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What do we call the property that if $a = b$, then $f(a) = f(b)$?

Wikipedia calls it "substitution property" but is that correct?

Xander Henderson
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Noob_Guy
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    Related (duplicate?): "Does 'Doing a thing to both sides of an equation' have a name?" ... While "Substitution property" seems appropriate, I just think of it as the "Treating-Equals-Equally-Yields-Equals Property". :) I'll note that the basic idea can be traced at least as far back as Euclid's "Common Notions" about addition and subtraction of equals. – Blue Oct 15 '22 at 09:47
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    See the introduction to Fitting's First-Order Logic and Automated Theorem Proving (1996) (pp. 9). Fitting writes "[Equality] obeys a substitutivity principle, one can 'substitute equals for equals.' That is, if we know that a = b, then the truth value of a statement is preserved if we replace some occurrences of a by occurrences of b. We will refer to this as the replacement property of equality. Further, in domains where it makes sense, one can add the same thing to both sides of an equality, and get another equality... – Jam Oct 15 '22 at 09:48
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    I'd call it ... ehr ... to be a function. I'm beginning to feeling dumb among these long comments. But my answer is still the same. – ajotatxe Oct 15 '22 at 09:52
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    I'd also like to add that it doesn't matter what you call a notion, provided you are clear and consistent. It may help to find commonly used terms, but provided you are able to express yourself, you aren't obligated to use exactly the same words as everyone else. Whether you call this a property of "substitution" or "replacement", it's the same thing and your reader understands that. Also, don't take Wikipedia as a guide for terminology. It's a mishmash of many different sources and (largely layman) editors. Just check its citations and you'll see what terms authors have used. – Jam Oct 15 '22 at 09:52
  • In Lean this is called congr_arg. – Ali Oct 15 '22 at 09:54
  • This is a special case of indiscernibility of identicals. – Zhen Lin Oct 15 '22 at 09:55
  • Duplicate https://math.stackexchange.com/questions/3189186 – Anne Bauval Oct 15 '22 at 10:08
  • Does this answer your question? Does "Doing a thing to both sides of an equation" have a name? – Jam Oct 15 '22 at 10:14
  • Another duplicate: https://math.stackexchange.com/questions/2882730/is-there-a-proof-that-performing-an-operation-on-both-sides-of-an-equation-prese – Daniel Schepler Oct 15 '22 at 17:53

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Well, I'd say, if you want to enter into philosophy, that two objects $x_1$ and $x_2$ can be distinguished only if there is some function $f$ that $f(x_1)\neq f(x_2)$. (For example, the 'subindex function', that would yield the $1$ and the $2$).

So the matter of your question, for me, is a pure axiom of logical reasoning.

ajotatxe
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  • Here is a source if anyone wants to explore the philosophical background behind this principle: SEP: The Identity of Indiscernibles. As an interesting sidenote, it can be considered to have first been formulated by Leibniz, one of the inventors of calculus. And calculus is very much a field where we would want to know if $x=y$ implies $f(x)=f(y)$ since, in a sense, we want the derivative to behave in a way that's the opposite! We want to see a discrepancy in a function's value at two near-coincident points: $f(x+h)=f(x) +hf'(x)$. – Jam Oct 15 '22 at 10:24
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this is the definition of a well defined function.

$(\forall x)(\forall y)(x=y\implies f(x)=f(y))$

A well defined function $f(x)$ returns a unique value for $x$.

ryaron
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