Wikipedia:Reference desk/Mathematics: Difference between revisions

 

Does the [[deduction theorem]] holds in [[provability logic|GL logic]]? Specifically, when '''Necessitation''' is defined as it is [[provability logic|here]], in that it can only be applied with no assumptions before the <math>\vdash</math> (so from <math>\{\phi\} \vdash \phi </math> one may ''not'' deduce <math>\{\phi\} \vdash \square\phi</math>). The article <ref>{{Cite journal|last=Hakli|first=Raul|last2=Negri|first2=Sara|date=2011-03-29|title=Does the deduction theorem fail for modal logic?|url=http://dx.doi.org/10.1007/s11229-011-9905-9|journal=Synthese|volume=187|issue=3|pages=849–867|doi=10.1007/s11229-011-9905-9|issn=0039-7857}}</ref> indicates the answer might be yes, but I have been told by someone whom I am inclined to believe that the answer is no. For some context, [https://plato.stanford.edu/entries/logic-provability/#ProvLogiPeanArit arithmetic soundness and completeness] mean that GL logic is in some way the same as talking about provability in [[Peano arithmetic|PA]], which (I believe) does have the deduction theorem. —&nbsp;[[User:Crh23|<span style="font-size: 110%; font-family:Garamond;">'''crh'''<span style="font-size: 80%">23</span></span>]]<small>&thinsp;([[User Talk:Crh23|Talk]])</small> 08:52, 11 June 2020 (UTC)