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Chapter 3: Problem 80

Let \(p, q\), and \(r\) represent the following simple statements: \(p:\) The temperature is above \(85^{\circ}\). q: We finished studying. \(r:\) We go to the beach. Write each symbolic statement in words. If a symbolic statement is given without parentheses, place them, as needed, before and after the most dominant connective and then translate into English. \(\sim(p \wedge q) \rightarrow \sim r\)

Short Answer

Expert verified
The statement \(\sim (p \wedge q) \rightarrow \sim r\) translates to: 'If it is not the case that both \(p: 'The temperature is above \(85^{\circ}\)' and \(q: 'We finished studying'\), then \(r: 'We will not go to the beach'\)'.

Step by step solution

01

Identify the variables

According to the given exercise, \(p\) represents 'The temperature is above \(85^{\circ}\)', \(q\) stands for 'We finished studying', and \(r\) signifies 'We go to the beach'.
02

Understand the structure

The symbolic statement is \(\sim (p \wedge q) \rightarrow \sim r\). The dominance of connectives in logical expressions is as follows: Negative (\(\sim\)) > Conjunction (\(\wedge\)) > Disjunction (\(\vee\)) > Conditional (\(\rightarrow\)) > Biconditional (\(\leftrightarrow\)). Thus, the given symbolic statement can be read as: 'If it is not the case that both \(p\) and \(q\), then not \(r\)'.
03

Translate the symbolic statement into words

Replace the variables (\(p\), \(q\), \(r\)) with their corresponding statements. So, this statement translates to: 'If it is not the case that both the temperature is above \(85^{\circ}\) and we finished studying, then we will not go to the beach.'

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Key Concepts

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

Logical Connectives
In symbolic logic, logical connectives are the building blocks that combine simple statements into more complex expressions. These connectives include:
  • Negation ( \( \sim \) ): Reverses the truth value of a statement. If a statement is true, its negation is false and vice versa.
  • Conjunction ( \( \wedge \) ): This means "and." A statement compounded with a conjunction is true only if both parts are true.
  • Disjunction ( \( \vee \) ): Representing "or," it is true if at least one of the statements is true.
  • Conditional ( \( \rightarrow \) ): Expresses dependency with "if... then," and is false only when the antecedent is true and the consequent is false.
  • Biconditional ( \( \leftrightarrow \) ): Indicates equivalence, meaning both statements must share the same truth value to be true.
Adding logical connectives allows us to form complex relationships within statements, making them pivotal in constructing logical analyses or proofs.
Conditional Statements
A conditional statement set up a relationship between two statements using the "if... then" structure. In symbolic form, it is expressed with the connective (\( \rightarrow \)), emphasizing that one event happens under the condition of another.
For example, the expression \( \sim (p \wedge q) \rightarrow \sim r \) fits this structure, where:
  • \( \sim (p \wedge q) \): Acts as the antecedent or the condition, describing a situation "not both p and q."
  • \( \sim r \): Serves as the consequent, indicating that "r is not the case."
The conditional statement means if the first part is true (though combined negatively with both p and q), it leads to the outcome that we do not proceed with r.
Negation
Negation is one of the simplest, yet powerful, logical connectives. It is used to deny the truth value of a given statement. Represented by the symbol \( \sim \) in symbolic logic, negation flips the truth; if something is initially true, its negation makes it false, and vice versa.
In the context of our symbolic statement \( \sim (p \wedge q) \rightarrow \sim r \), negation is applied twice:
  • \(\sim (p \wedge q) \): This reflects "It is not the case that both the temperature is above \(85^\circ\) and we finished studying."
  • \(\sim r \): Meaning "We will not go to the beach."
The presence of negation switches the anticipated outcomes, leading to different interpretations and results in logical statements.
Translation of Symbolic Statements
Translating symbolic expressions into plain language can clarify the logic behind complex statements. The process involves replacing symbols and letters with understandable phrases.
Consider the symbolic expression: \(\sim (p \wedge q) \rightarrow \sim r \).The steps are:
  • Identify each letter:
    • \(p\): "The temperature is above \(85^\circ\)."
    • \(q\): "We finished studying."
    • \(r\): "We go to the beach."
  • Break down the logic:
    • Negation and conjunction (\(\sim (p \wedge q)\)) become "It is not the case that both these are true."
    • Conditional ( \( \rightarrow \) ) relates the non-occurrence of the first condition to the negation of the second event (\(\sim r\)).
  • Translate fully to: "If it is not the case that both the temperature is above \(85^\circ\) and we finished studying, then we will not go to the beach."
This method ensures symbolic logic can be effectively communicated and applied in practical scenarios.

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Most popular questions from this chapter

Translate the argument below into symbolic form. Then use a truth table to determine if the argument is valid or invalid. It's wrong to smoke in public if secondary cigarette smoke is a health threat. If secondary cigarette smoke were not a health threat, the American Lung Association would not say that it is. The American Lung Association says that secondary cigarette smoke is a health threat. Therefore, it's wrong to smoke in public.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If I am at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I am at the beach, then I feel refreshed.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) We criminalize drugs or we damage the future of young people. We will not damage the future of young people. \(\therefore\) We criminalize drugs.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If you tell me what I already understand, you do not enlarge my understanding. If you tell me something that I do not understand, then your remarks are unintelligible to me. \(\therefore\) Whatever you tell me does not enlarge my understanding or is unintelligible to me.

Use Euler diagrams to determine whether each argument is valid or invalid. All dancers are athletes. Savion Glover is an athlete. Therefore, Savion Glover is a dancer.
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