Defining Analyticity
A definition of analyticity is offered and its merits discussed.
A sentence is analytic iff it expresses a necessary proposition.
Some clarification of the scope of the definition and of the terms used in the definition.
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.
Why its a good idea to define analyticity in terms of necessity.
A Formal Model
This discussion may be viewed as a sequel to a formal model which I knocked up some years ago Modelling the Triple-Dichotomy and the discussion there may be worth reading to provide further background. The idea of the formal model was to show the analyticity of the identification of analyticity and necessity by actually proving it. However, once the definitions are framed the identity is too obvious to warrant proof of any kind. The informal definition given here is essentially the same as that in the formal model, the main difference being that here analyticity is explicitly defined in terms of necessity, and reasons are given why that is a good idea. In the consequences connections with some of Kripke's ideas are discussed, which provide additional reasons (which I did not appreciate when I wrote the formal model) for defining analyticity in this way.
Quine and Kripke

This is part of a philosophical position in which the analytic/synthetic distinction assumes the central role which it had in Hume's philosophy (and later in Logical Positivism). In the present historical context, such a position requires repudiation of incompatible views promulgated by Quine and Kripke, and inevitably it is in that light which this must be viewed.

Quine sometimes backed off outright rejection of the analytic/synthetic dichotomy, claiming instead that the distinction could serve no philosophical purpose. The definition given here is intended to give a concept of analyticity which does plainly serve two purpose. These are not novel, they are two purposes which have been prominent in positivist thought since Hume.

  • Firstly, to make intelligible the epistemological distinction between the a priori and the a posteriori.
  • Secondly, to secure an analytic connection between analyticity and necessity, resulting in the elimination of necessity de re (i.e. of a certain kind of metaphysics, encompassing the kind made fashionable by Kripke).

The Definition
A sentence is analytic iff it expresses a necessary proposition.
Proposition, Necessity

The kind of necessity at stake here is of course that which philosophers term "logical" necessity, which unlike general usage of the term "necessary" is a term-of-art in analytic philosophy. More specifically this is to be understood here as paraphrased "true in all possible worlds", but not necessarily involving a metaphysical commitment to the existence of possible worlds.

The definition relates to truths, defining analytic truth in terms of necessary truth, which could of course be done without mentioning propositions. However the mention of propositions is in fact useful. The word proposition stands here for the meaning of a sentence in context sufficient for its disambiguation. Exactly what that might be, and whether this is conceived as some abstract or metaphysical entity or as a manner of speaking which can be interpreted without ontic commitment is left as dependent on the particular language in question (and on the manner in which the semantics of the language is understood). The implicit constraint however is that the meanings of sentences must be the kind of thing which can be said to be necessary, and hence which incorporate in some way the truth conditions determining the truth value of the proposition in every possible world.

Generalisation to Other Truth Values

It does not help for present purposes to generalise this definition to all truth values, and for that reason I have put forward the simpler rendition. The generalisation might go along the lines "a sentence has analytically the truth value t iff it expresses a proposition which is necessarily t", however I would be tempted myself if entering into such detail to render it formally, which I would do along the lines of Modelling the Triple-Dichotomy modifying that formal model by making the definitions of analyticity and necessity less independent (i.e. defining the former in terms of the latter).

Clarification of the Definition
Some clarification of the scope of the definition and of the terms used in the definition.
Terms of Art
analytic, necessary, proposition are all intended here with meanings particular to their application in philosophy. About which I make some preliminary remarks:
analytic is being defined, and therefore should be understood as here defined. It is nevertheless intended that this be broadly in line with at least some of its previous usage, and in particular, broadly in line with the idea of expressing a 'relation between ideas' occurring in David Hume.
necessary should be understood as logically necessary, which in turn is to be understood as 'true in every possible world' (without necessarily involving commitment to the existence of 'possible worlds' as metaphysical entities) The nominalist may wish to construe a logically necessary truth as one whose truth may be established without knowledge of the particular world in the context of which it is to be evaluated (except possible of the meaning of the language in which it is expressed and any context necessary for disambiguation).

a proposition is to be understood as the meaning of a sentence in sufficient context for disambiguation. Since this is an aspect of the semantics of a language, we allow that the notion of proposition be language relative.

The notion of analyticity is not applicable to all languages, it is applicable only to indicative sentences in descriptive or abstract languages. By descriptive is here meant a language intended for saying something about the material world, whereas an abstract language may refer only to abstract entities and not to the material world. The character of such languages, and the constraint on the applicability of the concept of analyticity may be seen in the nature of the propositions which are their meanings.

The essence of a proposition is to convey information about what the world is like by excluding certain possibilities, i.e. by identifying a collection of possibilities among which the actual world may be found, and a complementary collection among which the actual world may not be found. It is not intended that the trivial cases be excluded (necessary and contradictory propositions). Nor is it intended that the truth value of a proposition be known in every possible world, it may be that in some possible worlds it has no truth value (whether this is possible depends on the language in question).

It is required however, that the semantics of a language is couched in terms of a notion of proposition adequate to encompass all that is determinate in the language about the truth value of the proposition in the various possible worlds envisaged in the semantics. The is essential for logical necessity to be an operator on propositions, and ensures that insofar as a proposition is necessary, this information is embodies in the semantics of the language so that a sentence expressing that proposition will be analytic.

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.
Predicate contained Subject

Kant's definition of analyticity was:

"In all judgements in which the relation of a subject to the predicate is thought, this relation is possible in two different ways. Either the predicate B is part of the subject A, as something which is (covertly) contained in this concept A; or B lies outside the concept A, although it does indeed stand in connection with it. In the one case I entitle the judgement analytic, in the other synthetic"
Kant defines the concept of analyticity in terms of a semantic characteristic of subject-predicate judgements (though he has another definition in terms of proof). It is clear that he does not intend the concept to be exclusive to such judgements because he also says:
"I take into consideration affirmative judgements only, the subsequent application to negative judgements being easily made"
and it may therefore be thought reasonable to go one step further and regard the definition as covering any judgement logically equivalent to some subject predicate judgement.

It is easy to show that every judgement is equivalent to one in subject predicate form. One is therefore tempted to believe that Kant's criterion is equivalent to "true in virtue of meaning". The proof of the first claim is easy (see: Abstract Semantics). (I need to review that material to ensure that I get the second as well.)

Narrower Definitions
There have probably been many alternative definitions of analyticity which are intentionally narrower than the one proposed. Some authors have been taken to be defining a concept (sometimes the concept) of analyticity when the word "analytic" is not used.
True in Virtue of Meaning

In the twentieth century the idea that analyticity should best be defined as truth in virtue of meaning was put forward by various positivist thinkers. In Quine's "Two Dogmas of Empiricism" there is an excellent informal explanation of this conception of analyticity. It is not my present purpose to defend this definition (I think it needs no defence), but rather to explain how the definition above, in terms of necessity, can be understood as a minor refinement of it.

A proposition is the meaning of a sentence (given sufficient context to disambiguate the sentence). To say that a sentence expresses a proposition in some given context is simply to say that in that context the meaning of the sentence is that proposition.

When a proposition is asserted the truth of the assertion depends in general upon which possible world is actual, this is just a way of saying that its truth value depends upon what is the case, in particular on whether the condition it embodies is the case. When the necessity of a proposition is asserted, this assertion (that of necessity) does not depend upon what is the case, for it is asserted that the proposition will be true in every case.

Proof Theoretic Definitions
Some definitions, including one of Kant, and Frege's define analyticity according to how analytic statements are to be demonstrated. I disregard such definitions for two reasons:
  1. It is useful to make analyticity an unambiguously semantic concept, so that different ways of characterising the division marked by Hume's fork can be distinguished by distinct terminology.
  2. Gödeals incompleteness results ensure that the full scope of "relations between ideas" which Hume envisages (which includes mathematics) cannot be delimited by a formal deductive system.
This is not to say that question of how analyticity relates to derivability is not important, but that will be covered after settling on a definition.
Some Extra Bits
Completeness of Semantics

Couching the definition of analyticity explicitly in terms of necessity forces the notion of meaning relative to which analyticity is considered to be the same as the notion of meaning embodied in the propositions which are the subjects of modal operators.

It would of course be the case that if analyticity were to be judged by an incomplete semantics, then in some cases sentences would fail to be analytic simply because there was not sufficient information in the semantics to establish their truth. This would be the case even if the truth values of the proposition expressed by the sentence were completely determinate and the proposition was thus determined to be necessarily true.

Insofar as defined in terms of "truth in virtue of meaning" the concept of analyticity should therefore be defined as truth in virtue of the full meaning incorporating all that is determinate about the truth conditions of the sentence. We can even refer to Quine's explication at the end of "Two Dogmas" for his confirmation of that principle. If the sentence is necessary, this is a fact about its truth conditions which must be incorporated in the meaning relative to which analyticity is to be understood. This falls out naturally, if analyticity is defined in terms of necessity, and subject to the caveat on the meaning of meaning these two definitions say the same thing.

Metaphysics and Semantics

Further to this consideration of the necessity that the semantics in terms of which analyticity is judged be complete there is the question of the domain of the truth conditions and the supposed metaphysical character of logical necessity.

Logical necessity appears both to be semantic and to be metaphysical. It appears to be (and is generally held to be) a metaphysical concept possibly because in its most common definition "true in all possible worlds" the notion of "possible world" features crucially and is generally held to be a metaphysical concept.

However, logical necessity is a property of propositions, and propositions are the meanings of sentences. Since it is a property of semantic entities it must surely be semantic in character. Only one aspect of the proposition is relevant to the judgement of necessity, and that is the truth conditions. The truth conditions represent the truth values of the proposition under each possible circumstance, in every possible world. The possible worlds are therefore the domain of the truth conditions, considered as a function.

Thus we have it that in a definition of the semantics of a language there must be assigned to each judgement a proposition. What is a proposition? We don't need to know. For our present purposes we need only an abstract semantics, and for an abstract semantics we can chose convenient abstract representation for propositions which incorporate the information we need to have in a proposition. In particular we can chose some suitable representation for the truth conditions which form an essential part of the meanings of judgements. In doing so we must determine from our understanding of the subject matter of the language the domain of discourse which will be the domain of the truth conditions considered as a function.

In the case that the language is for talking about the world (rather than some mathematical domain or some fictional subject matter) then the determination of the domain of the truth conditions reflects the presumptions which are embedded into the language about the nature of the world. For some purposes it would be ideal if this were simply choice of a representatives for the possibilities which did not exclude any possible state of affairs, but in practice languages evolve for use in the real world and some of its features are assumed by our languages.

Thus, in the design of a language or in the activity of modelling the semantics of a natural language an activity is required which is similar to metaphysics. In the design case it amounts to deciding upon a metaphysic on the basis of which the language is offered, in the other case it is the discovery from the target language of that metaphysic which is essentially presupposed in the use of the language.

Thus, at the margin, some things become necessary as a result of language choices. If these are not really necessary, then we have a language which is only properly applicable in some possible worlds, those which satisfy the metaphysical presumptions. When applied in possible worlds which comply with the metaphysic those features will appear to be necessary when we might want to say that they are not really so.

Another aspect of the design of a language or of its semantic analysis is the assignment of meanings to names in the language. If a name is a rigid designator, then it is probable that the designation will be at least part of the meaning of the name (much more discussion in the area later). If some referring expression is not a rigid designator and its sense does not uniquely determine its reference in every possible world, then the reference will be contingent and will have to be fixed by the possible world. i.e. a possible world, just like an interpretation of a first order language in model theory, must stipulate the reference of names which are not rigid designators. (indefinite descriptions require some thought)

Some Merits
Why its a good idea to define analyticity in terms of necessity.
1. Defining Analyticity and Necessity Independently

If we attempt a formal analysis in which analyticity and necessity are independently defined, the definitions turn out to be the substantively identical.

The most significant difference is if analyticity is taken to be a property of statements and necessity of propositions, but as soon as the derivative application to the other category is included this difference melts away.

It therefore seems more sensible to make a merit of their interdefinability, and to define the property of statements in terms of the property of propositions.

3. Forcing a Single Conception of Semantics
The primary merit of defining analyticity in terms of necessity is that it forces the same conception of semantics to be used in both of these terms. It ensures that the information available to determine questions of necessity is also available for deciding analyticity. This prevents the separation of analyticity from necessity by hypothesising features of language which influence necessity but do not influence analyticity.
2. Semantics Versus Metaphysics

It is often said that analyticity is a semantic concept but that necessity is a metaphysical concept (and/or a modal concept), and that on that basis they must be distinct concepts and if coextensive this requires some demonstration.

This position to some extent begs the question against the nominalist, but ours may be said to beg the question against the essentialist..

Against this I argue:

  1. that as an operator on propositions (which are the meanings of sentences) necessity is clearly a semantic characteristic
  2. that an element of ``metaphysics'' may be seen in semantics, insofar as the formulation of a truth conditional semantics involves identification of the domain of discourse, the notion of possible world

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