Topic: HUMAN SEMANTICS
Compositionality Inductively, Co-inductively and Contextually
By Tim Fernando
To say that the meaning [[a]] of a term a is given by the meanings of a's parts and how these parts are combined is to state an equality[[a]] = : : : [[b]] : : : for b a part of a (1)
with meaning [[.]] appearing on both sides. (1) is commonly construed as a prescription for computing the meaning of a based on the parts of a and their mode of combination. As equality is symmetric, however, we can also read (1) from right to left, as a constraint on the meaning [[b]] of a term b that brings in the wider context where b may occur, in accordance with what Dag Westerstahl has recently described as "one version of Frege's famous Context Principle"
the meaning of a term is the contribution it makes to the meanings of complex terms of which it is a constituent. (Westerst ahl, 2004, p.3)
That is, if reading (1) left-to-right suggests breaking a term apart (and delving inside it), then reading (1) right-to-left suggests merging it with other terms (and exploring its surroundings). These complementary perspectives on (1) underly inductive and co-inductive aspects of compositionality (respectively), contrasted below by
(i) reviewing the co-inductive approach to the Fregean covers of Hodges (2001) anticipated in Fernando (1997)
(ii) inductively deriving a more recent theorem of (Westerst ahl, 2004) on the extensibility of compositional semantics closed under subterms.
Choosing between inductive and co-inductive approaches to (1) does not, by itself, determine the meaning function [[.]]. The ellipsis in (1) points to a broader notion of context capturing background assumptions that shape [[.]]. To square (1) with "dynamic" conceptions of meaning as context change (e.g. Heim,1983), we shall inject a certain notion of context c inside meanings, and not simply hang them outside [[.]] as subscripts, [[.]] = [[.]]c.