The Royal Swedish Academy of Sciences(26)

thepropositionsineverycontextmodel rst-orderlogic,underthefollowingde nitions:

⊥

∧ψ

∨ψ

= ψ

¬

x=Ay

x:A.

x:A. =========10 ×ψ[ +ψ] →ψ →0EqA(x,y) x:A [x:A ](13)

Thebracketisthususedtorectifytheoperations+andbecausethey

leadoutofpropositions.

Operationsde nedin(13)satisfytheusualrulesforintuitionistic rst-orderlogic,andtheresultingsystemisadependenttypetheorywith rst-orderlogicovereachtype.Itcanbedescribedcategoricallyastheinternallogicofaregularlcccwith nitesums.Thechiefdi erencebetweenthisformulationandmorecustomaryonesusingbothtypetheoryandpredicate

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