1.2. What is said: propositions

1.2.0. Overview

In 1.1.5, we noted the close relation between two properties of a deductive inference: it is a transition from premises to conclusion that is free of any risk of new error, and the information provided by its conclusion is already present in its premises. The relation between these properties points to a way of understanding the informational content of a sentence and provides the basis for a general picture of the function of language.

1.2.1. Truth values and possible worlds
First we look more closely at the concepts of risk and error involved in the idea of risk-free inference.

1.2.2. Ordering by content
When there is a risk-free inference from one sentence to another, the first may say the same thing as a second or it may say more by ruling out some possibility the second leaves open.

1.2.3. Contrasting content
Sentences may also be incompatible in the sense that each rules out any possibility in which the other is true or complementary in the sense of each is true in any possibility the other rules out—or they can be related in both these ways and have exactly opposite content.

1.2.4. Truth conditions and propositions
We can use these ideas to give an account of the content of the meaning of a sentence, of what it says.

1.2.5. Tautologies and absurdities
Two extremes in the ordering of sentences by content are sentences that say nothing and sentences that say too much to distinguish among possibilities.

1.2.6. Logical space and the algebra of propositions
Deductive logic can be seen as the theory of the meanings of sentences in the way that arithmetic is the theory of numbers.