LOGIC Chapter 3
LOGIC • Is the science of correct reasoning • Analyzes given information and, with a step-by-step process, moves from a known fact to a conclusion Deductive Reasoning!
LOGIC • Started with Greek philosopher Aristotle (384-322 BC), who studied verbal reasoning patterns • Was refined and put into symbols by the English mathematician George Boole (1815 – 1864), who as a result is considered the founder of computer science Logic is sometimes referred to as Boolean Algebra.
Statements, negations, and Quantified Statements Section 3.1
Statement • Is a complete, declarative sentence that is either true or false but not both at same time • A statement is NOT: • Opinion (Ex: Avatar is the best movie of all time.) • Question (Ex: Did you drive to school today?) • Exclamation (Ex: Help!) • Command (Ex: Read pages 1 though 100.)
EXAMPLE: Is this a statement? YES • London is the capital of England. • Go to the store please. • Did logic begin with Aristotle? • BHS Cardinals won Friday night’s game. • BHS is a great school. • Fire! • Shakespeare wrote The Hunger Games. NO NO YES NO NO YES
Notation • In symbolic logic, we use lowercase, cursive letters (such as p, q, r, and s) or lowercase, italic letters (such as p, q, r, and s) to denote our statements • For example, we would write p: Nashville is the capital of Tennessee.
Negation • Is a statement with the opposite truth value from the original statement • Is denoted with or ~ • Includes the word “not” when read • For example, we would write p: Obama is president. p: Obama is not president. q: Memphis is in Alabama. ~q: Memphis is not in Alabama. Read “not p” Read “not q”
EXAMPLE: Write the negation. p: Tennessee is a state. ~p: Tennessee is not a state. q: Today is Saturday. ~q: Today is not Saturday. r: July is not a month. ~r: July is a month. s: BHS does not have a swim team. ~s: BHS does have a swim team.
Quantifiers • Words such as all, some, and no (or none) are quantifiers. A statement containing one of these words is a quantified statement. • Quantifier Notation not or ~ all/every some Can be read as “There exists at least one…”
Another way to say the same thing is… For p: All A are B, most people want to say ~p: No A are B. This is NOT correct! p: All writers are poets. This is false, so ~ MUST be true. ~p: No writers are poets is also false, so it can NOT be the negation! Quantified Statements Start with the statement p: A are B. Then quantifiers would produce these ~p: A are not B. p: All A are B. ~p: Some A are not B. p: Some A are B. ~p: No A are B. It is not true that A are B. There are no A that are not B. Not all A are B. There exists at least one A that is a B. All A are not B.
Another way to say the same thing is… Quantified Statements Start with the statement p: A are B. Then quantifiers would produce these ~p: A are not B. p: All A are B. ~p: Some A are not B. p: Some A are B. ~p: No A are B. It is not true that A are B. There are no A that are not B. Not all A are B. There exists at least one A that is a B. All A are not B.
Example: Write the statement. p: People are honest. ~p: p: ~p: p: ~p: A B People are not honest. All people are honest. Some people are not honest. Also good: Not all people are honest. Some people are honest. All people are not honest. Also good: No people are honest.