1.2. What is said: propositions

1.2.0. Overview

In 1.1.5, we saw the close relation between two properties of a deductive inference: (i) it is a transition from premises to conclusion that is free of any risk of new error, and (ii) the information provided by the conclusion of a deductive inference is already present in its premises. The relation between these properties points to a way of understanding the informational content of a sentence.

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

1.2.3. Ordering by content
When there is a risk-free inference from one sentence to another, the first says everything the second does, but it may say more by ruling out some possibility the second leaves open.

1.2.4. Equivalence in content
Implication in both directions between sentences shows that each says everything the other does—that is, that they say the same thing.

1.2.5. The extremes of content
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.

1.2.7. Contrasting content
Other logical relations between sentences concern differences rather than similarities in content.

1.2.8. Deductive relations in general
The relations we have considered provide a complete collection of logical relations between two sentences, and certain connections among these relations can be depicted in a traditional diagram known as the square of opposition.