Willard Van Orman Quine

274 joseph s. ullian
the predicative proposals of Weyl and Wang, he speaks of himself
as “never tempted to embrace constructivism at the cost of trading
our crystalline bivalent logic for the fog of intuitionism” (RW 648).
Of intuitionism he has also written that it muddies “the distinction
between saying a sentence and talking about it” and that it “lacks
the familiarity, the convenience, the simplicity, and the beauty of
our logic” (PL 87).
What has caught Quine’s eye with less disfavor is the intriguing
mechanism of branching quantifiers. Is their exclusion from our logic
an arbitrary matter? In “Existence and Quantification” (p.109), he
attributes to Henkin consideration of sentences like this one:
Each thing bears P to something y and each thing bears
(1)
Q to something w such that Ryw.
It turns out that the two symmetric ways of rendering this in standard
quantificational form are not equivalent to each other, nor is either
equivalent to the natural construal of the branched
(2)
∀x ∃y
∀z∃w (Pxy.Qzw.Ryw),
a form recommended by Henkin to make (1)’s intended dependencies
explicit. 3
It has been debated whether English has sentences that are best
cast in terms of branching quantifiers. Barwise analyzed a broad array
of candidates and explored semantics for them. 4 One was Hintikka’s
favorite example: “Some relative of each villager and some relative
of each townsman hate each other.” Perhaps more seductive was
Gabbay and Moravcsik’s “Every man loves some woman (and) every
sheep befriends some girl that belong to the same club.” Barwise
concluded that “branching quantification does occur naturally in
English.”
Henkin noted that there is a more conventional way of setting
down a sentence like (1). It would be the conspicuously second-order
(3) ∃ f ∃g ∀x ∀z(Pxfx.Qzgz.Rfxgz).
This sentence quantifies over functions; (2) did not but departed from
the standard use of quantifiers. Might there be sufficient reason to
extend logic to include such branched schemata as (2) or even such
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