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Defining Necessity and Contingency

Overview:

Some ways of defining necessity and contingency are introduced and their merits discussed.
Introduction
We consider here three different conceptions of necessity, increasingly liberal in what they allow as necessary. The differences concern our attitude to the relationship between necessity and ontology.
Free Necessity
In which a necessary statements must be true in all domains of discourse. All matters of ontology are considered contingent.
Abstract Necessity
In which sense true statements about abstract entities may be considered necessary. The distinction between abstract and concrete ontology is crucial here.
Broad Necessity
In which necessary truths containing references to concrete entities are admitted, particularly where these are instances of necessary truths containing no concrete references.

Introduction:

We consider here three different conceptions of necessity, increasingly liberal in what they allow as necessary. The differences concern our attitude to the relationship between necessity and ontology.
A proposition is Logically Necessary, necessarily true or just plain necessary if the statement could not possibly be false. If it is true in all possible worlds. The concept depends therefore upon what we consider to be a possible world. The narrower our conception of possible world, the broader our conception of necessity.

The adjective logical is used to distinguish the concept from other kinds of necessity such as those induced by laws of nature or courts of law.

Even confining our considerations to logical necessity there are several alternative views on how tightly this concept should be defined. We here consider three alternatives differing in how tolerant they are of failures of reference:

Free Logical Necessity
In which a necessary statements must be true in all domains of discourse (example 4 only).
Abstract Logical Necessity
In which a failure of concrete reference will disqualify a candidate (examples 3 and 4).
Broad Logical Necessity
In which necessary truths containing references to material entities are admitted (examples 2,3 and 4).

Free Necessity:

This term, by analogy with free logic is necessity without any ontological commitment. A statement is freely necessary if it is true in every possible world even when possible worlds are allowed to have completely denuded ontologies.
Free Necessity
A statement is necessarily true (or again necessary simpliciter) if it is true in every possible world. How much help is to be found in this definition is moot. It does show the relationship between necessity and possibility tacity invoked in the previous paragraph, viz that these concepts are inter-definable.

A statement is necessarily false (or self contradictory) if it is false in every possible world.

If a statement is true in some, but false in other possible worlds then it is contingent. It is intended that all statements of a certain class are either necessary or contingent.

First Order Validity
In this most strict sense of necessity a possible world is similar to an interpretation in first order logic, and necessity is essentially first order validity. It takes the most radical view of the dictum that logic is devoid of subject matter, a statement being necessary only if true independently of the meaning of the "non-logical" symbols employed in it.

The term "free" is used because the notion of necessity here comes closest to that of validity in a free first order logic.

Abstract Necessity:

Abstract entities seem to live in a hinterland between necessity and contingency. Abstract Necessity recognises that they are discretionary rather than contingent, and provides that kind of necessity which can be attributed to the truths of arithmetic.
Abstract Necessity
With the definitions as given above for free necessity, any statement which fails in any possible world to have a truth value cannot possibly be necessary, even if is false in no possible world. The largest class of statements which is partitioned by the necessary/contingent dichotomy is the class of statements which have a truth value in every possible world.

This requirement may be thought to be implicit in a definiton of necessity couched as a property of propositions rather than statements.

Free necessity is particularly demanding since even if no reference to any contingent object is intended, as in the case of a pure set theory, statements may be denied necessity and rendered contingent.

If we want to regard as necessary statements which make reference to abstract entities, on the grounds that their lack of any reference to the material world ensures that if true they are true in every possible world, then we have the notion which is here called abstract necessity.

Possible World v. Domain of Discourse
Statements which are necessary in this sense suggest a gap between a possible world and a domain of discourse, which corresponds to a gap between necessity and first order validity. The gap arises from allowing the semantics of constants which refer to abstract entities to be fixed in the semantics, in the same way as the semantics of logical constants are determined. The semantics can be made clear by the introduction of the notion of a standard interpretation, and a revised definition of truth (simpliciter) as "true in every standard interpretation". The population of abstract entities is seen as part of a choice of meaning (for the language) rather than as being a factor which varies through the possible worlds.

Broad Necessity:

If we instantiate a general necessary truth such as a propositional tautology using a sentence which refers to some contingent entity, such as New York we get a statement which satisfies neither our definition of free nor of abstract necessity. Broad necessity is intended to accomodate such statements.
Contingent Reference
With the definitions as given, certain statements which we might like to consider necessary, for example, that it either is or is not raining in New York are still excluded.

They fail to meet the criteria, since in some possible worlds, viz those in which New York does not exist, the statement mail fail to be meaningful, and hence fail to have a truth value at all. On the other hand, we may think that propositional tautologies should be considered true even when their constituent atomic propositions turn out to be meaningless.

Broad Necessity
The notion of broad necessity encompasses this class of propositions by disregarding any possible worlds in which the names or noun phrases in the statement have no reference.

Insisting on models providing a reference for all names is done in first order logic, so this is no extension from the view of necessity in first order logic, but for some other logics and for natural languages it is a less trivial extension.

Semantic Conditions
In the case of abstract entities we have produced a notion of necessity which admits the intended abstract subject matter of the language into the ontology of every possible world, provided that the abstract semantics is consistent, so that failure of necessity does not arise because of abstract ontological presuppositions.

When we extend to protecting against non-existence of concrete entities the same considerations arise. What kinds of concrete entities can we presuppose?

Broad necessity admits semantics for contingent reference on exactly the same conditions, viz. that the semantics is consistent. The effect of this is to give cause for great care in interpreting necessary truths involving concrete references.


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