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A declarative statement that is either true or false, but not both.

An extension of propositional logic which includes quantifiers and predicates to express propositions involving objects and their properties.

Symbolized as $\forall$, denotes that the propositions within its scope are true for all elements in a domain.

Symbolized as $\exists$, denotes that there exists at least one element in the domain for which the proposition is true.

Symbolized as $\land$, it represents the logical 'AND' operation and is true if and only if both propositions are true.

Symbolized as $\lor$, it represents the logical 'OR' operation and is true if at least one of the propositions is true.

Symbolized as $\neg$, it represents the logical 'NOT' operation, inverting the truth value of the proposition it precedes.

Symbolized as $\rightarrow$, it represents the logical 'IF ... THEN ...' operation and is false only when the antecedent is true and the consequent is false.

Symbolized as $\leftrightarrow$, it is true if both propositions have the same truth value, whether true or false.

A form of argument that from propositions in the form '$p$ implies $q$' and $p$, infers $q$.

A form of argument that from propositions in the form '$p$ implies $q$' and the negation of $q$, infers the negation of $p$.

These laws explain how logical disjunction ($\lor$) and conjunction ($\land$) are transformed by negation: $\neg(p \land q) \equiv \neg p \lor \neg q$ and $\neg(p \lor q) \equiv \neg p \land \neg q$.

In a conditional statement, the contrapositive is the result of negating both the antecedent and the consequent and then flipping them. For $p \rightarrow q$, the contrapositive is $\neg q \rightarrow \neg p$.

The principle of bivalence states that every proposition takes exactly one of two truth values, true or false.

A logical argument is sound if it is valid and all its premises are true.

A logical system is complete if every truth of the system's language can be derived using the system's rules of inference.

An argument is valid if the conclusion logically follows from the premises, that is, if the premises are true then the conclusion must be true.

An unordered collection of distinct objects considered as a whole.

A relation that assigns to each element in a domain exactly one element in the codomain. Often denoted by $f: X \rightarrow Y$ where $f(x) = y$.

A collection of ordered pairs, typically defined as a subset of the Cartesian product of two or more sets.

A measure of the number of elements in a set.

Also called a 'one-to-one' function, an injective function is one where every element of the codomain is mapped by at most one element of the domain.

Also called an 'onto' function, a surjective function is one where every element of the codomain is mapped by at least one element of the domain.

A function is bijective if it is both injective and surjective, which means it forms a one-to-one correspondence between the domain and codomain.

If two entities are equal, then one can be replaced by the other in any expression without changing the truth value of the expression.

An equation that is true for all possible values of its variables.

A statement that is false due to logically containing an inconsistency or being in direct conflict with a basic assumption.

A formula or assertion that is true in every possible interpretation, essentially a universally true statement.

A form of argument in which a hypothesis is temporarily assumed to be true, and from this assumption, a conclusion is deduced.

The principle that from a contradiction, any statement can be proven, i.e., if a contradiction is true, then any and every proposition is true.

The set of entities over which variables and quantifiers in an expression or argument are meant to range.

A branch of logic that studies ways of combining or altering entire propositions, statements, or sentences to form more complicated propositions, statements, or sentences.

A set of symbols, rules for manipulating these symbols, and rules for specifying which strings of symbols form a valid expression.

A type of formal proof system that attempts to mimic natural reasoning, where the rules of inference are closely aligned with the norms of logical reasoning.

A sub-field of logic that represents logical expressions using symbols and variables rather than natural language.