Algebraic Semantics in Language and Philosophy (review)

Structure . . . when expressed mathematically, is abstract algebra. And this is how formal semantics can remain neutral about the question “what the objects really are” and still serve the needs of the philosopher [and linguist]: by providing an algebraic theory of the domain of entities in question’ (191).

That is the motivation for this book, which comprises fourteen of Link’s previously published papers, revised to form a coherent whole, plus an introduction and a mathematical appendix. All deal with the nature of plurality, and all analyze it using lattice theory or mereology (the modeling of division into parts).

L’s crucial claim, now generally accepted, is that plurals do not refer to sets. Rather, a plural predicate is true of certain combinations of individuals where these combinations form a lattice (i.e. above A, B, and C are A + B, A + C, B + C; above those is A + B + C).

A ‘proper plural’ (distributive) is true of every nonsingular node in the lattice above the appropriate individuals; thus if A, B, and C are students, then A and B are students, B and C are students, and A and C are students. (But not *A is/are students—A is not a plural node.) This parallels the behavior of mass nouns: If the first drop is water, and the second drop is water, then the combination of them is water.

A collective property is true only of the topmost node; thus if A, B, and C lifted the piano, it does not follow that A and B lifted the piano. Likewise, if the water in the bucket totals a gallon it does not follow that the first drop is a gallon.

That’s just the beginning of a very interesting subject. The reader should not feel compelled to read these fourteen papers in order, for several reasons. They were published separately, so each paper is, to some extent, self-contained. Later papers criticize positions taken in earlier papers. Some parts are too mathematical for casual reading: In formal logic (though all too seldom in formal linguistics), authors are expected to ‘deliver the goods’, so every formalism is worked out in detail to prove that it works as advertised.

Each paper begins with a lively introduction citing interesting linguistic data. All of the papers are worth reading, distributively as well as collectively.