Candidate No.: Y3898820 Word Count: 2,500 University of York Department of Philosophy BA Second Year Summative Assessment, Spring 2021/22 Intermediate Logic (PHI00096I) Essay (2,500 words) Question 2b: Is second order quantification adequately characterised as plural quantification? Introduction Against the standard semantics of second-order logic (hereafter SOL), under whose reading quantification is understood to be predicative and to range over properties, Boolos (1984) proposed that second-order quantification (hereafter SOQ) be construed as a counterpart for plural quantification (hereafter PQ); i.e., instead of saying ‘there is a class/ property...’ (SOQ as standardly interpreted), one says ‘there are objects...’. This essay concerns whether the PQ characterisation of SOQ is adequate. I will illustrate that (i) not only is PQ just one way of interpreting SOQ and therefore inadequate, (ii) it can also be a problematic reading of SOQ. However, my chief aim is to show that SOQ and PQ are different forms of quantification, and therefore, it is nonsensical to ask whether SOQ is adequately characterised as PQ, when SOQ is not even appropriately characterised by PQ. I will consider the question from three aspects which mark irreconcilable differences between SOQ and PQ: (1) semantics, (2) ontological commitments and (3) reference. In the first section, I will look at how, semantically, PQ is obtained by adding formulas onto SOQ and therefore does something SOQ does not. Afterwards I show that PQ brings about additional problems of ontological commitment that SOQ does not have. Lastly, I consider whether SOQ and PQ range over and/or refer to different entities, and whether that is an issue. Some Caveats When Boolos suggested the PQ characterisation of SOQ, he was talking about the latter’s particular form, monadic SOL (MSOL), not the entirety of SOL. In fact, MSOL can be fully expressed in the language of plural first-order (PFO) with a straightforward translation scheme (Linnebo, 2003) Naturally, his interpretation picks out objects instead of classes or properties, because the quantification is done first-order. Hence I dedicate significant portions of the essay to the discussion of the extent to which MSOL/PFO adequately characterises SOL. Another point we must consider is that PQ can be construed nominally or non-nominally . This is a metaphysical concern, of whether there exist abstract entities by the likes of universals. To quantify nominally is to deny that there can be such things as collections and classes, and that only the objects that make up these apparent universals are ‘real’. This is crucial because it is pertinent to the question when we think about what PQ and SOQ are ontologically committed to, and stretches also to the dilemmas concerning reference. On Boolos’s view, since he is not committed to classes and accounts for quantification in terms of the quantifying of objects (and not the plurality consisting of them), he is staunchly a nominalist 1. Semantics Semantically, MSOL differs from standard SOL. To illustrate this, particularly important to our purposes is the relations of satisfaction in the two semantics. In the standard semantics of SOL, we simply have an extension of the semantics in first-order logic. An aspect of satisfaction in standard SOL is as follows (Shapiro, 1991a) : If X n is a relation variable and <t> n is a sequence of n terms, then M,s ⊨ X n <t> n if the sequence of members of d denoted by the members of <t> n is an element of s(X n ) 1 M is a model as defined by < d,I > where d is the domain and I is the interpretation, and s is an 1 assignment on M Effectively, ∃ X Φ X would be satisfied in the model even if Φ holds of no members, i.e. holding only of the empty class. This would then invokes the ontological entity of classes, which Boolos wished to avoid. To do this, Boolos read the comprehension scheme ∃ X ∀ x(Xx ≡Φ (x)) in the following way (Shapiro, 1991b) : ‘either ¬ ∃ X Φ (x) or else there are some objects such that any object is one of them just in case Φ holds of it.’ Presented informally, this means that the relation of satisfaction would only hold when Φ is meaningful and denotes object(s). Thus we can say that the semantics has been altered to preclude the commitment to classes. Further, using MSOL, Boolos defines this relation to hold: M,s,R ⊨ Φ , where s is an assignment to only first-order variables and R is a second-order predicate in the metalanguage, which codes assignments to the second-order variables iff a second-order variable consists of the term 2 He goes on to use second-order quantifiers to be understood as plural quantifiers: M,s,R ⊨ ∃ V Φ iff ∃ X ∃ T( ∀ x(Xx ≡ Tp(V,x) ∧ ∀ u((u is a second-order variable other than V) → ∀ x(Tp(u,x) ≡ Rp(u,x))) ∧ M,s,T ⊨ Φ ). Talking in terms of classes, take p to be a pairing, the item p( U , u ) is in R iff u is in the class to be 2 assigned to U This is noticeable in the first quantifier in the formula. This is an additional clause to the standard semantics of SOL—so we can see that PQ needs to add a this new formula to standard SOL in order to quantify over plural objects. 2. Ontological Commitments As discussed in the previous sections, Boolos’s discovery was that MSOL/PFO can be characterised as PQ. But they exclude quantification which involve polyadic relationality, meaning great detriment to the applicability of the prima facie novel finding. As a result, there have been attempts to codify polyadic SOQ monadically (Rayo, Yablo, 2001) , so as to rectify this. Consider the following quantification: ∃ R (Rab ∧∀ x ∃ y ¬Rxy) Reading something like ‘there is a relation such that it obtains between a and b , and for all x there is some y that is not related in such a way to x .’ Notice that this cannot be expressed in Boolos’s preferred formulation ‘there are some... such that...’, unless we bring in ordered pairs (Rayo, Yablo, 2001) , interpreted as follows: ‘There are some pairs such that <a,b> is one of them, and for all x there is a y such that <x,y> is not one of them.’ Now this solution not goes against what Boolos cited as a merit of his discovery, namely its ontological economy (since on his reading PQ would not be committed to classes, unlike standardly-interpreted SOQ), but also exacerbates the issue: we are now committed to ordered pairs, a specified type of set, in addition to classes. Why classes, the ontological entity Boolos sought to avoid? Boolos says we need a way of accounting for the fact that it is possible to prove some results that hold no matter what set-theoretic predicate is used to replace some schematic predicate letter ( Linnebo, 2003) ; and to do so, we require a way to quantify into predicate positions. According to Parsons (1974) , while sets partially answer to the need to generalise on predicate positions, one still needs to postulate classes in order to state generalisations that cover all predicates in set theory, such as ones with a fixed interpretation (under unrestricted comprehension). Linnebo (2003) terms this the problem of Ontological Proliferation —that in addition to sets, another set-like ontological entity, i.e. classes, have to be postulated. While SOQ commits one to classes, an adequately universal PQ has to commit to ordered pairs as well. On the traditional interpretation, this means that PQ ranges over more ontological entities than SOQ. But is this really an issue? Prior (1971) disagrees, writing that we need not be ontologically committed to what something stands for, if we are non-nominalist about predicates. For example, one might say: He is something I am not—kind. We may treat ‘kind’ to be adjectival and non-nominal in force, just like ‘something’—predicates can be construed as ontologically non-committal, if we are not objectualist: here we are not saying that ‘he exemplifies kindness’, in which case one would be committed to the existence of the property ‘kindness’. This objection runs into a problem: it only works when we are non-nominalist. But, coming back to our subject matter, PQ, when it is combined with non-nominals, it disentangles the singular-plural and first-/second-order distinctions (Rayo, Yablo, 2001) which Boolos obfuscates. Consider: ∃ R i — somehow i things relate such that... R i x j y k — they j are so i related to them k It is clear that they are in SOL plural; grammatically, they are taking first-order plurals as arguments.What this de-nominalisation clarifies is that we must not conflate SOL with PQ. SOQ, as we can see, is clearly not PQ. So Boolos could not have adopted the objection along the lines of Prior’s, as it would lead him to this fatal consequence. 3. Reference Last but not least, let us consider the problems of reference if we were to characterise SOQ as PQ. I shall start by discussing the semantic difference between SOQ and PQ. It is widely accepted, along Fregean lines, that predicates refer to properties, not to objects. Whereas SOQ ranges over predicates and thus do not refer to objects, PQ ranges precisely over objects. We cannot simply translate the predicate to a term or a term to a predicate either, as the sense of the sentence would then be shifted (Trueman, 2021) Now nominalists might object to this by saying that predicates don’t refer—for them, satisfaction does not equate to reference (Trueman, 2021) ; i.e., when we say x is a Y , we are merely saying that ∀ Y(y satisfies x ↔ Yy) , but here there is no talk of reference. A notable proponent of this position is Quine, who suggested that there are no properties for SOL to quantify over—SOL merely quantifies over first-order objects, as we see in Boolos’s interpretation. This position is strengthened by the concept horse paradox : if properties are not in fact just objects, then no term could refer to a property, e.g. when we say something like ‘the property horse ’. But this need not be a fatal blow to the Fregean position which maintains that there is a distinction between predicates and terms (in turn also SOQ and PQ). Could we not simply rephrase the sentence ‘The property horse is a property’, into the following: ∃ Y ∀ x(x is a horse ↔ Yx) This seems to be a good translation, but only if all talk of properties can be translated smoothly in such a way. Trueman (2021) shows that this is not always the case, citing the sentence ‘No property is an object’ as an example, which leads to a paradoxical conundrum, which is unfortunately beyond the scope of the essay, for our present purposes. Could the two positions be reconciled? One might gain some insight into the matter by looking at Wright’s (2007) neutralist account of quantification —he writes that quantification by itself does not commit us to anything ontologically; he rejects Quine’s famous slogan that ‘to be is to be the value of a variable’ , which was ‘encouraged by a preoccupation with first-order quantification in a 3 language all of whose names refer’. To rectify this error, he proposes the Neutrality Principle : Quine’s idea that the ontological commitments of a theory are made fully manifest by the range 3 the kinds of quantification that a minimally adequate formulation of the theory requires ‘Quantification into the position occupied by a particular type of syntactic constituent in a statement of a particular form cannot generate ontological commitment to a kind of item not already semantically associated with the occurrence of that type of constituent in a true statement of that form.’ What this says is that we are not ontologically committed to anything further than what the truth conditions about the instances of quantification require. As an example, we can refer back to the first section in the essay: In standard SOL semantics, the relation of satisfaction, I recapitulate, is as follows: If X n is a relation variable and <t> n is a sequence of n terms, then M,s ⊨ X n <t> n if the sequence of members of d denoted by the members of <t> n is an element of s(X n ) And this implies that Φ would be a property/class, even if no object in the model in question satisfies it—we always have the empty class to satisfy it. This seems, then, to show that SOL commits us to classes. But on the neutralist interpretation, this needs not be the case. If Φ does not hold of any members in the model, i.e. it is not instantiated, it would not be true to say that ‘something is Φ ’, and therefore we are not committed to Φ —by extension classes, because they must have instantiations in order for statements about them to have a truth value. In other words, quantifiers are merely devices ‘for generating statements whose truth-conditions will correspond in ways specific to the kind of quantifier involved to those of the statements quantified into—the instances’ (Wright, 2007) Applying this to SOL, we can say that it does not differ from first-order logic in terms of the ontological commitments we have to make in quantificational statements: all SOQ is committed to is already present in its first-order predecessor. Perhaps we can take this approach and make this about reference instead of ontological commitments. We can say that predicates refer only when instantiated—when there are objects which satisfy its condition(s). While this sounds like it would be in the nominalist’s favour, it could also be interpreted otherwise: that neither terms nor predicates could refer on their own—in which case, it deprives objects of their special status (their ‘reality’) conferred by nominalists, and therefore, to the non-nominalist’s advantage, predicates and terms are at least on par with each other in terms of ontological and referential precedence. This actually bears a striking resemblance to the Fregean Context Principle; evidently, the neutralist principle is not strictly an interpretation in the nominalist’s favour. Conclusion Here I sum up what the considerations in the three sections came to: 1. Semantically, PQ, which uses MSOL, requires an additional formula to be affixed in order to be derived from standard SOL; PQ is derived from SOQ, not vice versa, so we cannot say that second order quantification can be characterised as plural quantification. 2. Ontologically, contrary to Boolos’s suggestion, PQ commits us to more and stricter things than SOQ if we wanted to apply MSOL to SOL—ordered pairs, in addition to classes. Attempts to salvage this run into more problems than we begun with. Attempting to transcribe the quantification MSOL into SOL results in a horrendous ontological proliferation, a result Boolos certainly did not want; we must then accept that SOQ is not to be characterised by PQ. 3. Referentially, we ran into a stalemate between nominalist and non-nominalist positions: for the former, predicates merely quantify over first-order terms, and therefore PQ adequately characterises SOQ; the latter disagrees and insists on a distinction between terms and predicates, the implication of which is that PQ must also be separated from SOQ—they are different forms of quantification. I thereby conclude that SOQ cannot be adequately characterised as PQ, because: I. PQ does something that standard SOQ does not do (point 1 above). II. The language of PQ, MSOL, cannot be made into pure SOL (point 2 above). III. I have shown in sections 1 and 2 that it seems unviable that predicates quantify over first-order terms. This means that the nominalist’s position in point 3 cannot be defended even if they can give a satisfactory account of reference—it is not enough. In addition, SOQ and PQ are different forms of quantification—for this reason, it is nonsensical to ask whether the former is adequately characterised by the latter. Bibliography Boolos, George. “To Be Is to Be a Value of a Variable (or to Be Some Values of Some Variables).” The Journal of Philosophy , vol. 81, no. 8, Journal of Philosophy, Inc., 1984, pp. 430–49, https://doi.org/10.2307/2026308. Linnebo, Øystein. “Plural Quantification Exposed.” Noûs , vol. 37, no. 1, Wiley, 2003, pp. 71–92, http://www.jstor.org/stable/3506205. Parsons, Charles. “Sets and Classes.” Noûs , vol. 8, no. 1, Wiley, 1974, pp. 1–12, https://doi.org/ 10.2307/2214641. Prior, A. 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