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Chapter 1: Problem 74

Let \(p, q,\) and \(r\) be simple propositions with \(t(p)=1, t(q)=0.3,\) and \(t(r)=\) \(0.5 .\) Compute the truth value of each, where \(s^{\prime}\) denotes the negation of the statement \(s\) $$p \wedge q^{\prime}$$

Short Answer

Expert verified
The truth value of the compound proposition \(p \wedge q^{\prime}\) is False or 0, given the truth values \(t(p) = 1\), \(t(q) = 0.3\), and \(t(r) = 0.5\).

Step by step solution

01

List given values for p, q, and r

The exercise provides us with the truth values of three simple propositions: - \(t(p) = 1\) (True) - \(t(q) = 0.3\) (Not a standard truth value, as it is neither 0 or 1. We will assume that any non-zero value will be considered True) - \(t(r) = 0.5\) (Same as for \(q\), we will assume any non-zero value is True)
02

Identify the negation of q

We are asked to find the truth value of the compound proposition: $$p \wedge q^{\prime}$$ This means we need to perform a logical conjunction (AND) operation between the truth values of propositions \(p\) and the negation of proposition \(q\) (denoted as \(q^{\prime}\)). Since \(t(q) = 0.3\), and we are assuming non-zero values as true, the proposition \(q\) is true. Therefore, the negation of \(q\) (denoted as \(q^{\prime}\)) will be false, or: $$t(q^{\prime}) = 0$$
03

Perform the conjunction (AND) operation

Now that we have the truth values for both \(p\) and \(q^{\prime}\), we can compute the truth value of the compound proposition $$p \wedge q^{\prime}$$ Recall that the output of a conjunction (AND) operation is true if and only if both input values are true. In this case, we have: $$t(p) = 1 \text{ (True)}$$ $$t(q^{\prime}) = 0 \text{ (False)}$$ Since not both input values are true, the output of the conjunction (AND) operation will be false: $$t(p \wedge q^{\prime}) = 0 \text{ (False)}$$ Thus, the truth value of the compound proposition $$p \wedge q^{\prime}$$ is False or 0.

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Key Concepts

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

Truth Values
In propositional logic, each statement, or 'proposition', is assigned a truth value which reflects its veracity. The standard truth values are binary: either 'true' or 'false', which are often represented as '1' for true, and '0' for false. The exercise mentioned a non-standard truth value of 0.3 for a certain proposition 'q'. This represents a break from classical propositional logic which confines to binary values only.

It's important to note, however, that in classical logic, only binary truth values are considered, and every proposition must be either entirely true or entirely false. This exercise appears to involve a non-classical approach, where values between 0 and 1 could indicate degrees of truth. However, for the purpose of solving the exercise, any non-zero value has been considered as 'true' in order to proceed with the logical operations.
Logical Conjunction
A logical conjunction is a basic operation in propositional logic, represented by the symbol \( \wedge \). It takes two propositions and assigns a truth value of 'true' if and only if both propositions are true. If either of the propositions is false, the conjunction is false.

The concept can be easily related to the word 'and' in everyday language. For instance, the compound statement 'I will go to the store and I will buy milk' is only true if both the individual occurrences - going to the store and buying milk - are true. Similarly, in propositional logic, if we consider two propositions \( p \) and \( q \), the conjunction \( p \wedge q \) is true only when \( t(p) = 1 \) and \( t(q) = 1 \). Any other combination would result in a false conjunction.
Negation of a Proposition
The negation of a proposition, denoted by a prime symbol \( ^\prime \) or tilde \( \sim \) , is essentially the logical inverse of the original proposition's truth value. If the original proposition is true, then its negation is false; conversely, if the original proposition is false, its negation is true. This operation turns 'true' into 'false' and 'false' into 'true'.

In the context of the exercise, given that the non-standard truth value \( t(q) = 0.3 \) was considered true, its negation \( q^\prime \) becomes false. The logic behind this is that when we negate something that is true, it becomes false, and vice versa. This inversion is a critical aspect of logical operations and helps in constructing more complex logical expressions.

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Most popular questions from this chapter

Exercises \(65-78\) deal with propositions in fuzzy logic. Let \(p, q,\) and \(r\) be simple propositions with \(t(p)=1, t(q)=0.3,\) and \(t(r)=\) 0.5 . Compute the truth value of each, where \(s^{\prime}\) denotes the negation of the statement \(s\) . $$ p \wedge q $$

At the bus terminal, Ellen overheard the following conversation between two baseball fans, L and M: L: I like the Yankees. M: You do not like the Yankees. You like the Dodgers. L: We both like the Dodgers. Does fan L like the Yankees? Who likes the Dodgers?

A third useful quantifier is the uniqueness quantifier \(\exists ! .\) The proposition \(\left(\exists^{\prime} x\right) P(x)\) means There exists a unique (meaning exactly one) \(x\) such that \(P(x) .\) Determine the truth value of each proposition, where UD \(=\) set of integers. $$(\forall x)(\exists ! y)(x+y=4)$$

Exercises \(65-78\) deal with propositions in fuzzy logic. Let \(p, q,\) and \(r\) be simple propositions with \(t(p)=1, t(q)=0.3,\) and \(t(r)=\) 0.5 . Compute the truth value of each, where \(s^{\prime}\) denotes the negation of the statement \(s\) . Let \(p\) be a simple proposition with \(t(p)=x\) and \(p^{\prime}\) its negation. Find each. $$ t\left(p \wedge p^{\prime}\right) $$

Exercises \(65-78\) deal with propositions in fuzzy logic. Let \(p, q,\) and \(r\) be simple propositions with \(t(p)=1, t(q)=0.3,\) and \(t(r)=\) 0.5 . Compute the truth value of each, where \(s^{\prime}\) denotes the negation of the statement \(s\) . $$ (p \vee q) \wedge\left(p^{\prime} \vee q\right) $$
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