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My set theory book has this sentence.(It was written in my native language. I translated these. Sorry for poor English.)

When $x$ is assigned as the value of the independent variable of the function $f$, then the value of the dependent variable is denoted by $f(x)$.

And the following sentence is in the footnote.

If the independent variable of $f$ is $x$, we get a ridiculous expression "When $x$ is assigned as the value of $x$..." We can solve this problem by distinguishing between syntactical and semantical variables.

I can not understand this footnote. What is the difference between semantical and syntactical variables?

amoogae
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1 Answers1

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A variable is a syntactical "object".

Thus, from a syntactical point of view we replace a variable occurring into a formula (an expression of the Language) with another term of the language.

Terms of the language are "names" that - when interpreted - will denote objects of the "universe" of the interpretation (its domain).

Thus, we may say :

When we assign to the variable $x$ the value $t$ as its reference (denotation), the function referenced (denoted) by $f$ will have $f(t)$ as value.

In a nutshell, the two "views" : the syntactical and the semantical, are linked by the relation of reference : assigning to a term of the language and object of the "world" as its "meaning".

The relation of reference between name and object is not the same with the operation (internal to the language) of substitution of a term with another one.

Mauro ALLEGRANZA
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  • Thanks. So in your answer, did you denote the syntactic variable by $x$ and the semantic variable by $t$ for distinguish? – amoogae Jun 10 '19 at 15:32
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    @amoogae - IMO, we have no "semantical" variables. We have an "object language" where we have expressions like $x+2$. In order to "speak" of a number, we need again a term $t$ (how else ? how can we speak of something without using terms of the Language ?) Thus, what we do is : assign to the variable $x$ a value $t$ and compute the value of the function $f$ for the input value $t$. – Mauro ALLEGRANZA Jun 10 '19 at 15:46