A New Logic Of Presupposition Projection
SALT 18
Benjamin R. George
A New Predictive Theory of Presupposition Projection
We propose a theory that derives the projections of presuppositions in the arguments
of functions from the (static, bivalent, extensional) truth-conditional semantics of the
functions, yielding fine-grained predictions about projection behavior and addressing some
empirical and theoretical concerns about dynamic accounts of presupposition projection.
Some problems for a theory of projection
Theories of presupposition projection in the dynamic semantics tradition (e.g. [2])
have generally made few predictions about how truth-conditional meaning can combine
with projection behavior - there are many functions on CCPs that correspond to the
truth-function of logical conjunction, but only one makes good projection predictions for
natural language conjunction (see [4] for related concerns). These theories usually resort to
case-by-case stipulation, and so make no predictions about projection under functions that
they don’t mention explicitly. Further, new data have called into question the empirical
claims of some earlier work on projection. In particular, presuppositions in restrictors of
quantifiers often project weakly or not at all (for most speakers, (1) pesupposes either
nothing at all or that there is at least one employed female topologist), and different
quantifiers give rise to different inferences about presuppositions in their nuclear scope
(for example (2-a) and (2-b) entail (2-e) while (2-c) and (2-d) apparently do not1).
(1)
Every topologist who dislikes her employer drinks.
(2)
a. None of these students have stopped smoking.
b. Each of these students has stopped smoking.
c. More than three of these students have stopped smoking.
d. Exactly three of these students have stopped smoking.
e. Each of these students has at some point smoked.
Sketch of the theory
A third truth value, written #, represents presupposition failure - sentences with un-
true presuppositions have value #, and predicates that make presuppositions map items
that lack the presupposed properties to #. Functions are defined over domains on non-
presuppositional values - the action of functions on presuppositional arguments comes from
the machinery defined below. For truth-functions, the semantics understands the value of
# as uncertain between the alternatives 0 and 1.2 If both choices yield the same output,
given the other arguments, that is the output of the function.3
This yields the strong Kleene [3] trivalent semantics for connectives; we augment it by
further requiring that, considering the arguments of a function in order, no argument can
have presuppositional content that rules out the possibility that the output is 1 - if some
argument violates this rule, we set the output to #. Formally, we consider the set of items
presupposition-equivalent to an argument, where 0 and 1 are mutually presupposition-
1This general pattern is consistent with the statistical generalizations of [1], although it must be
admitted that in that study speakers inferred a universal about half the time even from (2-d).
2That is, 0 and 1 are the alternatives for # - alternatives for predicates are discussed below.
3For example, (0 and #) takes the value 0, since (0 and 0)=(0 and 1)=0), but if the alternatives
produce different outputs, the function returns # (so (1 and #) takes the value #.
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equivalent and # is equivalent only to #, and two predicates are equivalent if they map
exactly the same entities to #. For an argument x preceded by arguments y1, ...yk (the
list may have length 0), we require that one of the following criteria holds:
(i) There are no non-presuppositional u1, ..., um such that f(y1, ..., yk, u1, ..., um) = 1,
evaluated under the rule defined above.4
(ii) There are some x presupposition-equivalent to x and some non-presuppositional
values z1, ..., zn, such that f(y1, ..., yk, x , z1, ..., zn) = 1 (again under the strong Kleene
definition of function application).5
We adopt a new notion of function ‘deployment’, distinct from the function application
already considered, written f[w] instead of f(w), where w is an argument list. If either (i)
or (ii) is satisfied, then f[w] = f(w) as defined perviously. Otherwise, f[w] = #.6
To extend this to quantifiers, we define the set of (non-presuppositional) alternatives
for a presuppositional predicate p. The set of alternatives depends on which entities that
matter for the function f with respect to the argument position of p.7 For quantifiers
the entities that matter when evaluating the restrictor are all entities, and those that
matter when evaluating the nuclear scope are all the entities in the restrictor. Let p0/# be
such that p0/#(x) = 1 iff p(x) = 1 and 0 otherwise, and let p1/#(x) be 0 when p(x) = 0
and otherwise 1. If p maps at least one entity that matters for f to 1, then the set of
alternatives for p is {p0/#}; otherwise, it is {p0/#, p1/#}.
We now get a weak presupposition for (1) - if there is even one topologist who dislikes
an entity by which she is employed, then the restrictor denotation p maps something
to 1, so its only alternative is p0/#; the requirement that the presuppositions allow an
output of 1 is satisfied, so unless p0/# is empty, the sentence is true iff every entity in
p0/# drinks. Considering the nuclear scope q of (2-a), suppose some student has never
smoked and consider any q presupposition-equivalent to q. If q maps any student to 1,
then evaluating at the alternative q 0/# will make no output 0, and if q maps all students
to 0, then q 1/# and q 0/# are both alternatives for q , and these produce different results
with no so evaluation yields #: no choice of q gives us an output of 1, so if there are any
never-smoking students then the truth value of (2-a) is #. If the student former smokers
are three, the system makes (2-d) true, since the only alternative we need to consider maps
all never-smokers to 0. We also predict universal presuppositions for (2-b) but not (2-c).
References
[1] Chemla, E. 2007. ‘Presuppositions vs. Scalar Implicatures’, XPRAG.
[2] Heim, I. 1983. ‘On the Projection Problem for Presuppositions’, WCCFL 2.
[3] Kleene, S. 1952. Introduction to Metamathematics. North Holland, Amsterdam.
[4] Soames, S. 1989. ‘Presupposition’. Handbook of Philosophical Logic IV.
4This is the case where 1 can’t be the output but the presuppositions of x are not to blame.
5This is the case where the presuppositional content of x does not rule out the output being 1.
6This gives us asymmetry of projection for and but not for or - an empirically appealing contrast.
7A formal definition of this notion of mattering is possible, but not in the allotted space.
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