In logic a branching quantifier,[1] also called a Henkin quantifier, finite partially ordered quantifier or even nonlinear quantifier, is a partial ordering[2]
\langleQx1...Qxn\rangle
of quantifiers for Q ∈ . It is a special case of generalized quantifier. In classical logic, quantifier prefixes are linearly ordered such that the value of a variable ym bound by a quantifier Qm depends on the value of the variables
y1, ..., ym−1
bound by quantifiers
Qy1, ..., Qym−1
preceding Qm. In a logic with (finite) partially ordered quantification this is not in general the case.
Branching quantification first appeared in a 1959 conference paper of Leon Henkin.[3] Systems of partially ordered quantification are intermediate in strength between first-order logic and second-order logic. They are being used as a basis for Hintikka's and Gabriel Sandu's independence-friendly logic.
The simplest Henkin quantifier
QH
(QHx1,x2,y1,y2)\varphi(x1,x2,y1,y2)\equiv\begin{pmatrix}\forallx1\existsy1\ \forallx2\existsy2\end{pmatrix}\varphi(x1,x2,y1,y2).
It (in fact every formula with a Henkin prefix, not just the simplest one) is equivalent to its second-order Skolemization, i.e.
\existsf\existsg\forallx1\forallx2\varphi(x1,x2,f(x1),g(x2)).
It is also powerful enough to define the quantifier
Q\geqN
(Q\geqNx)\varphi(x)\equiv(\existsa)(QHx1,x2,y1,y2)[\varphi(a)\land(x1=x2\leftrightarrowy1=y2)\land(\varphi(x1) → (\varphi(y1)\landy1 ≠ a))].
Several things follow from this, including the nonaxiomatizability of first-order logic with
QH
| 1 | |
| \Sigma | |
| 1 |
The following quantifiers are also definable by
QH
(QLx)(\varphix,\psix)\equiv\operatorname{Card}(\{x\colon\varphix\})\leq\operatorname{Card}(\{x\colon\psix\})\equiv(QHx1x2y1y2)[(x1=x2\leftrightarrowy1=y2)\land(\varphix1 → \psiy1)]
(QIx)(\varphix,\psix)\equiv(QLx)(\varphix,\psix)\land(QLx)(\psix,\varphix)
(QCx)(\varphix)\equiv(QLx)(x=x,\varphix)
The Henkin quantifier
QH
Hintikka in a 1973 paper[6] advanced the hypothesis that some sentences in natural languages are best understood in terms of branching quantifiers, for example: "some relative of each villager and some relative of each townsman hate each other" is supposed to be interpreted, according to Hintikka, as:[7] [8]
\begin{pmatrix}\forallx1\existsy1\ \forallx2\existsy2\end{pmatrix}[(V(x1)\wedgeT(x2)) → (R(x1,y1)\wedgeR(x2,y2)\wedgeH(y1,y2)\wedgeH(y2,y1))].
which is known to have no first-order logic equivalent.[7]
The idea of branching is not necessarily restricted to using the classical quantifiers as leaves. In a 1979 paper,[9] Jon Barwise proposed variations of Hintikka sentences (as the above is sometimes called) in which the inner quantifiers are themselves generalized quantifiers, for example: "Most villagers and most townsmen hate each other."[7] Observing that
| 1 | |
| \Sigma | |
| 1 |
| 1 | |
| \Pi | |
| 1 |
Hintikka's proposal was met with skepticism by a number of logicians because some first-order sentences like the one below appear to capture well enough the natural language Hintikka sentence.
[\forallx1\existsy1\forallx2\existsy2\varphi(x1,x2,y1,y2)]\wedge[\forallx2\existsy2\forallx1\existsy1\varphi(x1,x2,y1,y2)]
where
\varphi(x1,x2,y1,y2)
denotes
(V(x1)\wedgeT(x2)) → (R(x1,y1)\wedgeR(x2,y2)\wedgeH(y1,y2)\wedgeH(y2,y1))
Although much purely theoretical debate followed, it wasn't until 2009 that some empirical tests with students trained in logic found that they are more likely to assign models matching the "bidirectional" first-order sentence rather than branching-quantifier sentence to several natural-language constructs derived from the Hintikka sentence. For instance students were shown undirected bipartite graphs - with squares and circles as vertices - and asked to say whether sentences like "more than 3 circles and more than 3 squares are connected by lines" were correctly describing the diagrams.[7]