Restating the Dichotomies
Overview
A concise description of the necessary-analytic-a priori/contingent-synthetic-a posteriori distinction.
Context setting descriptions of the kind of language and semantics to which these concepts are applicable.
A proposition is necessary if it is true in every possible world.
A definition of analyticity is offered and its merits discussed.
We should expect an a priori justification for a necessary proposition and an a posteriori justification for a contingent proposition.
Preliminaries
Context setting descriptions of the kind of language and semantics to which these concepts are applicable.
Descriptive Language, Indicative Sentences

The principle concepts defined are intended to mark a fundamental and objective division into two kinds of certain kinds of sentence in certain kinds of language.

These are indicative sentences in descriptive languages.

Language is descriptive if it may be used to convey information about "the world" in a particular way. This is done using indicative sentences each of which says something about how the world might be by describing some condition which might or might not obtain. A full understanding of such a sentence consists in knowing in principle how to determine whether the condition described holds in any particular situation, or state of affairs. "In principle" here is intended to steer the reader away from expecting too much from such an understanding. It is not required, for example, that given any description of some state of affairs one can easily determine whether the condition is satisfied. The difference is similar to that between being well-defined and being decidable. The former suffices.

What is involved in an understanding is however not here at stake. What concerns is rather the information content in the meaning of indicative sentences (in context), which we require to encompass truth conditions determining its truth value in each possible circumstance. .

Possible Worlds
Possible worlds are often said to be metaphysical, but they are an essential feature of the semantics of descriptive languages.. The semantics of such languages encompasses truth conditions for statements made in the languages. These truth conditions assign truth values to the statements in every possible circumstance. The range of circumstances for which the truth values of a descriptive language are determined are the "possible worlds".
Meaning, Propositions

In an abstract sense, the semantics of a language assigns meaning to statements in the language, and this meaning may be thought of as that which synonymous statements have in common. We use the name "proposition" for the meaning of statements. Among the features shared by synonymous statements is the truth conditions of the statements, which are therefore part of the information content of the proposition expressed by a statement.

Exactly what a proposition is, we need not know, but an attempt to formalise the semantics of a language may involve a choice of representative for propositions (though some suitable meta-languages may permit propositions to be specified abstractly, without any nomination of any particular representatives).

Similar considerations apply to the notion of "possible world". The possible worlds are the domain of the truth conditions, they are the range of circumstance in which the truth values of statements are determined by the truth conditions. An abstract characterisation of the notion of possible world is a part of the semantics of a descriptive language, if the semantics of the language is codified this may involve choice of representatives for the possible worlds.

Necessary
A proposition is necessary if it is true in every possible world.
logical
The definition is of the notion of logical necessity, which is a term-of-art in philosophy rather than a notion in general currency. The definition should be taken as a proposal for usage, it is not claimed that it is equivalent to any previous usage. Of course the wording of the definition is not novel, but the intended interpretation of the concepts in terms in the definiens must of course be taken into account in any comparison.
proposition
The term proposition is used to refer to some statement as a mathematician might say "up to equivalence of meaning". Those without nominalistic qualms on the score of propositions may take it to be "the meaning" of some sentence given sufficient context for disambiguation. Those with greater qualms may consider it to be an equivalence class of sentences in context (again with sameness of meaning determining the class).
Defining Analyticity
A definition of analyticity is offered and its merits discussed.
Background
The Definition
A sentence is analytic iff it expresses a necessary proposition.
Clarification of the Definition
Some clarification of the scope of the definition and of the terms used in the definition.
Relation to Previous Definitions
The main point here is to argue that the given definition says almost exactly the same as "true in virtue of meaning", but also (though with not quite the same confidences) the same as Kant's semantic definition.
Some Merits
Why its a good idea to define analyticity in terms of necessity.
A Priori
We should expect an a priori justification for a necessary proposition and an a posteriori justification for a contingent proposition.
1. Introduction

The distinction between analytic and synthetic statements is a distinction of subject matter. A synthetic statement tells us something about the world, an analytic statement does not. It is natural to connect this semantic difference with an epistemic one. The manner in which we establish the truth of a statement should be connected with its content. If a statement has empirical content, then it will be true in some possible worlds and false in others, and to establish whether it is true we need to know something about which possible world corresponds to reality (or, is actual). By contrast, one cannot establish that a statement is necessary using a case which involves facts from any particular universe, one can at best thereby establish its truth in that particular world.

To make this point we use the concepts a priori and a posteriori which must be defined with care if they are to capture precisely the epistemic difference between analytic and synthetic statements.

2. Defining The A Priori

We define the concept of the a priori as a property of justifications, a justification being a case for belief in some proposition.

A justification is a priori if it makes no use of any information about the actual state of the universe (and hence we may say is equally relevant to every possible state of the universe). Information about the actual state of the universe may be thought of as contingent propositions, so an alternative formulation is that a justification is a priori if it makes no epistemic use of contingent propositions.

Any other justification is a posteriori, i.e. any justification which makes use of information about the actual state of the universe, i.e. which makes use of contingent propositions.

Some statements may be considered self evident, standing as their own justification. In that case they will be a priori if necessary and otherwise a posteriori. There are of course difficulties in deciding what propositions should be considered self evident and in deciding which of them are necessary and which contingent.

3. Knowledge

We are not here concerned with knowledge, and take no position on the meaning of that term, but, if knowledge is supposed to be justified, then it may be said to be a priori if it has a justification a priori and otherwise a posteriori.

Note that in this conception of a priori knowledge, the manner in which we discover the proposition is immaterial, only the manner of justification is relevant.

Note also that we are concerned with the justification of the proposition, not of any statement expressing the proposition. We may say that a statement expressing the proposition is justified a priori if the proposition is justified a priori, but note that the establishment of the connection between a statement and the proposition it expresses is irrelevant to the epistemic status of either.

4. Facts Excluded from Consideration
In considering the epistemic status of a statement, the following two kinds of factual information should be disregarded (i.e. should not render the statement a posteriori).
  1. information needed to establish which proposition is expressed by the statement
  2. information which played some role in the discovery of the proposition, but which does not feature in its justification
The first category is of particular relevance to the use of rigid designators, since in these cases knowledge of the thing designated is essential to the identification of the proposition expressed, and the manner of discovery (of the thing designated) is therefore immaterial to the epistemic status on this conception of a priority.
Afterthoughts

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