# What is the meaning of free variables?

In some books I see something like: phi(x1,...,xn) with free variables among x1,...,xn.

An example of this is on 2nd paragraph, page 25 of the book Knowledge in Action. I have included a snapshot of this page:

Does this mean all the free variables are only in x1,...,xn and perhaps some of x1,...,xn are not free? Then this would indicate some of them are bound, but I do not see a quantifier.

• Hi, welcome to Philosophy SE. Notational conventions vary, so please point out specific book and page where this is used and/or provide an extended quote. Some of xi might be specified as constants in the text, and some as variables. This happens when instantiation rules are used to remove quantifiers, for example. Jun 23, 2019 at 9:08
• Hi, thank you. I have edited my question to include a specific reference to a book. Jun 23, 2019 at 16:06

This is how the forallx text describes free and bound variables (page 223, section 26.3):

A bound variable is an occurrence of a variable x that is within the scope of either ∀x or ∃x.

A free variable is any occurrence of a variable that is not bound.

As an example to illustrate bound and free variables and the scope of quantifiers, they provide the following

∀x(E(x) ∨ D(y)) → ∃z(E(x) → L(z, x))

The scope of the universal quantifier ‘∀x’ is ‘∀x(E(x) ∨ D(y))’, so the first ‘x’ is bound by the universal quantifier. However, the second and third occurrence of ‘x’ are free. Equally, the ‘y’ is free. The scope of the existential quantifier ‘∃z’ is ‘(E(x) → L(z, x))’, so ‘z’ is bound.

Note how the ‘x’ is free in the consequent, but bound in the antecedent. Furthermore formulas need not be sentences in the context of the forallx text:

A sentence of FOL is any formula of FOL that contains no free variables.

It is important to be aware of the definitions of the logic text you are using and when reading a different text to note any differences one might find.

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf

• Thank you for the great explanation! I have edited my question and added a reference to a book. Jun 23, 2019 at 16:08
• I don't have Reiter's book. I wonder if you could quote the relevant part of the 2nd paragraph on page 25.@Fabiana Jun 23, 2019 at 16:13
• I added a snapshot. I hope it is clear Jun 23, 2019 at 16:26
• @Fabiana Unless "free" means something other than what I'm expecting from my answer, I would assume they are all free because they do not have quantifiers to bind them. Jun 23, 2019 at 16:41