nLab second-order logic





The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

(0,1)(0,1)-Category theory

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




Second-order logic(SOL) is an extension of first-order logic with quantifiers and variables that range over subsets of the universe of discourse, and hence is a higher-order logic of stage 2.

An important fragment is Monadic Second-order logic (MSOL), where second-order quantification is restricted to second-order unary relations between subsets i.e. MSOL quantifies only over set predicates e.g. X.φ(X)\forall X.\varphi(X) but not XY.φ(X,Y)\forall X\forall Y .\varphi(X,Y).

Second-order logic could be characterised as a first order theory with dependent types. There is a type VV called the domain of discourse, and for each term x:Vx:V, a dependent type 𝒫(x)\mathcal{P}(x) whose terms P(x):𝒫(x)P(x):\mathcal{P}(x) are propositions depending on xx.


As SOL permits characterisation of mathematical structures up to isomorphism, it is sometimes promoted as a contender to set theory for the foundations of mathematics (cf. references). However, characterising structures up to isomorphism is insufficient for category theory and higher category theory, as weak categories could only be characterised up to equivalence of categories, and weak ∞-groupoids could only be characterised up to homotopy equivalence.


Textbook account in view of the Curry-Howard correspondence:

  • Morten Heine Sørensen, Pawel Urzyczyn, Lectures on the Curry-Howard isomorphism, Studies in Logic 149, Elsevier (2006) [ISBN:9780444520777, pdf]

See also:

  • Wikipedia, Second-order logic

  • Stewart Shapiro, Do Not Claim Too Much: Second-Order Logic and First-Order Logic , Phil. Math. 3 no. 7 (1999) pp.42-64 . (pdf)

  • Jouko Väänänen, Second-order Logic and Foundations of Mathematics , BSL 7 no. 4 (2001) pp.504-520. (ps)

  • Jouko Väänänen, Second-order logic, set theory, and the foundations of mathematics , Ms. (pdf)

Last revised on December 30, 2022 at 21:20:48. See the history of this page for a list of all contributions to it.