Introduction to Mathematical Logic: Negation and Reverse Negation in Logic
Negation: The negation of a statement is a statement that asserts the opposite of the
original statement. It is denoted as ~p or ¬p.
Double Negation: A double negation is a negation of a negation. It is logically equivalent
to the original statement. (e.g. ~~p ≡ p)
Identifying Non-Statements in Logic
A non-statement is a statement that is grammatically correct but has no definite truth
value.
Examples of non-statements include questions, commands, and exclamations.
Conjunction in Logical Connectives
Conjunction: A logical connective that connects two statements such that the whole
statement is true when both statements are true.
Notation: p ∧ q
Understanding Statements and Truth Values
A statement is a sentence that is either true or false, but not both.
Truth value: the truth or falsehood of a statement.
Open Statements and Their Implications
Open statement: a statement with one or more free variables. It cannot be classified as
true or false until variables are assigned specific values.
Implications: Universally quantified open statements.
Bi-Conditional and Double Implication
Bi-Conditional: A logical connective that connects two statements such that the whole
statement is true if and only if both statements have the same truth value.
Notation: p ↔ q
Types of Statements and Their Examples
Tautology: a statement that is always true. (e.g. p ∨ ~p)
Contradiction: a statement that is always false. (e.g. p ∧ ~p)
Contingency: a statement that is neither a tautology nor a contradiction.
Determining Truth Values of Statements
Use a truth table to determine the truth value of complex statements.
Introduction to Mathematical Logic
Negation: The negation of a statement is a statement that asserts the opposite of the
original statement. It is denoted as ~p or ¬p.
Double Negation: A double negation is a negation of a negation. It is logically equivalent
to the original statement. (e.g. ~~p ≡ p)
Identifying Non-Statements in Logic
A non-statement is a statement that is grammatically correct but has no definite truth
value.
Examples of non-statements include questions, commands, and exclamations.
Conjunction in Logical Connectives
Conjunction: A logical connective that connects two statements such that the whole
statement is true when both statements are true.
Notation: p ∧ q
Understanding Statements and Truth Values
A statement is a sentence that is either true or false, but not both.
Truth value: the truth or falsehood of a statement.
Open Statements and Their Implications
Open statement: a statement with one or more free variables. It cannot be classified as
true or false until variables are assigned specific values.
Implications: Universally quantified open statements.
Bi-Conditional and Double Implication
Bi-Conditional: A logical connective that connects two statements such that the whole
statement is true if and only if both statements have the same truth value.
Notation: p ↔ q
Types of Statements and Their Examples
Tautology: a statement that is always true. (e.g. p ∨ ~p)
Contradiction: a statement that is always false. (e.g. p ∧ ~p)
Contingency: a statement that is neither a tautology nor a contradiction.
Determining Truth Values of Statements
Use a truth table to determine the truth value of complex statements.
Introduction to Mathematical Logic
