In mathematical logic, a definition is treated as an abbreviation - a denotation which simplifies the discourse making it shorter. This is so much so that in a formal theory or a logic we can do without definitions. Thus, logicians did not seem to have studied the notion of definition. This is strange because what mathematicians produce is defintions and theorems. Could it happen that logicians focused only on theorems (deduction rules, proof, etc.) but total ignored definitions?
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2See some aspects regarding logical theory of definitions in this post and this post – Mauro ALLEGRANZA Jun 27 '14 at 14:40
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2See also Definitions – Mauro ALLEGRANZA Jun 27 '14 at 15:01
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@MauroALLEGRANZA, the clarification of these aspects sounded very useful. Some of your remarks support the idea I have about a logical theory or definitions, which I will share in my own answer below. – Ioachim Drugus Jun 29 '14 at 15:00
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You are right in saying that form a 'logical point of view' : "a definition is treated as an abbreviation". What we have to be sure is that the abbreviation is "well-defined", i.e.that, in order to introduce a new symbol denoting "inequivocally" an object satisfying the "property" $P$ (like e.g. $\emptyset$ in set theory), we have proved previously that $\exists ! xP(x)$. – Mauro ALLEGRANZA Jun 29 '14 at 16:26
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Denote the fact that a concept named $c$ is defined by a property $P(x)$ like this: $c$ = $\delta x P(x)$. Treat '$\delta$' as a "quantifier" which bounds the variable $x$ in the formula $P(x)$ similar to the $\lambda$ "quantifier" in the "$\lambda-calculus$". This "quantifier", call it "definer", – Ioachim Drugus Jun 30 '14 at 01:49
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Sorry for my previous comment - inadvertantly I saved it, before I finished my thought. What I wanted to say is, that it sounds plausible that a logical theory of definitions can be presented as a calculus similar to "$\lambda-calculus$". Denote the fact that an object named $c$ is defined by a property $P(x)$ like this: $c$ = $\delta x P(x)$. Treat '$\delta$' as a "quantifier" which bounds the variable $x$ in the formula $P(x)$, and indicates the variable which value is the defined. I know this is fuzzy, but I cannot put it down more precisely at this time. – Ioachim Drugus Jun 30 '14 at 02:10
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1Really, I think this is more like a programming problem in spirit than a math problem. – Aug 05 '14 at 00:02
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This sounds to be a complete answer to my question:
the theory of definitions is part of the theory of conservative extensions of theories.
See the following discussion in the forum FOM (Foundation of Mathematics): http://www.personal.psu.edu/t20/fom/postings/9810/msg00068.html:
The idea of conservative extension is a generalisation of the notion of definition. (and is used self-consciously in that way by some proof tools e.g. HOL) What distinguishes a "foundation system" from other logical systems is that in a foundation system it is possible to derive the main body of mathematics using only conservative extensions. (of course, people used to say "definitions")
Ioachim Drugus
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I've always called Godel numbers for predicates with one free variable that can be proven to be uniquely satisfied Definitions. I've read of various formal notions of definability in Model Theory so why not Proof Theory.
For example, if you have a proof predicate $\mathtt{Pf}$, then the set of Definitions relative to this proof predicate can be specified as:
$$ A := \{ \; \ulcorner \phi \urcorner \; | \; \mathtt{Pf}(\ulcorner \exists ! x \; \phi(x) \urcorner) \; \}. $$
Naturally, the next thing to do is consider the set of definitions for sets that do not contain their own definitions as members.
You get:
$$ B := \{ \; \ulcorner \phi \urcorner \; | \; \mathtt{Pf}(\ulcorner \exists! x \; \phi(x) \; \wedge \; \ulcorner \phi \urcorner \notin x \urcorner) \; \}. $$
One can actually write up a predicate and then Godel number it to get a definition for B. Let $\theta_B(x)$ denote this predicate.
Directly from the definition, one can prove that:
$$ \ulcorner \theta_B\urcorner \in B \leftrightarrow \mathtt{Pf}(\ulcorner \ulcorner \theta_B\urcorner \notin B \urcorner). $$
Further applying the definition, one can prove that:
$$ \mathtt{Pf}(\ulcorner \ulcorner \theta_B\urcorner \notin B \urcorner) \leftrightarrow \mathtt{Pf}(\ulcorner \neg \mathtt{Pf}(\ulcorner \ulcorner \theta_B\urcorner \notin B \urcorner) \urcorner ). $$
If $\mathtt{Pf}$ is a proof predicate for a theory that isn't too weak, then we would have that it is consistent if and only if $\mathtt{Pf}$ does not prove everything if and only if $\mathtt{Pf}$ does not prove it is consistent.
Therefore, we get:
$$ \mathtt{Pf}(\ulcorner \neg \mathtt{Pf}(\ulcorner \ulcorner \theta_B\urcorner \notin B \urcorner) \urcorner ) \leftrightarrow \mathtt{Pf}\text{ is inconsistent}. $$
Combining all of the equivalences, we get:
$$ \ulcorner \theta_B\urcorner \in B \leftrightarrow \mathtt{Pf}\text{ is inconsistent}. $$
Somehow, the different levels of provability allow you to avoid getting a contradiction like you do from Russell's Paradox.
Do you know of anything related to this? Thanks! :)
-----Addendum-----
I guess for any sentence $\psi$ where you can prove: $\psi \leftrightarrow \mathtt{Pf}(\ulcorner \neg \psi \urcorner)$.
You can also prove: $\psi \leftrightarrow \mathtt{Pf}\text{ is inconsistent}.$
Michael Wehar
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1This is unexpected treatment - interesting, but I did not see anything related to this. – Ioachim Drugus Sep 04 '14 at 20:19
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Oh, ok. Why is it unexpected and why is it interesting? If you would be willing to talk with me about this idea, I would greatly appreciate it! :) – Michael Wehar Sep 04 '14 at 23:32
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1This sounds to me interesting because I did not expect that definitions have to do with proof theory. But I don't know proof theory to share with you anything useful. Possibly, you can arrive at something really interesting if you try to apply this proof theory approach to so called theoretical terms (http://plato.stanford.edu/entries/theoretical-terms-science/) – Ioachim Drugus Sep 05 '14 at 14:49
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