The Royal Swedish Academy of Sciences(25)

Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content,

WhenwespecializethistoafunctiontypeA→Bweget

[A→[B]]=A→[B],[A→B]≤A→[B].(10)Forbracketsontheleft,itiseasytoseethat

A→[B]=[A]→[B]=[[A]→[B]].

TakingB=0thereforeyieldsthenoteworthy

¬A=¬[A]=[¬A].(12)(11)

Since¬Aisthusalwaysapropositionitisnaturaltoaskwhetherperhaps:

[A]=¬¬A?

Theanswerisingeneralnegative,sincetherearemanysimplemodelsinregularlccc’sinwhichdoublenegationclosureisnottrivialonmonos.5FirstOrderLogicviaBracketTypes

Independenttypetheorywiththetype-formingoperations,

0,1,[A],A+B,EqA,B,x:Ax:AB,

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