The Department of Logic & Philosophy of Science Colloquium Series presents
"Henkin Semantics for First-Order Logic"
with Aldo Antonelli, Professor, UC Davis
Friday, January 11, 2013
SOcial Science Tower, Room 777
In his 1950 dissertation, Leon Henkin showed how to provide higher-order quantifiers
with non-standard, or "general" interpretations, in which, for instance, second-order
quantifiers are taken to range over collections of subsets of the domain that may
fall short of the full power-set. In contrast, first-order quantifiers are usually
regarded as immune to this sort of non-standard interpretations. Because of this asymmetry,
the semantics for first-order quantifiers is ordinarily taken to be determined by
the selection of a first-order domain of objects. The asymmetry is particularly evident
from the point of view of the modern theory of generalized quantifiers, according
to which a first-order quantifier is construed as a predicate of subsets of the domain.
For example, the first-order existential quantifier is taken to denote the collection
of all non-empty subsets, the quantifier "there are exactly k" is taken to denote
the collection of all k-membered subsets, etc. But the generalized conception still
views first-order quantifiers as predicates over the full power-set, while the possibility
that they, similarly to their second-order counterparts, might denote arbitrary collections
of subsets has gone mostly unnoticed. This talk introduces a Henkin-style semantics
for arbitrary first-order quantifiers, exploring some of the resulting properties,
and emphasizing the effects of imposing various further closure conditions on the
second-order component of the interpretation. Among other results, Antonelli will
show by a model-theoretic argument that in certain cases the notion of validity relative
to models satisfying the closure conditions is axiomatizable. Finally, although the
talk is mainly devoted to laying the technical groundwork, he will touch upon some
of the philosophical insights that can be gained from the consideration of non-standard
interpretations, especially as regards issues of semantic determinacy of first-order
quantifiers and their role in expressing existence claims and ontological commitment.
For further information, please contact Patty Jones, email@example.com or 949-824-1520.