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.