describe how logical expressions are evaluated and what their truth values reveal.

Each definition includes an intuitive explanation, formal properties, worked examples, and links to detailed lesson pages. Use the search bar or category filters above to navigate.
SemanticsSyntax
AAbsorptionSemanticsAAntecedentSemanticsAAssignment (Valuation)SemanticsBBiconditionalSemanticsCCompound PropositionSyntaxCConditional (Implication)SemanticsCConjunctionSyntaxCConjunctive Normal Form (CNF)SyntaxCConsequentSemanticsCContingencySemanticsCContradictionSemanticsCContrapositiveSemanticsCConverseSemanticsDDisjunctionSyntaxDDisjunctive Normal Form (DNF)SyntaxEElementary PropositionSyntaxIInverse (of Conditional)SemanticsLLaw of Excluded MiddleSemanticsLLiteralSyntaxLLogical ConnectiveSyntaxLLogical EquivalenceSemanticsNNegationSyntaxNNon-contradictionSemanticsPPropositionSyntaxSSatisfiabilitySemanticsTTautologySemanticsTTruth TableSemanticsWWell-Formed Formula (WFF)Syntax
Semantics(17)
Syntax(11)
28 of 28 terms
Semantics
Conditional (Implication)BiconditionalAntecedentConsequentConverseContrapositiveInverse (of Conditional)Logical EquivalenceTautologyContradictionContingencySatisfiabilityTruth TableAssignment (Valuation)AbsorptionLaw of Excluded MiddleNon-contradiction
Syntax
PropositionElementary PropositionCompound PropositionWell-Formed Formula (WFF)LiteralLogical ConnectiveNegationConjunctionDisjunctionDisjunctive Normal Form (DNF)Conjunctive Normal Form (CNF)

28 terms

Syntax

(11 items)

Proposition

A declarative statement that carries exactly one truth value: true or false, but not both
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intuitionexamplesrelated concepts
A proposition is the atomic unit of logical reasoning — a complete claim whose truth can be evaluated. Questions, commands, and exclamations are not propositions because they lack a definite truth value.
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Elementary Proposition

An atomic, indivisible statement PP that cannot be broken into simpler logical components
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intuitionpropertiesrelated concepts
Elementary propositions are the building blocks from which all compound expressions are constructed. They are denoted by single letters — PP, QQ, RR — and carry a truth value on their own, without internal logical structure.
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Compound Proposition

A proposition built by combining simpler propositions using logical connectives: φ:=P∘Q\varphi := P \circ Q where ∘∈{∧,∨,→,↔,¬}\circ \in \{\land, \lor, \to, \leftrightarrow, \neg\}
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intuitionexamplesrelated concepts
Compound propositions are constructed from elementary propositions and connectives according to formation rules. Their truth value is determined entirely by the truth values of their components and the connectives joining them.
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Well-Formed Formula (WFF)

An expression constructed according to the formation rules of propositional logic, ensuring unambiguous syntactic structure
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intuitionpropertiesrelated concepts
A WFF is a syntactically valid logical expression. Just as natural language has grammar, propositional logic has formation rules that determine which symbol sequences constitute proper formulas. An expression that violates these rules — like "∧P)(Q\land P )(Q" — is not a WFF and cannot be assigned a truth value.
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Literal

An atomic proposition or its negation: PP (positive literal) or ¬P\neg P (negative literal)
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intuitionexamplesrelated concepts
Literals are the simplest meaningful units in normal form representations. Every clause in CNF or DNF is built entirely from literals joined by a single connective type.
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Logical Connective

An operator that combines or modifies propositions to form compound propositions: ¬\neg, ∧\land, ∨\lor, →\to, ↔\leftrightarrow
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intuitionpropertiesrelated concepts
Connectives are the glue of propositional logic. Each one defines a truth function — a rule that determines the truth value of the compound expression from the truth values of its components.
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Negation

A unary connective ¬P\neg P that reverses the truth value of PP
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intuitionpropertiesrelated concepts
Negation flips true to false and false to true. It is the only standard unary connective in propositional logic.
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Conjunction

A binary connective P∧QP \land Q that is true only when both PP and QQ are true
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intuitionpropertiesrelated concepts
Conjunction corresponds to "and" in natural language. The compound statement P∧QP \land Q requires both components to hold simultaneously.
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Disjunction

A binary connective P∨QP \lor Q that is true when at least one of PP or QQ is true
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intuitionpropertiesrelated concepts
Disjunction corresponds to the inclusive "or" in natural language. The compound statement P∨QP \lor Q holds whenever one or both components are true. It is false only when both are false.
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Disjunctive Normal Form (DNF)

A formula expressed as a disjunction of conjunctions of literals: (L1∧L2)∨(L3∧L4)∨…(L_1 \land L_2) \lor (L_3 \land L_4) \lor \ldots
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intuitionexamplesrelated concepts
In DNF, the top-level operator is OR, and each clause is a conjunction (AND) of literals. Every propositional formula can be converted to an equivalent DNF expression. DNF is obtained directly from a truth table by forming one conjunction per true row and joining them with OR.
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Conjunctive Normal Form (CNF)

A formula expressed as a conjunction of disjunctions of literals: (L1∨L2)∧(L3∨L4)∧…(L_1 \lor L_2) \land (L_3 \lor L_4) \land \ldots
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intuitionexamplesrelated concepts
In CNF, the top-level operator is AND, and each clause is a disjunction (OR) of literals. Every propositional formula has an equivalent CNF representation. CNF is widely used in automated theorem proving and satisfiability testing (SAT solvers).
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Semantics

(17 items)

Conditional (Implication)

P→QP \to Q: false only when PP is true and QQ is false; true in all other cases
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intuitionpropertiesrelated concepts
The conditional "if PP then QQ" asserts a dependency: whenever PP holds, QQ must also hold. When PP is false, the conditional is vacuously true — a promise not triggered cannot be broken.
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Biconditional

P↔QP \leftrightarrow Q: true when PP and QQ share the same truth value, equivalent to (P→Q)∧(Q→P)(P \to Q) \land (Q \to P)
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intuitionpropertiesrelated concepts
The biconditional "PP if and only if QQ" asserts that both propositions are logically tied: they are true together and false together. It combines both directions of implication into a single statement.
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Antecedent

In a conditional P→QP \to Q, the antecedent is PP — the hypothesis or "if" part
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intuitionexamplesrelated concepts
The antecedent is the condition that triggers the implication. When the antecedent is false, the entire conditional is vacuously true regardless of the consequent.
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Consequent

In a conditional P→QP \to Q, the consequent is QQ — the conclusion or "then" part
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intuitionexamplesrelated concepts
The consequent is the result claimed by the implication. The conditional is violated only when the antecedent holds but the consequent does not.
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Converse

The converse of P→QP \to Q is Q→PQ \to P — the conditional with antecedent and consequent swapped
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intuitioncommon errorsrelated concepts
Swapping "if" and "then" produces the converse. A conditional and its converse are generally not equivalent — the truth of one does not guarantee the truth of the other.
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Contrapositive

The contrapositive of P→QP \to Q is ¬Q→¬P\neg Q \to \neg P — always logically equivalent to the original
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intuitionpropertiesrelated concepts
The contrapositive negates both parts and swaps their positions. Unlike the converse, the contrapositive is guaranteed to have the same truth value as the original conditional. This equivalence underpins proof by contrapositive.
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Inverse (of Conditional)

The inverse of P→QP \to Q is ¬P→¬Q\neg P \to \neg Q — not logically equivalent to the original
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intuitioncommon errorsrelated concepts
The inverse negates both parts without swapping. It is logically equivalent to the converse (Q→PQ \to P), but not to the original conditional. Confusing a conditional with its inverse is a common reasoning error.
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Logical Equivalence

Two formulas φ\varphi and ψ\psi are logically equivalent (φ≡ψ\varphi \equiv \psi) if they have identical truth values under every possible assignment
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intuitionpropertiesrelated concepts
Logical equivalence means two expressions are interchangeable in any context without altering truth. It is the semantic counterpart of the biconditional: φ≡ψ\varphi \equiv \psi holds exactly when φ↔ψ\varphi \leftrightarrow \psi is a tautology.
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Tautology

A formula φ\varphi that evaluates to true under every possible truth assignment. Notation: ⊨φ\models \varphi or φ≡⊤\varphi \equiv \top
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intuitionexamplesrelated concepts
A tautology is true by virtue of its logical form alone, regardless of what its component propositions say about the world. The truth table for a tautology contains only T in the output column.
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Contradiction

A formula φ\varphi that evaluates to false under every possible truth assignment. Notation: φ≡⊥\varphi \equiv \bot
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intuitionexamplesrelated concepts
A contradiction is false by logical structure alone. No matter what truth values its variables take, the formula can never be satisfied. The negation of any tautology is a contradiction, and vice versa.
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Contingency

A formula that is neither a tautology nor a contradiction — it is true under some assignments and false under others
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intuitionexamplesrelated concepts
Most propositions encountered in practice are contingencies. Their truth value depends on the specific circumstances — unlike tautologies (always true) or contradictions (always false). A contingency is satisfiable but not valid.
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Satisfiability

A formula φ\varphi is satisfiable if there exists at least one truth assignment under which φ\varphi evaluates to true
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intuitionpropertiesrelated concepts
A satisfiable formula is one that can possibly be true — it is not a contradiction. Both tautologies and contingencies are satisfiable; only contradictions are unsatisfiable. Determining satisfiability (the SAT problem) is a foundational question in logic and computer science.
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Truth Table

A tabular listing of all possible truth value combinations for a formula's variables and the resulting truth value of the formula
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intuitionpropertiesrelated concepts
A truth table systematically evaluates a formula under every possible input. For nn propositional variables, the table has 2n2^n rows. It is the primary tool for checking whether a formula is a tautology, contradiction, or contingency, and for verifying logical equivalences.
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Assignment (Valuation)

A function vv that maps each propositional variable to a truth value: v:{P,Q,R,…}→{T,F}v: \{P, Q, R, \ldots\} \to \{T, F\}
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intuitionpropertiesrelated concepts
An assignment fixes the truth value of every variable in a formula, allowing the formula's overall truth value to be computed. Each row of a truth table corresponds to one assignment. A formula is satisfied by an assignment if it evaluates to true under that assignment.
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Absorption

P∧(P∨Q)≡PP \land (P \lor Q) \equiv P and P∨(P∧Q)≡PP \lor (P \land Q) \equiv P
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intuitionpropertiesrelated concepts
Absorption eliminates redundant structure. If PP already appears at the top level, wrapping it inside an additional disjunction or conjunction with another variable adds no information — the result simplifies back to PP.
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Law of Excluded Middle

P∨¬PP \lor \neg P is always true — every proposition is either true or false, with no third option
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intuitionpropertiesrelated concepts
This law asserts that there is no middle ground between truth and falsity in classical logic. For any proposition PP, either PP holds or ¬P\neg P holds. Together with non-contradiction, it establishes the binary nature of classical truth.
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Non-contradiction

¬(P∧¬P)\neg(P \land \neg P) is always true — no proposition can be simultaneously true and false
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Non-contradiction states that PP and ¬P\neg P cannot both hold at once. Any system that violates this principle collapses — from a contradiction, every statement becomes provable (the principle of explosion).
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