deduction

Deductions are formalizations of proofs. Informally, they can be viewed as arriving at a conclusion from a set of assumptions, via applications of some prescribed rules in a finite number of steps.

There are several ways to formally define what a deduction is, given a deductive system. We will explore some of them here.

Deduction as a Sequence

Given a deductive system, a deduction from a set $\mathrm{\Delta }$ of (well-formed) formulas can be thought of as a finite sequence of wff’s

$$A_1,A_2,\mathrm{\cdots },A_n$$

such that $A_i$ is either an axiom, an element of $\mathrm{\Delta }$, or as the result of an application of an inference rule $R$ to earlier wff’s $A_{i_1},\mathrm{\dots },A_{i_k}$, where . If $A_i$ is an element of $\mathrm{\Delta }$, we call $A_i$ an assumption or a hypothesis. This way of presenting a deduction is said to be linear.

Deductions presented this way are also called deduction sequences or proof sequences. Deductions in Hilbert-style axiom systems are presented this way.

Deduction as a Tree

Alternatively, a deduction can be thought of as a labeled rooted tree. The nodes are labeled by formulas, and for each node with label, say $Y$, its immediate predecessors (children) are nodes whose labels are, say $X_1,\mathrm{\dots },X_n$ which are premises of some inference rule $R$. Deductions presented this way are also called deduction trees or proof trees. In a deduction tree, the labels of the leaves are either the axioms, or assumptions (from some set $\mathrm{\Delta }$), and the label of the root of the tree is the conclusion of the deduction. Below is an example of a deduction tree:

$$\mathrm{.25.25}Z\mathrm{.75.75}Y{X}_1\mathrm{.75.75.85826772}{X}_2$$

Here, $X_1$ and $X_2$ are assumptions and $Y$ the corresponding conclusion of some (binary) rule $R$, and $Y$ is the assumption and $Z$ is the conclusion of some (unary) rule $S$. Inference rules may be specified as labels of the edges.

Deductions in deduction systems such as natural deduction or sequent calculus are presented this way. In those systems, the edges of the trees are replaced by horizontal lines, dividing premises and conclusions, and rules are sometimes specified next to the lines, in the following fashion:

  $X_1$         $X_2$    $R$                $Y$                 $S$                $Z$

Deducibility Relation

Given a deductive system $D$, if $B$ is the conclusion of a deduction from a set $\mathrm{\Delta }$, we write

$$\mathrm{\Delta }{⊢}_{D}B.$$

If $\mathrm{\Gamma }$ is another set of wff’s, we also write $\mathrm{\Delta },\mathrm{\Gamma }{⊢}_{D}B$ to mean $\mathrm{\Delta }\cup \mathrm{\Gamma }{⊢}_{D}B$. In addition, if $\mathrm{\Gamma }=\{A_1,\mathrm{\dots },A_n\}$, we write

$$A_1,\mathrm{\dots },A_n{⊢}_{D}B\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}\mathrm{\Delta },A_1,\mathrm{\dots },A_n{⊢}_{D}B$$

to mean $\mathrm{\Gamma }{⊢}_{D}B$ and $\mathrm{\Delta }\cup \mathrm{\Gamma }{⊢}_{D}B$ respectively. Furthermore, we may also remove the subscript $D$ and write

$$\mathrm{\Delta }⊢B$$

if no confusion arises. When $\mathrm{\Delta }⊢B$, we also say that $B$ is deducible from $\mathrm{\Delta }$. The turnstile symbol $⊢$ is also called the deducibility relation.

If a formula $A$ is deducible from $\mathrm{\varnothing }$, we say that $A$ is a theorem (of the corresponding deductive system). In symbol, we write

$${⊢}_{D}A\mathit{\hspace{1em}\hspace{1em}}\text{or}\mathit{\hspace{1em}\hspace{1em}}⊢A.$$

A property of $⊢$ is the following:

if $⊢A$ and $A⊢B$, then $⊢B$.

In other words, if $A$ is a theorem, and $B$ can be deduced from $A$, then $B$ is also a theorem, which conforms with our intuition. More generally, we have

if $\mathrm{\Delta }⊢A$ and $\mathrm{\Gamma },A⊢B$, then $\mathrm{\Delta },\mathrm{\Gamma }⊢B$.

Remark. We refrain from calling a deduction a proof. As we have seen earlier, deductions are mathematical abstractions of proofs. So they are mathematical constructed, viewed either as sequences or trees (of formulas). On the other hand, in the literature, “proof” is usually reserved to be used in the meta-language, as in the “proof” of Fermat’s Last Theorem, etc…

Title	deduction
Canonical name	Deduction
Date of creation	2013-03-22 19:12:53
Last modified on	2013-03-22 19:12:53
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	12
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03F03
Classification	msc 03B99
Classification	msc 03B22
Synonym	proof tree
Synonym	proof sequence
Synonym	hypothesis
Defines	deduction tree
Defines	deduction sequence
Defines	deducible
Defines	deducibility relation
Defines	assumption