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Bertrand Russell in Principles of mathematics (1903) presents the notion of such that as fundamental to logic and mathematics, and states that it is “undefinable”:

The Indefinables of Mathematics

Definition of pure mathematics

Pure Mathematics is the class of all propositions of the form “p implies q”, where (...) neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: implication, the relation of a term to a class of which it is a member, the notion of such that, (…)

Symbolic Logic

(…) we take as indefinables (...) the notion of such that. It is these three notions that characterize the class-calculus.

— Bertrand Russell, Principles of mathematics (1903)

As a natural language expression, such that is presumably well understood by all proficient speakers of the language, and Russell certainly was one. If x is such that Fx is true, then x has one or perhaps several possible values each of which makes Fx true. Russell himself says that "we may consider all the values of x which are such that ϕx is true. In general, these values form a class, and in fact a class may be defined as all the terms satisfying some propositional function." Seems clear enough to me.

I couldn't find any other place where Russell would have discussed this issue.

Given this, it seems to me that we have to assume that Russell meant something more restrictive than a simple dictionary definition, but then what did he mean exactly when he said that such that is "undefinable"?

Did any subsequent author address this issue, or perhaps provide a definition of such that in the way that Russell thought was impossible?

Thank you for any relevant scholarly references.

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  • Not used in modern logic. It is linked to predication (we may call it exemplification?) and thus the relation of an individual to a class. Jan 21 at 10:18
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    He meant it as what is now called primitive notion in a formal system, like lines and points in formal geometry that are also colloquially "understood". Primitives are not defined via other terms and their "meaning" is fixed "implicitly" by axioms instead. "Such that" morphed into the ℩ description operator in Principia, and later into Hilbert's ε-operator. The latter was still used by Bourbaki but then went out of fashion.
    – Conifold
    Jan 21 at 10:48
  • See para.23 (page 20): "The values of x which render a propositional function ϕx true are like the roots of an equation—indeed the latter are a particular case of the former—and we may consider all the values of x which are such that ϕx is true. In general, these values form a class, and in fact a class may be defined as all the terms satisfying some propositional function." Jan 22 at 7:04

3 Answers 3

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Well, basically Russell meant that the notion of "such that" is undefinable in terms of other logical or mathematical concepts. This means that it is impossible to provide a definition of "such that" using only the other concepts that are considered to be fundamental to logic and mathematics. He saw it a primitive concept that is used to define other logical and mathematical concepts. For example, the concept of a set can be defined as a collection of objects that satisfy a certain condition. The condition is expressed using the phrase "such that". For example, we might define the set of all even numbers as the collection of all integers x such that x is divisible by 2.

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  • Ok for the first paragraph of your answer, but the second? "This is because any definition of "such that" would have to use the concept of "such that" in order to be defined." Any definition?! Surely, you don't know that. I hope you don't mean that whatever Russell said was true. You would need to rephrase, perhaps that this is what he believed or had to assume after failing to find a proper definition. Jan 22 at 14:56
  • I agree, it's better to be removed at all
    – user68439
    Jan 22 at 18:38
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  1. Russell considers “implication” as a basic operator of two-valued logic, see Chapter I, section 14. All other logical operators can be defined with the help of “implication” and the logical constant “false”. E.g., “P and Q” can be defined as

    (P => (Q => false)) => false;

    see also Propositional calculus.

    Note that there is a certain choice which logical operator one takes as basic operator. Hence “implication” is not distinguished by this property.

  2. Moreover, Russell considers “such that” as the undefined basic construction of class formation: He requires the existence of the class of all objects which have a certain property, defined by a propositional function. Today this is named the axiom of class formation by comprehension, see Chap. I, section 23.

Note: That's the revised version of my answer - revised due to the comment of @PW_246.

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    Russell considers ‘such that’ to mean ‘implies’? That seems odd.
    – PW_246
    Jan 21 at 20:31
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    @PW_246 Thanks, I revised my answer.
    – Jo Wehler
    Jan 21 at 21:35
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Perhaps Russell is trying to steer clear of foundational paradoxes in set theory (such as the barber paradox) that would be involved if "such that" is interpreted naively as the set of objects satisfying a property (usually denoted by "|" or ":"), which would get us into paradoxes such as {x | x does not shave himself}. Naively interpreted, the clause "such that" may lead to paradoxes; but if it is declared to be a primitive notion, Russell is free to defend his piece by claiming that you misinterpreted his work by interpreting the clause "such that" too naively.

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  • The best way for Russell to preclude naïve interpretation would have been to provide the correct one rather than posit that there is none. Jan 22 at 14:49

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