Abstract
In “Properties and the Interpretation of Second-Order Logic” (Hale, Philos Math 21:133–156, 2013) Bob Hale develops and defends a deflationary conception of properties where a property with particular satisfaction conditions actually (and in fact necessarily) exists if and only if it is possible that a predicate with those same satisfaction conditions exists. He argues further that, since our languages are finitary, there are at most countably infinitely many properties and, as a result, the account fails to underwrite the standard semantics for second-order logic. Here a more lenient version of the view is explored, which allows for the possibility of countably infinite predicates understood as the product of linguistic supertasks. This enriched deflationist account of properties—the Infinitary Deflationary Conception of Existence—supports the standard semantics for models with countable first-order domains, and allows one to prove the categoricity of the second-order Peano axioms.
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Notes
Hale equates his deflationary approach to properties with so-called abundant conceptions, but prefers the former terminology since it emphasizes the existence conditions for properties rather than their profusion. Further, as we shall see, Hale’s reading the deflationist conception (as well as the alternative development of these ideas explored here) is less than fully ‘abundant’, since it does not imply the existence of a property for each arbitrary collection of objects. Hale’s deflationism should be carefully distinguished from minimalist views such as Thomas Hofweber’s internalism (Hofweber 2006). Although both views closely link existence claims regarding properties to the possible or actual existence of a corresponding predicate, internalism rejects the thought that such existence claims actually commit us to the existence of properties.
Hale refers to his earlier (Hale 2010) for a full defense of the deflationary approach to ontology.
It is important to note that on the deflationary conception object existence, like property existence, is a matter of what expressions might exist, not a matter of what expressions (contingently) do exist!
The astute reader will note that there is an asymmetry to our formulation of the deflationary account of properties and the deflationary account of objects—the former involves (or is associated with) a total function, while the latter involves a partial function. This is as it should be, however. If a predicate possibly exists, then the corresponding property exists at every possible world (although the extension of the property might vary from world to world, and be empty at some worlds). The possible existence of a singular term does not guarantee the existence of a corresponding object at every world, however.
The additional condition that the possible singular term, had it existed, would figure in true atomic statements has disappeared from this way of formulating the idea. Since in what follows we shall only be concerned with the inference from the first bullet-point to the second—i.e., that necessarily, for any object that exists, there is a possible singular term that would denote that object had it (the singular term) existed—this simplification is harmless.
Hale actually argues for an intensional account of property individuation, where two distinct properties might hold of the same objects, and might even necessarily hold of the same objects (see Hale 2013, pp. 139–144). Since the present concern is determining whether, and under what conditions, the deflationary account might provide ‘enough’ properties to validate the second-order comprehension schema, or support the standard semantics for second-order logic (or both), however, I ignore this complication.
Of course, two distinct predicates might stand for the same property, and, given the intentional individuation of properties outlined in Hale (2013), more than one property might be co-extensive with the same predicate (and perhaps even necessarily so). The point is that a single predicate cannot stand for two non-coextensive properties.
Since the point of the present essay is to (correctly) extend Hale’s philosophical ideas, rather than criticize those points where he gets the mathematics wrong, I leave it to the interested reader to identify the exact point at which Hale’s reasoning goes awry. See below, however, for some additional observations.
For example, a model containing countably many objects, and whose properties correspond to the finite and co-finite subsets of those objects, would suffice.
This paragraph owes much to helpful comments from Stewart Shapiro and an anonymous referee.
After all, a single object language can be studied from the perspective of many different metalanguages, with vastly different expressive resources.
A supertask is a task that requires performing a countable infinity of distinct sub-tasks within a finite period of time. A hypertask is a task that requires performing an uncountable infinity of tasks within a finite period of time.
Hale generously credits the present author with bringing the issue of infinitary expression and supertasks to his attention in the same footnote that contains his objections to such an extension of the view. He notes that “the whole issue deserves much fuller investigation and discussion” (Hale 2013, p. 146, footnote 29). The present essay is, needless to say, an attempt to provide some initial investigation, and to prompt fuller discussion, in the manner called for.
Of course, such a more rigorous approach might have mathematical benefits, but that is another matter!
Thanks are due to an anonymous referee for emphasizing the importance of this clarification, and apologies are due to the same referee for failing to address this concern in exactly the manner he or she would have preferred.
Care needs to be taken here: If we expand the language by adding a function \(\mathcal {F}(x)\) that takes the Gödel code of a predicate as argument, and outputs the satisfaction condition for that predicate (and there seems to be no good reason to object to such a function), then we can diagonalize on:
$$\begin{aligned} (\exists X)(SC_{prop}(X, \mathcal {F}(y)) \wedge \lnot X(x)) \end{aligned}$$to obtain a predicate \(\Phi (y)\) such that:
$$\begin{aligned} (\forall z)[\Phi (z) \leftrightarrow (\exists X)(SC_{prop}(X, \mathcal {F}(\ulcorner \Phi (y) \urcorner )) \wedge \lnot X(z)) \end{aligned}$$In short, we can construct a Russell/Liar-like predicate \(\Phi (y)\) such that, for any object \(a\), \(\Phi (y)\) holds of \(a\) if and only if there is a property with the same satisfaction conditions as \(\Phi (y)\) yet which fails to hold of \(a\). But by definition, any two things with the same satisfaction conditions hold of the same things (in every possible world!). So, in particular, we have
$$\begin{aligned} (\forall X)(SC_{prop}(X, \mathcal {F}(\ulcorner \Phi (y) \urcorner )) \rightarrow (\forall z)(X(z) \leftrightarrow \Phi (x))) \end{aligned}$$Contradiction.
Of course, this is not all that surprising: satisfaction conditions (and our \(SC_{(\dots )}\) predicates holding of them) are semantic notions, and it is well-known that naïve formulation of the principles governing semantic notions can lead to paradox. Thus, a more careful and precise formulation of the principles governing satisfaction conditions would involve applying whatever prophylactic one prefers in the more well-known instances of this sort of phenomenon (i.e., plug in one’s favored solution to the Liar paradox here: my own preference can be found in Cook 2007a, 2009). Given the straightforward nature of the arguments given below, however (and, in particular, the fact that they involve no diagonalization or other paradox-prone constructions), we can safely ignore this problem for the time being. Thanks go to an anonymous referee for pointing out this contradiction, and the need to address it.
We also assume the \(n\)-ary analogues of CP for all \(n \in \mathbb {N}\). For example, the binary version can be written as:
$$\begin{aligned} \text {CP}^2: \;&\Box (\forall \Sigma ^2)(\forall X^2)(\forall \ulcorner \Phi (y_1, y_2) \urcorner )((SC_{prop}(X^2, \Sigma )\wedge SC_{pred}(\ulcorner \Phi (y_1, y_2) \urcorner , \Sigma ^2)) \\&\rightarrow \Box (\forall z_1)(\forall z_2)(X(z_1, z_2) \leftrightarrow \Phi (x_1, x_2))) \end{aligned}$$with superscripts indicating the arity of properties and satisfaction conditions (a notational detail we shall suppress when arity is clear from context).
Additionally, it is worth noting that CP is ‘ungrammatical’ as written, since it includes both ‘mention’ of the predicate in question (i.e. \(\ulcorner \Phi (x)\urcorner \)) and its ‘use’ in the final biconditional clause. This could be rectified by introducing a satisfaction predicate \(Sat(x, y)\) such that:
For any predicate \(\ulcorner \Phi (y) \urcorner \):
$$\begin{aligned} (\forall x)(Sat(\ulcorner \Phi (y) \urcorner , x) \leftrightarrow \Phi (x)) \end{aligned}$$and then formulating the Coextensionality Principle as:
$$\begin{aligned} \text {CP}: \;&\Box (\forall \Sigma )(\forall X)(\forall \ulcorner \Phi (y) \urcorner )((SC_{prop}(X, \Sigma )\wedge SC_{pred}(\ulcorner \Phi (y) \urcorner , \Sigma )) \\&\rightarrow \Box (\forall z)(X(z) \leftrightarrow Sat(\ulcorner \Phi (y) \urcorner , x))) \end{aligned}$$The simplification introduced in the ‘incorrect’ formulation is harmless in the present (somewhat informal) context, however, and avoids the introduction of paradox-prone notions such as \(Sat(x, y)\). See footnote 17 above for more discussion of paradoxes, and how to deal with them, within the present approach.
Note that we do not assume that, for every satisfaction condition, there is a predicate or property that has that satisfaction condition—that is, we do not assume (and do not need):
$$\begin{aligned}&\Box (\forall \Sigma )(\exists \Pi )(SC_{prop}(X, \Sigma )\\&\Box (\forall \Sigma )(\exists \ulcorner \Phi (y) \urcorner )(SC_{prop}(\ulcorner \Phi (y) \urcorner , \Sigma ) \end{aligned}$$The adoption of the Weak Object Principle, rather than some stronger principle that arguably captures the entirety of the deflationary conception of objects in the same sense as Hale’s Property Principle captures the entirety of the deflationary conception of properties, allows us to sidestep, in a technical sense, the need to deal with cases where terms fail to refer.
Note that we do not need separate existence principles analogous to SCEP\(_{prop}\) and SCEP\(_{pred}\) for denotation conditions, since WOP entails that, for any object, there exists a denotation condition ‘picking out’ that object.
Hale prefers talk of possibilities rather than worlds, where the former (unlike traditional treatments of the latter) need not be assumed to be fully determinate or maximal. But he notes (and I agree) that this point is orthogonal to the concerns of both his and the current essay—see (Hale (2013), p. 135, footnote 5).
Hale talks here of absolute necessity, rather than logical necessity, and defines absolute necessity in terms of the counterfactual conditional as follows:
where the universal quantifier ranges over all propositions unrestrictedly. Thus, Hale’s absolute necessity is not necessarily the same as the notion of logical necessity being mobilized here. But, although absolute necessity and logical necessity might not be everywhere equivalent, I see no reason to think that they will fail to agree when the case at issue is the possible existence of linguistic items, and every reason to think they will be ‘locally’ equivalent in this case. Also, the reader unconvinved by Hale’s brief argument can find longer, more technically complex arguments that S5 is the logic of logical necessity in Burgess (1999) and Halldén (1963).
See (Hale (2013), pp. 135–136) for the original argument.
The parenthetical qualification is important: the Finite Compossibility of Expressions Principle does not require that it is the same predicates or singular terms that exist at a single possible world, but merely a sequence of predicates and terms with pairwise the same satisfaction or denotation conditions.
Of course, I make no assumption that conceivability its a reliable—or even useful—guide to possibility. In fact, conceivable or not, I believe that the ‘situation’ ruled out by the Finitary Predicate Construction Principle is, in fact, logically impossible. Thus, the point of the Finitary Predicate Construction Principle is not to restrict attention to only a certain kind of possible world (i.e those that satisfy the principle), but rather to highlight a particularly important observation regarding all possible worlds.
Providing such necessary and sufficient conditions would require, amongst other things, that we determine the exact location of the boundary that separates logical from non-logical vocabulary, which is both an extremely difficult task, and one that seems somewhat outside the remit of the present essay.
Note that this we could modify this step of the proof to prove a stronger, modal formulation of the full comprehension schema, along the lines of Corollary 2.4 above. Details are left to the interested reader.
As noted at the end of the introductory section, I shall use infinitary expressions of this sort in an intuitive manner, although the arguments here and below can be straightforwardly adapted in standard infinitary logics such as those studied in Keisler (1971).
Although I have left the specification of \(Logic(\mathcal {F})\) rather informal in order to maximize the general applicability of the results below, one can easily precisify this notion by identifying the infinitary logical operations with those logical operations definable in some preferred infinitary language (e.g, \(\mathcal {L}_{\omega ,\omega }\)).
As usual, this proof can be modified to provide a stronger, modal formulation of the theorem. Details are left to the interested reader.
Of course, the Infinitary Deflationary Conception of Existence does seem to require an uncountably infinite class of possible worlds if hypertasks are disallowed, since we need enough worlds to ‘spread out’ the various supertasks required for the vast number of distinct predicates whose possibility is entailed by the view. I see no reason to think that this sort of proliferation is at all problematic, however. At any rate, the defender of the deflationary account of objects who also believes that the (standard, classical) real numbers exist—and Hale is such a defender, see Hale (2000)—is already in a similar position: if continuum-many reals exist, then by the deflationary conception of objects continuum-many distinct singular terms must be possible. Either they are jointly possible, and there is a single world where a linguistic hypertask has been completed, or they are not jointly possible, and are spread out amongst continuum-many distinct possible worlds.
Note that the observation that the second-order semantics supported by the Infinitary Deflationary Conception of Existence satisfies the infinitary comprehension schema does not require that the language we use when quantifying over the first- or second-order domains of models in that semantics include the infinitary resources required to formulate this infinitary form of comprehension. On the contrary, this is the surprising power of Theorem 3.6—that it shows that the mere possibility of infinitary logical constructions entails that our own world contains an abundance of properties that cannot be defined merely in terms of the finitary resources available locally (assuming, of course, that we are in fact contingently restricted to finitary expressions in the actual world).
Further, if all that mattered were the possible or actual existence of formal languages with appropriate predicates, then we could easily retain full semantics tout court merely by pointing out that nothing prevents infinitary formal languages with formulas of arbitrary infinite length.
Note that this point, if correct (and I am not actually completely convinced that it is) does not rule out the logical possibility of hypertasks in general, but only rules out the possibility of linguistic hypertasks!
Of course, this particular sort of instruction schema will only work for formulas where the symbols are ordered in a simple \(\omega \)-sequence. But any countably infinite sequence can be embedded in a finite interval of the reals, and hence can be given a similar, although in some cases more complicated, set of instructions.
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Acknowledgments
A huge thanks is due to Bob Hale, who tolerated many conversations and e-mails regarding this material (both before and after Hale 2013 was accepted for publication). He was, as always, extremely helpful and extremely patient, and provided many insightful suggestions that greatly improved the material above and its organization. Thanks are also due to Stewart Shapiro and Chris Menzel for helpful comments, and to a generous and insightful audience at the Between First- and Second-Order Logic workshop at UniLog 2013: The World Congress on Universal Logic in Rio de Janeiro, Brazil, where an early version of these ideas was presented.
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Cook, R.T. Possible predicates and actual properties. Synthese 196, 2555–2582 (2019). https://doi.org/10.1007/s11229-014-0592-1
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DOI: https://doi.org/10.1007/s11229-014-0592-1