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deductive system in nLab

Deductive systems

Context

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

homotopy levels

semantics

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

Deductive systems

Definition

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 1⋯J nJ \frac{J_1 \quad \cdots \quad J_n}{J}

    If n=0n=0, a step is often called an axiom. In set theory, if n>0n \gt 0, a step is usually called an axiom schema.

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

Example

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, x⋅e=xx\cdot e = x and x=y⋅x −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=a′b=b′a⋅b=a′⋅b′ \frac{a=a' \quad b=b'}{a\cdot b = a'\cdot b'}

where aa, bb, etc. are metavariables. 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 n⊢J. 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.

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

Last revised on May 25, 2024 at 16:23:06. See the history of this page for a list of all contributions to it.