Definition Formalization Related Compound Statements Examples Implications Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � , then logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � , then � conclusion � .” logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � , then � conclusion � .” 2. But we must be careful: There are three entities that have truth values associated with them: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � , then � conclusion � .” 2. But we must be careful: There are three entities that have truth values associated with them: The hypothesis logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � , then � conclusion � .” 2. But we must be careful: There are three entities that have truth values associated with them: The hypothesis, the conclusion logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � , then � conclusion � .” 2. But we must be careful: There are three entities that have truth values associated with them: The hypothesis, the conclusion and the implication (the “if-then” statement) itself . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � , then � conclusion � .” 2. But we must be careful: There are three entities that have truth values associated with them: The hypothesis, the conclusion and the implication (the “if-then” statement) itself . 3. At this stage, we must focus on the implication, that is, on the connection between hypothesis and conclusion logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 1. Ideally, every theorem in mathematics and every valid inference is of the form “If � hypothesis � , then � conclusion � .” 2. But we must be careful: There are three entities that have truth values associated with them: The hypothesis, the conclusion and the implication (the “if-then” statement) itself . 3. At this stage, we must focus on the implication, that is, on the connection between hypothesis and conclusion, not on the hypothesis and conclusion themselves. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 4. True statements must imply true statements. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 4. True statements must imply true statements. For example, “If n is a prime number that is greater than 2, then n is odd” is a true statement. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 4. True statements must imply true statements. For example, “If n is a prime number that is greater than 2, then n is odd” is a true statement. 5. Ideally, we never slip away from dealing with true statements. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 4. True statements must imply true statements. For example, “If n is a prime number that is greater than 2, then n is odd” is a true statement. 5. Ideally, we never slip away from dealing with true statements. But sometimes a false hypothesis is used logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 4. True statements must imply true statements. For example, “If n is a prime number that is greater than 2, then n is odd” is a true statement. 5. Ideally, we never slip away from dealing with true statements. But sometimes a false hypothesis is used, inadvertently or deliberately. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 4. True statements must imply true statements. For example, “If n is a prime number that is greater than 2, then n is odd” is a true statement. 5. Ideally, we never slip away from dealing with true statements. But sometimes a false hypothesis is used, inadvertently or deliberately. 6. Logic must be robust enough to stay internally consistent. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 4. True statements must imply true statements. For example, “If n is a prime number that is greater than 2, then n is odd” is a true statement. 5. Ideally, we never slip away from dealing with true statements. But sometimes a false hypothesis is used, inadvertently or deliberately. 6. Logic must be robust enough to stay internally consistent. 7. In the following examples, we illustrate what false hypotheses can do. The following examples are not useful mathematics. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 8. False statements can imply false statements. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Definition Formalization Related Compound Statements Examples Which Implications are True? 8. False statements can imply false statements. For example, if we assume 1 > 2, then we can derive − 1 > 0 by subtracting 2 on both sides. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Implications
Recommend
More recommend
