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I am trying to respond to a comment from a reviewer, where he/she suggests that there might be definitions which have a causal component.

Suppose I have the following totally made up definition (where x stands for some person): To be wise is to be kind and intelligent.

D1: ∀x[Property(x, wisdom) = Property(x, intelligence) ^ Property(x, kindness)]

Suppose then that I for some reason wanted to impose the additional restriction in the definiens that wisdom causes intelligence and kindness (again totally made up):

D2: ∀x[Property(x, wisdom) = Property(x, intelligence) ^ Property(x, kindness) ^ Causes(wisdom, intelligence) ^ Causes(wisdom, kindness)]

For me, definitions and causal relations seem like fundemantally different concepts, e.g. that definitions are identity relations, and that relata have to be distinct in causal relations. But, on the other hand, a definition might involve some causal assumptions, e.g. influenza is a disease with properties x, y, z, caused by a virus.

Are definitions of type D2 acceptable? Any help is greatly appreciated!

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    Note : if you are defining a "property of individuals", then you must have : for every x, Wise(x) = Kind(x) and Intelligent(x). – Mauro ALLEGRANZA Mar 25 at 13:28
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    Assuming that you are trying to define Wisdom, the second def uses the term to be defined on both sides : definiens and definiendum. – Mauro ALLEGRANZA Mar 25 at 13:31
  • @MauroALLEGRANZA Updated my question with quantifiers. Yes, I am curious whether there is an issue of circularity here or not. On one hand, as you suggest, wisdom, appears on both sides. On the other hand the truth value of Property(x, wisdom) might differ from the truth value of e.g. Causes(wisdom, intelligence). Am I misunderstanding something here? – JCR Mar 25 at 13:35
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    You can remove the circularity by replacing wisdom with a variable y and treating the whole formula as an implicit definition: wisdom is y such that... Of course, as always with implicit definitions, the existence question comes up. But that is as it should be, the definition assumes that there is such a thing that causes kindness and intelligence, and it may turn out that there is not. Perhaps the issue you are having with involving causality in definitions, it is no longer obvious that the definiendum is well-defined. That has to be argued. But the issue is not specific to causality. – Conifold Mar 25 at 17:52
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    It isn't circular, but the notation is odd, you are trying to make it explicit. What you have is an existential claim ∃y∀x(Property(x, intelligence) ^ Property(x, kindness) ^ Causes(y, intelligence) ^ Causes(y, kindness)). If it holds then "wisdom" is defined by instantiating y. If you want to be slick you can use Hilbert's epsilon operator: εyP(y) reads "some y such that P(y)". If the above formula is ∃yP(y) then you can write Wisdom = εyP(y). – Conifold Mar 25 at 20:25
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It does make sense, albeit in a more elaborate way. Definitions and relations, causal or otherwise, are not fundamentally different, in fact, relations are often the building blocks of definitions. However, the definitions involving them are more complicated than simple "A is B and C together" that involve only properties. The definitions involving relations are implicit, the claim that the relation holds is not guaranteed. We have to argue that an object/property satisfying all the specified relations exists. In general, for a binary realation we will have something like ∃y∀xR(x,y). If we can prove that this is true, even better, if we can prove that such a y is unique, then we can say that the y is defined by this implicitly. For example, if R(x,y) means "x divides y" then this implicitly defines the number 0, the only number that everything else divides. But if 0 is not available, say, if we restrict to positive integers, then it defines nothing at all. As mathematicians say, it is not well-defined.

With the OP example we have ∃y∀xR(x,y)=∃y∀xProperty(x, intelligence) ∧ Property(x, kindness) → Property(x, y) ∧ Causes(y, intelligence) ∧ Causes(y, kindness)). In words, "there is a property that everybody who is kind and intelligent has, which causes the kindness and the intelligence". If there is, indeed, a unique such property we can agree to call it "wisdom". A slick way to notate this is to use the Hilbert's epsilon operator ε: εyP(y) reads "some y such that P(y)". Now one can write Wisdom = εy∀xR(x,y), i.e. "wisdom is that property which everybody who is kind and intelligent has, and which causes the kindness and the intelligence".

  • Awesome! This solves my question. – JCR Mar 25 at 22:31
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Of course causal components belong in some definitions. Causatives are an important concept in linguisics. While good definitions can be hard to produce, there are many words which I think almost everyone would conclude have a causal component.

For example, ask any native English speaker to define "kill" and they'll almost certainly define it along the lines of "cause to die". Now it can't be reduced to just that; a better definition might be "to perform an action that is the primary cause of a death". But I think it's clear that "causing" is central to the meaning of "kill". It's not enough just to say that the lexical relation between "kill" and "die" is a causal relation because the meaning of one word is not reducible to that relation. Both "kill" and "murder" have a causal lexical relation to "die" but they have non-synonymous meanings, and each word's definition must therefore include the concept of "causality" in order to characterise the kind of causality it denotes.

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To respond to this query within the Spinozistic frame we need to move away from suppositions and synthetically constructed abstractions and discuss only objects/things in reality. In other words we must remove all hypothetical situations as in the statement, 'Suppose someone says, etc'. Spinoza insisted that the ONLY accurate definitions MUST include the cause of the item to be defined and must include the potential for all of its derivative properties to be deduced.

So, the definition of a circle should read; a line with one endpoint fixed and the other free. This definition, once you picture it in your mind's eye, includes the motion which allows for the movement of the free endpoint of the line forming endless numbers of circles. From this free movement we can also then derive the properties of a circle, like it's radius, circumference, etc.

Hope this is not too distracting or confusing. It will admittedly take some time to absorb.

Another example would be to define- gravity as the force describing mutual attraction among all bodies in the universe. Best of luck. Charles M. Saunders

  • Read more about Spinoza in 'Letters to No One in Particular' available on Amazon and free on Kindle – Charles M Saunders Mar 26 at 1:33

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