Difference between revisions of "Semantics of terms (FOL)"

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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)

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 ==Definition==
 Given a [[first order language]], {{M|\mathscr{L} }} and a {{link|model|FOL}}, {{M|(\mathbf{M},\sigma)}} of {{M|\mathscr{L} }} also, the {{link|semantics|FOL}} of a {{link|term|FOL}}, {{M|t\in\mathscr{L}_T}}, which we denote by {{M|t_{\mathbf{M}[\sigma]} }} is [[inductive definition|defined inductively]] as follows{{rMLFFISWL}}:
-# If {{M|x}} is a {{link|variable symbol|FOL}} then: {{M|1=x_{\mathbf{M}[\sigma]}=\sigma(x)}}
+# If {{M|x}} is a {{link|variable symbol|FOL}} then: {{M|1=x_{\mathbf{M}[\sigma]}:=\sigma(x)}}
-# If {{M|c}} is a {{link|constant symbol|FOL}} then: {{M|1=c_{\mathbf{M}[\sigma]}=c_\mathbf{M} }} (recall {{M|c_\mathbf{M} }} denotes {{M|I(c)}} where {{M|I}} is an {{link|interpretation|FOL}})
+# If {{M|c}} is a {{link|constant symbol|FOL}} then: {{M|1=c_{\mathbf{M}[\sigma]}:=c_\mathbf{M} }} (recall {{M|c_\mathbf{M} }} denotes {{M|I(c)}} where {{M|I}} is an {{link|interpretation|FOL}})
 # If {{M|f}} is an ''[[arity|{{n|ary}}]]'' {{link|function symbol|FOL}} and {{M|t_1,\ldots,t_n\in\mathscr{L}_T}} are {{link|term|FOL|s}} then: {{M|1=(ft_1\cdots t_n)_{\mathbf{M}[\sigma]}:=f_\mathbf{M}((t_1)_{\mathbf{M}[\sigma]},\ldots,(t_n)_{\mathbf{M}[\sigma]})}} (recall {{M|f_\mathbf{M} }} denotes {{M|I(f)}} where {{M|I}} is an {{link|interpretation|FOL}})
 ==See next==
 * {{Link|Semantics of logical connectives|FOL}}