How can we best understand the notion of necessity?
The currently standard way to understand the notion of necessity is provided by David Lewis’ “possible worlds” semantics. However, for reasons that I will enumerate below, however popular and convenient this semantics, it does not give us the best means of understanding necessity. The only viable alternative to the flawed “possible worlds” semantics, is the concept of “analyticity” and the notion of necessity in terms of “Truth by definition”. In this essay, I will support that conclusion.
Possible Worlds Semantics
In his 1973 work “Counterfactuals“(a), David Lewis provided a rigorous definition of “possible worlds”. His main proposal was that possible worlds should be the proper basis from which to understand counterfactual conditionals. Along the way, he defined “necessity” as follows:
“For any possible world (i) and sentence P, the sentence ‘necessarily P’ is true at the world (i) if and only if for every world (j), such that (j) is accessible from (i), P is true at (j).”
[I have altered the symbols for reasons of reproductive simplicity]
Lewis, Counterfactuals, pg 5.
Since that seminal work, the language of “possible worlds” has become the standard way of speaking about necessity. Commonly, in the literature, you see possible worlds used to define (logical) necessity as follows –
“necessarily P” is true iff P is true in all (logically) possible worlds
Certainly, the possible worlds semantics makes it appear easy to discuss necessity. But this easy simplicity masks a number of key underlying difficulties.
Firstly, possible worlds are not inspectable. Whether they are abstract concepts as many maintain, or real existents as Lewis maintains, they are not causally connected with this, the actual world. We do not actually pretend to really inspect any other possible world than our current world. So the question of whether some statement (de dicto necessity), or a predication of some particular (de re necessity) is true “in all possible worlds”, is not a question of empirical investigation. Whether the statement (or predication) is necessarily true must be determined on the basis of factors that are known to us here in this actual world. What we claim is true or not in all possible worlds is so purely by stipulation.
Secondly, the problem of trans-world identity is not resolved. When trying to decide whether a statement like “Necessarily, Aristotle was human” is true or not, possible worlds semantics would propose (in principle at least) that we examine every possible world to see if in worlds where Aristotle exists, Aristotle was human. But just what is it that picks out an instance of “Aristotle” in any other possible world than this actual world. Some philosophers (Kripke(b) in particular) have suggested that trans-world identity is established by an identity of origins. So any existent in some possible world that had the same parents (same sperm and egg? same genetic code?) would be the “Same” Aristotle. But this runs into difficulties. One is that the hypothesis leaves problematic the question of the identicality of the origins – and so forth with infinite circularity. Another is that usually we would intuitively judge the statement “possibly, Aristotle had six fingers” as true. But if the trans-world counterpart had the same parents, and human developmental processes are the same, then contrary to our intuitive judgement, it would in fact not be possible. On the other hand, if human developmental processes are allowed to freely flex, then it would also be equally possible that Aristotle was a wombat. But this is a possibility we intuitively judge as false, on the assumption that a wombat originating from the “Same” origin as Aristotle, would not be the same “Aristotle” that we refer to in this world. Kripkean rigid designation not withstanding, we intuitively withhold “Sameness” from something that is so different from the Aristotle in this world, that he/it would be unrecognizable in another. Alternatively, if we allow the human development processes to flex only enough to render false the statement “possibly, Aristotle was a wombat” while allowing the sentence “possibly Aristotle had six fingers” to be true, then we are letting our intuitive responses govern the sense of trans-world identity. Clearly our judgements of necessity and possibility are prior to, and not dependent on the possible worlds semantics. Obviously, we do not inspect “all possible worlds” in anything like an empirical sense. Rather we do it conceptually, on the basis of what we stipulate about the meaning of the concepts being employed.
Thirdly, necessity is a matter of linguistic definition, not a matter of how things are. Statements of necessity are not really about reality. Necessity is not a question of empirical investigation. Just what is meant by saying “Necessarily P” or “Necessarily A is F”? If we take the possible worlds semantics seriously, we would mean “under all conceivable ways in which the world might be different, it is true that P, or that A is F.” We are talking about the concepts we employ in stating “P” and claiming that those concepts are such that “P” remains true regardless of how the world turns out or might be different. To say “Necessarily P” is to say that “P is true, come what may”. So P must be true in virtue of the nature of the concepts employed in P, regardless of how reality turns out to be.
The statement “necessarily, all bachelors are male” is true because of how we define our terms, independently of how the rest of the world may be. Likewise, “necessarily, 2 plus 2 equals 4” is true by definition. (Or more stringently – deducible by defined rules of inference from (Peano’s?) defined axioms of arithmetic). Statements which are necessary, which are true in all possible worlds, are true because that is the way that we have created the concepts we are using.
Axioms are definitions of a sort. And the rules of formal logical deduction are also based on defined axioms. As a result, any statement that is either an axiom, or is logically deducible from axioms by way of defined rules of inference can be said to be “True by definition”. If “2 plus 2 equals 4” is a logical consequence of Peano’s axioms of arithmetic, then it is a statement “True by definition”. The meaning of any word is a definitional association with the respective “cognitive pigeon-hole.” So statements that are “True by definition” are “True in virtue of the meaning of the words employed.” The question of whether “Aristotle is necessarily human” is true or not depends on how we have defined the words “Aristotle” and “human”. If we conceive of “Aristotle” as a cluster of descriptions that includes the proposition that he was a human being, then it is true that “Aristotle was necessarily human”. If we conceive of “Aristotle” as a cluster of descriptions that includes say “The greatest Greek philosopher”, but does not include that he was a human being, then we allow it to be possible that Aristotle was not human (possibly he was a wombat). Determining whether any statement “necessarily P” (or “F is necessarily G”) is true or not is thus a matter of investigating the meaning of the words employed.
Godel’s Incompleteness Theorems showed that “no consistent theory of arithmetic is finitely axiomatizable” only if it is assumed that arithmetic is “complete”. Specifically, the incompleteness theorem states that given a first order axiomatization of arithmetic (such as the first-order Peano axioms), there exist arithmetical statements which cannot be either proved or disproved using those axioms. But so what if there are arithmetical statements that are undecidable? It might matter to mathematicians. But it does not matter to philosophy or linguistics. Mathematics is an artificial language that we have constructed (defined) in order to describe certain features of reality. As we discover new areas where the existing capabilities are not adequate to our needs, we establish additional axioms/definitions and explore their logical consequences. If those consequences “fit” the world we are trying to describe, then the axioms/definitions work. Otherwise, we go back and try again. All that is needed, surely, is that any arithmetical statement that we are pragmatically interested in using can be proved to be either true or false.
(I think it also remains to be established just what it would mean for an undecidable statement within mathematics to be judged “True” even if it could not be proved “True” within the language of mathematics. Surely that would involve applying an extra-mathematical notion of truth? Would that not just mean that there is a gap between what can be proved in mathematics and the world we are trying to describe with that language? And why should we have any special (i.e. non-pragmatic) problem with such a gap?)
Fourthly, we simply do not think in terms of possible worlds. Necessity is a notion that is in common use in the general population. To be sure, philosophy has tried to formalize the notion to be more specific that that in general use. But the intent of that formalization is not to create a new and exotic concept. Rather it is to retain as grounds for any formalization the common usage of the notion. The significance of this is that, with the exception of trained philosophers, no one thinks of necessity in terms of possible worlds. Although possible worlds semantics has proved very useful in talking about necessity in a non-specific sense, whatever is necessary is so because it simply has to be that way to make sense of the words that we use.
And finally, we had the concept of necessity long before the semantics of possible worlds. Indeed, the theory of possible worlds is parasitical on the notion of necessity. (Necessity and possibility are inter-definable.) We may not have been able to talk about necessity as easily, prior to the advent of the possible world semantics. But the literature, prior to David Lewis and Robert Stalnaker, is full of discussions of necessity de re and necessity de dicto. Discussions that demonstrate a sound conception of necessity and possibility independent of possible worlds language.
Analyticity
The specific term “analytic” was initially introduced by Kant in his Critique of Pure Reason:
“In all judgments in which the relation of a subject to the predicate is thought (if I only consider affirmative judgments, since the application to negative ones is easy) this relation is possible in two different ways. Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case, I call the judgment analytic, in the second synthetic.”(4a)
According to Kant, all analytic judgments rest on the principle of non-contradiction. “For since the predicate of an affirmative analytic judgment is already thought beforehand in the concept of the subject, it cannot be denied of the subject without contradiction.”(4b) And “analytic judgments are therefore those in which the connection is through identity.”(4c)
The next advance in our understanding of the concept of analyticity was provided by Frege as a consequence of his efforts to develop what we now think of as modern symbolic logic(5). In the process of carefully specifying the details of a “formal” language of logic (what we now call the axiomatic predicate calculus), he set down an account of what are called “logical constants” – “and”, “or”, “not”, “all”, and the like. Separating the logical constants from “referring expressions” – words that, unlike the logical constants, refer to things in the world – he permitted the classification of “logical truths”. Logical truths are those statements that are true in virtue of the meaning of the logical constants, no matter what referring expressions appear in the statement. Hence “all X are X” (eg. “all doctors are doctors”) is a logical truth. Unfortunately, Frege’s concept of a logical truth does not cover such common statements as “all X are Y” (eg. “all ophthalmologists are doctors”).
So Frege appealed to the idea of definitions, assuming that a definition preserves meaning between the definiendum and the definiens. With this addition, Frege explains non-logical analyticity as statements that can be converted into strict logical truth by replacing definienda with the appropriate definiens – or synonyms for synonyms. By treating a definition as a stipulation of what is synonymous with what, we can greatly expand the scope of English sentences that can be called analytic. This is especially so if we expand the scope of acceptable definitions beyond the direct stipulations of an explicit definition, to include ostensive, implicit, and conventional definitions. Any English statement that can be re-phrased in this way into a “logically true” statement of symbolic logic can therefore also be called “analytic”.
Frege’s theory of analyticity seems to nicely capture the intuitive feel of “True in virtue of the meaning of the words”. It also, by way of synonymy, captures much of the “cognitive containment” theory of Kant. And in those cases where it seems to stumble, an argument could be made that it is due to an indeterminacy in the meaning of the words involved, or an insufficiency in the formal language(s) of modern symbolic logic, but not any inadequacy in the notion of synonymous translation.
And thus things stood until Quine’s “Two Dogmas of Empiricism“. In this brief essay, Quine presents what has become accepted as the definitive critique of the Analytic-Synthetic Dichotomy. The focus of Quine’s attack on the notion of analyticity is his claim that the concept of the analytic depends on a prior understanding of synonymy, and that there is no acceptable explanation of synonymy.
Quine has no difficulty with the class of analytic statements Frege called “logical truths”. Nor does he have a difficulty with explicitly stipulated synonymy. His problem lies with that class of statements that are claimed to be analytic but depend for that classification on an unacceptable presupposition of synonymy.
The end result of Quine’s analysis is that the notion of “analyticity” is dependent on the notion of “Synonomy” which in turn is dependent on the notion of “analyticity”. Hence, Quine’s conclusion that the notion of analyticity is sufficiently ill understood that it makes no sense to employ the classification for sentences that are not analytic in virtue of being a logical truth, or in virtue of a stipulated synonymy.
If, as Quine claims, there is no coherent and determinate property that is picked out by the classifying label “analytic” (or, since Quine has demonstrated their correlativity, the label “Synonymous”), then that can only be because there are no facts of the matter as to whether a sentence is analytic (synonymous) or not.
“But, now, how can there fail to be facts about whether any two expressions mean the same — even where these are drawn from within a single speaker’s idiolect, so that no questions of interlinguistic synonymy arise? Wouldn’t this have to entail that there are no facts about what each expression means individually? Putting the question the other way: Could there be a fact of the matter about what each expression means, but no fact of the matter about whether they mean the same?” Boghossian(6), Pg 370
In order to support his conclusion that synonymy is not possible, Quine must maintain that while there is a fact of the matter about what each sentence means, no two sentences can ever possibly mean the same thing. This position only works on the assumption that the meaning of any one sentence is given by the entirety of ones belief network (the first of any pair of sentence appearing in the belief-network of the second, and vice versa). Quine’s “definitive critique” of the analytic-synthetic distinction is therefore driven by a prior commitment to meaning-holism. Given the undisputed philosophical usage of “analytic”, Quine’s critique of the analytic-synthetic distinction can be viewed as a reductio ad absurdum proof that his meaning-holism is the wrong theory of meaning. Or, more leniently, one can view his critiques of the specific explanations of “analyticity” that he examined as showing that they are flawed in some fashion, without demonstrating that no explanation of analyticity is possible.
Conclusion
We clearly make use of the terms “analytic” and “necessary” with a fair degree of consistency. Because these terms are used to classify sentences, it is reasonable to suppose that whatever explanation for them is eventually deemed satisfactory will be couched in concepts relevant to a theory of the meaning of these sentences. When we add to this the observation that there is not yet any generally accepted theory of the meaning of sentences, it is perhaps not too surprising that there is no generally accepted explanation for analyticity and necessity. The failure of philosophers to have discovered a generally accepted theory of meaning for these terms is not a demonstration that no such theory is possible.
What is needed is a theory of meaning that will provide a suitable explanation for “Synonymy” that is not as strict as Quine demands. For example, Quine’s concept of synonymy is too strict to permit Kant’s notion of “cognitive containment” – it does not permit an explanation of why such sentences as “anything green is extended” are deemed analytic. There is no sense of “Sameness of meaning” that will allow us to translate that statement into a logical truth. What is needed is a looser conception of synonymy that will allow for both the notion of “containment”, and for a broader acceptance of the role of definitions in the establishment of acceptable synonymy. An alternative theory of meaning, like a set-theoretic rendition of an intentional description-cluster theory of meaning similar to that of Searle’s theory of proper names(7), offers hope of providing a satisfactory explanation of analyticity in terms of meanings.
Despite its admitted inadequacies, the best way to explain the concept of analyticity remains –
A sentence S is analytic iff it can be paraphrased in to a logical truth by replacing “cognitive synonyms” for “cognitive synonyms” – where a “cognitive synonym” for one term is another that “means the same thing” in a general sense.
And given this notion of analyticity, we can then best understand necessity in terms of –
A sentence P is necessary (or F is necessarily G) iff it is analytic in the stipulated sense.
Notes & References
(a) Lewis, David; Counterfactuals, Blackwell Publishing, Oxford, England. 1973. ISBN 0-631-22425-4
(b) Kripke, Saul A. Naming and Necessity. Harvard University Press. Cambridge, Massachusetts. 1980. ISBN 0-674-59845-8.
(1) Quine, V.O. Two Dogmas of Empiricism; downloaded from the Internet August 8, 2008 from URL=<http://www.ditext.com/quine/quine.html> [Originally published in The Philosophical Review 60 (1951): 20-43. Reprinted in W.V.O. Quine, From a Logical Point of View (Harvard University Press, 1953; second, revised, edition 1961)]
(2) Bradley, Raymon D; A Refutation of Quine’s Holism, downloaded from the Internet, Jan 25 2009, URL=<http://www.sfu.ca/philosophy/bradley/intro_refutation_of_quine.htm>
(3) Grice, H.P. & Strawson, P.F. “In Defence of a Dogma”, The Philosophical Review, Vol. 65, No. 2. Apr. 1956, Pgs 141-158 downloaded from the Internet Sept 20, 2008, URL=<http://www.jstor.org/stable/2182828>
(4a) Kant, I. The Critique of Pure Reason, trans. by P. Guyer and A.W.Wood. Cambridge University Press, Cambridge, England. 1781/1998. §A:6-7.
(4b) Kant, I. Prolegomena to Any Future Metaphysics. The Library of Liberal Arts / Bobbs-Merrill Educational Publishing, Indianapolis, Indiana. 1950. ISBN 0-672-60487-7. pg 27
(4c) Kant, I. The Critique of Pure Reason, trans. by P. Guyer and A.W.Wood. Cambridge University Press, Cambridge, England. 1781/1998. §A:7 & B:11.
(5) Frege, G.
The Foundations of Arithmetic, 2nd revised ed., Blackwell Publishing, London, England. 1884/1980
“On Sense and Reference,” in P.Geach and M. Black (eds.), Translations from the Works of Gottlob Frege, Blackwell Publishing, Oxford, England. 1892a/1966. pp56-78.
“On Concept and Object,” in P.Geach and M. Black (eds.), Translations from the Works of Gottlob Frege, Blackwell Publishing, Oxford, England. 1892b/1966. pp42-55.
(6) Boghossian, P. “Analyticity Reconsidered” Nous, Vol. 30, No. 3 (Sep., 1996), pp. 360-391, downloaded from the Internet Sept 20, 2008. URL=<http://www.jstor.org/stable/2216275>
(7) Searle, J. R.
“Proper Names”, Mind, New Series, Vol 67, No 226, (Apr 1958), pp 166-173.
Speech Acts: An Essay in the Philosophy of Langauge. Cambridge: Cambridge University Press. 1969.
Intentionality: An Essay in the Philosophy of Mind. Cambridge: Cambridge University Press. 1983.
