Definition Smells

Signs that a definition probably needs work.

Uses the word "abstraction"
- Why is this a problem?
- See also AbstractionAddiction
- All words, without exception, are either abstractions or references (names). Perhaps you should redirect to OxygenAddiction?.
Uses other poorly defined words - It's sometimes said that a definition is as strong as its weakest word.
- I can see where this would be a problem - vagarity and such tends to accumulate unless used in certain manners. Of course, other words aren't 'poorly defined' simply because the reader 'poorly understands' them.
Too long
- Or "Too short", or "Too silly", or "Too serious"...
- When you can demonstrate that a definition is "Too long", it might be a meaningful 'smell'. At the moment, it is meaningless.
- See also KeepItSimpleStupid
- AsSimpleAsPossibleButNoSimpler is a fine principle, so long as your remember the last part.
- An excessively-long definition perhaps should be reclassified as a specification.
  - Why? There is certainly nothing in the denotation of either word that indicates that definitions are "short" and specifications are "long".
  - {Furthermore, define "long". Define "short".}
  - It may not be possible or useful to draw a hard line; it's continuous: the longer it is, the less useful it is to humanity. It should perhaps be split into multiple smaller definitions or renamed to "specification" or the like. Remember, definitions are tools. Tools can be good or bad, yet still be tools.
  - And how does changing it from a "definition" to "specification" help make it more useful if there are no changes to the actual definition?
  - If the number of trees grows large enough, then calling it a "forest" instead of a "grove" will be more useful to human readers.
  - I agree since both "grove" and "forest" include size in their definition. However, "definition" and "specification" don't, so they're not exactly analogous.
  - That would bring us to examine the definition of definition and specification. I doubt they are clear cut and overlap.
Only applies to a specific language
- A definition over a specific language, or at least initially that way (since languages can merge), can be quite precise and useful. Operational definitions in mathematical and linguistic fields will often be over a specific language.
- Perhaps we should distinquish between "working definition" and absolute definitions.
- Perhaps. But I'd reject your suggestion if you can't prove there is a meaningful difference. If operational definitions can become absolute merely by popularity and lack of competition, I don't reckon there is a meaningful difference.
- Isn't that Argument-by-Popularity? One has to be careful about relying on that, or at least state a context.
  - No. A definition isn't an argument, and all definitions are determined by popularity within a given group (which can be as small as one person).
Requires lots of math. This may make it a UselessTruth. More specifically, it may require buying into a mathematical model of the world or a piece of it. However, the existence of a model by itself does not by itself make it the only or best definition.
- Entirely wrong. A definition that requires a lot of math can be very precise and useful, especially when the definition is over a mathematical model. Further, it will never require you "buying into a mathematical model or a piece of it". That would only be true if someone is incapable of separating comprehension from belief.
- One can create God by defining a mathematical model for him/her/it. But, that does not necessarily make God exist. Religion only requires that a human-like being have access to control supernatural features. This can be a consistent model. But, that does not nec make it the RIGHT model. Consider The Matrix.
- Wrong. One cannot create God by defining a mathematical model. One can create a model, but models are not the reality so modeled. As to whether a model is a RIGHT model: models have a purpose in prediction. A sufficient condition for a model being a RIGHT model is that it provides for correct and falsifiable prediction. Because models are not reality, there is no such thing as "the" right model.
- In science we generally strive for the model with the best fit to reality. Whether there is One True Right model, we may never know. Whether it matters depends on whether we are trying to predict reality or explain it.
Not falsifiable or a tautology. That is, it should be possible to say yes or no whether a given thing fits the definition and at least some things should have a "no" answer.
- [For] any highly precise definition, a proposition that the definition applies in circumstance X should either be a tautology or a contradiction, depending on the X. E.g. "X is a list" should either be true by definition (a tautology) or false by definition (a contradiction). If this weren't the case, then the definition isn't the thing saying whether X is or is not an example thereof. THAT would be a fine example of 'poorly defined'.
Depends on subjective or difficult-to-verify concepts such as "intent".
- Wrong again. Intent and purpose are not subjective, nor particularly difficult-to-verify (no more so than "automobile" or "skateboard"). If you're complaining about concepts that can only be "verified" inductively, like "object" and "intent", then you shouldn't also complain about math, because only mathematical definitions can be verified deductively.
- Whether intent is "objective" or not is taken up in ObjectivityIsAnIllusion. Buried there, perhaps. It received more attention, I believe, in MostHolyWarsTiedToPsychology.
- And subjective definitions are not a problem either. Even "liking" something can be applied to good, objective use. When a definition is "subjective", then any propostion of the form "subject believes subjective over object" is objective.
- This is from the perspective of technical definitions, not really "street" definitions. For street definitions, subjectivity may be fine.
- I wasn't aware that definitions, technical or otherwise, possess 'perspective'. ;p Anyhow, if you mean "subjective" as provided by a dictionary - being wholly dependent upon the thoughts and interpretations and feelings of the thinking subject, then I'll agree that subjectivity is a 'smell' regarding technical definitions - it's rather difficult to formalize such things. OTOH, if you truly mean "subjective" as you use it in "ObjectivityIsAnIllusion", I'll feel need to reject your claim entirely.

I have a feeling that Top is making crap up again. He makes several claims above he only wishes had some validity. What... he does this about once a month? creates a new page of bullshit to direct people to when he wishes to defend his fallacy? Last time, IIRC, it was ObjectivityIsAnIllusion, and before that was TautologyMachine, I think. I should start keeping a list... or, on second thought, perhaps not. Tracking top's crap isn't exactly a good use of my time.

(Moved reply to ObjectivityIsAnIllusionContinued?)

Not falsifiable or a tautology. That is, it should be possible to say yes or no whether a given thing fits the definition and at least some things should have a "no" answer.
- Perhaps top means: "a proposition that the definition applies in a particular circumstance X ought to be falsifiable for at least some X". With this, I would agree. A definition is useless if it can't be used to 'finite' the scope of what you're talking about.

I'm adding this back in, because I'm going to disagree with both. It's not unusual to define something in order to show that it doesn't exist (E.g. "Largest Natural Number", "Ideal Voting System"). So even definitions that are "tautological" (as top would put it) are perfectly acceptable.

Yes, but "smell" generally implies that we are not seeking the oddball exceptions to the rule. Otherwise, the title would be "things definitions must have" or the like. I see no need to bring out such trivia, beyond a small footnote.
{I think you mean definitions that are "contradictory" are acceptable if only to prove they are contradictory. A definition D is only "tautological" if Forall X. D(X). And I agree with Top, here: definitions that refer to nothing or everything are a 'smell' in normal use.}
- So, what do you think of the definition for "Largest twin prime"? Does that definition smell? If yes, does it suddenly stop smelling if we prove that it exists? If no, does it start smelling if we prove that it doesn't?
- {"twin prime" is a fine definition, and is certainly not "tautological" (as top would put it). But "largest twin prime" is not a description you normally utilize, or that provides any real value other than trivia... indeed, until you know whether or not it exists, it's useless except in the sense of asking the question as to whether or not it exists. If we someday prove it exists, perhaps we'll find more use for it.}
- {Consider something simpler: Let's call an integer a crime if it is simultaneously composite and prime. Now, it is rather trivial to prove that there are no crime numbers. It's also rather trivial to prove, independently, that the property of being or not being a crime integer is objective. Finally, we know a few useful properties of crime numbers, such as being indivisible and possessing at least two factors. But does this definition 'smell'? I'd say yes. And we can even prove it smells. Since we can prove that there are no crime numbers, we can also prove that talking about their properties or otherwise attempting to utilize them is pointless and quite useless. And being useless is, itself, a 'smell'.}
- Now let's consider the hypothetical case that the non-existence of crime numbers has far reaching effects in mathematics. (I.e. we are often referring to their non-existence when proving other things). Chances are, we would be using that definition of crime.
- {Point. Though one could just as easily refer to the proof and continue on with words 'composite' and 'prime'. There wasn't any significant gain in defining a new word: crime numbers.}
- That's true enough. I think I'm ready to let this lie. Normally, I wouldn't have even raised this much fuss, except that I've encountered enough situations where it (requiring falsifiability) would be inconvenient to remove, and at least one person on this Wiki is using it as a RedHerring.

In the end, a definition only has to do two things,

Tell you what meets that definition.
Tell you what doesn't meet that definition.
(Note: This does not mean that determining whether a particular object meets/doesn't meet the definition needs to be computable, or even accessable)

Usually, though we also want it to

match some concept that we have in our head.
not be confused with what has already been defined.

By this, we can see that poorly defined words are usually a problem (but, not always. Set theory never defines what a set is. Euclidean geometry never defines what points, lines, and planes are. Etc.). Length would be a problem only if an equivalent shorter one was available. Requiring a specific language, math; not being falsifiable; or depending on abstraction/intent are definitely not problems.

I think everyone agrees that "points" are a UsefulLie. Poor definitions tend to be useless lies or a UselessTruth. A useful truth would be the ideal, but is probably a rare beast.

{Every set theory individually defines what a 'set' is, usually constructively through a finite collection of axioms. And Euclidean geometry defines points, lines, and planes in a similar manner - they gain existence and meaning based upon the axioms utilized to describe them. And Top is wrong: like all definitions, points aren't propositions and therefore cannot be "lies". That leaves them at just being "Useful". Top, I'll say it again: definitions are not propositions. Definitions, no matter how good or how poor, are neither lies NOR truths.}

Every set theory does not define what a 'set' is. Let's say you have two models for your set theory (and I don't know of any interesting set theories that don't have at least two), the axioms can't be used tell which one of the two are the "real" sets. If they are both sets, then you should be able to take a set from one model, a set from the other, and take their union. You can't. Likewise with 'points', 'lines', and 'planes' in Euclidian Geometry.

{Individual set theories are mathematical models already. There is little need to model the set theories. A single set theory IS a set of axioms, and the 'primitive' in a set theory tends to be 'element'; 'set' is defined 'implicitly and constructively by virtue of axioms over collections elements. If you discuss taking the union of sets from two different set theories, that can often be done (if there is a homomorphism from one set theory to another), but it doesn't change that each set-theory implicitly defines 'set'. And 'points', 'lines', and 'planes' in Euclidean Geometry are defined by reference-identity and the axioms that relate them.}

A model is something that allows you to map all the statements in a language to the truth values. Individual set theories are not models, and there is just as much need to model set theories as there is to model any other part of mathematics. To hopefully make it clearer, I'm not talking about finding the equivalent set in the first model of the set from the second, taking its union with the set from the first. I'm talking about directly taking the union of the set from the first model and the set from the second. The result of such a union is (most likely) not a member of either model.

{Can you reference a set-theory that is not a model? All those set theories I've a recollection of reading ARE mathematical models by virtue of listing their axioms and formation rules, which map the set-descriptions to legal/illegal (and thus, implicitly, define 'set'). These axioms generally provide a specific rule for unions in terms of the underlying elements.}

None of them are models. A theory (in this sense) is a subset of all the statements in a language. A model is a mapping from all the statements in a language to the truth values that satisfies a theory (maps to true for every statement in the theory and preserves the meanings of the logical connectors). Less formally, a theory is what is provable, while a model is what is true. Any "interesting" set theory will have models that disagree about the truth of some statements. For example, there are models of ZFC where the continuum hypothesis is true, and models where it isn't.

While there is some truth to the idea that 'set' is implicitly defined by set theory, I find it unsatisfying. It's kind of like defining 'red' as 'a color'. It's incomplete. We can get away with this incompleteness because nothing in set theory actually depends on what a 'set' is.

{On review, you are correct about the models... though, trivially, every theory comes with a free model called 'is-a-legal-sentence'.}

{The truth that 'set' is implicitly defined by a set theory is of import. In discussions I've often found people assuming that 'set theory' depends upon the definition of 'set' simply because it has 'set' in its name. Same with 'type theory' and 'type system' with 'type'. This assumption is often so deeply ingrained that they have difficulty grasping any other possibility. Recognizing that it is often the other way around, that a 'set theory' is essentially a theory that implicitly defines a 'set' and that a 'type system' implicitly defines 'type', and that there is no dependency requiring that 'set' or 'type' be defined first, can be a rather profound revelation. Thus, I emphasize it a bit here.}

{And as far as defining 'red' as 'a color' - that's not too far from complete. Physiologically, red is just 'a color'. At least in the sense of human vision, you could 'define base_color = red | green | blue', then define images as being patterned collections of colors based upon spatial triggering of the rods and cones in our eyes. All other colors are mixtures of the three.}