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Quine's predicate functorese has been proposed as a "feature-placing" language for ontological nihilism (Strawson, Azzouni, Dasgupta, Diehl). This is often used to eliminate objects and individuals from ontology. Assuming that works, does predicate functorese still maintain the existence of primitive relations even without objects?

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TL; DR Not according to Quine, and not according to common alternatives. However, while existence of relations is ruled out by Quine, one can reconcile PFL (predicate functor logic) with both existence and non-existence of relations on alternative views.

Quine's criterion and PFL

Quine's criterion, "to be is to be a value of a quantifiable variable", applies specifically to classical FOL. PFL, like any other language, has to be paraphrased into FOL before its ontological commitments can be judged. As he stressed in Algebraic Logic and Predicate Functors:

"When a theory is given the usual quantificational form, the things that the theory accepts as existing are indeed the things that it accepts as the values of its variables of quantification. If a theory is given another form, moreover, there is no sense in asking what the theory accepts as existing except as we are in a position to say how to translate the theory into the usual quantificational form."

So, by Quine's lights, PFL is useless to nihilists. Since it is paraphrasable into the usual FOL with object variables it can not absolve them of commitment to objects. A good discussion of Quine's own reconciliation of his criterion with PFL is Collins's chapter Quine on Ontological Commitment in Light of Predicate-Functor Logic.

PFL without paraphrase

What if we keep the criterion but, pace Quine, reject the paraphrase part? That would still not entail commitment to relations. Unless one intends to quantify over predicates (relations), having named predicates in the language does not entail ontological commitment to them. So existence of relations does not follow.

However, it has to be said, such partial adoption of Quine's conception is a cheat. There is no quantification over anything in PFL, it is like propositional logic in this regard, but that is because an equivalent end is accomplished there by other means (the cropping function, a.k.a. the derelativization functor). We are starting to see the wisdom of Quine's paraphrase-first requirement. Without it one can escape, or impose, ontological commitments by mere technical trickery. But this raises the question of what makes classical FOL so special that it should be uniquely revealing as to theory's ontological commitments. According to Collins, Quine does not give a fully satisfactory answer to this question, nor is a principled one likely to emerge:

"As things stand, treating FOL as privileged depends upon dubious considerations. Indeed, if we adopt a generic criterion of ontological commitment in terms of predicate satisfiability, as Quine suggests, then there is not so much a symbolic criterion of ontological commitment, as simply a commitment to the idea of a universe of entities, whose role as truthmakers becomes perspicuous in FOL, but is strictly independent of FOL... Thus, we need not worry about nihilism or absolutism as being engendered by a choice of logical form, for serious ontological commitment is only dimly tracked, if at all, by a choice of logical form."

Fictionalism and nihilism

Of course, one can reject Quine's procedure in its entirety, and many do, but then they have to come up with an alternative way of determining what does or does not contribute to ontology. Otherwise, the question of what our language commits us to take as existing is empty. One radical response is that it commits us to nothing at all, everything can be talked about "as if". This is the position of fictionalists. If a nihilist wishes to adopt this position then, strictly speaking, she has no need for PFL, already the presence of objects in FOL is merely the presence of useful fictions. But PFL does better in a Wittgensteinian sense: it does not just say that objects are superfluous in a fictionalist caption, it shows it.

This said, nothing forces object nihilists to go full fictionalism. Absent Quine's criterion, one is free to advance non-linguistic (metaphysical, scientific or pragmatic) arguments in favor of picking what does and does not exist, and it does not have to be determined by linguistic form. Proponents of relational ontologies typically do not reify all relations, only those "grounded" in science, for example. Here is Collins's surmise:

"The speaker of predicate-functorese effectively talks in predicates. Imagine, however, that she was introduced to the concept of a subject. This concept offers her a choice. On the one hand, she may posit a univocal subject — the Absolute — and let the predicates just be modifiers of it. It is unclear to me if such would capture what Spinoza and various idealists thought, but no matter: the Absolute would make true our truths. On the other hand, she could go nihilist by way of appending a pleonastic subject to her predicates. On this view, every sentence would have the form familiar from ‘It’s raining’: It is such that F, in general. The idea of rendering all of our claims into feature-placement claims, such as weather reports, often goes by the name of ontological nihilism, and can be endorsed independently of the virtues of predicate-functor logic. My present claim, following Burgess, is not that ontological nihilism should be accepted, but simply that it is a position a speaker of predicate-functorese may adopt."

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  • “ Unless one intends to quantify over predicates (relations), having named predicates in the language does not entail ontological commitment to them. So existence of relations does not follow.” Suppose a different criteria for existence -“mind independence- were used. Under this criteria and not Quines, does the existence of relations follow from PFL? May 28, 2021 at 12:44
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    @GhostRocket For something to follow from PFL the criterion has to be linguistic, i.e. it has to confer existence based on syntax. Whether a name can be a value of a variable is such a criterion. Mind independence, on the other hand, is a matter of interpretation, not syntax. The syntax of PFL has nothing to say about mind independence, so such criterion cannot be applied to PFL at all. Users are free to interpret its predicate names as existing relations, or not, or interpret some as existing and others as not.
    – Conifold
    May 28, 2021 at 13:42
  • Thanks, that helps a lot. “ On the one hand, she may posit a univocal subject — the Absolute — and let the predicates just be modifiers of it. It is unclear to me if such would capture what Spinoza and various idealists thought, but no matter: the Absolute would make true our truths.” What does that mean? May 28, 2021 at 13:48
  • This is where I am getting conceptually stuck: “While its combinatory principles operate differently from those of familiar natural languages, this should not create any deep difficulty in learning it.“ Why are those combinatory rules not considered relations? May 28, 2021 at 14:54
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    @GhostRocket It means that PFL can be interpreted as describing relations of a single object, Absolute, to itself. It is redundant then to indicate this object explicitly, and a language with predicate symbols only is natural. But asserting Absolute's existence allows to interpret all sentences as true in the usual FOL way. "Combinatory principles" refer not to relations but to what replaces variables and quantification in PFL (cropping function), the principles of composing and parsing its expressions. And there is no learning difficulty because they are translatable into the familiar ones.
    – Conifold
    May 28, 2021 at 23:09
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In order for predicate functorese logic (PFL) to have the same expressiveness and utility of first order language (FOL) without ontic commitment of any objects as quantified bound variables suitable for ontological nihilists, the existence of primitive relations between objects have to be maintained in compositional predicates (functors) and in fact it's one the main ideas for Quine to invent this logic due to his famous ontological parsimony position. An example to preserve primitive relations can be referenced here:

consider the sentence ‘Every integer has a successor.’ Using Z for ‘integer’ and S for ‘successor,’ we will regiment this in FOL as ¬∃x(Zx ∧ ¬∃ySyx). Applying the conversion procedure, we get νρκZνρS, which, by our pronunciation guide... so our first example would correspond to ... ‘It is not (integering and not just successoring).’

So here objects (integers and successors) are replaced by primitive relational predicates integering and successoring. Another example for a single object without relation is the sentence ‘There's a cat’, and the nihilist would say ‘It is catting’ with PFL interpreting existence of object as some more abstract feature predicate as a distinctive relation to some other feature like dogging. Finally besides the above primitive relations preserving syntax, a rich semantics interpretation provided from a metalanguage is also important to accomplish nihilist's goal.

One difficult issue concerns how to give a nihilistically acceptable semantics for predicate functorese. Existing proposals for such a semantics will be unacceptable because they use a domain-theoretic—ultimately set-theoretic—model theory. To avoid this objection, a suitably enriched feature-placing language might provide a metalanguage for predicate functorese. But giving this is not the task for this paper... Finally, it would seem that PF has all the hallmarks of a speakable language: it has a clear compositional structure in accordance with a few simple rules. While its combinatory principles operate differently from those of familiar natural languages, this should not create any deep difficulty in learning it. Thus, the prospects for using PF as the nihilist’s fundamental language appear good.

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  • That would mean Azzouni’s claim that such a language totally eliminates al relations is false, correcr? May 28, 2021 at 11:00
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    @GhostRocket I'm not clear about the reference of above Azzouni's claim. Azzouni is a nominalist about abstract (math) object and argues that nothing would change if mathematical objects ceased to exist in a reference of Azzouni's student's paper here (philarchive.org/archive/MARNWS). He's a realist about physical objects while denying relations ontic commitment. Like math structualism trying to use category/functor theory as universal math foundation, PFL has similar spirit to treat predicate functorial relation as ontic instead of usual subject which turned to fictional "it" in PFL. May 28, 2021 at 20:27
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    @GhostRocket having said above, of course for a nihilist there's no ontic commitment about anything by definition, any (formal) language/logic can only express like a poet's metaphors simply for the convenience of communication and nothing is really committed. So after getting rid of the existential and other quantifiers/variables, predicate functors just syntactically better fit nihilists, relations are expressed more "ontic like" than objects which simply treated as an abstract imaginary "it", you can even use imaginary number "i" if you like. May 28, 2021 at 20:47
  • Whether Azzouni is truly a nominalist is a matter of debate (Otavio Bueno labled his position deflationary nominalism). But this much is clear: He denies the existence of both relations and individuals/objects, leaving only "feature". Azzouni's claim is the only thing left to describe the world is a feature-placing language using predicate functor logic. May 29, 2021 at 1:20
  • Is the ontic nihilist position considered common in modern philosophy? Are there any good reasons for rejecting it? May 29, 2021 at 1:21

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