What are free variables and what does it mean for a statement to contain one?

Question

I am puzzled a bit. I read the wiki page and an introductory book on logics, but I can't quite grasp it yet. The place where i came across them is Van Inwagens 'Material Beings'.

Consider the following three sentences:

(1) x=x

(2) ∃x x=x

(3) ∀x x=x

It seems to me that they are all equally true.

Also, I can't see the difference between the next two sentences:

(4) If x happens, I will get mad (free)

(5) ∃x If x happens, I will get mad (bound)

The only difference I see is to (6):

(6) ∀x If x happens, I will get mad (bound)

Clearly, in (6) im constantly mad and in (5) not.

A good answer would explain the difference between (4) and (5), I feel that that explanation would help me along in understanding what it means for a sentence to contain a free variable.

Edit: there is now a follow up question here

Answer 1

I apologize for the change of notation, but it will simplify things in the long run. Consider sentences (1-6):

╔════════════╦══════════════╗
║ (1) x=x    ║ (4) Hx->M    ║
║ (2) ∃x:x=x ║ (5) ∃x:Hx->M ║
║ (3) ∀x:x=x ║ (6) ∀x:Hx->M ║
╚════════════╩══════════════╝

Notice the difference between the columns: in the first the sentential function is "x = x", in the second it is "if x happens, I will get mad" ("Hx -> M"). Briefly, some informal definitions:

Sentential functions are functions which, when their arguments are supplied, return a declarative sentence. Declarative sentences are expressions that can be given truth-conditions and thus can be evaluated to true or false.

Let g(x) denote "x = x", and h(x) denote "if x happens, I will get mad" ("Hx -> M"), to obtain:

╔═════════════╦═════════════╗
║ (1) g(x)    ║ (4) h(x)    ║
║ (2) ∃x:g(x) ║ (5) ∃x:h(x) ║
║ (3) ∀x:g(x) ║ (6) ∀x:h(x) ║
╚═════════════╩═════════════╝

This table is just to show how formally similar (1-3) and (4-6) are. Now let's get to the problems.

Claim 1. (1-3) are all equally true.

Expression (1) cannot be true, because it's a sentential function; sentential functions don't have truth-conditions (see the definition above). Sentence (3) says that everything is self-identical, so it's pretty uncontroversially true. Sentence (2) says that there exists something which is self-identical. In order for (2) to be true, at least one thing has to exist, while (3) is true for even empty domains.

In sum: (1) has no truth-conditions, (2) is true in all non-empty domains, and (3) is true in all domains.

Claim 2. (5 & 6) have different truth-conditions.

Exactly! For the same reason (2 & 3) have different truth-conditions.

Question. What's the difference between (4 & 5)?

As I said in my response to your first claim: (1) has no truth-conditions. The same is true for (4); it too has no truth-conditions. (5), like (2), however, does have truth-conditions: (5) is true if and only if: there exists some x such that either: x is not happening or you're getting mad.

Hope that makes things a little clearer.

Answer 2

First difference:

• In some sense statement (4) is incomplete, if x indeed is a variable, then the truth value of (4) is undefined.

• However, statement (5) does have a well defined truth value (even if it is unknown) even though the statement contains a variable.

a second difference:

• (5) is logically entailed by one (or more) true statements of (4) for different x's.

• However, one can know/assert/prove a statement of form (5) without knowing/identifying/proving, for any particular 'x', a statement of the form (4).

Answer 3

Variables are a useful concept in various contexts. Their usefulness arises from their ability to refer to objects by a more or less arbitrary name or symbol. In first order logic, the need for this ability arises for the quantifiers ∃ and ∀. Also the lambda abstraction λ needs this ability, and other examples less closely related to logic are probably easy to find. There are alternatives like de Bruijn indices, which allow to avoid the usage of variables.

Like a bound variable, a free variable refers to an object. Depending on context, more or less assumptions about such an object are made. In the context of first order logic, the only assumption is that the referenced object exists. So in this context, free variables are nearly the same thing as constants, except for the fact that they serve a different purpose. Therefore, the treatment and exact interpretation of (free) variables is often more restricted and slightly more subtle than the treatment of constants.

For example in Ebbinghaus et. al, officially only the variables "v_0, v_1, v_2, ..." exists, despite the fact that the text normally uses "x, y, z, ..." when talking about variables. The funny thing is that "x, y, z, ..." can mean any of "v_0, v_1, v_2, ...", so that the text has to sometimes explicitly exclude the case "x=y" (the text uses "\equiv" for equality between objects). The text further defines the subset "L^S_n" of the language "L^S" for the formulas which only contain "v_0, ..., v_{n-1}" freely. S-structures and S-interpretations are two separate notions, where an S-interpretation contains a concrete assignment of objects to the variables "v_0, v_1, v_2, ...". For an S-structure, you can ask whether it could satisfy a given set of S-formulas, or whether a set of S-formulas will be satisfied by the S-structure for all possible variable assignments.

As another example, my references for universal algebra explicitly specify "X" as a finite or countable infinite set, whose elements are called variables, and explicitly tag this "X" to any set of terms or formulas. What this has in common with the previous example is that there are at most countable infinite different variables. Still, in universal algebra, the free variables are also use as a sort of generic element, although not through abuse of notation but through an explicit construction of corresponding free algebras.

Answer 4

Sentences with free variables are like the indefinite integrals in calculus, until they're bound to some interval they do not have a well-defined value, and similarly such a sentence until it is bound has no well-defined truth value, that is, in a sense it is incomplete. But note that this does not mean that they are not valuable, in the same way that an indefinite integral is valuable in calculus. A free variable should be thought of a 'generic' variable, or a placeholder.

Informally we can say that the difference between free & bound variables is that for the free variables we do not know what kind of values the variable takes, for example take the sentence:

x=x,

Now is x an integer, a real number, a cat, infinity or the universe? We do not know as we haven't been told. A free variable is bound when this information is available, for example:

(for all x in the integers) x=x

This is a bound sentence as we now know that x must be an integer, similarly

(for all x in {cats in Europe}) x is a mammal

This is bound since x must be a cat that is in Europe.

Now the first-order calculus introduces the quantifiers 'for all' & 'there exists', and for these to make any sense we also must have variables (there are none in the propositional calculus) and one can distinguish sentences in which all variables are bound to a quantifier and those that are not.

Now, we can distinguish two further cases:

1. at least some but not all variables are bound

2. and all variables are bound.

It turns out that of these two cases it is the second that is of the most direct interest. This is because these are the sentences that have well defined truth values under an 'interpretation' that assigns values to x. For example:

2x=x

has no well defined truth value as we do not know what the value of x should be, but:

(for all x in the integers) 2x=x

is obviously false, and

(there exists x in the integers) 2x=x

is true since we can choose x to be zero.

Answer 5

Your question amounts to asking what the difference is between saying:

1. "it is the same thing as it"

2. "something is the same thing as itself"

3. "everything is the same thing as itself"

Answer:

2 and 3 express different propositions but 1 expresses no proposition at all. It is not "well formed", this means that it is not a sentence in first order logic. Stack exchange is not the place to learn first order logic, buy a book by Quine. Intuitively though, it is obvious that 1 is meaningless because it has not been completed and turned into either 2 or 3.

To answer your Q directly: A free variable in a sentence is a variable in a sentence for which it has not been specified whether the concept of All or the concept of SOME is to apply to it. So in your example: 'x=x' contains two occurrences of the same free variable 'x' because it has yet to be specified whether the concept of All is to apply to the 'x', and so be turned into 'Ax x=x', or whether the concept of SOME is to apply to the x and so be turned into 'Ex x=x'