deductive system


Deduction and Induction

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 rulecomposition of classifying morphisms / pullback of display mapssubstitution
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, (idemponent) 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



Deductive systems


In logic, type theory, and the foundations of mathematics, a deductive system (or, sometimes, inference system) is specified by

  1. A collection of judgments, and
  2. A collection of steps, each of which has a (typically finite) list of judgments as hypotheses and a single judgment as conclusion. A step is usually written as
    J 1J nJ \frac{J_1 \quad \cdots \quad J_n}{J}

    If n=0n=0, a step is often called an axiom.

Usually, one generates the steps by using rules of inference, which are schematic ways of describing collections of steps, generally involving metavariables.


This use of the terminology “deductive system” is not completely standard, but it is not uncommon, and we need some name by which to refer to this general notion.


In the concrete algebraic theory of groups, the judgments are formal equations between terms built out of variables and the symbols ee, \cdot, and () 1(-)^{-1}. Thus, for instance, xe=xx\cdot e = x and x=yx 1x = y \cdot x^{-1} are judgments.

The rules of inference express, among other things, that equality is a congruence relative to the “operations”. For instance, there is a rule

a=ab=bab=ab \frac{a=a' \quad b=b'}{a\cdot b = a'\cdot b'}

where aa, bb, etc. are metavariable?s. Substituting particular terms for these metavariables produces a step which is an instance of this rule.

Proof trees and theorems

A proof tree in a deductive system is a rooted tree whose edges are labeled by judgments and whose nodes are labeled by steps. We usually draw these like so:

J 1J 2 J 3J 4 J 5 J 6 \array{\arrayopts{\rowlines{solid}} \array{\arrayopts{\rowlines{solid}} J_1 \quad J_2 \\ J_3} \quad \array{\arrayopts{\rowlines{solid}} J_4 \\ J_5} \\ J_6 }

(To draw such trees on the nLab, see the HowTo for a hack using the array command. For LaTeX papers, there is the mathpartir package.)

If there is a proof tree with root JJ and no leaves (which means that every branch must terminate in an axiom), we say that JJ is a theorem and write

J.\vdash J.

More generally, if there is a proof tree with root JJ and leaves J 1,,J nJ_1,\dots, J_n, we write

J 1,,J nJ. J_1, \dots, J_n \;\vdash\; J.

This is equivalent to saying that JJ is a theorem in the extended deductive system obtained by adding J 1,,J nJ_1,\dots,J_n as axioms.


This use of \vdash to express a statement about the deductive system should be distinguished from its use in particular deductive systems as a syntactic ingredient in judgments. For instance, in sequent calculus the judgments are sequents, which are sequences of statements connected by a turnstile \vdash. Similarly, in type theory and natural deduction one often uses \vdash inside a single judgment when that judgment is of a hypothetical sort. However, when using a logical framework, these two meanings of \vdash become essentially identified.

Formal systems

Depending on the strength of the metalanguage used to define the judgments and steps, simply having a deductive system does not in itself necessarily yield an effective procedure for enumerating valid proof trees and theorems. Deductive systems which do yield such an enumeration are sometimes referred to as formal systems. For example, Gödel’s incompleteness theorems are statements about formal systems in this sense. It is worth keeping in mind that more general deductive systems are considered in proof theory and type theory, typically because by side-stepping these coding issues one can give a simpler account of computational phenomena such as cut-elimination. A well-known example of such a so-called “semi-formal system” is first-order arithmetic? with the ω-rule?, used by Schütte in order to simplify Gentzen’s proof that the consistency of first-order arithmetic may be reduced to well-foundedness of the ordinal ϵ 0\epsilon_0.

Examples and special cases

Revised on April 28, 2016 13:53:27 by Mike Shulman (