6. How would you present the following propositions and their negations using logical quantifiers? Also, interpret the negations in words. i) The politician can fool all the people all the time. ii) Every real number is the square of some real number. iii) There is a lawyer who never tells lies.

To present the propositions and their negations using logical quantifiers, we'll use standard notation:

- \( \forall \) (for all)
- \( \exists \) (there exists)
- \( \neg \) (not)

Let's analyze each proposition:

### i) The politician can fool all the people all the time.

**Proposition:** "The politician can fool all the people all the time."

**Logical Notation:** \( \forall p \, (F(p)) \)

- \( p \): a person
- \( F(p) \): the politician can fool person \( p \)

**Negation:** "It is not true that the politician can fool all the people all the time."

**Logical Notation:** \( \neg \forall p \, (F(p)) \)

**Equivalent to:** \( \exists p \, (\neg F(p)) \)

**Interpretation in Words:** "There exists at least one person whom the politician cannot fool."

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### ii) Every real number is the square of some real number.

**Proposition:** "Every real number is the square of some real number."

**Logical Notation:** \( \forall x \, \exists y \, (x = y^2) \)

- \( x \): a real number
- \( y \): a real number such that \( x = y^2 \)

**Negation:** "It is not true that every real number is the square of some real number."

**Logical Notation:** \( \neg \forall x \, \exists y \, (x = y^2) \)

**Equivalent to:** \( \exists x \, \neg \exists y \, (x = y^2) \)

**Interpretation in Words:** "There exists at least one real number that is not the square of any real number."

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### iii) There is a lawyer who never tells lies.

**Proposition:** "There is a lawyer who never tells lies."

**Logical Notation:** \( \exists l \, \forall t \, (T(l, t)) \)

- \( l \): a lawyer
- \( t \): a statement or situation
- \( T(l, t) \): lawyer \( l \) tells the truth in situation \( t \)

**Negation:** "It is not true that there is a lawyer who never tells lies."

**Logical Notation:** \( \neg \exists l \, \forall t \, (T(l, t)) \)

**Equivalent to:** \( \forall l \, \exists t \, (\neg T(l, t)) \)

**Interpretation in Words:** "Every lawyer tells lies in some situation."

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Would you like further details or have any questions? Here are some related questions:

1. How do logical quantifiers differ from each other?
2. What is the significance of the order of quantifiers in a logical statement?
3. Can you provide an example of a real number that is not the square of any real number?
4. What are the common logical operators used in mathematical logic?
5. How does the negation of a universal quantifier differ from the negation of an existential quantifier?
6. What is the role of logical quantifiers in mathematical proofs?
7. Can a proposition be expressed using different combinations of quantifiers and still be equivalent?
8. How can one determine the truth value of a statement involving multiple quantifiers?

**Tip:** When interpreting logical statements, pay close attention to the order of quantifiers, as it can significantly change the meaning of the proposition.

Explore logical quantifiers and propositions with detailed examples. Learn how to present propositions and their negations using logical quantifiers. Suitable for advanced students and anyone interested in mathematical logic.

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