Semantics of terms (FOL)

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Definition

Given a first order language, $\mathscr{L}$ and a model, $(\mathbf{M},\sigma)$ of $\mathscr{L}$ also, the semantics of a term, $t\in\mathscr{L}_T$ , which we denote by $t_{\mathbf{M}[\sigma]}$ is defined inductively as follows^[1]:

If $x$ is a variable symbol then: $x_{\mathbf{M}[\sigma]}:=\sigma(x)$
If $c$ is a constant symbol then: $c_{\mathbf{M}[\sigma]}:=c_\mathbf{M}$ (recall $c_\mathbf{M}$ denotes $I(c)$ where $I$ is an interpretation)
If $f$ is an $n$ -ary function symbol and $t_1,\ldots,t_n\in\mathscr{L}_T$ are terms then: $(ft_1\cdots t_n)_{\mathbf{M}[\sigma]}:=f_\mathbf{M}((t_1)_{\mathbf{M}[\sigma]},\ldots,(t_n)_{\mathbf{M}[\sigma]})$ (recall $f_\mathbf{M}$ denotes $I(f)$ where $I$ is an interpretation)