How do we say that two logically equivalent definitions are not semantically equivalent?

This is a question of terminology (and also relates to philosophy).

Very often we have a certain term, such as "differentiable function", and then we have a definition:

(1) def. A differentiable function is a function such that ...

Then we have a theorem stating

(2) theorem. A function is differentiable iff ...

If this theorem has a correct proof, then we could also have made (2) into a definition, and state that (1) and (2) are "equivalent" definitions.

However, we cannot simply assume that they are equivalent. This has to be proven.

Now my question is one of terminology: Is there an accepted simple term or phrase that captures the fact that "these two definitions are logically equivalent, but they are not semantically the same definition so that their logical equivalence requires a proof"?

For example, I'm thinking of something like "these two semantically independent definitions are equivalent".

A comparison with analytical philosophy might clarify: Frege's concept of "sense" and "reference": it may be that the Evening star and the Morning star are in fact the same particular planet Venus (they have the same reference), but as they are perceived by us, they have different identities (they have different senses). It then requires (physical) proof that they have the same reference. Similarly, our definitions may have the same reference (they refer to the same subset of functions, i.e. are equivalent), but they do not have the same sense (the definitions are different in terms of the meaning they convey when we read them). Hence it requires (mathematical) proof that the two definitions with different 'senses' (meanings), in fact are logically equivalent (have the same 'reference').

• I don't think it's correct to call them both a definition. One is the definition and one is just a statement. Of course the property used in the definition is itself a statement about the definiendum. Second the morning star is not a giant ball of burning hydrogen, it's the planet Venus (usually, the term is occationally used for other planets as well or even the star Sirius). – skyking Apr 11 '17 at 5:09
• Thanks for the correction on Venus. On your first point: I do often come across textbooks that speak of "alternative definitions" for the same thing. of course if we choose 1 definition, the others become theorems, but we could have chosen one of the theorems as definitions instead. In any case, I think this is not critical to the core of my question. – user56834 Apr 11 '17 at 5:25
• Imo if two definitions are equivalent, all that says to me is that we have a choice as to which we will take as our definition, and which will be a statement about some object. I don't know about a word for this situation though – Andres Mejia Apr 11 '17 at 5:31
• Relevant : Beth definability. – Mauro ALLEGRANZA Apr 11 '17 at 5:59
• @Programmer2134 Thanks for explaining. So, I am using semantic in the precise sense that it is used in logic, not in a philosophical way, and I am pointing out that you're mis-using semantic in that setting. In philosophy, sense and reference may both be semantic concepts (at least, if one believes that different English phrases may analytically have the same sense; I'm doubtful that such analytic truths exist at all and so is Quine). But in logic, "semantic" refers to the semantics of the formulas, and the two formulas have the same semantics. – 6005 Apr 11 '17 at 6:08

You could have said that they are equivalent definition, but that may not as strict as only one of them would be a definition and the other being a theorem.

If we want to be strict the most concise way is to decompose the definition and theorem first. They are of the form:

$$\phi\overset{def}\equiv\psi_{def}$$ $$\phi\Leftrightarrow\psi_{thm}$$

what one would do is to start with a theorem that says that $\psi_{def}\equiv\psi_{thm}$, if needed there can be a lot of statements that are equivalent. At this point we are only talking about equivalent statements. After this we introduce the definition (which can use any chosen statement).

What I've seen is that the parts of a definition as above has it's separate name. One call $\phi$ the definiendum and $\phi_{def}$ the differentiae. In similar way we would call $\phi$ and $\psi_{thm}$ the sides of the equivalence (of the theorem). That way we can say that the sides of the theorem are equivalent to the differentiae of the definition (or isolated that $\psi_{thm}$ is equivalent with the differentiae). However I find that somewhat clumsy and the term is not that usually used in mathematics anyway.

Extended comment: I have sometimes seen definitions presented in the following alternate manner:

(1) Theorem. For a function $f$, the following are equivalent: ____, ____, ____.

(2) Definition. A function is differentiable if it satisfies any one of the equivalent conditions in Theorem (1).

Such a style is nice because it ties the definition to all the equivalence theorem, and immediately allows using all the equivalent conditions interchangeably. Conceptually, it prevents the reader from thinking of differentiable as a single statement and instead a concept that has a number of equivalent formulations. (I don't know a term for this type of presentation; however, maybe it is along the lines of what you're asking for.)

On the other hand, there are important cases in mathematics where different words are used for different definitions, by convention, despite the fact that the definitions end up being equivalent. For example, we conceptually distinguish between holomorphic and analytic complex functions. Similarly in logic we distinguish the single turnstile $\vdash$ (proof-theoretic entailment) from the double turnstile $\vDash$ (semantic entailment), due to the fact that there is a priori no reason for them to agree, and then we prove a big "soundness and completeness" result stating their equivalence for some system.

There are also cases where one definition generalizes, and another does not, so it's important to use the one that generalizes. For example, I do not like when some authors define a compact set of real numbers as one that is closed and bounded; such a definition sets the reader up for failure later on in understanding what compact means in a general topological space.

• I was about to write something in this direction as well. The advantage of this style of definition is also that it motivates the definition. Why should we introduce the term "normal subgroup" for a subgroup $N$ such that the weird property $gNg^{-1}=N$ holds? But if we first learn that (a) $gNg^{-1}=N$ for all $g$, (b) $N$ is the kernel of a homomorphism defined on $G$, (c) the cosets $gN$ are a group; are equivalent properties of dubgroups of $G$, then we are motivated to coin a name for these subgroups. And we are encouraged to sometimes uses (b) instead of (a) for a simpler proof, say. – Hagen von Eitzen Apr 11 '17 at 6:20
• Very insightful point about that style of definition. About my question: you formulate that "there is a priori no reason for them to agree" (about semantic and proof-theoretic entailment). That is indeed what my question is referring to: I am looking for some clear and precise way of stating this ("a priori" seems a kind of vague, elusive term, even after having read about Kant's philosophy of knowledge, though I understand practically what you mean of course). – user56834 Apr 11 '17 at 6:42
• @Programmer2134 I agree a priori is vague, but it may be the best I can think of. In practice good mathematical writing will simply explain what is going on to the reader. For example, "there are several different ways to define this, and we can prove that they are all equivalent". Or, "we are interested in the case where ___. Several conditions are equivalent to this:" etc. – 6005 Apr 11 '17 at 21:49