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We learn First-order logic, and we learn about terms at Formula rule (https://en.wikipedia.org/wiki/First-order_logic#Formation_rules)

My questions is:
We learn that each language have terms: constants, functions, variables,...
Variable $x$ can be only a constant? or it can other things? like if we have constant $C$ and function $f(t_1)$ ($t_1$ is a term), so $x$ can be equals to $f(C)$? or $x$ can be only equals to $C$?

I don't understand it well...

Thank you!

1 Answers 1

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Variables and constants are symbols.

The set $Var$ of variables of the language and the set $Cons$ of constants of the language must be disjoint :

$Var \cap Cons = \emptyset$.

Constants are "names" used to denote objects of the domain of interpretation, like e.g. $0$ for the number zero.

Variables are used to write quantified formulas, like e.g. $\forall x (x \ge 0)$.

"Complex" terms are built-up with variables, constants and function symbols (from a new set $Fun$ of symbols): $+$ is a binary function symbol and with it we can write e.g. the term : $x+0$.

When we interpret a formula, we have to assume a domain (a collection of "objects") like e.g. the set $\mathbb N$ of natural numbers and suitable interpretations for constants and function symbols.

The variable must be interpreted according the formal specifications of the semantics of the language: in a nutshell, the formula $\forall x (x \ge 0)$ is true in $\mathbb N$ because all natural numbers are non-negative.

A variable in an open formula, like e.g. $x=0$ acts as a pronoun (compare with: "it is red"): the formula per se has no meaning.

In order to give meaning to the formula we have to assign a "temporary" denotation to $x$: the way to do this is defined by the semantical specifications.

We may choose different examples, with the "universe" of humankind as domain of the interpretation.

Consider now the open formula:

"$x$ is father of Abel".

If we assign to the varibale $x$ Adam as denotation, the resulting sentence is true; if instead we assign Cain to $x$, the resulting sentence is false.

If we quantify the formula, we have : "$∀x (x$ is father of Abel)", that is false, and "$∃x (x$ is father of Abel)", that is true.

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    First of all thank you for your answer, but I asked something else: Lets look at the formula you wrote: $\forall x (x \ge 0)$, what $x$ can be? What types of things I can insert to $x$? e.g. I can say that $x=f(0)$ or that only $x=0$? – 2017-01-02
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    @Yoar - it depends on the interpretation you are working with: if e.g. $\mathbb N$, then the variables stay for *numbers*. – 2017-01-02
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    Thank you! and you can give me please an example for something that $x$ is not only a number? That $x$ can be a function (or something like this...). – 2017-01-02
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    Yes, I understand this, but I'll try to be more specific: If $L=\{\approx,r,s,c\}$ where $r,s$ are one term functions and $c$ is constant. at the formula $\forall x (r(x))$ what $x$ can be?? It's legal to say: $x=r(s(c))$?? Thank you!! – 2017-01-02
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    @Yoar - the language has *variables* (usually an infinite supply of them : $x_i$) among its symbols. A variable is a variable: neither a constant not a function symbol. – 2017-01-02
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    Yes, but variable $x$ can store a value, right? So my question is - which (type of) values $x$ can store or contain? – 2017-01-02
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    @Yoar - It sounds more like you're unclear on how the universal instantiation rule of inference works. If you have a formula $\forall x(\varphi(x))$, you can place *any term* in place of $x$; this means it could be any other variable, any constant, any function symbol applied to a variable or constant, any function symbol applied to a function symbol that has been applied to a variable or constant, etc. I think your idea of a variable "storing a value" is an imprecise and unhelpful one. – 2017-01-02
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    @MaliceVidrine, you right... And this is why I asked it here, to learn abot it better :-) When you wrote: "_you can place any term in place of $x$..._" - you answer my question! Thank you! – 2017-01-02
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    @Yoar - sorry, I cannot rewrite the above answer in a comment... A variable stay for "objects" of the domain of the interpretation, and thus its "value" depends on te interpretation. $\forall x \varphi(x)$ "means" that **all** objects of the interpreattions are "$\varphi$"s, so it depends on the chosen interpetation: if we use $\mathbb N$ and $\varphi$ stay for "is Even", then the formula means : "all natural numbers are even", that is plainly false. If the interpretation is the *humankind* and $\varphi$ stay for "is mortal", then the formula means "all humans are mortal", that is true. – 2017-01-02