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Quine's famous thesis about ontological commitment is roughly the following: there exist only entities which fall under the domain of quantification of our theory and that can be the values of variables of true sentences in a first order language.

For instance: "there is an x such that x is an elephant" commits ourselves to the existence of things to which the predicate "elephant" correctly applies.

I do not understand why the sentence shouldn't commit us to properties, too.

If we are dealing with first order logic, "there is an x such that x is an elephant" is true iff there is a thing in the domain that belongs to the subset of elephants.

So we should be committed to the subset "elephants", which we can identify as a property.

I do not understand why, whereas "there's an x that has elephant-hood" should commit us to a property, the paraphrase with "x is an elephant" avoids the problem.

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    Because you cannot quantify over predicates, which is what properties appear as in first order logic. Subsets are in Quine's ontology, but they are only poor man's surrogates for properties, not properties as such. Predicates with very different intensions may happen to pick out the same subset, creatures with kidneys and creatures with hearts in Quine's own example.
    – Conifold
    Apr 11 at 7:36
  • Thanks for the answer. It still does not convinces me that the existence of properity is dispenable for the correct paraphrase. When we say "There is a white dog" (Quine example) it is true that we are not quantifying over whitness or dogness, yet the truth condition for that statement is (in FOL) there there is an element of the domain that belongs to the set of whitness and the set of dogness. Committing ourselves to such a truth condition isn't commiting ourselves to properites?
    – PwNzDust
    Apr 11 at 7:41
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    No. There are no sets of whites or of dogs as such because there is no unrestricted comprehension in FOL. The corresponding predicates pick out white and dog subsets in pre-existing ambient sets, but only on a set by set basis. And the predicates themselves, which are the actual "truth conditions", are purely linguistic constructs in Quine's framework. In other words, they reflect our ways of grouping objects together based on pragmatic tools and purposes rather than on carving nature at the ontological joints. Quine is a nominalist about properties and relations.
    – Conifold
    Apr 12 at 4:37
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Quine was generally rather nominal than realistic towards the relative abstract within the context of a predicate as described here. A predicate is a sentence that contains a finite number of quantified variables range over non-logical objects and becomes a statement when specific values are substituted for the variables.

And of such entities Quine canvassed an influential mistrust: a mistrust based, initially, on their mere abstractness—though Quine himself later, under pressure of the apparent needs of science, overcame his phobia of the abstract—but also on the ground that they seem to lack clear criteria of identity—a clear basis on which they may be identified and distinguished among themselves. It was the latter consideration which first led Quine to propose that the range of the variables in higher-order logic might as well be taken to be sets—abstract identities no doubt, but ones with a clear criterion of identity given by the axiom of extensionality—and then eventually to slide into a view in which "second-order logic" became, in effect, a misnomer—unless, at any rate, one regards set theory as logic.

Followers of Hilbert have continued to quantify predicate letters, obtaining what they call higher-order predicate calculus. The values of these variables are in effect sets; and this way of presenting set theory gives it a deceptive resemblance to logic .... set theory's staggering existential assumptions are cunningly hidden now in the tacit shift from schematic predicate letters to quantifiable set variables.

So as you rightly concerned, later Quine allowed sets (including abstract properties such as whiteness, elephanthood, etc) as variables in a predicate sentence but mainly in higher-order logic. In strict and classic FOL, variables should remain as concrete objects to keep its "first order only" simplicity in a formal way.

A natural language sentence involving (abstract) properties as relative adjective such as "There is a white dog." cannot be strictly expressed in FOL simply as ∃d in D White(d) ∧ Dog(d), since predicate adjectives are not the exactly same kind of thing as second-order predicates such as color.

Btw, here's another source confirming Quine's anti-realistic view about his ontological commitment:

Ontological parsimony can be defined in various ways, and often is equated to versions of Occam's razor, a "rule of thumb, which obliges us to favor theories or hypotheses that make the fewest unwarranted, or ad hoc, assumptions about the data from which they are derived." Glock regards 'ontological parsimony' as one of the 'five main points' of Quine's conception of ontology.

FOL is a framework in which one can only quantify over elements of the domain of discourse. In second-order logic one is allowed to quantify over subsets of the domain of discourse. If you really want to substitute a set value ({dog A, dogB, dog C}) to a quantified variable in FOL with same nominal level (easy situations), modern plural quantification (per Boolos 1984 and Lewis 1991) is the theory that an individual variable x may take on plural, it can give FOL the power of set theory, but without any "ontological commitment" to such objects as sets in line with Quine's view (bear in mind ultimately logic wants to avoid set theory and be free of set related paradoxes).

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  • Thanks for the answer. If in FOL we can't express anything about properties, it semestre to me that truth condition for "there is an x that is F" should committ us toghe existence of a set (the F things) and to an element of the domani that belong to that set. This happening because our semantic theory specifie that that sentence is true iff there is an F element in the domani, and to be true that there is an F element is the domani is just for that element to belong to the F-subset
    – PwNzDust
    Apr 12 at 5:32
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    If u once read and accept Leiniz's Monadology, u'll comfortably accept an object (substance) is just a bundle of properties. Also from philosophy of language POV, it's semantically wrong to use one property F as a proper name of x in ur example. So I won't even write a sentence like x (relatively concrete) is F (relatively abstract property). "is" should be regarded as a proper name in any semantically correct sentence. U can of course say "there's x having F", then just translate to FOL as ∃x in X s.t. F(x). FOL just cannot express more complicated cases where F is only an adjective. Apr 12 at 5:52
  • Thanks, I think I had badly constructed the example. So, for instance, if a nominalist that accepts the FOL semantic analysis of declarative sentence were to utter: "There is at least one apple", he should be committed to the existence not only of a particular thing that is an apple, but also to the collection of all the apples that actually exist. Isn't that problematic for the nominalist?
    – PwNzDust
    Apr 12 at 6:00
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    If u really want to substitute a set value ({dog A, dogB, dog C}) to a quantified variable in FOL with same nominal level (easy situations), modern "plural quantification" is the theory that an individual variable x may take on plural (en.wikipedia.org/wiki/Plural_quantification), it can give FOL the power of set theory, but without any "ontological commitment" to such objects as sets (logic wants to get rid of set theory related paradoxes). The classic expositions are Boolos 1984 and Lewis 1991. Apr 12 at 19:20
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    I think u are right in your sense, u can treat truth value as an attached boolean value like in most computer languages it's a primitive type, not identical with any element belonging to any set as in computer languages we also have other (most likely advanced) type variable to reference those elements or sets. Any statement is true just "corresponds" to an existent element under the often invoked Correspondence theory of truth in philosophy... Apr 12 at 21:09

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