# Need clarification regarding **free variables** and **ground sentences**

I have been reading Logical Fundations of Artificial Intelligence by Michael R. Genesereth, and have questions regarding some of his paragraphs on Page 20.

1.

A variable can also occur as a term in a sentence without an enclosing quantifier. When used in this way, a variable is said to be free, whereas a variable that occurs in a sentence and in an enclosing quantifier is said to be bound. ....

Does the definition of free variables mean the sentence is without an enclosing quantifier or the variable appearing in the sentence is without one?

2.

If a sentence has no free variables, it is called a closed sentence. If it has neither free nor bound variables, it is called a ground sentence.

Does being a ground sentence mean that the sentence has no variable at all? If no so, which kinds of variables could it contain? Are there some examples?

• DO NOT SHOUT. It's rude. – gen-ℤ ready to perish Sep 16 '17 at 21:18
• @ChaseRyanTaylor I'm sorry, but I don't understand, how could one "shout" on the web? – Yan Yang Sep 16 '17 at 21:32
• ALL CAPS conventionally indicates yelling. To show emphasis, surround your text in one asterisk, two asterisks, or three asterisks on either side – gen-ℤ ready to perish Sep 17 '17 at 1:30
• @ChaseRyanTaylor Thank you! I didn't know lol – Yan Yang Sep 17 '17 at 4:23
• It doesn't work in the headline. But I think it's okay this way. – Yan Yang Sep 17 '17 at 4:24

1. It means that the variable is without an enclosing quantifier. Thus, in $\forall x (x < y)$, the variable $x$ is bound, while $y$ is free.
2. Yes, it means it has no variables at all. A language may still have constants to form sentences. For example, if a language has the constant $0$ and the function $+$, you could form the sentence $0+0=0$.