There are multiple methods for naming things with more than two characteristics after their properties. A simple approach is to fix one characteristics to come first. However this yields at least two different naming methods. For instance,
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*--------------------------------------|----------------------------------*
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Propositional FirstOrder SecondOrder
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*---------|----------* *--------|--------* *--------|--------*
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Propositional.Int Propositional.Cl FirstOrder.Int FirstOrder.Cl SecondOrder.Int SecondOrder.Cl
and
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*--------------------------|----------------------------*
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Int Cl
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*--------------------*-----------------* *------------------*-----------------*
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Int.Propositional Int.FirstOrder Int.SecondOrder Cl.Propositional Cl.FirstOrder Cl.SecondOrder
Technically, the barriers between logics with different expressive powers Propositional / FirstOrder / SecondOrder seems large (because new syntactical structures (quantifiers, modalities, ...) must be introduced), while that between different logical philosophies Int / Cl appears small (to make intuitionistic into classical, one need only add LEM). Therefore, option 1 seems more appropriate.
However, mathematically, the distinction in logical philosophy seems more significant than the minor difference in expressive power. While many texts treat first-order logic as an extension of propositional logic, this is not the case with classical/intuitionistic. (Personally, I consider it a mistake to view classical logic and intuitionistic logic literally, as systems describing the same objects. Rather, as done by negative translation, intuitionistic logic should be regarded as a more detailed version of logic that embeds classical logic.) From this perspective, option 2 is preferable.
Currently, these are mixed up. Which one should be adopted?
There are multiple methods for naming things with more than two characteristics after their properties. A simple approach is to fix one characteristics to come first. However this yields at least two different naming methods. For instance,
and
Technically, the barriers between logics with different expressive powers
Propositional/FirstOrder/SecondOrderseems large (because new syntactical structures (quantifiers, modalities, ...) must be introduced), while that between different logical philosophiesInt/Clappears small (to make intuitionistic into classical, one need only add LEM). Therefore, option 1 seems more appropriate.However, mathematically, the distinction in logical philosophy seems more significant than the minor difference in expressive power. While many texts treat first-order logic as an extension of propositional logic, this is not the case with classical/intuitionistic. (Personally, I consider it a mistake to view classical logic and intuitionistic logic literally, as systems describing the same objects. Rather, as done by negative translation, intuitionistic logic should be regarded as a more detailed version of logic that embeds classical logic.) From this perspective, option 2 is preferable.
Currently, these are mixed up. Which one should be adopted?