Question

Which of the following is an open sentence? (z)((∼Yb ↔ Ya) ∨ (∃x)(Fx & Gx)) (y)((Cy...

Which of the following is an open sentence?

(z)((∼Yb ↔ Ya) ∨ (∃x)(Fx & Gx))

(y)((Cy ∨ (v)Dy) → ∼Yy)

((x)(Px & Myx) ↔ (y)(Cy ∨ Dy))

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Homework Answers

Answer #1

A sentence is open if and only it has at least one free variable. Otherwise it is closed.

A free variable is one that is not bound. So there will be no quantifier (there exists, for all, etc.) for a free variable.

(1) The sentence is open since while x and z are bounded (due to the quantifiers 'there exists' and 'for all' [(z) means 'for all z']), the variables a and b are free.

(2) The sentence is not open since the only variable y is bounded (due to the quantifier 'for all' [(y) means 'for all y'])

(3) The sentence is not open since all the variables x and y are bounded (due to the quantifier 'for all' [(x) means 'for all x' and (y) means 'for all y']).

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