Construct a predicate logic expression to represent the statement: "Every person who loves someone is loved by someone."
user
Sure, I can help construct the predicate logic expression for you.

First, let's break down the English sentence into its predicates and subject-predicate relationships:

1. Person: The subject
2. Loves: The predicate (middle term)
3. Someone: The predicate (middle term)
4. Loved by: The predicate (middle term)

Using logical symbols and variables, the predicate logic expression of the statement is:

For all x (Person(x) and loves(x, y) implies loved_by(y, x)),

where 'x' represents any person, 'y' represents any other entity, 'loves' is a binary predicate or relation for loving, and 'loved_by' is also a binary predicate or relation for being loved by.

This expression means that for any person 'x', if 'x' loves someone (entity 'y'), then that person 'y' will also love 'x' (person).

So, the complete expression would be:

∀x[Person(x) celebratedBy(x, P) -> C∧A]]

Here, 'P' stands for the Person's name/variable, 'C' stands for being Loved By condition, and 'A' is for being Loved, both are also predicates.
assistant
Construct a predicate logic expression to represent the statement: "Every person who loves someone is loved by someone."
user
Let's use the following predicates and variables to represent the statement:

P(x): x is a person
L(x, y): x loves y

Now we can construct the predicate logic expression:

∀x[(P(x) ∧ ∃y(P(y) ∧ L(x, y))) → ∃z(P(z) ∧ L(z, x))]

This expression reads as "For all x, if x is a person and there exists some person y such that x loves y, then there exists some person z such that z loves x." which represents the given statement.
assistant