This might be a request for further philosophical clarification, rather than a mathematical question. You know, we are working with two kind of quantifiers $\forall x$ and $\exists x$. Somehow, we decided this language is enough for us. I can see that. For example, I can express further things like "There exists more than three elements in this set that satisfies predicate P". For example, I can try to express this as

$$\exists x, y, z, w .P(x) \land P(y) \land P(z) \land P(w)$$

I think this captures my initial statement. If I can find 4 elements that P is true for all of them, then I can say there exists more than three elements in this set that satisfies P. You see though, I needed to use 4 auxiliary variables to express this and if I wanted to say "there are more than 100 elements", I could not find enough symbols to identify my variables.

In a way, what I am trying to say is, yes two quantifiers are powerful to express whole of these kind of things but they are not nice to use when things get ugly. It is, as if, predicate logic is a designed language just like programming languages and somewhere along the way designers decided 2 quantifiers are enough to express mathematics.

My question is, why we have 2 quantifiers? Who decided that it is best? Why we don't have further syntax to express other notions? For example, I could explain "there exists more than 100 elements in the set that satisfies predicate P" as $$\exists (> 100)x.P(x)$$

why we don't have something like this? I can see that mathematics is the collective work of many geniuses and there is definitely a reason why my above suggestion is not used in mathematics. But what is the reason for things being as they are?

Edit: There are clarifications and suggestions for further notation in the comments. But what I am really asking is, what is the "design rationale" you should have in mind when deciding on available quantifiers for your logic? There are logics beside predicate logic and we might devise new ones in the future. How mathematicians tackle this problem of making available "just enough" quantifiers? For example, do they seek minimal number of quantifiers that makes their logic complete, or some similar things?

  • $\begingroup$ $\exists!$ is commonly used to denote "there exists exactly one". And you can use any quantiviers you want, as long as you define them. I do believe, however, that there aren't many besides $\exists$ and $\forall$ that applies as generally, and thus is as useful. This means that other quantifiers just don't get the exposure that they need in order to gain traction and become popular enough to be standard. $\endgroup$ – Arthur Feb 26 '18 at 11:24
  • $\begingroup$ In your symbolization of "there exist more than three elements" one needs also to say that $x,y,z,w$ have no two of them equal. $\endgroup$ – coffeemath Feb 26 '18 at 11:28
  • $\begingroup$ You can write $|\{y\in x\mid P(x)\}|>100$ for that. No quantifiers needed. Actually I like it that we can do it with one quantifier. $\forall x P(x)$ can be written as $\neg\exists\neg P(x)$. Other notations can be added based and defined on one essential one. Nice situation. $\endgroup$ – drhab Feb 26 '18 at 11:29
  • $\begingroup$ Hope what I am "really" trying to ask is clear with the edits. $\endgroup$ – meguli Feb 26 '18 at 11:34
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    $\begingroup$ "Who decided that it is best?" Wasn't this decided at the 1934 World Council of Bishops of Logic in Bevagna? More seriously - nobody makes these decisions: each author can choose their own logical system based on taste and convenience. So if the two-quantifier system became predominant, it was some combination of historical inertia and convenience for authors, rather than a closed-room decision between a council of logicians. $\endgroup$ – Carl Mummert Feb 26 '18 at 13:29

If you just want quantifiers for counting there are things like $\exists^{=n}x$, $\exists^{>n}x$,$\exists^{\leq n} x$ which say that there exists exactly n, more than n and less than or equal to n elements such that.

As you seem to be aware of, the counting quantifiers meaning can already be expressed in predicate logic. If we wanted to include all counting quantifiers we would suddenly go from 2 to an infinite amount of quantifiers. It sure works, but having a system containing an infinite amount of redundant axioms seems a bit ... bad.

Note though that already having both $\forall x$ and $\exists x$ is a bit redundant since $\forall x \varphi$ is equivalent with $\neg \exists x \neg \varphi$, so we could actually make do with only a single quantifier if we wanted. The reason why we usualy include both quantifiers is because it good for translation purposes.

Now there are also other logics where you allow more quantifiers (which can not be expressed using $\forall$ and $\exists$) such as Least Fixpoint Logic, and there are logics where you allow less quantifiers, such as $FO^{<s}$. These are useful logics which have their applications in different contexts

One can ask why we are almost always using "normal" predicate logic when doing reasonings in mathematics, and one of the reasons for that I would argue is the fact that, According to Linströms theorem, it is the logic which satisfies both the Compactness theorem and the Löwenheim-Skolem theorem. If you do not know what these theorems say it is fine. These are fundamental theorems which essentially say how nice certain basic aspects of the logical reasoning are.

I hope this, somewhat is what you sought in an answer.

  • $\begingroup$ I think I need to look into those two theorems to really make the connections in my mind. But in the case these are fairly advanced mathematics, can you add your philosophical understanding of those results in a simplified manner so that I can refer and compare? $\endgroup$ – meguli Feb 26 '18 at 12:09

Using the minimum number possible is indeed one of the factors. In fact, many authors just use one quantifier (often the universal one) and view the other one as just a definition in terms of the quantifier that was included. For many proofs, every symbol in the logic requires its own case, and so reducing the number of logical symbols reduces the length of proofs that have to be written up. There was also a historical preference for minimalism - some authors prefer not to have redundant axioms, and so having a large number of redundant quantifiers might also have seemed avoidable.


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