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# Statements and Quantifiers

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Statements and Quantifiers
Chapter 1 Section 3-1 Statements and Quantifiers

Statements A statement is defined as a declarative sentence that is either true or false, but not both simultaneously.

Statements Decide if a Statement The zip code for Folsom is 95630
Where are you going today? Some numbers are positive.

Compound Statements Compound statement: formed by combining two or more statements. Connectives: used to form compound statements; and, or, not, and if…then

Example: Compound Statements
Decide whether each statement is compound. a) If Amanda said it, then it must be true. The gun was made by Smith and Wesson. I read the Sacramento Bee and the Wall Street Journal.

Negations Statement: “Max has a valuable card”
Negation: “Max does not have a valuable card.” The negation of a true statement is false and the negation of a false statement is true.

Example: Forming Negations
Give a negation of each inequality. Do not use a slash symbol.

Symbols Connectors Connective Symbol Type of Statement and Conjunction
Disjunction not ~ Negation

Example: Translating from Symbols to Words
p: George Bush is President q: A rectangle is a 3 sided object. a) p q d) ~(~q) b) p q e) ~p q c) ~q f) ~( p ~q) Determine if T or F and translate into compound statements

Negations of Quantified Statements
All do. Some do not (not all do) Some do. None do (all do not) Some do not None do. Existential Quantifiers: Some ---- at least one --- there exists Universal Quantifiers: None / All

Example: Forming Negations of Quantified Statements
Form the negation of each statement. Some cats have fleas. Some cats do not have fleas. No cats have fleas.

Sets of Numbers Natural (counting) {1, 2, 3, 4, …}
Whole numbers {0, 1, 2, 3, 4, …} Integers {…,–3, –2, –1, 0, 1, 2, 3, …} Rational numbers May be written as a terminating decimal, like 0.25, or a repeating decimal like 0.333… Irrational {x | x is not expressible as a quotient of integers} Decimal representations never terminate and never repeat. Real numbers {x | x can be expressed as a decimal}

Analyzing Arguments with Euler Diagrams
Chapter 1 Section 3-5 Analyzing Arguments with Euler Diagrams

Logical Arguments Made up of……..
premises (assumptions, laws, rules, widely held ideas, or observations) conclusion argument: premises together with a conclusion

Valid and Invalid Arguments
Valid argument: if the fact that all the premises are true forces the conclusion to be true. Invalid: Not valid. It is called a fallacy.

Euler Diagrams Several techniques can be used to check the validity of an argument. One of these is a visual technique based on Euler Diagrams.

Euler Diagrams for Quantified Statements
Technique for determining the validity of arguments whose premises contain the words all, some, and no. Euler Diagrams for Quantified Statements

Euler Diagrams and Arguments
Make an Euler diagram for the first premise. Make an Euler diagram for the second premise on top of the one for the first premise. The argument is valid if and only if every possible diagram illustrates the conclusion of the argument. If there is even one possible diagram that contradicts the conclusion, this indicates that the conclusion is not true in every case, so the argument is invalid.

Example: Using an Euler Diagram to Determine Validity
Is the following argument valid? All cats are animals. Figgy is a cat. Figgy is an animal.

Example: Using an Euler Diagram to Determine Validity
All cats are animals. Figgy is a cat. Figgy is an animal. Animals Cats x Now try Sec 2.5 #1 x represents Figgy.

Example: Using an Euler Diagram to Determine Validity
Is the following argument valid? All sunny days are hot. Today is not hot Today is not sunny.

Example: Using an Euler Diagram to Determine Validity
All sunny days are hot. Today is not hot Today is not sunny. Hot days x Sunny days Now try Sec 3.5 #6 x represents today

Example: Using an Euler Diagram to Determine Validity
Is the following argument valid? All cars have wheels. That vehicle has wheels. That vehicle is a car.

Example: Using an Euler Diagram to Determine Validity
All cars have wheels. That vehicle has wheels. That vehicle is a car. Things that have wheels x ? Cars x ? Now try Sec 3.5 #3 x represents “that vehicle”

Example: Using an Euler Diagram to Determine Validity
Is the following argument valid? Some students drink coffee. I am a student . I drink coffee .

Example: Using an Euler Diagram to Determine Validity
Some students drink coffee. I am a student . I drink coffee . People that drink coffee x ? Students Now try Sec 3.5 #9 x?

Example: An argument can have a true conclusion yet be invalid
Is the following argument valid? All cars have tires. All tires have rubber . All cars have rubber . This is p. 132 #18 27

Example: An argument can have a true conclusion yet be invalid
All cars have tires. All tires have rubber . All cars have rubber . Rubber items Tires Cars This is p. 132 #18 28

Example: An argument can have a true conclusion yet be invalid
Is the following argument valid? Quebec is northeast of Ottawa. Quebec is northeast of Toronto . Ottawa is northeast of Toronto . This is similar to Sec 3.5 #21 29

Euler Diagrams and the Quantifier “SOME”
All people are mortal Some mortals are students. Therefore, some people are students. Step 1: Make an Euler diagram for the first premise. All people are mortal. Step 2: Make an Euler diagram for the second premise on top of the one for the first premise.

Step 3. The argument is valid if and only if every possible diagram
continued Step 3. The argument is valid if and only if every possible diagram illustrates the conclusion of the argument. The arguments conclusion is: Some people are students. Can you think of another way to draw the diagram for the second premise? The diagram does not show the “people” circle and the “students” circle intersecting with a dot in the region of intersection. The conclusion does not follow from the premises. The argument is invalid.

Analyzing Arguments with Truth Tables
Chapter 1 Section 3-6 Analyzing Arguments with Truth Tables

Truth Tables In section 3.5 Euler diagrams were used to test the validity of arguments. These work well with simple arguments but may not work well with more complex ones. If the words “all,” “some,” or “no” are not present, it may be better to use a truth table than an Euler diagram to test validity.

Example: Truth Tables (Two Premises)
Is the following argument valid? If the door is open, then I must close it. The door is open. I must close it.

Example: Truth Tables (Two Premises)
If the door is open, then I must close it. The door is open. I must close it. Solution Let p represent “the door is open” and q represent “I must close it.”

Example: Truth Tables (Two Premises)
Premise and premise implies conclusion Now Construct the truth table.

Valid Argument Forms Modus Ponens Modus Tollens Disjunctive Syllogism
Reasoning by Transitivity

Invalid Argument Forms (Fallacies)
Fallacy of the Converse Fallacy of the Inverse

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