Logic What is logic? Logic is the science of how to evaluate arguments and reasoning. Critical thinking is a process of evaluation which uses logic to separate truth from falsehood, reasonable from unreasonable beliefs. If you want to better evaluate the various claims, ideas, and arguments you encounter, you need a better understanding of basic logic and the process of critical thinking. These aren't trivial pursuits; they are essential to making good decisions and forming sound beliefs about our world. What is Logic? What is Critical Thinking? Strategies and Skills for Critical Thinking, Using Logic By Austin Cline
Logic and Geometry Why do we study logic in Geometry? Geometric proofs use logical reasoning and the definitions and properties of geometric figures and terms to state definitively that something is always true.
What is a mathematical sentence? A mathematical sentence is a complete idea that has a truth value. It is called a closed sentence. A sentence whose truth value is uncertain is called an open sentence. Questions and phrases are not mathematical sentences since they cannot be judged to be true or false.
Identify each: Sentence or Not a sentence Sentence - true "A triangle contains 180˚.” “A Golden Retriever is a type of cat.” Sentence -false "Did you get an A in Math?" Not a sentence – a question "All kittens" Not a sentence – a phrase “The Big Bang Theory is the best comedy of all time.” Not a sentence – an opinion
An open sentence is a sentence which you can not tell the truth value. It may contain a variable. "x - 3 = 8" is an open sentence – without knowing the value of “x”, you cannot judge it to be true or false. "It is my favorite subject." is an open sentence– without knowing what “It” represents, you can not judge it to be true or false.
“Jennifer Lawrence is the actress who played Bella in Twilight." -- a closed sentence– it is a false statement. An closed sentence is a sentence which you can tell the truth value. “A parallelogram has diagonals that bisect each other.” -- a closed sentence – it is a true statement.
Negation Negation is the process of changing the truth value of a statement, usually by adding the word NOT or taking away the word NOT. Example Statement: Mrs. Russolello likes the classroom cold. This is atruestatement. Negation: Mrs. Russolello does NOT like the classroom cold. This is a false statement.
Negation with Quantifiers • Negation involving quantifiers such as • All/Every • Some • None Require different rules to change the truth value.
Negation with Quantifiers Example: Statement: All students have blonde hair. True or False? False Negation: NOT all students have blonde hair. True or False? True OR Negation: Some students do not have blonde hair. True or False? True OR Negation: It is not the case that all students have blonde hair. True or False? True This is NOT the negation: No students have blonde hair. True or False? False – the truth value did not change; therefore it is not the negation!
Compound Statements A compound statement is a statement with two or more ideas connected in one sentence. • Conjunction – AND is the connecting word The slope represents the rate of change in a linear equation AND the y-intercept represents the starting value. • Disjunction – OR is the connecting word Pat is a boy OR Pat is a girl.
Conjunction Rule • Both simple statements must be TRUE for the statement to be TRUE • Truth Table Where P represents the first simple statement and Q represents the second simple statement.
Disjunction Rule • Both simple statements must be FALSE for the statement to be FALSE • Truth Table Where P represents the first simple statement and Q represents the second simple statement.