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deduction∗
CWoo†
2013-03-22 3:29:23
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 ∆ of (well-formed) formulas
can be thought of as a finite sequence of wff’s
A1 , A2 , · · · , An
such that Ai is either an axiom, an element of ∆, or as the result of an application
of an inference rule R to earlier wff’s Ai1 , . . . , Aik , where i1 , . . . , ik < i. If Ai
is an element of ∆, we call Ai 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 X1 , . . . , Xn 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 ∆), and the label of the
root of the tree is the conclusion of the deduction. Below is an example of a
∗ hDeductioni created: h2013-03-2i by: hCWooi version: h42134i Privacy setting: h1i
hDefinitioni h03F03i h03B99i h03B22i
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deduction tree:
X1
X2
Y
Z
Here, X1 and X2 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:
X1
X2
Y
Z
R
S
Deducibility Relation
Given a deductive system D, if B is the conclusion of a deduction from a set ∆,
we write
∆ `D B.
If Γ is another set of wff’s, we also write ∆, Γ `D B to mean ∆ ∪ Γ `D B. In
addition, if Γ = {A1 , . . . , An }, we write
A1 , . . . , An `D B
and
∆, A1 , . . . , An `D B
to mean Γ `D B and ∆ ∪ Γ `D B respectively. Furthermore, we may also
remove the subscript D and write
∆`B
if no confusion arises. When ∆ ` B, we also say that B is deducible from ∆.
The turnstile symbol ` is also called the deducibility relation.
If a formula A is deducible from ∅, we say that A is a theorem (of the
corresponding deductive system). In symbol, we write
`D A
or
` 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 ∆ ` A and Γ, A ` B, then ∆, Γ ` B.
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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...
References
[1] A. S. Troelstra, H. Schwichtenberg, Basic Proof Theory, 2nd Edition, Cambridge University Press, 2000
[2] D. Dalen, Logic and Structure, 4th Edition, Springer, 2008
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