Semantics of terms (FOL)

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Definition

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

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

See next

• Semantics of logical connectives
  • Semantics of formulas