natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
basic constructions:
strong axioms
In logic, type theory, and the foundations of mathematics, a deductive system (or, sometimes, inference system) is specified by
If $n=0$, 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 $e$, $\cdot$, and $(-)^{-1}$. Thus, for instance, $x\cdot e = x$ and $x = y \cdot x^{-1}$ 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
where $a$, $b$, etc. are metavariable?s. Substituting particular terms for these metavariables produces a step which is an instance of this rule.
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:
If there is a proof tree with root $J$ and no leaves (which means that every branch must terminate in an axiom), we say that $J$ is a theorem and write
More generally, if there is a proof tree with root $J$ and leaves $J_1,\dots, J_n$, we write
This is equivalent to saying that $J$ is a theorem in the extended deductive system obtained by adding $J_1,\dots,J_n$ 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.
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 $\epsilon_0$.