A posteriori necessities

The usual examples of a posteriori necessities are identities between kinds and objects under two descriptions, at least one of which involves a contingent mode of presentation, such as water (presented as “the stuff in this pond”, say) and H₂O.

Such a posteriori necessities are certainly interesting. But we should not assume that these exhaust the scope of all a posteriori necessities.

For instance, Thomas Aquinas was committed to the existence of God being an a posteriori necessity: he held that necessarily God existed, but that all a priori arguments for the existence of God failed, while some a posteriori ones, like the Five Ways, succeeded.

For another theistic example, let p be an unprovable mathematical truth. Then p is, presumably, not a priori knowable. But God could reveal the truth of p, in which case we would know it a posteriori, via observation of God’s revelation. And, plausibly, mathematical truths are necessary.

For a third example, we could imagine a world where there is an odd law of nature: if anyone asserts a false mathematical statement, they immediately acquire hideous warts. In that world, all mathematical truths, including the unprovable ones, would be knowable a posteriori.

"The" natural numbers

Benacerraf famously pointed out that there are infinitely many isomorphic mathematical structures that could equally well be the referrent of “the natural numbers”. Mathematicians are generally not bothered by this underdetermination of the concept of “the natural numbers”, precisely because the different structures are isomorphic.

What is more worrying are the infinitely many elementarily inequivalent mathematical structures that, it seems, could count as “the natural numbers”. (This becomes particularly worrying given that we’ve learned from Goedel that these structures give rise to different notions of provability.)

(I suppose this is a kind of instance of the Kripke-Wittgenstein puzzles.)

In response, here is a start of a story. Those claims about the natural numbers that differ in truth value between models are vague. We can then understand this vagueness epistemically or in some more beefy way.

An attractive second step is to understand it epistemically, and then say that God connects us up with his preferred class of equivalent models of the naturals.

Marriage and natural kinds

On a New York Times blog, Wedgwood has offered an interesting argument for same-sex marriage. The argument is focused on what he calls "the 'social meaning' of marriage", which "consists of the understandings and expectations regarding marriage that almost all members of society share." He notes that this meaning "cannot include any controversial doctrines", but rather must include only uncontroversial assumptions about the nature of marriage. He then argues that same-sex couples have the same interest in having access to an institution that yields the same understandings and expectations as heterosexual couples do. Therefore, the social meaning needs to be shifted in such a way as to give access to marriage to same-sex couples. (This quick summary doesn't do full justice to Wedgwood's rich piece.)

But now compare the social meaning of "marriage" to the social meaning of "water", defined by the understandings and expectations regarding water that almost all members of society share.

There is indeed such a social meaning of water. And this social meaning of water may well help fix the meaning of the word "water". But it does not fix the meaning on its own, as we see from Twin Earth cases. On Twin Earth, suppose, there are beings like us, with a language like ours, and they have a substance that behaves just as water does, and hence obeys the same ordinary understandings and expectations that water does, but its chemical structure is instead XYZ.[note 1] The Twin Earthers' word "water" does not refer to H₂O, but to XYZ, even though their word "water" has the same social meaning as our word "water" does. Thus, the meaning of the word "water" is not exhausted by the social meaning of water.

If we focus on social meaning as central to meaning, we're going to say something like: "The word 'water' refers to that natural kind in the vicinity of the word's users that best fits with the users' 'understandings and expectations.'" In other words, the meaning of "water" isn't fixed by the social meaning alone: it is fixed by social meaning plus facts about what natural kinds are really exemplified in the vicinity of the speakers. Moreover, water will have non-obvious essential properties, such as that its molecules are a union of two atoms with atomic number 1 with one atom with atomic number 8. We aren't going to find these properties by examining the social meaning of water.

Now go back to marriage. If "marriage" is analogous to "water", the social meaning of marriage does not by itself fix the meaning of the word "marriage". Rather, "marriage" refers to that natural kind of relationships in the vicinity of the speakers that in fact best fits with the language-users' understandings and expectations. But then we would expect marriage to have all sorts of essential properties that go beyond the social meaning. And it could turn out that the only natural kind of relationships in the vicinity that fits with our understandings and expectations also has the essential property of being a union of a man and a woman.

I take it that Wedgwood is assuming that marriage is unlike water, that the meaning of marriage is exhausted by its social meaning rather than there being a mind-independent natural kind for the word "marriage" to refer to. But whether this is so is the most fundamental question in the debate, and he gives no case his view on it.

A less deep problem with Wedgwood's line of thought is that, as he himself acknowledges, a part of the shared understanding of marriage is that sexual activity is a normal aspect of marriage. But if same-sex sexual activity is not permissible, then same-sex couples do not have the same interest in an institution that comes along with an assumption of normative sexuality, since it is generally not in one's interest that people assume one is doing something impermissible (whether or not one is actually doing it). Thus, Wedgwood's argument requires that same-sex sexual activity is permissible. But his typical philosophical or theological opponent will deny this permissibility.