Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
The following argument shows that the premises "If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on," "If the sailing race is held, then the trophy will be awarded," and "The trophy was not awarded" imply the conclusion "It rained." Identify the reason at each step of the arguments. 1. The trophy was not awarded 2. If the sailing race is held, then the trophy will be awarded 3. The sailing race was not held 4. The sailing race was not held or the lifesaving demonstration did not go on 5. It's not true that the sailing race will be held and the lifesaving demonstration will go on 6. If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on 7. It's not true that it did not rain or it was not foggy 8. It rained and it was foggy 9. It rained

The following argument shows that the premises "If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on," "If the sailing race is held, then the trophy will be awarded," and "The trophy was not awarded" imply the conclusion "It rained." Identify the reason at each step of the arguments. 1. The trophy was not awarded 2. If the sailing race is held, then the trophy will be awarded 3. The sailing race was not held 4. The sailing race was not held or the lifesaving demonstration did not go on 5. It's not true that the sailing race will be held and the lifesaving demonstration will go on 6. If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on 7. It's not true that it did not rain or it was not foggy 8. It rained and it was foggy 9. It rained

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please help solve this question. 

### Logical Reasoning and Argument Analysis The following argument demonstrates that the premises "If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on," "If the sailing race is held, then the trophy will be awarded," and "The trophy was not awarded" imply the conclusion "It rained." Let’s break down the reasoning step by step. 1. **The trophy was not awarded** - **Reasoning**: This is a given fact in the argument. 2. **If the sailing race is held, then the trophy will be awarded** - **Reasoning**: This premise links the occurrence of the sailing race to the awarding of the trophy. 3. **The sailing race was not held** - **Reasoning**: From 1 and 2, if the trophy was not awarded, then the sailing race could not have been held (modus tollens). 4. **The sailing race was not held or the lifesaving demonstration did not go on** - **Reasoning**: This is derived from the definition of the event occurrences. If either the sailing race or lifesaving demonstration did not happen, this disjunction holds. 5. **It’s not true that the sailing race will be held and the lifesaving demonstration will go on** - **Reasoning**: From 4, we have at least one event not happening, so the conjunction of both events happening is false. 6. **If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on** - **Reasoning**: This is a given premise that sets a conditional relationship dependent on the weather conditions. 7. **It’s not true that it did not rain or it was not foggy** - **Reasoning**: Since the conjunction of both events happening is false (from 5), the antecedent in the conditional in premise 6 must also be false (denying the antecedent). 8. **It rained and it was foggy** - **Reasoning**: Negating the disjunction (from 7) gives us the conjunction of both conditions being true. 9. **It rained** - **Reasoning**: This is a simplification of the conjunction derived in 8. ### Conclusion Thus, the logical steps and premises provided conclude that the
Transcribed Image Text:### Logical Reasoning and Argument Analysis The following argument demonstrates that the premises "If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on," "If the sailing race is held, then the trophy will be awarded," and "The trophy was not awarded" imply the conclusion "It rained." Let’s break down the reasoning step by step. 1. **The trophy was not awarded** - **Reasoning**: This is a given fact in the argument. 2. **If the sailing race is held, then the trophy will be awarded** - **Reasoning**: This premise links the occurrence of the sailing race to the awarding of the trophy. 3. **The sailing race was not held** - **Reasoning**: From 1 and 2, if the trophy was not awarded, then the sailing race could not have been held (modus tollens). 4. **The sailing race was not held or the lifesaving demonstration did not go on** - **Reasoning**: This is derived from the definition of the event occurrences. If either the sailing race or lifesaving demonstration did not happen, this disjunction holds. 5. **It’s not true that the sailing race will be held and the lifesaving demonstration will go on** - **Reasoning**: From 4, we have at least one event not happening, so the conjunction of both events happening is false. 6. **If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on** - **Reasoning**: This is a given premise that sets a conditional relationship dependent on the weather conditions. 7. **It’s not true that it did not rain or it was not foggy** - **Reasoning**: Since the conjunction of both events happening is false (from 5), the antecedent in the conditional in premise 6 must also be false (denying the antecedent). 8. **It rained and it was foggy** - **Reasoning**: Negating the disjunction (from 7) gives us the conjunction of both conditions being true. 9. **It rained** - **Reasoning**: This is a simplification of the conjunction derived in 8. ### Conclusion Thus, the logical steps and premises provided conclude that the
### Logic Steps and Rules This image represents a list of steps and their associated logical rules. On the left side, there is a column of dropdown buttons labeled with different steps. Here is the detailed transcription and explanation. **Step Dropdowns:** - Step 5 - Step 9 - Step 4 - Step 2 - Step 8 - Step 6 - Step 3 - Step 7 - Step 1 Each step can be associated with a logical rule, listed on the right side of the image. **Logical Rules:** 1. **Premise** 2. **Modus ponens** 3. **Modus tollens** 4. **Addition** 5. **Simplification** 6. **De Morgan's Law** Each logical rule mentioned here is fundamental in the study of logic, helping in constructing and deconstructing logical arguments. They can be defined as follows: 1. **Premise:** A statement or proposition from which another is inferred or follows as a conclusion. 2. **Modus ponens (Affirming the antecedent):** If 'P implies Q' (P → Q) and P is true, then Q must also be true. 3. **Modus tollens (Denying the consequent):** If 'P implies Q' (P → Q) and Q is false, then P must also be false. 4. **Addition:** If P is true, then P or Q (P ∨ Q) is also true. 5. **Simplification:** If P and Q is true (P ∧ Q), then P is true. 6. **De Morgan's Law:** The equivalence of the negation of a conjunction to the disjunction of the negations. (\(\neg(P \land Q)\) is equivalent to \((\neg P) \lor (\neg Q)\)). These rules and steps are essential tools for those studying formal logic or involved in mathematical proofs, helping to establish the validity of arguments systematically.
Transcribed Image Text:### Logic Steps and Rules This image represents a list of steps and their associated logical rules. On the left side, there is a column of dropdown buttons labeled with different steps. Here is the detailed transcription and explanation. **Step Dropdowns:** - Step 5 - Step 9 - Step 4 - Step 2 - Step 8 - Step 6 - Step 3 - Step 7 - Step 1 Each step can be associated with a logical rule, listed on the right side of the image. **Logical Rules:** 1. **Premise** 2. **Modus ponens** 3. **Modus tollens** 4. **Addition** 5. **Simplification** 6. **De Morgan's Law** Each logical rule mentioned here is fundamental in the study of logic, helping in constructing and deconstructing logical arguments. They can be defined as follows: 1. **Premise:** A statement or proposition from which another is inferred or follows as a conclusion. 2. **Modus ponens (Affirming the antecedent):** If 'P implies Q' (P → Q) and P is true, then Q must also be true. 3. **Modus tollens (Denying the consequent):** If 'P implies Q' (P → Q) and Q is false, then P must also be false. 4. **Addition:** If P is true, then P or Q (P ∨ Q) is also true. 5. **Simplification:** If P and Q is true (P ∧ Q), then P is true. 6. **De Morgan's Law:** The equivalence of the negation of a conjunction to the disjunction of the negations. (\(\neg(P \land Q)\) is equivalent to \((\neg P) \lor (\neg Q)\)). These rules and steps are essential tools for those studying formal logic or involved in mathematical proofs, helping to establish the validity of arguments systematically.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS