Abstract
Definitions are important tools for our attempt to construct an intelligible imagine of reality. Regarded as such, there are interesting epistemological clues about them to consider: What does a legitimate definition look like? Or: Is there a privileged form that definitions should have? How do we find good definitions? Or: How do we know that a given definition is a good one? Traces of the debate on these, and similar other questions are ubiquitous in the history of philosophy. The aim of this note, however, is not to give a historical account on this matter. Rather, on the basis of some very recent work of a proof–theoretical nature, I plan to address those issues in the context of a discussion concerning circular definitions. A definition is circular in the sense of this paper if the very concept that one is defining is used in the condition defining it. Despite their peculiar character, circular definitions are neither rare, nor easy to dispense with: it turns out that they affect in a significant way the ordinary life, as well as the philosophically interesting level of speech. That is the very reason why they have slowly gathered consideration from scholars in recent times, and have become matter of debate. Circular definitions break the traditional schema that can be used to give an account for ordinary definitions, therefore they raise the problem whether they are legitimate or should be avoided. The goal of this paper is to discuss the issue of circular definitions, illustrate the problematic features connected to it, and present recent developments in the logical research on the topic that help providing them with an arguably plausible justification.
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Notes
- 1.
On a similar basis, one can argue that to set up a term definition has different philosophical implications from carrying out a proof: the latter for Aristotle also comprises a declaration of existence of the “object” of it (see Charles 2010b, p. 141).
- 2.
For a rather exhaustive picture of the quite complicated issue of definitions in classical Greek philosophy, the reader is referred to the relevant chapters of Charles (2010c) that, as it might be clear already, I have used myself to obtain the crucial information I gathered for the sake of this introductory section.
- 3.
Plato’s “method of division”, based upon going through sequences of opposite features, for instance, is presented as marking the difference between the early dialogues of his, where Socrates’ attempts in solving quests end in aporia, and late dialogues where the solution of the problem originating the discussion is more frequent (see, for instance, Brown (2010) and Gill (2010)). Aristotle’s dialecthic is similarly seen as an attempt to examine the problem of predication (which is one of the critical issues of definitions, philosophically speaking) via questions about opposites (see Chiba 2010). In addition, it is suggested that the relationship between definitions and proofs in Aristotle might be a source useful to clarify in which cases one can claim that the essence of things has been isolated, i.e. good definitions have been set up (see Charles 2010a, §3, in particular).
- 4.
The reader interested to some more elaborate view in this respect, can be referred to the quite comprehensive taxonomy of types of definitions provided by Anil Gupta in (2015).
- 5.
Timo Beringer is responsible for having spotted and make this example of actual circularity available to me.
- 6.
As is well known, we can assume that \(\mathcal {L}^{-Q}\) has names for every element of |M| in all cases that are relevant to the present discussion without loss of generality.
- 7.
- 8.
This view has been supported, for instance, by Gupta and Belnap (1993).
- 9.
The key result related to the existence of fixpoints for an infinite sequence of sets is the theorem by Tarski and Knaster according to which every monotone sequence admits fixpoints. The result applies up to a certain extent to sequences of hypotheses determined by circular definitions in the sense of this paper, whenever the definiendum occurs only positively in the definiens (an occurrence of a predicate P in the logical formula φ(x, P) being positive if P occurs in the scope of an even number of negations).
- 10.
- 11.
To be more precise, provided that a model M for \(\mathcal {L}^{-P}\) is given, then one inductively defines (M, H i)⊧θ for every hypothesis H, i ∈ I index set, and θ from \(\mathcal {L}\), as the smallest relation that extends validity in M for sentences of \(\mathcal {L}^{-P}\), by interpreting occurrences of the form \(P(\overline {a})\), \(\neg P(\overline {a})\) according to H i (in particular, by putting: \((\mathbf {M},H_i)\models P(\overline {a})\) if and only if a ∈ H i, and \((\mathbf {M},H_i)\models \neg P(\overline {a})\) if and only if a∉H i).
- 12.
That is, i ≺I i + 1, and for every j ∈ I such that i ≺I j, then i + 1 ≼I j holds.
- 13.
I am using here the symbol ⊃ for material implication, saving the usual → for a new conditional connective to be discussed below.
- 14.
For a much more comprehensive investigation related to the goal of this paper and the specific content of this section, I refer the reader to Bruni (2017).
- 15.
In more precise terms, the idea of the extended validity relation explained in Footnote 11 gives rise to a proper semantics which one can view as stratified into layers. The notion of regular hypotheses from Sect. 7.4 is made relative to their period leading, for every \(n\in \mathbb {N}\), to the notion of n-regular hypothesis in a model M of the ground language (i.e., \(\mathcal {L}^{-P}:=\mathcal {L}\setminus \{P\}\)) that applies to any hypothesis H i (which is a subset of the domain of M) for which H i = H i+n is the case. This is used, in turn, to define the notion of n-validity for formulas of the full language \(\mathcal {L}\), as the property of any θ for which there exists \(m\in \mathbb {N}\) such that (M, H i+m)⊧θ holds for every n-regular hypothesis H i. The family of calculi \((\mbox{\textsf {C}}_n)_{n\in \mathbb {N}}\) from Gupta and Belnap (1993, ch. 5) is such that, for each \(n\in \mathbb {N}\), C n is sound and complete with respect to this notion of n-validity (hence, every theorem of C n is n-valid over any ground model M, and every n-valid formula is provable in C n). The special case for n = 0 of the n-validity relation applies to any formula of \(\mathcal {L}\) for which there exists \(m\in \mathbb {N}\) such that (M, H i+m)⊧θ holds for every hypothesis H i (since H i = H i+0 holds by definition).
- 16.
I am presenting here \(\mathcal {L}(I)\) as a single language, whereas there were actually a family of them, \((\mathcal {L}(I)_n)_{n\in \mathbb {Z}}\), in Bruni (2013). This was motivated then by seeing each language \(\mathcal {L}(I)_n\) as providing the corresponding theory HC n I speak of below with its own linguistic base. Looking back, this dependence of languages on the parameter \(n\in \mathbb {Z}\) was not really necessary, and it can be dropped as I do here for the sake of readability. I would like to thank an anonymous referee for giving me the opportunity of re-thinking this matter while I was trying to make up for a related comment she made on a previous draft of the paper.
- 17.
The acquainted reader would notice here that this is the same as adding the step conditionals as an “external feature” of the language \(\mathcal {L}^{sc}\) to formulas of \(\mathcal {L}^+\). In particular, →1 and ←1 do not occur in the definiens φ(x, P) of P. This is coherent with what was assumed in Bruni (2013) and enough to achieve the modest goal of this paper, but is less than what is needed for the broader aim pursued by Gupta and Standefer (see Gupta and Standefer 2017, p. 47 in particular).
- 18.
- 19.
For \(p,q\in \mathbb {Z}\), I use the standard notation |p − q| to use the modulus operation over integer numbers (hence, |p − q| = (p − q) if p ≥ q, |q − p| = (q − p) if q > p).
References
Asmus, C. (2013). Vagueness and revision sequences. Synthese, 190, 953–974.
Bolzano, B. (1996). Contribution to a better-grounded presentation of mathematics. In W. Ewald (Ed.), From Kant to Hilbert: A source book in the foundations of mathematics (pp. 174–224). Oxford: Clarendon Press.
Brown, L. (2010). Definition and division in Plato’s sophist. In D. Charles (Ed.), Definitions in Greek philosophy (pp. 151–171). Oxford/New York: Oxford University Press.
Bruni, R. (2013). Analytic calculi for circular concepts by finite revision. Studia Logica, 101, 915–932.
Bruni, R. (2017). Proof-theoretic analysis of step conditionals (In preparation).
Charles, D. (2010a). Definition and explanation in the posterior analytics and metaphysics. In D. Charles (Ed.), Definitions in Greek philosophy (pp. 286–328). Oxford/New York: Oxford University Press.
Charles, D. (2010b) The paradox in the meno and Aristotle’s attempts to resolve it. In D. Charles (Ed.), Definitions in Greek philosophy (pp. 115–150). Oxford/New York: Oxford University Press.
Charles, D. (Ed.). (2010c). Definitions in Greek philosophy. Oxford/New York: Oxford University Press.
Chiba, K. (2010). Aristotle on essence and defining-phrase in his dialectic. In D. Charles (Ed.), Definitions in Greek philosophy (pp. 172–199). Oxford/New York: Oxford University Press.
Feferman, S. (2010). The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968–2008. In R. Schindler (Ed.), Ways of proof theory (pp. 7–30). Frankfurt: Ontos Verlag.
Gill, M. L. (2010). Division and definition in Plato’s sophist and statestman. In D. Charles (Ed.), Definitions in Greek philosophy (pp. 172–199). Oxford/New York: Oxford University Press.
Gupta, A. (1997). Definition and revision: A response to McGee and Martin. Philosophical Issues, 8, 419–443.
Gupta, A. (2011). On circular concepts. In A. Gupta (Ed.), Truth, meaning and experience (pp. 95–134). Oxford/New York: Oxford University Press.
Gupta, A. (2015). Definitions. Stanford encyclopedia of philosophy, https://plato.stanford.edu/entries/definitions, revised on April 2015.
Gupta, A., & Belnap, N. (1993). The revision theory of truth. Cambridge: MIT Press.
Gupta, A., & Standefer, S. (2017). Conditionals in theories of truth. Journal of Philosophical Logic, 46, 27–63.
Martin, D. A. (1997). Revision and its rivals. Philosophical Issues, 8, 407–418.
McGee, V. (1997). Revision. Philosophical Issues, 8, 387–406.
Tarski, A. (1983). The concept of truth in formalized languages. In J. Corcoran (Ed.), Logic, semantics and metamathematics (2nd ed., pp. 152–278). Indianapolis: Hackett.
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Bruni, R. (2019). Addressing Circular Definitions via Systems of Proofs. In: Centrone, S., Negri, S., Sarikaya, D., Schuster, P.M. (eds) Mathesis Universalis, Computability and Proof. Synthese Library, vol 412. Springer, Cham. https://doi.org/10.1007/978-3-030-20447-1_7
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