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Chapter 1: Problem 6

Use rules of inference to show that the hypotheses 鈥淚f it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on,鈥� 鈥淚f the sailing race is held, then the trophy will be awarded,鈥� and 鈥淭he trophy was not awarded鈥� imply the conclusion 鈥淚t rained.鈥�

Short Answer

Expert verified
It rained because the hypotheses logically imply that it did.

Step by step solution

01

Define the Statements

Translate each part of the exercise into logical statements. Let: - \( p \): It rains - \( q \): It is foggy - \( r \): The sailing race will be held - \( s \): The lifesaving demonstration will go on - \( t \): The trophy will be awarded
02

Express the Hypotheses in Logical Form

Using the notation from Step 1, express each hypothesis as a logical statement: 1. \((eg p \lor eg q) \rightarrow (r \land s)\) 2. \(r \rightarrow t\) 3. \(eg t\)
03

Apply Modus Tollens to Hypotheses 2 and 3

Using Modus Tollens on \(r \rightarrow t\) and \(eg t\), infer \(eg r\): - If \(r\) were true, then \(t\) would be true by hypothesis 2. - Since \(t\) is false (hypothesis 3), \(r\) must be false. - Conclusion: \(eg r\).
04

Apply Modus Tollens to Hypothesis 1

Using \((eg p \lor eg q) \rightarrow (r \land s)\) and \(eg r\): - If \(r\) is false, then \((r \land s)\) is also false. - Since hypothesis 1 implies \((r \land s)\), this means \((eg p \lor eg q)\) is false. - Conclusion: \((eg(eg p \lor eg q))\).
05

Simplify the Conclusion

Simplify \(eg(eg p \lor eg q)\) using De Morgan's Laws: - \(eg(eg p \lor eg q)\) is equivalent to \(p \land q\). - Conclusion: \(p \land q\).
06

Derive the Final Conclusion

From \(p \land q\), infer that \(p\) must be true. Therefore, it rained. - Conclusion: \(p\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logical statements
Logical statements are the building blocks of logical reasoning. They are sentences that are either true or false, but not both. In our exercise, we defined several logical statements based on given scenarios:
  • ewline Let \( p \): It rains
  • \( q \): It is foggy
  • \( r \): The sailing race will be held
  • \( s \): The lifesaving demonstration will go on
  • \( t \): The trophy will be awarded
By translating verbal scenarios into logical statements, we can manipulate and analyze them using formal logic rules.
modus tollens
Modus Tollens is a fundamental rule of inference in logic. It allows us to conclude the negation of a premise if an implication and its consequent are known to be true and false respectively. It follows this structure:
  • If \( A \rightarrow B \), and if \( eg B \) is true, then \( eg A \) is also true.
In our exercise, we used Modus Tollens twice:
  • From hypothesis 2 (\( r \rightarrow t \)) and hypothesis 3 (\( eg t \)), we inferred \( eg r \).
  • We then applied it again to hypothesis 1, together with \( eg r \), to infer that \( p \land q \).
Modus Tollens is powerful because it helps us deduce what must be false based on certain given truths.
De Morgan鈥檚 laws
De Morgan's Laws are essential for simplifying and transforming logical expressions. They show how to distribute negation inside and outside of parentheses involving \( \land \) and \( \lor \) operators. The laws are stated as:
  • \( eg (A \land B) \equiv eg A \lor eg B \)
  • \( eg (A \lor B) \equiv eg A \land eg B \)
In our exercise, we applied one of these laws to simplify our conclusion:
  • We had \( eg (eg p \lor eg q) \).
  • Applying De Morgan鈥檚 Law, we transformed it to \( p \land q \).
This simplification was crucial for arriving at the final conclusion that it rained.
hypotheses and conclusion
In logical reasoning, hypotheses are the starting assumptions or premised statements. Conclusions are the results deduced from these hypotheses using rules like Modus Tollens and De Morgan's Laws. In our exercise, we started with three hypotheses:
  • \( (eg p \lor eg q) \rightarrow (r \land s) \)
  • \( r \rightarrow t \)
  • \( eg t \)
From these, we deduced the conclusion \( p \): It rained. Each step involved applying logical rules systematically to reveal what must be true or false. This method ensures conclusions are logically sound and based on given information.

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Most popular questions from this chapter

Prove that there is no positive integer \(n\) such that \(n^{2}+\) \(n^{3}=100 .\)

Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), and R(x) be the statements 鈥渪 is a professor,鈥� 鈥渪 is ignorant,鈥� and 鈥渪 is vain,鈥� respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), and R(x), where the domain consists of all people. $$ \begin{array}{l}{\text { a) No professors are ignorant. }} \\ {\text { b) All ignorant people are vain. }} \\ {\text { c) No professors are vain. }} \\\ {\text { d) Does (c) follow from (a) and (b)? }}\end{array} $$

Suppose the domain of the propositional function \(P(x, y)\) consists of pairs \(x\) and \(y,\) where \(x\) is \(1,2,\) or 3 and \(y\) is \(1,2,\) or \(3 .\) Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \exists x P(x, 3)} & {\text { b) } \forall y P(1, y)} \\ {\text { c) } \exists y \neg P(2, y)} & {\text { d) } \forall x \neg P(x, 2)}\end{array} $$

Prove or disprove that you can use dominoes to tile the standard checkerboard with two adjacent corners removed (that is, corners that are not opposite).

Suppose that the domain of \(Q(x, y, z)\) consists of triples \(x, y, z,\) where \(x=0,1,\) or \(2, y=0\) or \(1,\) and \(z=0\) or \(1 .\) Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \forall y Q(0, y, 0)} & {\text { b) } \exists x Q(x, 1,1)} \\ {\text { c) } \exists z \neg Q(0,0, z)} & {\text { d) } \exists x \neg Q(x, 0,1)}\end{array} $$
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