Mathematical Logic and Reasoning Vocabulary

Vocabulary flashcards defining fundamental terms, connectives, quantifiers, equivalences, and argument forms in mathematical logic.

Logic

The study of the methods and principles used to distinguish correct from incorrect reasoning.

Proposition

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

Truth Value

The truthfulness or falsity of a statement.

Simple Statement

A logical statement that conveys a single idea.

Compound Statement

A statement formed by combining two or more simple statements using logical connectives.

Negation

A logical connective, represented by $\neg p$, that reverses the truth value of a proposition $p$.

Conjunction

A compound proposition, represented by $p \wedge q$, that is true only when both $p$ and $q$ are true.

Disjunction

A compound proposition, represented by $p \vee q$, that is true when $p$ is true, $q$ is true, or both are true.

Conditional Statement

A compound proposition, represented by $p \rightarrow q$, that is false only when $p$ is true and $q$ is false.

Antecedent

The hypothesis or initial statement $p$ in a conditional statement $p \rightarrow q$.

Consequent

The conclusion or resulting statement $q$ in a conditional statement $p \rightarrow q$.

Biconditional Statement

A compound proposition, represented by $p \leftrightarrow q$, that is true when both $p$ and $q$ have the same truth value.

Quantification

The process of specifying how many elements in a domain satisfy an open formula to make it a complete statement with a truth value.

Universal Quantification

A quantified claim, represented by $\forall x P(x)$, stating that every object $x$ in a given domain satisfies the condition $P(x)$.

Existential Quantification

A quantified claim, represented by $\exists x P(x)$, stating that at least one object $x$ in a given domain satisfies the condition $P(x)$.

Converse

The related conditional statement $q \rightarrow p$ formed by interchanging the antecedent and consequent of $p \rightarrow q$.

Inverse

The related conditional statement $\neg p \rightarrow \neg q$ formed by negating both the antecedent and consequent of $p \rightarrow q$.

Contrapositive

The related conditional statement $\neg q \rightarrow \neg p$ formed by negating and interchanging the antecedent and consequent of $p \rightarrow q$, which is logically equivalent to the original conditional.

Logical Equivalence

The relationship between two statements that always have identical truth values across all possible truth values of their component propositions.

De Morgan's Laws for Statements

Logical equivalences stating that $\neg(p \vee q) \equiv \neg p \wedge \neg q$ and $\neg(p \wedge q) \equiv \neg p \vee \neg q$.

Tautology

A compound statement that is always true regardless of the truth values of its simple component statements.

Self-Contradiction

A compound statement that is always false regardless of the truth values of its simple component statements.

Argument

A set of logical statements consisting of premises offered as reasons to accept a conclusion.

Premises

Statements offered in an argument as reasons or evidence to support the conclusion.

Conclusion

The main claim in an argument that is supposed to follow from the premises.

Valid Argument

An argument form in which it is impossible for all premises to be true and the conclusion to be false.

Invalid Argument

An argument form in which it is possible for all premises to be true while the conclusion is false.

Modus Ponens

A valid argument form given by: if $p \rightarrow q$ and $p$, then $q$.

Modus Tollens

A valid argument form given by: if $p \rightarrow q$ and $\neg q$, then $\neg p$.

Hypothetical Syllogism

A valid argument form given by: if $p \rightarrow q$ and $q \rightarrow r$, then $p \rightarrow r$.

Disjunctive Syllogism

A valid argument form given by: if $p \vee q$ and $\neg p$, then $q$.

Affirming the Consequent

A common invalid argument form given by: if $p \rightarrow q$ and $q$, concluding $p$.

Denying the Antecedent

A common invalid argument form given by: if $p \rightarrow q$ and $\neg p$, concluding $\neg q$.