Necessity, analyticity and the a priori

The following three pairs of concepts form a system of classification which plays a crucial role in philosophical theorising. The concepts of each pair are mutually exclusive and each pair is related to the others in a number of complex ways.

Necessity and contingency

The first pair is that of necessary truth and contingent truth. Something is contingently true if it is true but could have been false, or alternatively, something is contingent if it is the case but could have been otherwise. For instance it is contingently true that my name is Darren: my parents could have decided to call me something else. There are some features of the world however that we think could not have been otherwise: material objects could have different physical attributes to those they do—they might have been larger or smaller, for instance; but they could not fail to have any physical attributes whatsoever—they could not fail to be some size or other. That is a necessary fact about physical objects.

This distinction is a metaphysical distinction: it is to do with the way things must be in order to be the things that they are, as opposed to the way they happen to be but might have been otherwise.

The analytic and the synthetic

The second pair is that of analytic statement and synthetic statement. Consider the statement "All swans are white". Whether this statement is true or not depends upon the way the world really is, that is whether all swans really are white. Such statements are known as synthetic statements. Consider, however, the statement "All bachelors are unmarried men". If one knows the meaning of the constituent expressions of that statement, one knows in advance what its truth value is. (It would be patently absurd to go and find all the bachelors in the world and check whether they really are unmarried men in order to convince one's self of the truth of that statement). Such statements are known as analytic statements.

This is a logical/semantical distinction: it is about the way the truth value of a statement is determined. It is plausible to view the determination of the truth value of a statement to consist of two factors—on one hand the meaning of the constituent expressions of the statement (a linguistic component), and on the other hand the way the world is (a factual component). Analytic statements are those statements in which the factual component has shrunk to zero, and whose truth values are determined solely by the meanings of their parts.

The a priori and the a posteriori

The final distinction is between things which can be known a priori and things which can only be known a posteriori. Consider again the whiteness of swans. How could I establish the truth or falsity of this statement? It seems it could only be by empirical investigation: that is by going out into the world and finding out by collecting data and establishing evidence. Indeed the falsity of this statement was discovered precisely in that way—by the discovery of black swans. When a fact can only be established by empirical investigation, we say it can only be known a posteriori. Compare how we establish the truth of some arithmetical statement, for instance, 2+2=4. Once we have grasped the arithmetical concepts involved, like addition, equality, two and four, we can know that 2+2=4 by pure reflection. We certainly would not check to see whether our calculation was correct by gathering pairs of pairs and checking to see that each numbered four in total. We say that such facts can be known a priori.

This is an epistemological distinction: it is about the way knowledge of certain facts is acquired. A truth is known a priori if it is known independently of any experience of how things are in the world. A truth is known a posteriori if it can only be known as a result of experience.

Relations of ideas and matters of fact

Relations of Ideas Matters of Fact
Metaphysical Necessary Contingent
Semantical/Logical Analytic Synthetic
Epistemological A Priori A Posteriori
For example "2+2=4" "Not all swans are white"

The distinctions are clearly related. They appear to be coextensive, that is, true of the same things, as indicated in the above table. The two-fold distinction between, on the one hand, necessity, analyticity and a priority, and, on the other hand, contingency, syntheticity and a posteriority, and its significance, is powerfully brought to life in Hume's Enquiry:

All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there were never a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.

Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction, than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind. (§§20–21.)

Indeed we might think that it is because of relations between ideas that we are able to understand the notions of a priority and necessity. If there are a class of statements that are analytic, that is that express relations of ideas, then their truth value can be known to us independently of experience of the world, i.e. a priori, because their truth value is not determined by any facts about the world. And if their truth value is not determined by any facts about the world, then no matter how the world was different, their truth value would remain unchanged. Therefore they express necessary truths.

Dismantling the distinction between relations of ideas and matters of fact

However, our neat classification is perhaps far too neat. There are compelling reasons for thinking that the three distinctions may not be coextensive. It would after all require some further justification to believe that all necessary truths must be expressible by analytic statements which are known to be true a priori. The three distinctions, after all, are distinctions about different kinds of facts—facts about the necessary structure of the world, facts about meaning, and facts about how we acquire knowledge. Without further motivation it would be philosophically unwise to assume that the three distinctions always coincide in their classification of particular cases, no matter to what extent they do in the vast majority of cases.

Kant in his Critique of Pure Reason was perhaps the first philosopher to realise the importance and significance of distinguishing between metaphysical, semantical and epistemological facts. In particular, Kant believed that there was a class of truths that were synthetic but knowable a priori. He believed that the root of many of Hume's sceptical conclusions in his philosophy, concerning causation particularly, were due to the coarseness of his distinction between relations of ideas, and matters of fact, and offered the existence of synthetic a prioris as a diagnosis of what he took to be Hume's failure.

Saul Kripke, in Naming and Necessity, argues that some necessary truths are only discoverable a posteriori. Consider the astronomical discovery that the morning star is in fact the same heavenly body as the evening star. They are both, in fact, the planet Venus. Now this is an empirical discovery. No amount of a priori cogitation could possibly have revealed this fact to us. It is nevertheless, argues Kripke, a necessary fact. For statements of identity are guaranteed by logic to be true in every possible world. Everything is the thing it is and not some other thing. Every object is necessarily self-identical.


Necessary Contingent Analytic Synthetic A Priori A Posteriori
A Posteriori ? Yes ? Yes No
A Priori Yes ? Yes ?
Synthetic ? Yes No
Analytic Yes ?
Contingent No

The relationship between these three pairs of concepts is therefore much more complicated then had at first appeared. The only combinations that we know cannot occur are those indicated by a "No" in the above table, and that follows simply from the definitions of the concepts. The "Yes"s indicate uncontroversial combinations. The question marks are hotly debated.

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