Necessity and Contingency


We consider the notions of necessary and contingent partly through consideration of possible worlds, in order to explicate logical truth.
We compare three different definitions of necessity (and hence of contingency).
some necessary propositions
Four examplary propositions are presented, one for each kind of necessity considered, and one which is unequivocally contingent.
defining necessity and contingency
We then sketch three alternative kinds of necessity, between which the examples serve to discriminate.
possible worlds
Finally we consider the relationship between necessity and possible worlds, showing the conception of possible world which corresponds to each of our notions of necessity.


We compare three different definitions of necessity (and hence of contingency).
ignoring possible worlds
The role of possible worlds in explicating necessity is semantic. They are relevant because contingent propositions or sentences are supposed to have truth values which depend upon the how the world is and necessary propositions are thought of as the special limiting case where the value turns out true however the world might be (i.e. in every possible world). However, many formal languages were devised without there being any intention of speaking of material reality, having an intended subject matter which is wholly abstract. The designer of such a language may declare by fiat that statements have truth values which are independent of any possible world and hence must be necessary if true.
considering possible worlds
Even if our language is intended for making statements about the world special consideration must be given to abstract entities. Modern science is wholly dependent on the use of mathematics, and hence upon the use of a rich abstract ontology for constructing the mathematical models in terms of which scientific statements are couched. The logical status of abstract entities is quite different to that of concrete entities. Recognition of this leads to more complex ideas of what possible worlds are, with consequences for what statements turn out to be necessary.

Some Necessary Propositions:

Four propositions are put forward to facilitate comparison of the three alternative conceptions of logical necessity which we consider.
prop 1: "p or (not p)"
Proposition 1, where the symbols "or" and "not" are intended as logical disjunction and negation respectively and "p" is an arbitrary proposition, is plausibly true in all possible worlds. It doesn't appear to have any ontological implications.
prop 3: "Either the population of New York is greater than 1 million, or it isn't."
Proposition 3 would probably be thought meaningless if New York did not exist, and so it might not be true. But it cannot be false, it is an instance of the tautology expressed by proposition 1. We may therefore prefer not to say it is contingent, and we may be willling to consider it necessary.
prop 2: "2 + 2 = 4"
It is difficult to conceive any possibility that proposition 2 is false except that the sentence we use to express the proposition means something other than we normally mean by "2+2=4". Assuming the semantics of the language is fixed, the proposition may be thought necessary. There is embodied in the semantics an ontological presupposition which merits discussion.
prop 4: "The population of New York is greater than 1 million."
Proposition 4 is true in some possible worlds and false in others. It need not necessarily be true and hence it is not logically necessary and we say it is contingent.

Defining Necessity and Contingency:

Some ways of defining necessity and contingency are introduced and their merits discussed.
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 (e.g. prop 1)
In which a necessary statements must be true in all domains of discourse. All matters of ontology are considered contingent.
Abstract Necessity (prop 2)
In which sense true statements about abstract entities may be considered necessary. The distinction between abstract and concrete ontology is crucial here.
Broad Necessity (prop 3)
In which necessary truths containing references to concrete entities are admitted, particularly where these are instances of necessary truths containing no concrete references.

Possible Worlds:

Finally we consider the relationship between necessity and possible worlds, showing the conception of possible world which corresponds to each of our notions of necessity.
These thoughts on possible worlds are intended to help explicate the notion of logical necessity and the boundary between logic and science.
Sets of Sentences
Some philosophers have identified "possible worlds" with "maximal consistent sets of sentences". We briefly consider this before passing on to the more flexible idea of a possible world as an interpretation.
(free necessity)
If we identify "possible worlds" with "interpretation of a first order language", then we get a notion of logical necessity which corresponds to that of first order validity.
Standard Interpretations
(abstract necessity)
Restricting the notion of possible world to interpretations which are consistent with the intended meaning of abstract constants yields a more general notion of necessity which fits better with both the analytic and the a priori.
(broad necessity)
To talk about the world we need abstract (often mathematical) models. We get these by adopting some convenient but otherwise arbitrary abstract ontology, which we use for building abstract models of the real world. Models of the real world require no new concrete ontology but simply the introduction of new constants which are denote abstract surrogates for the real world.

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