The meaning of terms

As specified above, the terms of $\mathcal{L}_1$ are the constants and variables of $\mathcal{L}_1$. For $\mathcal{L}_0$, assigning a semantic value to any constant was done through the interpretation function of the structure $\mathbf{S}=\langle\mathbf{D}, I\rangle$. This will be retained, so that the value of a constant $\mathfrak{c}_{n}$ ($n \in \mathbb{N}$) relative to the structure $\mathbf{S}$, in other words (symbols), $I(\mathfrak{c}_{n})$, is just some member of $\mathbf{D}$. But again, though variables have a function similar to constants, their semantic value is only temporarily assigned, and is otherwise variable. Accordingly, we do not use the interpretation function $I$ to assign values to our variables, but a separate function $\mathbf{a}$ called a variable assignment.

The variable assignment function $\mathbf{a}$ assigns to each variable $\mathbf{x}_{n}$ ($n \in \mathbb{N}$) a member of $\mathbf{D}$, like $I$ does for constants. But unlike with $I$, we will introduce a device that will allow us to change the variable assignment inside a structure $\mathbf{S}$. The basis of this is what is called an $x$-variant assignment $\mathbf{a}'$ for an existing assignment $\mathbf{a}$. Unsurprisingly, $\mathbf{a}'$ is called an $x$-variant assignment of $\mathbf{a}$ because it differs from it in at most what it assigns to the variable $x$. (It should be noted that in usual mathematical fashion, $\mathbf{a}$ may be its own (trivial) $x$-variant assignment, in which case nothing is changed from the initial assignment.)

Let’s look at an example for this. Suppose you have a mathematical statement as follows: $$x \times 4 < 2 \times y$$ Here, we have multiple variables, so each assignment of values to the variables will assign a value to $x$, and to $y$. Suppose we assign the value $5$ to $x$ and $10$ to $y$. This will result in an incorrect statement, since $20<20$ is incorrect. So now take the $x$-variant assignment that assigns $4$ to $x$, but otherwise leaves the assignment as it was. This time, we get a correct statement, since $16 <20$. Of course, we can also take a $y$-variant assignment. And in general, not all $x$– or $y$-variant assignment will make the statement correct. For example, the $y$-variant assignment (to the initial one) where $y=5$ will clearly make it incorrect again.

We can make the above ideas more precise as follows:

Let’s return to the general picture. Since some terms of the language $\mathcal{L}_1$ are variables, it won’t be enough to work solely with the interpretation function $I$ to assign semantic values to our terms in $\mathcal{L}_1$ – we need a variable assignment too. Clearly, since these terms then go into forming atomic and complex formulas, we will need to relativize the semantic values of expressions of $\mathcal{L}_1$ both to a particular structure $\mathbf{S}=\langle\mathbf{D}, I\rangle$ and to a corresponding variable assignment $\mathbf{a}$ in $\mathbf{D}$ at any one time. We will introduce some new notation for this. In particular, if $X$ is any expression of $\mathcal{L}_1$, then its semantic value relative to a structure $\mathbf{S}=\langle\mathbf{D}, I\rangle$ and variable assignment $\mathbf{a}$ in $\mathbf{D}$ will be denoted: $$I(X)[\mathbf{a}]$$ Using this notation, we can easily specify what it means to assign a semantic value to a term of $\mathcal{L}_1$ as follows:

One thing you may notice here is that variables may ‘see’ more of the domain than the constants of a language. For example, since $I(\mathfrak{c}_{n})=n \times 2$, no odd number in the domain will have a constant that refers to it. On the other hand, a variable may be assigned the value of an odd number nevertheless. In the above case, since $\mathbf{a}(\mathbf{x}_{n})= n + 7$, if $n$ is even, then the variable assignment $\mathbf{a}$ will assign it an odd number as its value. As we will see soon when we introduce quantifiers, at times, they will be able to see more of the domain than our constants, and this way, they can express more facts about a structure. Indeed, the consequences of this are at the core of many foundational theorems in logic.