2.2. Proofs: analyzing entailment

2.2.0. Overview

We can get some insight into deductive logic by looking at basic principles of entailment, but more will come by looking at how these principles may be combined in proofs.

2.2.1. Proofs as trees
The simplest way of combining deductive principles takes the shape of a tree in which premises, premises from which these premises are concluded, and so on, grow and branch from the final conclusion.

2.2.2. Argument trees
The search for a proof can take the form of a tree connecting arguments whose validity is tied.

2.2.3. Derivations
Although writing a proof as a tree can make its structure very explicit, we will mainly use a compact notation that more closely matches the patterns that are used when deductive reasoning is put into words.

2.2.4. Rules for derivations
In the context of derivations, principles of entailment take the form of rules that direct the search for a proof.

2.2.5. An example
All derivations that involve conjunction alone share many features; we will look closely at one typical example.

2.2.6. Two perspectives on derivations
Derivations have aspects that reflect both conclusion trees and argument trees; the latter aspect will prove especially important.

2.2.7. More rules
Tautology and absurdity provide a first example of derivation rules for logical forms other than conjunction.

2.2.8. Resources
In order to plot a course in constructing a proof for a given conclusion, we need to keep track of not only the premises but also the conclusions that have already been reached.