That makes me crazy to think about it because my book and other pages on the web talks about free and bound variables without any definition. I think everything in mathematics has a definition so there must be a definition for it.

Also, I am taking a course called abstract mathematics talks about how to proof and logic. I am curious about if logic and these proof study have another name in mathematics.

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    $\begingroup$ Using GOOGLE, you’ll find answers to your question. Like math.helsinki.fi/logic/opetus/log1/…. $\endgroup$ – mathcounterexamples.net May 11 at 8:59
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    $\begingroup$ Yes an occurrence of a variable is bound if it is in the scope of a quantifier : the occurrence of $x$ in $\forall x P(x)$, otherwise is free : the occurrence of $y$ in $\forall x (P(x) \land Q(y))$. $\endgroup$ – Mauro ALLEGRANZA May 11 at 9:06
  • $\begingroup$ @MauroALLEGRANZA How is it like to be a bound and free variable ? $\endgroup$ – Smyra May 11 at 9:15
  • $\begingroup$ @Smyra Doesn't Mauro's definition satisfy you ? If so, please clarify what is still unclear. $\endgroup$ – Peter May 11 at 9:29
  • $\begingroup$ What I know so far, A free variable is just a standing letter for an object but since the object isn't clear we can free to assert any object for the free variable. It can change the meaning of a statement. A bound variable is only a letter it doesnt mean it stands for an object and even we could change the letter the meaning of a statement will not change. It has a connection with a set. But the free variable also has a connection with a set since we choose our objects for free variable according to it. I dont understand clearly the difference in some cases like this. $\endgroup$ – Smyra May 11 at 9:45

Free and bound variables are defined in the context of the syntax of first-order logic, considering terms (i.e. "names" for objects) and formuals (i.e. statements).

The formal definition of the set $\text {FV}(φ)$ of free variables of a formula $φ$ is defined by :

all the variables occurring in a term or atomic formula are free.

$\text {FV}(¬φ) = \text {FV}(φ)$;

$\text {FV}(φ ∨ ψ) = \text {FV}(φ) \cup \text {FV}(ψ)$, (and the same for the other binary connectives);

$\text {FV}(∀xφ) = \text {FV}(∃xφ) = \text {FV}(φ) \setminus \{ x \}$.

A variable that is not free is bound.

A formula $φ$ is called closed if $\text {FV}(φ)=\emptyset$.

In formula $\forall x P(x)$, the variable $x$ is bound.

In formula $\forall x R(x,y)$ the variable $x$ is bound while the variable $y$ is free.

A closed formula, when interpreted, expresses a sentence, i.e. has a definite tuth value.

$\forall n (n \ge 0)$ is true in $\mathbb N$, while $\exists n (n < 0)$ is false in it.

What is the truth value of a formula with a free variable, like e.g. $(x > 0)$ ?

It depends... It depends on the value we assign to the variable $x$.

A free variable acts as a pronoun of natural language : its reference must be identified according to the context : if I say "it is red", the truth vale of the statement depends on the object I'm pointing at with my finger : the red book or the blue pen on my table.

In the same way, there are ways (defined by the formal semantical specifications of the first-order language : see variable assignment function) to give "temporary" reference to free variables of a formula.

Consider the formula :


if we substitute to $x$ the (name for the) number $3$, we obtain a true sentence (i.e. $3+2=5$).

If instead we substitute to $x$ the (name for the) number $4$, we obtain a false sentence (i.e. $4+2=5$).

A formula with free var is called "open" because it has no (fixed) meaning : it is "open to" different interpretations; in order to give it meaning, we have to transform it into a sentence (i.e. a closed formula).

  • $\begingroup$ It is a high class answer! Thank you! $\endgroup$ – Smyra May 11 at 15:48

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