What are examples of analytic a posteriori knowledge?

There is the analytic/synthetic distinction and the a priori/a posteriori distinction. These two distinctions form four types of knowledge:

• analytic a priori
• synthetic a priori
• analytic a posteriori
• synthetic a posteriori

Kant thought analytic a posteriori is self-contradictory. But, some philosophers (e.g. Stephen Palmquist) treat it as valid.

What are examples of analytic a posteriori knowledge? What are their refutations?

Kripke has some examples in his book Naming and Necessity. The proposition Hesperus is Phosphorus (the evening star is the morning star, both being what we call Venus) is one of them. Kripke finds this to be analytic a posteriori because there once was a time in which people thought of Hesperus and Phosphorus as two different stars, later on they found out that they we're actually the same planet. In this way they necessarily point to the same object but this has been found out through the empirical evidence.

• I will add that all such examples I am aware of tend to make use of the same situations of temporal semantics-in-syntax (i.e. definitions). When meanings change on experience, you can have analytic a posteriori when the definition allows proof reasoning to be self-evident. Some have argued that definitions don't change the meaning in existing statements (you are instead defining something new), but those kinds of arguments cannot be brought all the way to primitives without assuming infallible classification. Commented Mar 8, 2013 at 0:32
• Kripke argue that "hesperus is phosphorus" is a case of a posteriori necessity, not analycity. Logical empiricists conflated necessity and analycity, but not Kripke. Commented Nov 26, 2014 at 11:02

Analytic a posteriori claims are generally considered something of a paradox. First, let's recall that an analytic proposition's truth is entirely a function of its meaning -- "all widows were once married" is a simple example; certain claims about mathematical objects also fit here ("a pentagon has five sides.")

So, an analytic proposition is 'inherent' in a way that isn't the case for a synthetic proposition. Consider Kant's own example of a synthetic proposition: "all bodies are heavy." The reason this isn't analytic is that the predicate ("heavy") isn't 'contained' in the subject ("body"), as it would be for the claim, e.g., that a square is four-sided. There is an inherent 'ease' with analytic claims, since all one needs to do to know it is "extract" the predicate from the subject.

Now, the a priori/a posteriori distinction is about whether we know something from experience. This may seem similar but it is distinct from the analytic-synthetic question (which again is about whether the subject contains the predicate or not); note that many a priori claims are also synthetic. Perhaps the clearest examples of a priori claims are mathematical expressions (`2+3=5`).

Finally, let's consider the problematic hybrid you have asked after. A proposition that's analytic a posteriori would contain the predicate within the subject (as 'triangle' contains 'three sides') but would only be justifiable based on experience. Kant thought this category was paradoxical, as he thinks you never need to resort to experience to justify analytic claims.

However, some modern critics like Stephen Palmquist have argued that in fact philosophy requires these aposterior analytic claims to function in its characteristic 'hypothetical' mode:

To begin with, the impossibility of analytic a posteriori knowledge is generally considered to be 'quite evident' [P5:182-3]: indeed, it is a nonsensical contradiction in terms for those who equate 'analytic' and 'a priori' [see Ap. IV]. Even though Kant argues against those who identify analyticity and apriority [e.g., in Kt1:1-10], he joins them in dismissing this class of knowledge with only a brief explanation: 'it would be absurd to found an analytic judgment on experience. Since, in forming the judgment, I must not go outside my concept, there is no need to appeal to the testimony of experience in its support' [Kt1:11; cf. Kt2:268 and Kt4:12]. There are, however, a few theorists who do regard the analytic a posteriori as providing the best description of certain types of knowledge.[20] Notwithstanding Kant's lack of concern for this class of knowledge, I shall argue in IV.3 that certain aspects of his philosophy can best be understood by reinterpreting them in terms of the analytic a posteriori. At this point, though, it will suffice to say that we should expect such knowledge, if it is possible, to have its validity grounded in some way in experience (a posteriori), and yet also to proceed by making inferences solely on the (analytic) basis of an application of the laws of logic to the concepts or propositions involved.

You can read Palmquist's whole book here. (This section appears in Chapter Four.)

While this question is concerned with explaining and exemplifying the notion of analytic a posteriori knowledge, Saul Kripke suggests his version of what should be necessary a posteriori knowledge and provides examples. Do note the difference between Kripke's suggestion and what is asked for in the original question, as Necessity and Analyticity are not the same thing.

... Aprioricity, analyticity, and necessity have since been more clearly separated from each other ... Kripke argued that there are necessary a posteriori truths, such as the proposition that water is H2O (if it is true). According to Kripke, this statement is necessarily true (since water and H2O are the same thing, they are identical in every possible world, and truths of identity are logically necessary) and a posteriori (since it is known only through empirical investigation). Following such considerations of Kripke and others (such as Hilary Putnam), philosophers tend to distinguish more clearly the notion of aprioricity from that of necessity and analyticity.

Now, this next paragraph (from the same article and consecutive to the one above) could suggest why this may be relevant to our question regarding analytic a posteriori:

Kripke's definitions of these terms, however, diverge in subtle ways from those of Kant. Taking these differences into account, Kripke's controversial analysis of naming as contingent and a priori would best fit into Kant's epistemological framework by calling it "analytic a posteriori".

And this is the footnote of the last paragraph:

Stephen Palmquist, "A Priori Knowledge in Perspective: (II) Naming, Necessity and the Analytic A Posteriori", The Review of Metaphysics 41:2 (December 1987), pp.255-282. See also "A Priori Knowledge in Perspective: (I) Mathematics, Method and Pure Intuition", The Review of Metaphysics 41:1 (September 1987), pp.3-22. In this pair of articles, Palmquist demonstrates that the context often determines how a particular proposition should be classified. A proposition that is synthetic a posteriori in one context might be analytic a priori in another.

I would consider idiomatic communication, such as the usage of prepositions, to be a good example. A picture hangs "on" the wall just because of the meaning that inheres in the preposition "on", yet this meaning is has validity only by experience with the idiom itself, hence is a posteriori.

Further it is only by experience with the idiom that I would associate the picture on the wall with the cup that is "on" my desk or the athlete who is "on" her game. But that fact seems to have little to do with the fact that in each instance "on" continues to have a sort of analytically derivable fitness to the situation.

I think one could argue that the statement "All bodies are extended" is both analytic and only known a posteriori. If we define a "body" as something which extends into temporal/spacial reality, then by definition all bodies are extended. Yet now we can ask, how do we know this? Simply by the way we have defined the words "body" and "extension". But this doesn't really answer the question. How do we know what bodies are? How do we come to know extension? This is tricky. Imagine someone who has no ability to use his or her senses. Do you think this person would be able to grasp the concept of a body? Or of extension? To me this seems doubtful. How can one understand what a body without experiencing the world? How could have a distinct idea of some object if they had no understanding of temporal/spacial reality? I very much doubt this would be possible. In order to understand bodies and extension, one would first have to have a clear picture of time and space. This can happen only through experience.

Though, for some reason largely forgotten, there is an extensive and thorough critique of Kant in Physical Realism by Thomas Case. On page 353, he says: "Such is the outline of a realistic theory of self evident analytical judgments a posteriori..." People ought to consult this book if they are having trouble understanding the subject matter.

"Such is the outline of a realistic theory of self evident analytical judgments a posteriori, of which the points are, first, that such judgements are not always about names and conceptions, but also about objects of sense and reason; secondly, that we discover the objects by general reasoning from sense, by perfect abstraction apprehend a simple kind of object, and analyze it into subject and predicate by, not from, the principles of identity and difference, or contradiction, a posteriori; thirdly, that analytical judgements are self-evident to one who has abstracted the objects, universal without exception, and convertible; and, fourthly, that analytical judgements about objects of reason in the abstract are sometimes principles of science." (353-354)

I'll have to go along with Poincare in the notion that nothing new comes out of logic. Anayltic a-priori is a myth, because a person's premise lies directly within the problem, and as soon as the premise is realized - the conclusion becomes based on posteri knowledge. Thus, the person's conclusion relies on posteri knowledge. Perhaps, the Synthetic process of intuition, is the closest thing to a-priori, but that is only implicitly independent of previous knowledge.