Statements and Propositions

A statement is a sentence that conveys an idea and claims it to be true or false. Examples: The earth is round. Lincoln is the capital of Nebraska. Vinegar and baking soda will react chemically. Oswald did not kill JFK.

A proposition captures the meaning that the statement conveys, independent of the wording, grammar, or language. That is, a proposition is the idea itself, not the way the idea is phrased.

The same proposition can be expressed in multiple ways. Example: “The Earth goes around the Sun.” and “Our planet orbits the Sun.” both express the same proposition (Earth revolves around the Sun) but use different wording.

Often we use the terms “proposition” and “statement” as synonyms, but it is important to recognize that they have different meanings.

Simple Statement

A simple statement is a simple sentence that does not contain a negated term.

Examples: I like school. The sun rises in the east. The angles of an equilateral triangle all equal 60 degrees.

Compound Statement

A compound statement is any proposition that has two or more simple statements or at least one negated simple statement.

Examples of compound statements

• It is not true that all professionals are ethical. (negated)
• Good unit practice saves time, and good unit practice leads to fewer mistakes. (two simple statements connected with an “and”)

Six Kinds of Statements

In propositional logic, there are six ways to write statements

The first way is a simple statement: John is a freshman.

The next five way use these five operators: and, or, if-them, if-and-only if, not).

In propositional logic simple statements are represented with a letter, and the five operators are represented with symbols

Propositional Logic Operators

Operator	Symbol	Reading	Example
Negation	¬	Not	¬P (Not P)
Conjunction	∧	And	P ∧ Q (P and Q)
Disjunction	∨	Or	P ∨ Q (P or Q)
Conditional	→	If-Then	P → Q (If P then Q)
Biconditional	↔︎	If and Only If	P ↔︎ Q (P if and only if Q)

Propositional Logic Operators

Operator	Symbol (LaTeX)	Reading	Example
Negation	¬	Not	¬P (Not P)
Conjunction	∧	And	P∧Q (P and Q)
Disjunction	∨	Or	P∨Q (P or Q)
Conditional	→	If-Then	P→Q (If P then Q)
Biconditional	↔︎	If and Only If	P↔︎Q (P if and only if Q)

Propositional Logic Operators

Operator	Symbol (LaTeX)	Reading	Example
Negation	¬	Not	¬P (Not P)
Conjunction	∧	And	P∧Q (P and Q)
Disjunction	∨	Or	P∨Q (P or Q)
Conditional	→	If-Then	P→Q (If P then Q)
Biconditional	↔︎	If and Only If	P↔︎Q (P if and only if Q)

Propositional Logic Operators

Operator	Symbol (LaTeX)	Reading	Example
Negation	¬	Not	¬P (Not P)
Conjunction	∧	And	P∧Q (P and Q)
Disjunction	∨	Or	P∨Q (P or Q)
Conditional	→	If-Then	P→Q (If P then Q)
Biconditional	↔︎	If and Only If	P↔︎Q (P if and only if Q)

Operator	Symbol (LaTeX)	Reading	Example
Negation	¬	Not	¬P (Not P)
Conjunction	∧	And	P ∧ Q (P and Q)
Disjunction	∨	Or	P ∨ Q (P or Q)
Conditional	→	If-Then	P → Q (If P then Q)
Biconditional	↔︎	If and Only If	P ↔︎ Q (P if and only if Q)