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Chapter 1: Problem 31

Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) \(\forall x \exists y \forall z T(x, y, z)\) b) \(\forall x \exists y P(x, y) \vee \forall x \exists y Q(x, y)\) c) \(\forall x \exists y P(x, y) \wedge \exists z R(x, y, z) )\) d) \(\forall x \exists y(P(x, y) \rightarrow Q(x, y))\)

Short Answer

Expert verified
a) \(\exists x \forall y \exists z eg T(x, y, z)\)b) \(\exists x \forall y eg P(x, y) \wedge \exists x \forall y eg Q(x, y)\)c) \(\exists x \forall y eg P(x, y) \vee \forall z eg R(x, y, z)\)d) \(\exists x \forall y (P(x, y) \wedge eg Q(x, y))\)

Step by step solution

01

- Negate Statement a

Given statement: \[\forall x \exists y \forall z \, T(x, y, z)\] Apply negation to the entire statement: \[eg(\forall x \exists y \forall z \, T(x, y, z))\] Use logical equivalences to move negation inward: \[\exists x eg(\exists y \forall z \, T(x, y, z))\] \[\exists x \forall y eg(\forall z \, T(x, y, z))\] \[\exists x \forall y \exists z \, eg T(x, y, z)\]
02

- Negate Statement b

Given statement: \[\forall x \exists y \, P(x, y) \vee \forall x \exists y \, Q(x, y)\] Apply negation to the entire statement: \[eg(\forall x \exists y \, P(x, y) \vee \forall x \exists y \, Q(x, y))\] Use De Morgan's laws: \[eg(\forall x \exists y \, P(x, y)) \wedge eg(\forall x \exists y \, Q(x, y))\] Move negation inward: \[\exists x \forall y \, eg P(x, y) \wedge \exists x \forall y\, eg Q(x, y)\]
03

- Negate Statement c

Given statement: \[\forall x \exists y \, P(x, y) \wedge \exists z \, R(x, y, z)\] Apply negation to the entire statement: \[eg(\forall x \exists y \, P(x, y) \wedge \exists z \, R(x, y, z))\] Use De Morgan's laws and move negation inward: \[eg(\forall x \exists y \, P(x, y)) \vee eg(\exists z \, R(x, y, z))\] Final result: \[\exists x \forall y \, eg P(x, y) \vee \forall z eg R(x, y, z)\]
04

- Negate Statement d

Given statement: \[\forall x \exists y (P(x, y) \rightarrow Q(x, y))\] Apply negation to the entire statement: \[eg(\forall x \exists y (P(x, y) \rightarrow Q(x, y)))\] Move negation inward: \[\exists x \forall y \, eg(P(x, y) \rightarrow Q(x, y))\] Use logical equivalences to address implication: \[\exists x \forall y \, (P(x, y) \wedge eg Q(x, y))\]

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Key Concepts

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

logical equivalences
Logical equivalences are statements that hold true under the same conditions, allowing expressions to be substituted for one another without changing the logical value. Understanding logical equivalences is vital to simplifying complex logical expressions. For instance, the negation of a universal quantifier (\(\forall x P(x)\)) is equivalent to the existential quantifier of the negation (\(eg\forall x P(x) \rightarrow \exists x eg P(x)\)). By recognizing these patterns, we can easily manipulate and simplify logical statements.
predicate logic
Predicate logic extends propositional logic by dealing with predicates and quantifiers. Predicates are functions that return a true or false value for given inputs. For example, in the statement \(\forall x P(x)\), \(P(x)\) is a predicate. Predicate logic allows for more expressive statements about objects and their properties. Using quantifiers like \(\forall\) (for all) and \(\exists\) (there exists), we can make general or specific claims about sets of objects, leading to more powerful logical reasoning.
De Morgan's laws
De Morgan's laws are fundamental rules in logic, named after Augustus De Morgan. These laws relate the conjunction (\(\wedge\)) and disjunction (\(\vee\)) operators through negation. They state that the negation of a conjunction is equivalent to the disjunction of the negations and vice versa. Formally:

- \(eg (P \wedge Q) \equiv (eg P) \vee (eg Q)\)
- \(eg (P \vee Q) \equiv (eg P) \wedge (eg Q)\)
These transformations are crucial for simplifying expressions and for transformations in negation problems. They make it easier to move negations inward and deal with complex statements.
quantifiers in logic
Quantifiers are symbols used in predicate logic to specify the number of instances that a predicate applies to. The two most common quantifiers are the universal quantifier (\(\forall\)) and the existential quantifier (\(\exists\)).

- The universal quantifier \(\forall x\) means 'for all x' and indicates that a predicate holds for every element in the domain of discourse.
- The existential quantifier \(\exists x\) means 'there exists an x' and signifies that there is at least one element in the domain for which the predicate holds true.
When negating statements with quantifiers, their interaction must be considered. For example, \(eg \forall x P(x)\) translates to \(\exists x eg P(x)\), and \(eg \exists x P(x)\) translates to \(\forall x eg P(x)\). These negations play a crucial role in logical transformations.

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Most popular questions from this chapter

Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), R(x), and S(x) be the statements 鈥渪 is a baby,鈥� 鈥渪 is logical,鈥� 鈥渪 is able to manage a crocodile,鈥� and 鈥渪 is despised,鈥� respectively. Suppose that the domain consists of all people. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), R(x), and S(x). a) Babies are illogical. b) Nobody is despised who can manage a crocodile. c) Illogical persons are despised. d) Babies cannot manage crocodiles. e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Use quantifiers to express the distributive laws of multiplication over addition for real numbers.

Translate these statements into English, where \(C(x)\) is " \(x\) is a comedian" and \(F(x)\) is " \(x\) is funny" and the domain consists of all people. $$ \begin{array}{ll}{\text { a) } \forall x(C(x) \rightarrow F(x))} & {\text { b) } \forall x(C(x) \wedge F(x))} \\ {\text { c) } \quad \exists x(C(x) \rightarrow F(x))} & {\text { d) } \exists x(C(x) \wedge F(x))}\end{array} $$

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Someone in your class can speak Hindi. b) Everyone in your class is friendly. c) There is a person in your class who was not born in California. d) A student in your class has been in a movie. e) No student in your class has taken a course in logic programming.

Determine whether this argument, taken from Kalish and Montague [KaMo64], is valid. If Superman were able and willing to prevent evil, he would do so. If Superman were unable to prevent evil, he would be impotent; if he were unwilling to prevent evil, he would be malevolent. Superman does not prevent evil. If Superman exists, he is neither impotent nor malevolent. Therefore, Superman does not exist.
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