Complexity and Hierarchy in Truth Predicates

Glanzberg, Michael

doi:10.1007/978-94-017-9673-6_10

Michael Glanzberg⁷

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 36))

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Abstract

In this paper, I speak in favor of hierarchies in the theory of truth. I argue that hierarchies are more well-motivated and can provide better and more workable theories than is often assumed. Along the way, I sketch the sort of hierarchy I believe is plausible and defensible. My defense of hierarchies assumes an ‘inflationary’ view of truth that sees truth as a substantial semantic concept. I argue that if one adopts this view of truth, hierarchies arise naturally. I also show that this approach to truth makes it a very complex concept. I argue that this complexity helps motivate hierarchies. Complexity and hierarchy go together, if you adopt the right view of truth.

Versions of this material were presented at the Seventh Barcelona Workshop on Issues in the Theory of Reference, University of Barcelona, June 2011, the Truth at Work Conference, Institut d’Histoire et de Philosophie des Sciences et des Techniques, Université Paris-1 and Ecole Normal Supérieure, June 2011, and Birkbeck College, University of London, February 2012. Thanks to all the participants at those events for very helpful discussions. I am especially grateful to Eduardo Barrio, Jc Beall, Alexi Burgess, Marcus Giaquinto, Øystein Linnebo, and two anonymous referees for comments and discussion of previous versions of this paper.

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Notes on the Computational Aspects of Kripke’s Theory of Truth

Article 05 November 2016

Paradoxes, Contradictions and Circularity

The Innocence of Truth in Semantic Paradox

Article 22 April 2019

Notes

1.
This way of thinking about the nature of truth and its role in solutions to the paradoxes comes from joint work with Jc Beall (e.g. Beall and Glanzberg 2008), though Beall himself prefers a very different set of options to the ones I endorse here.
2.
The transparency view is a descendent of the disquotational view of truth associated with Leeds (1978) and Quine (1970). Of course, there are a number of other versions of deflationism, which are importantly different, but this sample view will suffice for our purposes here.
3.
The usual provisos apply here, about sentences in contexts.
4.
This view of meaning is closely associated with Davidson (e.g. Davidson 1967), but it is also part of the long tradition in philosophy of language from Frege to Carnap to Montague and beyond. Of course, like all philosophical views, it is controversial, and conceptual role or inferentialist approaches to meaning deny it. Indeed, deflationism of the sort described by Field (1986) also denies it. Though Davidson endorses the close connection between truth and meaning, he holds a very different view of the place of reference in semantics, as we see in Davidson (1977, 1990).
5.
I have in mind the correspondence theory in something like the form it appeared in work of Russell (e.g. Russell 1912) and Moore (e.g. Moore 1953). The sort of theory I advocate here perhaps has more in common with the sort of correspondence discussed by David (1994).
6.
The terminology is somewhat unfortunate, as Tarski already appropriated the term ‘semantic’ for his semantic conception of truth. Alas, it is not easy to say just what Tarski had in mind by that. Depending on what he did have in mind, my use of the term may or may not overlap with his. In previous versions of this work, I used the term ‘complex view’ of truth, but that proves confusing when we come to discuss complexity results below.
7.
See the large literature in the philosophy of language on tacit knowledge, including such contributions as Chomsky (1980), Davies (1987), and Higginbotham (1989).
8.
Notions of reflection have appeared in the literature on truth, though often with relatively little discussion. For instance, Kripke’s famous remark about “some later stage in the development of natural language, one in which speakers reflect on the generation process leading to the minimal fixed point” (Kripke 1975 p. 80) seems to be gesturing towards the sort of reflection I have in mind. I have discussed this idea in my (2006), and in a somewhat different form in my (2004c).
9.
I will be moving back and forth between proof-theoretic and definability-theoretic perspectives. For proof theory, it will sometimes matter that \(\mathcal{L}\) is the language of arithmetic, though we will rarely get into enough technical detail to see this. Definability theory often prefers to work with purely relational structures and replace functions with relations; but again, we will not get into enough details to see this.
10.
When thinking about the semantic properties of a language like the ones we speak, we should probably focus on the intended interpretation, and so perhaps for \(\mathcal{L}\) we should be working with \(\mathbb{N}\) rather than an arbitrary model. Occasionally, we will need to know the model is reasonably nice, but for the most part, we will not be concerned with which structure it is. We should also note that the mathematical definition of truth is a mathematical representation of a concept with empirical applications (as Etchemendy (1988) and Soames (1984) reminded us).
11.
I follow the notational conventions of Halbach (2011). They are mostly standard. \({}^{\circ}\) is the evaluation function for terms, which is definable in PA. Recall that the language of PA has no predicates other than identity, and hence the form of the axioms below is specific to PA. Minor changes can accomodate other sorts of languages.
12.
In many cases, I shall talk about formal theories without going into full details of their expositions, but this case is central enough, and illustrative enough, that the details seem to be worth mentioning.
13.
There are some limits to what these sorts of models of reflection capture. For one thing, we may well learn about \(\mathcal{L}\) and PA more explicitly than we learn our natural languages. Neither approach fully addresses the question of how reference and satisfaction are fixed for a language. This is illustrated by the fact that CT does not rule out non-standard models. Generally, these are good theories of how truth works, but by no means complete theories of intentionality.
14.
For discussion of these sorts of results, see for instance Feferman (1991) or Halbach (2011). These results are proof-theoretically somewhat delicate; for instance, as is well-known, if we weaken the induction schema of CT we get back a conservative extension of PA.
15.
This result is quite general, and not really specific to arithmetic. Moschovakis (1974) proves a general version only assuming what he calls an ‘acceptable structure’. Some assumptions are needed; for instance, the result fails for recursively saturated structures (cf. Barwise 1975).
16.
This is what Horsten (2011) calls the power of the compositional theory of truth. As he notes, it is a surprising fact that we gain in arithmetic strength simply by adding semantic axioms.
17.
I have argued (Glanzberg 2002, 2004a, 2006) that in the kinds of cases at issue for the paradox, reflection invariably does change the topic.
18.
Which admittedly had precursors in the literature, such as van Fraassen (1968, 1970).
19.
The role of hypotheses in the process is highlighted by the revision theory of truth (Gupta and Belnap 1993).
20.
For extensive discussion of these and other theories, see Cantini (1996), Halbach (2011), and Horsten (2011). Feferman’s work was circulated well before publication, and was reported in part by McGee (1991) and Reinhardt (1986).
21.
Feferman (1991) also presents an alternative version of KF which employs a different, and stronger, way of treating schemas. The result is equivalent to ramified analysis up to Γ₀.
22.
Many find ω-inconsistency a reason to reject FS. For an interesting discussion, see Barrio (2006).
23.
This raises the question of whether the concept of truth is too complex to be grasped implicitly by all speakers, as the semantic view of truth requires. I do not have space to pursue this issue in depth, but let me quickly note that a great deal of work in cognitive science suggests we do have implicit grasp of complex concepts. A nice example is the concept of causation, which children have in some form starting as young as 6 months. The pressing question as I see it is not whether we can have implicit grasp of complex concepts, but how we can. The literature on perception of cause raises interesting questions about modularity and innateness of this concept. See, for instance Carey (2009) and Scholl and Tremoulet (2000).
24.
Or at least, almost. As I mentioned in Sect. 10.1, we would have a perfect theory of truth as applied to one language, which has been the main concern of work on the paradoxes. We still might like to understand better the way truth works across languages.
25.
I discussed this in fairly general semantic terms in Glanzberg (2001), in model-theoretic terms in Glanzberg (2004a), and in proof-theoretic terms, focusing on FS, in Glanzberg (2004c).
26.
See the papers in Beall (2008) for many different perspectives on revenge paradoxes.
27.
I have defended the contextualist aspects of my view in Glanzberg (2001, 2004a, 2006). As I mentioned, contextualism helps with the development of the particular sort of hierarchical view I prefer, but generally, the step from any hierarchical view to contextualism is very small. One need only accept that reflection (or whatever else generates the hierarchy) takes place in real time as we work with and reason about our concepts, and so takes place within contexts. Contexts thus serve to index the stages of reflection. This is the core of the view I have defended, and I believe underlies other contextualist views such as those of Parsons (1974).
28.
A theory close in spirit to mine, but using very different resources, is developed by Barwise and Etchemendy (1987). Iterated Kripke constructions are also discussed by Field (2008), and briefly in the older discussions of Burge (1979) and the postscript to Parsons (1974).
29.
There is something special about ‘stepping back’ and reflecting about the semantics of your own language that triggers hierarchies. I investigated some aspects of this in a number of papers (e.g. Glanzberg 2006). A very different view is presented in Gauker (2006).
30.
See McGee (1991) for a nice presentation of the details, and Halbach (1995) for an in-depth exploration.
31.
For those who find this example dated, John Dean was White House Counsel to Richard Nixon, and went on to testify against Nixon at the Senate Watergate Hearings.
32.
In the background here is the issue of absolute generality, as hierarchies one way or another deny the possibility of expressing absolute generality. The attitude I am taking here about generalizations is much the same as the one I took about absolute generality in Glanzberg (2004b, 2006). The other papers in Rayo and Uzquiano (2006) will give a good indication of the state of that debate.
33.
The careful discussions of hierarchies in Field (2008) and Halbach (2011) come to conclusions about these objections somewhat similar to mine, but disagree sharply on how well we can live with the problems we do find.
34.
The connection between ‘reflection’ and ‘reflection principle’ is probably no accident, and I am certainly exploiting it. This is the ‘uniform reflection principle’. For more on reflection principles, see Feferman (1962), and Kreisel and Lévy (1968).
35.
It is natural to ask if we can get a single complete theory by iterating this process to a suitable end-point. Results in this area are quite subtle, but substantially negative. See the papers of Feferman (1962), Feferman and Spector (1962), and Visser (1981).
36.
I discussed the connection with Hilbert’s program, and some specific issues about revenge paradoxes, at greater length in Glanzberg (2004c).
37.
Actually, I think a good story can be told here. Insofar as accounts like the iterative conception of set help to identify our concept of set, then we can observe how the multiple theories all express the iterative conception. Thus, I believe we can explain why we really have one concept.
38.
I thus depart from the position I took in Glanzberg (2004c). I do still hold most of what I said there; particularly, that the comparison with Kreiselian stratification helps to show why the hierarchical nature of truth is unobjectionable. But, in that paper, I was more optimistic about how close the analogy between the Kreiselian case and truth could be drawn.
39.
Again, there are formal results to back this up. A natural deflationist analog to CT simply uses the Tarski biconditionals rather than the CT axioms. This theory is a conservative extension of PA, as Halbach (2011) discusses.
40.
Though I do not have the space to pursue the matter here, let me mention briefly that I suspect this is why Field (2008) endorses a hierarchy of determinacy operators but not a hierarchy of truth predicates. There are some important issues here, but in very crude terms, if you adopt the semantic perspective, not only can you offer the defense of truth hierarchies I have, you will also find determinacy operators to look a lot like they try to capture a notion of truth. If you adopt Field’s own transparency view of truth, on the other hand, thy are clearly distinct conceptually, and the hierarchy seems unnatural and unmotivated for truth.

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Northwestern University, Chicago/Evanston, USA
Michael Glanzberg

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Philosophy, ILLC & AUC - University of Amsterdam, Amsterdam, The Netherlands
Theodora Achourioti
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Henri Galinon
Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Barcelona, Barcelona, Spain
José Martínez Fernández
Department of Mathematics and Department of Philosophy, University of Bristol, Bristol, United Kingdom
Kentaro Fujimoto

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Glanzberg, M. (2015). Complexity and Hierarchy in Truth Predicates. In: Achourioti, T., Galinon, H., Martínez Fernández, J., Fujimoto, K. (eds) Unifying the Philosophy of Truth. Logic, Epistemology, and the Unity of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9673-6_10

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