Relation to sentences

Use of the word proposition in logic refers to an idea that is new in this century. The English usage is from Russell's reading of Frege. To explain propositions it is customary to talk first about declarative sentences in some natural language such as English. A sentence is an aggregation of words which expresses a complete thought, and when that thought is an assertionassertion, e.g. 0 < 1, then the sentence is declarative. The thought expressed is the sense of the sentence. We are interested in the conditions under which we can assert a sentence or judge it to be true.

Logic is not concerned directly with the nature of natural language and sentences. It is a more abstract subject. The abstract object corresponding to a sentence is a propositionproposition. As [27] says ``...a proposition as we use the term, is an abstract object of the same general category as class, number or function.'' He says that any concept of truth-valuetruth value is a proposition whether or not it is expressed in a natural language.⁸

This definition from [45] as explained by [26] will suffice even for the varieties of constructive logic we will consider. We can regard truth-values themselves as abstractions from a more concrete relationship, namely that we know evidence for the truth of a formula; by forgetting the details of the evidence, we come to the notion of a truth-value. We say that the asserted sentence is true. Thus, when we judge that a sentence or expression is a proposition, we are saying that we know what counts as evidence for its truth, that is, we understand what counts as a proof of it.

It is useful to single out two special propositions, say $\mathbf{\top}$ for a proposition agreed to be true, accepted as true without further analysis. We can say it is a canonical true proposition,canonically true proposition a generalization of the concrete proposition 0 = 0 in ${{\mathbb{N} }}$. We say that $\top$ is atomically true. Likewise, let $\mathbf{\bot}$ be a canonically false proposition, a generalization of the idea 0 = 1 in ${{\mathbb{N} }}$; it has no proof.