"Impredicative" definitions in mathematics

Question

In this blog post, the following definition of an "impredicative definition" is offered:

A definition is said to be impredicative if it defines an object E by means of a quantification over a domain of entities which includes E itself. An example: the standard definition of the infimum of a set X is impredicative. For we say that y = inf(X) if and only if y is a lower bound for X, and for any lower bound z of X, z ≤ y. And note that this definition quantifies over the lower bounds of X, one of which is the infimum itself (assuming there is one).

The point is made that some mathematicians think this type of definition is as bad as a circular definition. But how can that be? One just has to establish that there exists at least one lower bound for X to make this definition "good". Two things can happen: either a lower bound does not exist, and then there is no infimum because there is no lower bound, or there is at least one lower bound, and then the definition of infimum is operative, and doesn't seem circular to me. Is the concern that this definition hides a premise that one needs to establish the existence of at least one lower bound first? This would seem trivial to resolve, rather than rejecting the definition wholesale as "impredicative".

If that definition was changed to:

Given a set X, if (1) there exists lower bounds of X, (2) lower bounds can be ordered and (3) y is a lower bound for X such that (4) for any lower bound z of X, z ≤ y (there exists a y such that for all lower bounds z of X, z ≤ y) , then y is called inf(X)

would it sill be impredicative? The various existence requirements are clearly stipulated in the premises, and only if they are all met, can we call one of the lower bounds the "infimum". But up to the conclusion the existence of the "infimum" itself is not assumed, so there seems to be no circularity.

Answer 1

One aspect of the impredicativism, here, is the role of the concept of upper/lower bounds in the theory of real numbers. On the predicativist scheme of things, ℝ doesn't really exist as a completed set, or is a proper class (and then we have the amusing fusion of CH and the inflation of the Continuum to the size of a proper class, although predicativists would not be minded to describe the "situation" quite like that). It is not possible, then, by predicativist lights, to identify real numbers strictly with functions involving the relevant boundary phenomena, but the identification in terms of Cauchy sequences is preferable to the extent that a predicativism-friendly/adjacent account can be given in these terms, whereas the Dedekind-cut characterization is less palatable, here. (See also this essay about the subject.)

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ADDENDUM: the relative noncircularity of the OP's example

In retrospect, I can see a way to take the OP's example either predicatively without reluctance, or at least as somehow in a vague zone between rigorous predicativity and rigorous circularity. The reference to "a given set" can be thought of as "if we were given a set" such that it has an infimum, what are the intrinsic properties of infimae, by their definition, here? Or, then, when we cross the generic thought of some set with the generic description "has an infimum," what do we get?

No need, then, to assume an ante rem reason why we would know what it would be like if there was a set with an infimum. It is said that ante rem realism would allow for objective impredicativity, so a different take on questions about mathematical ontology would leave us with just the seemingly-circular-description method of identifying them. For if-then-ism instead, we need not make the transcendental assumptions of realists or intuitionists (or even semiotic, if not abstract, formalists), so in a sense, the appearance of impredicativity in the OP's example can be seen through, to the generic predicativity of the intended meaning. (Recall that intuitionists do not typically reject the explosion argument scheme in toto, though from what I've been reading, they do not accept this scheme on the basis of intuitionism's own subtheory of disjunction, but for some other reason.)

Answer 2

Using the term once in the sentence does not fundamentally eliminate the circularity, which is actually a necessary condition of the definition, but to argue that presumes some knowledge of the act of definition.

The question you ask after is one about the nature of definition, so I'm going to formulate a response in regards to your question within the framework provided by Robinson's Definition in one of two good philosophical works devoted to the philosophy of definition itself. Inherent in Robinson's definition are some metaphilosophical positions that are, in and of themselves, dispubtable, such as the notion that speech acts that are definitional imperatives necessarily preclude the object proposition from have truth conditional, and so on. Therefore, you can certainly challenge the following account, but what you are asking about is what is the sense we should make of self-reference in definition, and starting with Robinson's analysis is a good start to find an answer that satisfies you.

Robinson offers a taxonomy of definition such that he divides the act of definition into two taxa, one related to linguistic activities, and one related to conceptual activities. In his estimation, the latter is called a real definition, which he laments should be called a definition at all. From pg. 149:

The inventors of the notion of definition, Socrates and Plato, were obviously thinking only of the definition of things and not at all of the definition of words. The search for the definition of piety in Plato's Eurthypro is certainly an inquiry about the thing piety, not about the word 'piety'.

(NB the use-mention distinction that presumes the syntax-semantic division.) So, it is obvious, that if one dichotomizes in such a fashion, the the definition of infimum presented is not inherently an exercise in examining language, but of exploring in terms of necessity and sufficiency that which can be applied to a context to determine existential quantification. Thus, we have infimum as explanandum and the proposition as explanas. This should set off a red flag, because in an ordinary language conception, definiendum and definiens are the terminology used. Thus, we have an entire terminology of definition that minds the syntax-semantic distinction. This is important, because it draws a nuance necessary to give descriptivist insight into the purpose of the circularity inherent in impredicativity which seems to serve a pragmatic function of sorts despite it being logically tautological, an important Wittgensteinain theme on language use.

So what does Robinson mean when he uses the term real definition as opposed to word-thing definitions like lexical definitions (descriptivist acts of definition) and stipulative definitions (prescriptivist acts of definition)? Turns out, he argues there are no fewer than 12 distinctions in real definition that are occurring in positing a real definition. While it's beyond the scope of this post to flesh them out, the example you provided prima facie fits his stipulation of a speech act that is an analytical real definition. Onwards.

He describes definitions in this chapter as psychological speech acts, and not logical structures. To wit, he posits that there are real definitions which go to establish an abstraction exists (Real Definition as Abstraction, p.170) and then goes on to explain that the existential quantification of an abstraction then necessitates analysis and synthesis into a pre-established linguist framework. Thus, an analytical real definition takes a name of an abstraction, and then uses predicates to conjoin the name of the abstraction into the framework. Thus to use a set in its own definition is about taking a name of an abstraction, and then expressing criteria to clarify the language-game that goes into existential quantification. That's exactly what your example does. Thus, on the LHS of the analytical definition, we have the abstraction, but on the RHS of the definition, we have the abstraction in a model. (The mathematical logical characterization is mine, not his.)

So, a set defined in terms of itself needs to be tautological because it is a form of predication that is explicit about the relationships that are conjoined to the set itself. How could one predicate necessary conditions for existence without invoking the concept to begin with. And that's why it's explanandum and not definiendum. A analytical real definition explains a concept using semantic grounding, it doesn't offer a shallow syntactic substitution as in word-thing definitions.

A simpler way to explain it might be in the parlance of propositional calculus. A word-thing definition functions analogously to using logical equivalence to eliminate or reduce terms syntactically, where as using an analytical real definition would mean strictly introducing a predicated semantics which provides semantic grounding. In mathematics, this happens all of the time. We call it introducing rigor. So, the impredicativity of a definition of the infimum of a set in terms of the set itself occurs because prior to the introduction of the arithmetic constraint on a lower bound, there is an ambiguity of sorts that needs to be resolved. In fact, there's a name for this among us word mongerers, it's called a precising definition:

A precising definition is a definition that contracts or reduces the scope of the lexical definition of a term for a specific purpose by including additional criteria that narrow down the set of things meeting the definition

Thus, we have the infimum of a set naively and intuitively, and we have it rigorously using a metric space.

The moral of the story is that you are attempting to hide the circularity inherent in a precising definition by elliding over the term in your reformulation which works syntactically, but the circularity inherent is not a question of syntax, it is fundamentally bound to the semantics. So, I say no.

Answer 3

Attempt 2

Alright, Frank, I'll provide a second answer, since you didn't find my first answer persuasive so far.

You claim "some mathematicians" find the impredicativity in the definition of the infimum you provided to be objectionable as a form of circular definition and then provided an alternative form of definition as a proposed solution. In my previous answer and subsequent back and forth with you, I have counterasserted that the impredicativity is not only not objectionable, but necessary since the rigor added to the term 'infimum' is a precising definition that establishes through the addition of a metric criterion articulated on a well-ordered set of lower bounds. So...

The Example versus Your Definition

You are confusing the impredicativity in the definition with the labeling of the definition. The impredicativity inherent in your example extends not from the label or name or mention 'infimum', but from the self-reference in the definition (or better the explanation) of what an infimum is by invoking the set X on both sides of the logical biconditional. Your solution is simply to take the converse half of the biconditional and move the mention 'infimum' to the RHS from the LHS. Let's simplify to make the predicates and logical operators simpler to see:

Example as biconditional: y is the infimum of X <--> y is the greatest lower bound of X Frank's rewrite as converse: If y is the greatest lower bound of X, then y is the infimum of X.

Have you really said anything that wasn't said in the first statement? No, all of you have done is eliminated the conditional and kept the converse from the original biconditional. The example is 'P IFF Q' and you've just simplified to 'Q THEN P'. There is no substantial modification of semantics in either P or Q, is there? That's because the impredicativity in the example isn't a function of the syntax or the logical implication between predicates, but of the shared semantics of the predicates themselves. What is self-referential in the example is that the terms '(element) y' and '(set) X' occur in BOTH the first and second predicates of the proposition, so your attempt to eliminate circularity by moving 'infimum' from the LHS to the RHS doesn't remove impredicativity at all PRECISELY because the self-referential portions haven't been modified. To boot, one can write a logically equivalent statement of the example by swapping the location of everything before IFF with everything after. That is, 'P IFF Q' is logically equivalent to 'Q IFF P'. Let's do that with your rewrite:

Given a set X, if AND ONLY IF (1) there exists lower bounds of X, (2) lower bounds can be ordered and (3) y is a lower bound for X such that (4) for any lower bound z of X, z ≤ y, y is called inf(X)

See? By adding 'AND ONLY IF' in your definition and removing 'then', your definition goes back from being the converse to the biconditional form of the original itself (though you seemed to hedge your language by explicitly conducting existential quantification in (1) which is presumed in the example). Now, let's get to the heart of the impredicativity itself which has nothing to do either with explicit existential predication or the logical consequence between the predicates. The impredicativity stems from the the invocation of 'element y' and 'set X' in both the first and second predicates. That is where the self-reference occurs.

Semantic Compositionality and Synonymy

Let's review the two predicates in naturalish language and functional syntax:

P: y is the infimum of X
[infimum(y,X)]
Q: y is the greatest lower bound of X
[glb(y,X)]
Example: y is the infimum of X if and only if y is the greatest lower bound of X
[infimum(y,X) <-> glb(y,X)]
Your Rewrite: if y is the greatest lower bound of X then y is the infimum of X
[glb(y,X) -> infimum(y,X) ]

See the impredicatvity now? It's not the mention or location of 'infimum', it's the invocation of (y,X) as part of both predicates! This is the self-reference that some might object too. But, like I've said, this impredicativity is not a bug, it's a feature. It allows us to establish a psychological equivalence (or mathematical identity relation if you prefer) between mention 'infimum' and the conditions to establish the glb using the notation '≤'. And that is important in real analysis, because it adds the rigor of a metric space to an intuitive notion of greatest lower bound, which can feel uneasy because to talk about a greatest lower anything feels a bit paradoxical, doesn't it? Same exact process for supremum, and we now are clear on the abstractions infimum and supremum and can move on to monotonicity and other abstractions.

Now, as a brief extension to address one objection, we can finish up our exploration to gain an insight into why the impredicativity is a necessary part of the act of precising (as in precising definition) by noting that:

CLAIM 1 'infimum' is a synonym for 'lower bound'

does have a defense. There are two senses to synonym, one which demands an exact substitution and one that allows an approximate substitution. For instance, take a look at all the synonyms for infer (thesaurus.com). Any thesaurus entry will be the same. It will introduce semantically similar items as well as semantically identical items. In this case, you are right to object that it is not an identity. But similarity is itself used even in aritmetic as an approximative identity relation. For example consider another claim:

CLAIM 2 y₁≈y₂ and y₁=y₂ are identity relations

Is a wholly acceptable proposition as long as one accepts there are approximative and strict notions of identity. The mere fact that there are distinct symbols for each essentially confirm the acceptability of CLAIM 2. So, to claim that 'infimum' is a synonym for 'lower bound' is quite meaningful because it allows the claimant to initially indicate in natural language of the underlying semantic similarity, and then to specify additional criteria to move the approximate identity to one of strict identity using the addition of the well-orderedness and existence of a greatest member of the set of all lower bounds. That is the quintessence of rigorizing mathematics! In natural language:

EXAMPLE IN NATURAL LANGUAGE: An infimum is a lower bound that is the greatest of all lower bounds.

And viola! There we have what is going on in this highly technical example to begin with. We are saying that there exists this thing 'infimum' that is not only a lower bound, but also the greatest when compared with all the others.

Summary

So, does your definition serve as an adequate response to "those" who complain about circularity? No, because:

1. There is no problem from circularity in the first place (My entire real analysis textbook is filled with this type of analytical real definition (according to Robinson).
2. Your attempt to remove self-reference doesn't remove self-reference. It only weakens the logic relation (setting aside your strengthening of the definition by making explicit existential quantification).