The word or can be interpreted in two distinct ways:
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The document discusses the principles of logical propositions, including conjunctions, disjunctions, and conditionals. It provides examples of truth values for different types of statements and highlights logical equivalence among propositions. Additionally, it explains how the structure of 'if...then' statements can be transformed into converses, inverses, and contrapositives and evaluates their truth values.
The word or can be interpreted in two distinct ways:
- 2. Example (cont) Because the double negation has the same truth value as the original proposition, we conclude that the scientist believes it likely that there is an association between low-level radiation and cancer among younger workers.
- 3. Propositions are often joined with logical connectors—words such as and, or, and if…then. Example: p = I won the game. q = It was fun. Logical Connectors Logical Connector and or if…then New Proposition I won the game and it was fun. I won the game or it was fun. If I won the game, then it was fun.
- 4. p q p and q T T T T F F F T F F F F Given two propositions p and q, the statement p and q is called their conjunction. It is true only if p and q are both true. Symbol: And Statements (Conjunctions)
- 5. Example Evaluate the truth value of the following two statements. a. The capital of France is Paris and Antarctica is cold. b. The capital of France is Paris and the capital of America is Madrid. Solution • a. The statement contains two distinct propositions: The capital of France is Paris and Antarctica is cold. Because both propositions are true, their conjunction is also true.
- 6. Example Evaluate the truth value of the following two statements. a. The capital of France is Paris and Antarctica is cold. b. The capital of France is Paris and the capital of America is Madrid. • b. The statement contains two distinct propositions: The capital of France is Paris and the capital of America is Madrid. Although the first proposition is true, the second is false. Therefore, their conjunction is false.
- 7. • An inclusive or means “either or both.” • An exclusive or means “one or the other, but not both.” The word or can be interpreted in two distinct ways: In logic, assume or is inclusive unless told otherwise. Two Types of Or
- 8. p q p or q T T T T F T F T T F F F The Logic of Or (Disjunctions) Given two propositions p and q, the statement p or q is called their disjunction. It is true unless p and q are both false. Symbol:
- 9. Example Consider the statement airplanes can fly or cows can read. Is it true? Solution The statement is a disjunction of two propositions: (1) airplanes can fly; (2) cows can read. The first proposition is clearly true, while the second is clearly false, which makes the disjunction p or q true. That is, the statement airplanes can fly or cows can read is true.
- 10. A statement of the form if p, then q is called a conditional proposition (or implication). It is true unless p is true and q is false. p q if p, then q T T T T F F F T T F F T Proposition p is called the hypothesis. Proposition q is called the conclusion. The Logic of If . . . Then Statements (Conditionals)
- 11. • p is sufficient for q • p will lead to q • p implies q The following are common alternative ways of stating if p, then q: q is necessary for p q if p q whenever p Alternative Phrasings of Conditionals
- 12. Conditional: Converse: Inverse: Contrapositive: If it is raining, then I will bring an umbrella to work. If I bring an umbrella to work, then it must be raining. If it is not raining, then I will not bring an umbrella to work. If I do not bring an umbrella to work, then it must not be raining. If p, then q If q, then p If not p, then not q If not q, then not p Variations on the Conditional
- 13. Two statements are logically equivalent if they share the same truth values. Logical Equivalence p q notp notq if p, then q if q, then p (converse) if notp, then notq (inverse) if not q, then notp (contrapositive) T T F F T T T T T F F T F T T F F T T F T F F T F F T T T T T T logically equivalent logically equivalent
- 14. Example Consider the true statement if a creature is a whale, then it is a mammal. Write its converse, inverse, and contrapositive. Evaluate the truth of each statement. Which statements are logically equivalent? Solution The statement has the form if p, then q, where p = a creature is a whale and q = a creature is a mammal. Therefore, we find • converse (if q, then p): if a creature is a mammal, then it is a whale. This statement is false, because most mammals are not whales.
- 15. Example • inverse (if not p, then not q): if a creature is not a whale, then it is not a mammal. This statement is also false; for example, dogs are not whales, but they are mammals. • contrapositive (if not q, then not p): if a creature is not a mammal, then it is not a whale. Like the original statement, this statement is true, because all whales are mammals.
- 16. Example Note that the original proposition and its contrapositive have the same truth value and are logically equivalent. Similarly, the converse and inverse have the same truth value and are logically equivalent.