nLab
coherent logic

Context

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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecompositionsubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator 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

homotopy levels

semantics

Coherent logic

Definitions

Coherent logic is a fragment of (finitary) first-order logic which allows only the connectives and quantifiers

A coherent formula is a formula? in coherent logic.

A coherent sequent is a sequent of the form φψ\varphi \vdash \psi, where φ\varphi and ψ\psi are coherent formulas, possibly with free variables x 1,,x nx_1,\dots,x_n.

In full first-order logic, such a sequent is equivalent to the single formula

x 1,,x n(φψ) \forall x_1, \dots, \forall x_n (\varphi \Rightarrow \psi)

(in the empty context). Of course, this latter formula is not coherent, but this shows that when we deal with coherent sequents rather than merely formulas, it can also be thought of as allowing one instance of \Rightarrow and a string of \foralls at the very outer level of a formula.

Coherent logic (including sequents, as above) is the internal logic of a coherent category. The classifying topos of a coherent theory is a coherent topos.

Examples

  • Any (finitary) algebraic theory is coherent.

  • A good example of a coherent theory that is not algebraic (in any of the usual senses, although it comes from algebra) is the theory of a local ring; a similar example is the theory of a discrete field.

  • The theory of a total order is coherent, though also not algebraic. The theory of a partial order is essentially algebraic, but the totality axiom x,y(xy)(yx)\vdash_{x,y} (x\le y) \vee (y\le x) is coherent but not essentially algebraic.

  • The theory of a linear order is (seemingly) not coherent: the “connectedness” axiom (xy),(yx)(x=y)(x\nless y), (y\nless x) \vdash (x=y) is not coherent since negation is not allowed in coherent formulas. We can express one outer negation, however, as in the irreflexivity axiom (x<x)(x\lt x)\vdash \bot.

Properties

Coherent logic has many pleasing properties.

  • Every finitary first-order theory is equivalent, over classical logic, to a coherent theory. This theory is called its Morleyization and can be obtained by adding new relations representing each first-order formula and its negation, with axioms that guarantee (over classical logic) these relations are interpreted correctly (using the facts that (PQ)(¬PQ)(P\Rightarrow Q) \dashv\vdash (\neg P \vee Q) and (x,P)(¬x,¬P)(\forall x, P) \dashv\vdash (\neg \exists x, \neg P) in classical logic). See D1.5.9 in Sketches of an Elephant, or Prop. 3.2.8 in Makkai-Paré.

  • By (one of the theorems called) Deligne’s theorem, every coherent topos has enough points. In particular, this applies to the classifying toposes of coherent theories. It follows that models in Set are sufficient to detect provability in coherent logic. By Morleyization, we can obtain from this the classical completeness theorem for first-order logic?. See for instance 6.2.2 in Makkai-Reyes.

  • Coherent logic also satisfies a definability theorem: if a relation can be constructed in every Set-model of a coherent theory TT, in a natural way, then that relation is named by some coherent formula in TT. See chapter 7 of Makkai-Reyes or D3.5.1 in Sketches of an Elephant.

  • It follows that if a morphism of coherent theories (i.e. an interpretation of one coherent theory in another) induces an equivalence of their categories of models in SetSet, then it is a Morita equivalence of theories (i.e. induces an equivalence of classifying toposes, hence an equivalence of categories of models in all Grothendieck toposes). This is called the conceptual completeness theorem; see 7.1.8 in Makkai-Reyes or D3.5.9 in [Sketches of an Elephant]]. (Note, though, that two coherent theories can have equivalent categories of models in SetSet without being Morita equivalent, if the former equivalence is not induced by a morphism of theories; see for instance D3.5.4 in the Elephant.)

However, here is a property which one might expect coherent theories to have, but which they do not.

  • The category of models of a coherent theory (in Set) is always an accessible category (this is true more generally for models of any logical theory). However, it need not be finitely accessible (i.e. ω\omega-accessible). An example is given in Adamek-Rosicky 5.36: the theory of sets equipped with a binary relation \prec such that for all xx there exists a yy with xyx\prec y. Then the model (,<)(\mathbb{N},\lt) of this theory does not admit any morphism from an (ω\omega-)compact object. (However, many coherent theories do have a finitely accessible category of SetSet-models.)

Variations

  • Sometimes coherent logic is called geometric logic, but that term now more commonly used for the analogous fragment of infinitary logic which allows disjunctions over arbitrary sets (though still only finitary conjunctions). See geometric logic.

  • Occasionally the existential quantifiers in coherent logic are further restricted to range only over finitely presented types.

References

Revised on March 1, 2012 15:53:59 by Urs Schreiber (82.172.178.200)