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fixing unified table
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spamegg1 committed Oct 17, 2024
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Taken from [Computational Trilogy](https://ncatlab.org/nlab/show/computational+trilogy), it's actually more of a quadrilogy (or tetralogy).

| logic | set theory | category theory | type theory |
|:-----:|:----------:|:---------------:|:-----------:|
| proposition | set | object| type|
| predicate| family of sets| display morphism| dependent type|
| proof| element| generalized element| term/program|
| cut rule || composition of classifying morphisms / pullback of display maps | substitution|
| introduction rule for implication|| counit for hom-tensor adjunction| lambda |
| elimination rule for implication || unit for hom-tensor adjunction| application|
| cut elimination for implication|| one of the zigzag identities for hom-tensor adjunction| beta reduction|
| identity elimination for implication || the other zigzag identity for hom-tensor adjunction | eta conversion|
| true | singleton | terminal object/(-2)-truncated object| h-level 0-type/unit type|
| false| empty set | initial object| empty type |
| proposition, truth value | subsingleton| subterminal object/(-1)-truncated object| h-proposition, mere proposition|
| logical conjunction | cartesian product| product | product type|
| disjunction| disjoint union (support of)| coproduct ((-1)-truncation of)| sum type (bracket type of)|
| implication| function set (into subsingleton)| internal hom (into subterminal object)| function type (into h-proposition)|
| negation | function set into empty set| internal hom into initial object| function type into empty type |
| universal quantification | indexed cartesian product (of family of subsingletons) | dependent product (of family of subterminal objects)| dependent product type (of family of h-propositions)|
| existential quantification| indexed disjoint union (support of)| dependent sum ((-1)-truncation of)| dependent sum type (bracket type of)|
| logical equivalence | bijection set | object of isomorphisms | equivalence type|
|| support set| support object/(-1)-truncation| propositional truncation/bracket type |
||| n-image of morphism into terminal object/n-truncation| n-truncation modality|
| equality | diagonal function/diagonal subset/diagonal relation| path space object| identity type/path type |
| completely presented set | set| discrete object/0-truncated object| h-level 2-type/set/h-set|
| set| set with equivalence relation| internal 0-groupoid| Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation |
|| equivalence class/quotient set | quotient| quotient type |
| induction|| colimit | inductive type, W-type, M-type|
| higher induction|| higher colimit| higher inductive type|
| -|| 0-truncated higher colimit| quotient inductive type |
| coinduction|| limit | coinductive type|
|| preset || type without identity types |
|| set of truth values| subobject classifier| type of propositions|
| domain of discourse | universe| object classifier| type universe |
| modality || closure operator, (idempotent) monad| modal type theory, monad (in computer science)|
| linear logic || (symmetric, closed) monoidal category| linear type theory/quantum computation|
| proof net|| string diagram| quantum circuit |
| (absence of) contraction rule|| (absence of) diagonal| no-cloning theorem|
||| synthetic mathematics| domain specific embedded programming language|
| **logic** | **set theory** | **category theory** | **type theory** |
|--------------------------------------|--------------------------------------------------------|-----------------------------------------------------------------|---------------------------------------------------------------------------------------|
| proposition | set | object | type |
| predicate | family of sets | display morphism | dependent type |
| proof | element | generalized element | term/program |
| cut rule | | composition of classifying morphisms / pullback of display maps | substitution |
| introduction rule for implication | | counit for hom-tensor adjunction | lambda |
| elimination rule for implication | | unit for hom-tensor adjunction | application |
| cut elimination for implication | | one of the zigzag identities for hom-tensor adjunction | beta reduction |
| identity elimination for implication | | the other zigzag identity for hom-tensor adjunction | eta conversion |
| true | singleton | terminal object/(-2)-truncated object | h-level 0-type/unit type |
| false | empty set | initial object | empty type |
| proposition, truth value | subsingleton | subterminal object/(-1)-truncated object | h-proposition, mere proposition |
| logical conjunction | cartesian product | product | product type |
| disjunction | disjoint union (support of) | coproduct ((-1)-truncation of) | sum type (bracket type of) |
| implication | function set (into subsingleton) | internal hom (into subterminal object) | function type (into h-proposition) |
| negation | function set into empty set | internal hom into initial object | function type into empty type |
| universal quantification | indexed cartesian product (of family of subsingletons) | dependent product (of family of subterminal objects) | dependent product type (of family of h-propositions) |
| existential quantification | indexed disjoint union (support of) | dependent sum ((-1)-truncation of) | dependent sum type (bracket type of) |
| logical equivalence | bijection set | object of isomorphisms | equivalence type |
| | support set | support object/(-1)-truncation | propositional truncation/bracket type |
| | | n-image of morphism into terminal object/n-truncation | n-truncation modality |
| equality | diagonal function/diagonal subset/diagonal relation | path space object | identity type/path type |
| completely presented set | set | discrete object/0-truncated object | h-level 2-type/set/h-set |
| set | set with equivalence relation | internal 0-groupoid | Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation |
| | equivalence class/quotient set | quotient | quotient type |
| induction | | colimit | inductive type, W-type, M-type |
| higher induction | | higher colimit | higher inductive type |
| - | | 0-truncated higher colimit | quotient inductive type |
| coinduction | | limit | coinductive type |
| | preset | | type without identity types |
| | set of truth values | subobject classifier | type of propositions |
| domain of discourse | universe | object classifier | type universe |
| modality | | closure operator, (idempotent) monad | modal type theory, monad (in computer science) |
| linear logic | | (symmetric, closed) monoidal category | linear type theory/quantum computation |
| proof net | | string diagram | quantum circuit |
| (absence of) contraction rule | | (absence of) diagonal | no-cloning theorem |
| | | synthetic mathematics | domain specific embedded programming language |

## Work in progress

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