1.4. General principles of deductive reasoning

1.4.0. Overview

All the deductive properties and relations of sentences can be seen as special cases of a single relation. We will look at this relation and also see how to study the full range of deductive logic by way of entailment and a couple of auxiliary ideas.

1.4.1. A closer look at entailment
Entailment will be at the heart of our study and we will begin by looking in some detail at a couple ways of formulating its definition.

1.4.2. Division
It will be useful to have a special term for the kind of pattern of truth values that entailment rules out.

1.4.3. Conditional exhaustiveness
Although entailment does not encompass all the concepts of deductive logic, there is a similarly defined relation that does.

1.4.4. A general framework
All the deductive properties and relations we will consider can be expressed in terms of conditional exhaustiveness and expressed in a way that corresponds directly to definitions of them.

1.4.5. Reduction to entailment
Although conditional exhaustiveness provides a way of thinking about deductive properties and relations, entailment is way that they are most naturally established, and we need to consider how this can be done.

1.4.6. Laws for entailment
The ideas behind the reflexivity and transitivity of implication provide the core of the general principles that hold for entailment by any number of premises.

1.4.7. Duality
The specific principles concerning ⊤ and ⊥ display a kind of symmetry that we will also find in principles for other logical forms.