Interpreting propositional formulas with quantifiers in the English language

Question

This is a follow up to an earlier posting of mine. I think I am beginning to understand the answers suggested, but I am not fully there yet.

My confusion arises from the inconsistency in the symbolic representation for the existential and universal quantifiers: one uses conjunction and the other uses implication. I wonder why the rationale provided for the universal quantifier does not hold for the existential quantifier and vice-versa.

So let me ask the questions again but from a different angle.

Given:

R(x): x has taken a course in Rhetoric
S(x): x is a student in this class

How do we translate the following symbolic expressions into English?

I): ∃x (S(x) ∧ R(x))

II): ∃x (S(x) → R(x))

III): ∀x (S(x) ∧ R(x))

IV): ∀x (S(x) → R(x))

Answer 1

Again you need to be fully aware of all background knowledge including its unstated domain of discourse. By English conversational convention, the domain seems all students (in your school), then we may translate as follows (of course not the only way):

I): There's a student in this class who has taken a course in Rhetoric.

II): There's such a student if he or she is a student in this class then this student has taken a course in Rhetoric.

III): Every student (in your school) is a student in this class and has taken a course in Rhetoric.

IV): All students in this class have taken a course in Rhetoric.

Like most things in math, the actual math symbolic part is usually mechanical and understandable, the trick is epistemic awareness or not.

Answer 2

Let's assume for the moment that the domain of discourse is everything that exists. Here are a few ways to translate each of the sentences.

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I) There's a student in this class who has taken rhetoric.

I) There exists an entity $x$ such that $x$ is a student in this class and $x$ has taken rhetoric.

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The sentence (II) is not very natural.

II) There exists an entity $x$ such that it is not the case that $x$ is a student in this class or $x$ has taken rhetoric or both.

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The sentence (III) is extremely strong.

III) Every entity is a student who has taken rhetoric.

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IV) Every student in this class has taken rhetoric.

IV) For every entity $x$, if $x$ is a student in this class, then $x$ has taken rhetoric.

IV) For every entity $x$, it is not the case that $x$ is a student in this class or $x$ has taken rhetoric or both.

Answer 3

Given:

R(x): x has taken a course in Rhetoric
S(x): x is a student in this class

How do we translate the following symbolic expressions into English?

I): ∃x (S(x) ∧ R(x))

There is someone who is a student in this class and has taken a course in rhetoric.

(We assume here that the the domain of discourse is all of humanity.)

II): ∃x (S(x) → R(x))

There is someone who, if they are a student is this class, then they have taken a course in rhetoric.

III): ∀x (S(x) ∧ R(x))

Everyone is a student in this class and has taken a course in rhetoric.

IV): ∀x (S(x) → R(x))

Everyone, if they are a student in this class, then they have taken a course in rhetoric.

Answer 4

Domain of discourse: the set of all persons.

1. $$∃x \big(S(x) ∧ R(x)\big)$$ Some Student in this class has taken Rhetoric.

2. $$∃x \big(S(x) → R(x)\big)$$ $$∃x \big(\lnot S(x) \lor R(x)\big)$$ (these two formulae are first-order equivalent)

   Some person either is not a Student in this class, or has taken Rhetoric.

3. $$∀x \big(S(x) ∧ R(x)\big)$$ Every person is a Student in this class who has taken Rhetoric.

4. $$∀x \big(S(x) → R(x)\big)$$ Every Student in this class has taken Rhetoric.