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In mathematical logic or other formal systems, what is the definition of a definition, formally?

If "A" is defined as "B", what is the definition of "A" like? Does it involve both "A" and "B" (e.g. "A := B"), or just "B"?

For example, on p126 in §3. Extensions by Definitions in VIII Syntactic Interpretations and Normal Forms in Ebbinghaus' Mathematical Logic, suppose that $S$ is a (non-logical) symbol set,

3.1 Definition. Let $\Phi$ be a set of $S$-sentences.

(a) Suppose $P \notin S$ is an $n$-ary relation symbol and $\phi_P(v_0, ... , v_{n-1})$ an $S$-formula. Then we say that $$ \forall v_0, .... \forall v_{n-1} (P v_0 ... n_{n-1} \leftrightarrow \phi_P(v_0, ... , v_{n-1})) $$ is an $S$-definition of $P$ in $\Phi$.

Which shall I call as an $S$-definition of $P$ in $\Phi$:

  1. $ \forall v_0, .... \forall v_{n-1} (P v_0 ... n_{n-1} \leftrightarrow \phi_P(v_0, ... , v_{n-1})) $?

    • Is it circular to define $P$ in terms of itself?

    • Is an $$-definition of $$ in $Φ$ an interpretation of symbol $P$ as an $S'$-sentence? (as part of a syntactic interpretation of $S'$ in $S'$ itself?)

    • Is the appearance of $P$ in its own definition $∀ _0,....∀ _{−1}(_0...v_{−1}↔_(_0,...,_{−1}))$, in the same sense as the appearance of $A$ in $:=$?

  2. $\phi_P(v_0, ... , v_{n-1})) $ ? (I guess that $P$ is defined as $\phi_P(v_0, ... , v_{n-1})) $ in $\Phi$.)

  3. $\phi_P$? (Compare that to the second: $P$ itself doesn't involve variables)

See How does this definition define a symbol $P$ outside the symbol set $S$ as a $S$-sentence?

Thanks.

Tim
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    In formal logic, a definition usually means a shorthand. That is, there is a way to write all statements without using the definition, they just become longer. Sometimes much longer. – Thomas Andrews Aug 19 '20 at 20:26
  • So which shall I call as an -definition of in Φ? – Tim Aug 19 '20 at 20:27
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    What is a definition of a definition of a definition? – Asaf Karagila Aug 19 '20 at 21:55
  • In systems of natural deduction, you usually have an assumption (or premise) rule to introduce statements without any justification. And you have rules to discharge them and introduce '$\implies$' or '$\neg$'. AFAIK there is no formal "definition" rule to introduce definitions, whatever they may be. They seem to be an informal notion--a certain kind of assumption (or premise) with certain informally defined characteristics. – Dan Christensen Aug 19 '20 at 23:48
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    As @ThomasAndrews said, in formal logic a definition is usually just a shorthand notation. See this post for a simple motivating example, and this post for a more formal explanation. Dan is totally wrong about there being no formal rule to introduce definitions; it is trivial to enhance any deductive system for FOL with suitable rules for that. – user21820 Aug 20 '20 at 12:43
  • https://en.m.wikipedia.org/wiki/Extension_by_definitions – Mykola Pochekai Sep 04 '20 at 14:42

2 Answers2

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Below I'll first try to describe the process in a more intuitive way, then address your worries about circularity. I suspect the latter point may actually be more helpful, so feel free to read the second section first - and in particular, the highlighted motto there will I think be quite helpful.

(Re: your final comment, the definition is $(1)$ - the thing which tells you how the new symbol behaves, in terms of the old symbols you already have and understand.)

The key phrase here is "expansion by definitions."

Intuitively, we have in mind the following process:

  • Starting with a signature $S$ and some set $\Phi$ of $S$-sentences, we become a little annoyed by inefficiencies: there are some things which we can talk about using $S$-formulas but only in a roundabout way. Think for example about the language of set theory, $\{\in\}$: we can express things like "$x$ is the Cartesian product of $y$ and $z$" in this language, but only via annoyingly long formulas. (It's a good exercise to handle the previous example - using, say, Kuratowski pairs.)

  • So given our really complicated formula $\varphi(x_0,...,x_{n-1})$, we want to whip up a new theory which is basically the same as $\Phi$ except that it additionally has an "abbreviation" for $\varphi$.

  • First, this means we want to enlarge our language: rather than work with $S$ we want to work with $S\cup\{R\}$ for some $n$-ary relation symbol $R\not\in S$ which we intend to serve as an abbreviation for $\varphi$.

  • Now we have to define a theory in this larger language. This theory should subsume what we already have (that is, $\Phi$), should correctly dictate the behavior of $R$ (that is, say that it's an abbreviation for $\varphi$), and shouldn't do anything else. This leads us to consider the new theory $$\Phi':=\Phi\cup\{\forall x_0,...,x_{n-1}(R(x_0,...,x_{n-1})\leftrightarrow \varphi(x_0,...,x_{n-1})\}.$$

The passage from $S,\Phi$, and $\varphi$ to $S\cup\{R\}$ and $\Phi'$ is an expansion by definitions. We have some serious redundancy here: in a precise sense, $\Phi'$ is really no better than $\Phi$. (Formally, $\Phi'$ is a conservative extension of $\Phi$ in the strongest possible sense: every model of $\Phi$ has exactly one expansion to a model of $\Phi'$.) This isn't surprising. We already knew we could express the thing we cared about via $\varphi$, we just wanted to be able to do so more quickly.

Incidentally, note that this suggests a natural "maximally efficient" version of any theory: just add new symbols for every formula! This is called Morleyization, and is occasionally useful (although usually kind of silly).

OK, now what about the circularity you are worried about?

First, note that "$R$" itself is just a symbol. The new sentence we're adding isn't really a definition of $R$, but rather a definition of the meaning of $R$, or if you prefer a rule governing the behavior of $R$.

More seriously, circularity is never an issue in FOL! The key idea is the following, which I think is an important departure from the intuitions one might bring in from programming:

A set of first-order sentences doesn't create things, it describes things.

Specifically, a set of first-order sentences $\Phi$ carves out a particular class of structures, those about which it is an accurate description. For instance, the possibly-dangerous-looking sets $$\{\forall x(P(x)\leftrightarrow P(x))\}$$ and $$\{\forall x(Q(x)\leftrightarrow \neg Q(x))\}$$ are perfectly circle-free; they're just vacuous (= hold of every structure) and contradictory (= hold of no structure) respectively.

user21820
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Noah Schweber
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  • Thanks. (1) Is an $$-definition of $$ in $Φ$ an interpretation of symbol $P$ as an $S'$-sentence? (as part of a syntactic interpretation of $S'$ in $S'$ itself?) (2) What other books, similar to Ebbinghaus' Mathematical Logic, talk about "expansion by definitions", and/or "syntactic interpretation"? – Tim Aug 19 '20 at 21:45
  • Re: (1), yes - although some sources will refer to $\varphi_P$ itself as the definition, and this won't be a big deal (logic generally suffers from a lot of mildly-confused terminology). Re: (2), practically any book about model theory will do this - Hodges, for example. – Noah Schweber Aug 19 '20 at 21:46
  • Do you call "syntactic interpretation" as "syntactic interpretation" or some other alias? My suspicion came from Peter Smith's "Finally in Part A, there’s rather ill-written chapter on normal forms, on extending theories by definitions, and (badly explained) on what the authors call ‘syntactic interpretations’." – Tim Aug 19 '20 at 21:47
  • @Tim Depends on context - either "syntactic interpretation" or simply "interpretation." – Noah Schweber Aug 19 '20 at 21:48
  • @Tim Really though I'd worry about terminology only after you're comfortable with some approach to the topic - e.g. the Ebbinghaus/Flum/Thomas one. Once you understand one presentation well, switching between presentations will be fairly trivial. In particular, I'd say the main issue to focus on here is the appearance of circularity. – Noah Schweber Aug 19 '20 at 21:51
  • Regarding to your second to last comment: Is a syntactic interpretation an interpretation/structure? – Tim Aug 19 '20 at 21:59
  • $ \forall v_0, .... \forall v_{n-1} (P v_0 ... n_{n-1} \leftrightarrow \phi_P(v_0, ... , v_{n-1})) $ is an $$ -definition of $$ in $Φ$. Does that mean that every use/occurrence of $P$ can be replaced with $ \forall v_0, .... \forall v_{n-1} (P v_0 ... n_{n-1} \leftrightarrow \phi_P(v_0, ... , v_{n-1})) $? For example, I have difficulty to see that $P$ in $P v_0 ... n_{n-1}$ can be replaced with $ \forall v_0, .... \forall v_{n-1} (P v_0 ... n_{n-1} \leftrightarrow \phi_P(v_0, ... , v_{n-1})) $. – Tim Aug 19 '20 at 23:00
  • Let us continue this discussion in chat. – Noah Schweber Aug 19 '20 at 23:16
  • Is the appearance of $P$ in its own definition $ \forall v_0, .... \forall v_{n-1} (P v_0 ... n_{n-1} \leftrightarrow \phi_P(v_0, ... , v_{n-1})) $, in the same sense as the appearance of $A$ in $A:=B$? – Tim Aug 20 '20 at 11:58
  • @Tim Sort of, but not quite in my opinion. The expression "$A:=B$" does two things at once: it adds a new symbol to our working vocabulary, namely "$A$," and it then adds a new rule to our existing set of rules, namely "$A=B$." These two steps are more clearly separated in our current context: the former corresponds to picking out an unused symbol "$P$" and adding it to our language, while the latter is the definition itself. Incidentally, though, I think this kind of scrutiny can be somewhat misleading: (cont'd) – Noah Schweber Aug 26 '20 at 19:21
  • it singles out sentences of a particular form as somehow "special" (and this is why it can raise worries of circularity if we're not careful). It's much better to think in terms of descriptions: supposing I see someone add a new relation to my structure, how can I tell whether that relation is actually functioning as shorthand for a particular formula of interest? One answer is: iff the new structure satisfies $\forall \overline{v}(P(\overline(v))\leftrightarrow\phi_P(\overline{v}))$. Consequently adding that sentence to our existing theory will restrict attention to models where this happens. – Noah Schweber Aug 26 '20 at 19:25
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We have a signature $S$ and we extend it to $S':=S\cup\{P\}$.

The $S$-definition of $P$ is the $S'$-formula $$\forall v_0\dots v_{n-1}: Pv_0\dots v_{n-1}\leftrightarrow \phi_P(v_0,\dots,v_{n-1})$$ which can be formally handled as an extra axiom to the given $S$-theory we're working with, thus producing an equivalent $S'$-theory, in which the symbol $P$ can be used as abbreviation for the formula $\phi_P$.

For example, the below formula is the definition of the usual ordering relation $\le$ of nonnegative integers in the language $(0,+)$: $$\forall x,y:\ x\le y\,\leftrightarrow\,\exists z: x+z=y$$

Berci
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    Thanks. Is it circular to define in terms of itself? – Tim Aug 19 '20 at 21:05
  • Is an -definition of in Φ an interpretation of symbol P as an S'-sentence? (as part of a syntactic interpretation of S' in S' itself? c.f. p120) – Tim Aug 19 '20 at 21:09
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    Well, $P$ here is just a symbol, we're defining it internally into a given formal theory. Introduce a symbol, add an axiom with its meaning, but otherwise it will be a logically equivalent theory. – Berci Aug 19 '20 at 21:10