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When we use 'terms' such as words it is generally clear however, if we have a formula:

And I write:

'x+1=2 is true for x=1' is this 'using' or 'mentioning'?

If a formula contains variables, it has no truth value or particular meaning, how do I 'use' a free variable?

When using variables, how much of it is the syntax of the string or the semantics of the variable? Is it possible to 'use' them like words?

Another question is that of intent, If I write:

Some would say 'two is a prime number', is this an example of use or 'mention'?

I am using the 'assertion' (string of symbols), yet I do not wish to personally make the assertion it just make the reader aware of the assertion, in this case I'd say as long as the 'assertion' is being communicated it is 'use' but I'm not sure.

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  • If we write "Formula 'x+1=2' has one free variable" we are asserting a property of the formula and not the result of an arithmetical operation. Thus, it is MENTION.
    – Mauro ALLEGRANZA
    Commented Dec 15, 2022 at 6:39
  • If we agree that truth is a property of sentences, when we write 'x+1=2 is true for x=1' we are asserting a property of the (interpreted) formula. Thus, it is MENTION. We are "describing" in plain language Tarski's formulation: "The assignment 1 → x satisfies the formula ‘(x+1=2)’ ".
    – Mauro ALLEGRANZA
    Commented Dec 15, 2022 at 6:49
  • @MauroALLEGRANZA if I were simply to give the formula such as: x+1=2 as given to communicate for example that the value of x is 1, or to use a constraint, that would be 'use'?
    – Confused
    Commented Dec 15, 2022 at 10:14
  • If we use a formula, this is use; if we quote it (using quotation marks) this is mention.
    – Mauro ALLEGRANZA
    Commented Dec 15, 2022 at 10:21
  • @MauroALLEGRANZA often people write x+1=2 is true for x=1, as you point out this is a mention, but they dont use quotes, is this just poor notation?
    – Confused
    Commented Dec 15, 2022 at 10:29

2 Answers 2

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In general, the term "use" refers to the act of employing something for a specific purpose or in a specific context. When we use words or formulas, we are employing them to convey a particular meaning or to perform a specific function.

In the case of the formula "x+1=2", it is being used to express the fact that, for a particular value of x (in this case, 1), the formula is true. This is an example of using the formula, as it is being employed to convey a specific meaning.

As for free variables, they can be used in a similar way to other variables. For example, if we have a formula with a free variable x, we can use that formula to represent a relationship between x and other variables in the formula. For example, the formula "x+1=2" can be used to represent the relationship between x and the other variables in the formula (in this case, 1 and 2).

In terms of the syntax and semantics of variables, the syntax refers to the way in which the variables are written or arranged, while the semantics refers to the meaning of the variables. When using variables, both the syntax and the semantics are important. For example, in the formula "x+1=2", the syntax of the formula (i.e., the way in which the variables and operators are arranged) conveys the mathematical operation being performed, while the semantics of the variables (i.e., the meaning of the individual symbols) determines the specific values being used in the formula.

As for the question of intent, whether something is considered "use" or "mention" can depend on the context and the intention of the speaker or writer. In the case of the statement "Some would say 'two is a prime number'", it could be considered either use or mention depending on the context and the speaker's intention. If the speaker is simply mentioning the statement without expressing their own opinion on its truth or falsehood, it could be considered a mention. However, if the speaker is using the statement to make a point or to express their own opinion, it could be considered a use. Ultimately, the determination of whether something is "use" or "mention" depends on the context and the speaker's intent.

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Long comment

If we write "Formula 'x+1=2' has one free variable" we are asserting a property of the formula and not the result of an arithmetical operation. Thus, it is MENTION.

If we agree that truth is a property of sentences, when we write 'x+1=2 is true for x=1' we are asserting a property of the (interpreted) formula.

Thus, it is MENTION; we are "describing" in plain language Tarski's rule: "The assignment 1 → x satisfies the formula ‘(x+1=2)’ ".

Usual mathematical practice is semi-formal, because humans are intelligent being that can use implicit information from the context in order to correctly understand statements. If I'm uttering "Rose has four letters" there is no need to explicitly insert quotation marks because we can understand that I'm speaking of "rose" and not of Rose (a girl).

In mathematics, we use formulas to express mathematical facts: as already discussed, 1+1=2 express a computation.

The issue with a formula with free variables is slightly more convoluted, but the basic principle stands. In order to perform a computation with a formula like x+1=2 we have to decide what is the value of x.

So when we give the instruction: assign to variable x the value 1 into formula "x+1=2" we are mentioning the formula. After the "substitution", what we get is the new formula 1+1=2 that we will use to perform the computation.

It is interesting to note that also philosopher can sometimes be sloppy.

Leibniz expressed the paramount principle defining identity with the dictum: Eadem sunt quorum unum potest substitui alteri salva veritate.

As noted by contemporary logician Alonzo Church (Introduction to Mathematical Logic (1956), page 300, footnote 302):

"In this form there is a certain confusion of use and mention: things are identical if the name of one can be substituted for that of the other without loss of truth

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  • We could ask, do we ever really 'use' formulas with free variables, or simply 'mention' them, then, when we assign we get a formula like '1+1=2' that can then be 'used' in order to express and perform the calculation? It makes no difference but it's an interesting idea, for example most algebraic things we do are at the 'synactic' level so do we really ever 'use' the formulas until the very end when we assign and then subsitute? It is a tiresome distinction notheless, and I think sometimes it is best no to think about.
    – Confused
    Commented Dec 15, 2022 at 10:58
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    @Confused - in algebraic manipulation, variables are quite always used to express generality: ax+by+c=0 is a recipe to build numerical expression: it means that what we will say is not depending on the specific values of parameters.
    – Mauro ALLEGRANZA
    Commented Dec 15, 2022 at 11:02
  • so to say "the number ax+by+c=0" is a 'use' because it describes objects in a generalised way? Whereas to say the "formula ax+by+c=0" is a 'mention' in the 'use' the formula is being discussed as an object so we can consider it a 'use'? In terms of generality we can go from formula to formula however we can still see a 'use' in treating the variables as objects.
    – Confused
    Commented Dec 15, 2022 at 11:04
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    @Confused - yes, usually formula above "describes" a line in the x-y plane.
    – Mauro ALLEGRANZA
    Commented Dec 15, 2022 at 11:05

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