deduction (original) (raw)

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

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 Δ), and the label of the root of the tree is the conclusion of the deduction. Below is an example of a deduction tree:

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 R Y S Z

Deducibility Relation

Given a deductive system D, if B is the conclusion of a deduction from a set Δ, we write

If Γ is another set of wff’s, we also write Δ,Γ⊢DB to mean Δ∪Γ⊢DB. In addition, if Γ={A1,…,An}, we write

to mean Γ⊢DB and Δ∪Γ⊢DB respectively. Furthermore, we may also remove the subscript D and write

if no confusion arises. When Δ⊢B, we also say that B is deducible from Δ. 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

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 Δ⊢A and Γ,A⊢B, then Δ,Γ⊢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…

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