Now, however, let’s take Quine head on, and grant him a first-order regimentation. What ontological commitments does a sentence like ‘Everybody loves somebody’ carry? It is tempting to say that it is committed to lovers, but is it admissible to ignore the fact that the sentence really contains a relation, and not the predicate ‘is a lover’ ? It also leaves out the beloved.18
One might also consider a theory whose sole axiom says: ‘There is something that is taller than something’, or formalised: ‘∃x∃y T xy’. What is this theory committed to? Whatkinds of things is it committed to? The criterion is convincing enough when it comes to one-place predicates. ‘Something is red’ carries ontological commitment to red things, but it does not seem at all straightforward what happens when we are dealing with relations. Maybe we have to say that we are committed to that kind of things that are taller than some other thing, and things that have something taller than them?
To generalise the point, let us consider the following theories. T1, to start with,
has the following three axioms:
18(Resnik, 1988) makes this latter point, and also comes up with further complications. See also
A1.1 ∀xRxx
A1.2 ∀x∀y(Rxy ⊃Ryx)
A1.3 ∀x∀y∀z((Rxy∧Ryz)⊃Rxz)
What are the ontological commitments of T1? We would commonly say that T1
characterises ‘R’ as an equivalence relation. To say that the theory is committed to things that stand in an equivalence relation to each other, however, quantifies over relations: There is an equivalence relation in which these things stand to each other. This should not be acceptable for Quine. It is hard to see how the quantification over relations in this statement about the ontological commitment of T1could be
paraphrased away, short of repeating the axioms. If just repeating the axioms is fine, though, why not do so generally, and hence also in the case of higher-order sentences? It would then be difficult to see why we had to reformulate second-order theories in the language of set theory to expose the ontological commitment.
In order to emphasises the point, consider T2:
A2.1 ∀x∀y∀z((Rxy∧Ryz)⊃Rxz) A2.2 ∀x∃yRxy
A2.3 ∀x∀y∀z((Rxy∧Rxz)⊃Ryz)
It is very difficult to formulate the ontological commitment of this theory and avoid both of the following two “traps”: Giving an explication of the ontological commit- ment in terms of the relation that is characterised (“a relation that is transitive, serial, and euclidian”) – Quine would not want to say that this first-order theory is committed to the existence of relations19 – and not merely saying something like:
“The theory is committed to things such thatϕ”, where ϕis (more or less) a repe- tition of the axioms.
Might it be that the ontological commitment of polyadic theories is to kinds of tuples of things, rather than just kind of things as in the monadic case? A dyadic relation might just be construed as a predicate applying to an ordered pair, a triadic one as a predicate applying to triples, and so on. For Quine, this must be unac- ceptable as well. His paradigm for philosophical explication, we read in Word and Object, is the Wiener-Kuratowski definition of an ordered pair.20 ‘hx, yi’ is defined as ‘{ {x},{x, y} }’, a set of sets. It should not turn out to be the case, though, that not just higher-order logic, but polyadic first-order logic already commits a theory to sets. Quine’s conception of relation-symbols is that of many-place predicates. They are as neutral as the common, one-place predicates. These do not refer to properties or attributes, and those do not refer to relations. Both properties and relations are abstract universals that Quine rejects. This means, however, that it cannot be in a Quinean spirit to adjust the criterion such that the commitment is to, e.g., kinds of pairs. Two-place predicates are merely true or false of things they apply to, just like one-place predicates. No further ontological commitment arises than to the kind of things that are in the range of the quantifiers, those that have to exist in order to make such a sentence true. This, however, is precisely the problem: What kinds of things, exactly, have to exist, in order to make a sentence containing a two-place predicate true, turns out be difficult to figure out.
Note, that the difficulty does not arise, as it might seem at first glance, from the fact that no intuitive interpretation for ‘R’ is given in the two sample theories above. First of all, such an interpretation is not needed for monadic theories. A
theory containing ‘∃xF x’ as a theorem is committed to Fs – whatever they might be. It would be strange indeed if such an intuitive interpretation is required for the polyadic, but not the monadic case. Also, consider T3, with ‘M’ standing for the
dyadic relation ‘is married to’: A3.1 ∀x∀y(M xy ⊃M yx) A3.2 ∀x¬M xx
A3.3 ∃x∃y∃z(M xy∧M xz∧y6=z)
A3.4 ∀x∀y∀z((M xy∧M xz∧y6=z)⊃ ¬∃v(M zv∧v 6=x))
To say that the ontological commitment is to married people does not do justice to the case. People are not always married in the way described by the axioms. It seems, however, that something ontologically interesting is said, and that is that thiskind of people are married in a certain way, the way that is specified by [A3.1]– [A3.4]. This might pre-systematically be better expressed by saying that marriage is organised in a certain way, the T3-way: symmetric, irreflexive, polygamic, hare-
matic.21 It is hard to see a way to couch any of this in terms of Quine’s criterion of ontological commitment that would be acceptable to Quine, and still allow for a discrimination against higher-order languages.
T3, of course, is false of now-a-days western countries (at least if we take marriage
to be the legal institution), but might well be true of societies in other times and places. It does not matter, though, if it is not: The ontological commitment of it must not depend on the way the worldis. Rather, it tells us what the worldhas to be like, or more specifically, what kinds of things have to exist in order for it to be
21Not necessarily sexist, though. T
3leaves open that members of any gender can be married to
true. This last specification, however, is the one that creates the problems. Quine’s criterion for ontological commitment works best when we are dealing with one-place predicates only. This is hardly surprising, since the criterion is meant to tell us what kinds of things there have to be in order for the theory to be true. One-place predicates are the lexical items we use when we talk about kinds of things.
Could this not be just shrugged off? Why not bite the bullet and say that T3
has no ontological commitment? The axioms of the theory still say something that we can talk about. Such a strategy would render the best part of the project of a criterion of ontological commitment pointless. In order for the statements of T3 to
be true certain things have to exist; this much is clear. If a criterion of ontological commitment cannot deliver the answer what those things are, with what right is it called a criterion of ontological commitment? It becomes clearer how pressing the problem is when we consider T4 as an example:
Suc1 ∃!x∀y¬Sxy (zero)
0 =df ıx(∀y¬Sxy) (definition)
Suc2 ∀x∃!ySyx (functionality)
Suc3 ∀xyz((Szx∧Szy)⊃x=y) (uniqueness)
Sum1 ∀xy∃!zAxyz (functionality)
Sum2 ∀xAx0x (zero)
Sum3 ∀xyzv[(Szy∧Axzv)⊃ ∀w(Axyw⊃Svw)] (successor)
Prod1 ∀xy∃!zP xyz (functionality)
Prod2 ∀xP x00 (zero)
Prod3 ∀xyzv[(Szy∧P xyv)⊃ ∃!w∃!u(P xyw∧Awxu)] (successor) Ind [Φ0∧ ∀x(Φx⊃ ∃y(Sxy∧Φy))]⊃ ∀xΦx (induction)
T4 can obviously be recognised as a formulation of Peano Arithmetic without func-
tion symbols or one-place predicates. Note that the definition of zero is for the convenience of easier legibility only, and can of course be eliminated. In order to do without it, replace Sum2, Prod2, and Ind by the following:
Sum20 ∀xyz(¬Syz ⊃Axyx) Prod20 ∀xyz(¬Syz ⊃P xyy)
Ind0 [(∀xy(¬Sxy ⊃Φ(x))∧ ∀x(Φ(x)⊃ ∃y(Sxy∧Φ(y)))]⊃ ∀xΦ(x)
If one wanted to take the bullet-biting way out, and accept that theories that do not figure one-place predicated do not have ontological commitment,T4 would also
have to be considered void of ontological commitment. This should be a highly unwelcome result. It would mean that with T4 we could have arithmetic as an
ontological free lunch.