Difference between revisions of "Semantics of terms (FOL)"
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(Created page with "{{Stub page|grade=A|msg=Created to save me sifting through notes or scouring PDFs, needs fleshing out}} __TOC__ ==Definition== Given a first order language, {{M|\mathscr{L...") |
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==Definition== | ==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}}: | 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}}) | # 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}}) | ||
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==See next== | ==See next== | ||
* {{Link|Semantics of logical connectives|FOL}} | * {{Link|Semantics of logical connectives|FOL}} | ||
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Contents
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]
- 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)
- 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)
