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There're variations of logical expressions. e.g.

$\forall a \exists b(P(a,b))$

is also written as

$\forall a \exists b:P(a,b)$

or

$\forall a \exists b.P(a,b)$

or even

$(\forall a \exists b)[P(a,b)]$

So I wonder if there is an official (or say well-accepted) formulation on logical expression forms.

kakakali
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    The answer I very much want to give is "it doesn't matter, they're all equivalent, it's like asking if there is an official font, why are you asking". Why are you asking? – Patrick Stevens May 22 '20 at 18:16
  • @PatrickStevens Well, I'm afraid that without formulation, the same notation may have different meanings (based on the author's preference). Though the examples here are common and easy to read, not all expressions are clear enough e.g.. Also, a formulation may give a guide of efficiency and clarity. Even proper font would help as they're clear, beautiful, and reader-friendly. And I think this is why the publishers would provide authors templates. – kakakali May 22 '20 at 18:30
  • @PatrickStevens - If you write it, I'll upvote it. – Taroccoesbrocco May 22 '20 at 18:33
  • @kakakali - But you are asking if there is one font that is official (or well-accepted) for all the publishers! – Taroccoesbrocco May 22 '20 at 18:35
  • @kakakali In real life, people will often prefer to write things using a natural language, not symbols, simply because it's easier to understand. It's only really when discussing proof theory, foundational logic/set theory, or introductory mathematical courses that I've seen lots of quantifiers like that. – Patrick Stevens May 22 '20 at 18:45
  • @Taroccoesbrocco Emm, changing font never changes the meaning but different notations may have a difference. At least, a consistent language is helpful so I want to know if there is one. It's natural people have various perceptions and I would like to listen to them to find useful recommendations and guides. – kakakali May 22 '20 at 18:46
  • @PatrickStevens Yeah, I agree to you with this and I do have read related questions on StackExchange (though I also understand those who support logical expressions). The problem is, for me, I found that logical expression is much more clear and simple. So I write them in private (I also write in natural languages when I show them to others). – kakakali May 22 '20 at 18:53
  • Mathematicians choose different notations by their preferences. The reason might be because of convenience, or one notation shows something (like intuition or how to calculate) than the others, or even just a convention. However, I can certainly say that notations on quantifiers are just a matter of preference. There is no difference in meaning. – Hanul Jeon May 22 '20 at 19:48
  • (Side note: What I have seen most frequently is parenthesizing formulas after quantifiers: like $\forall x\exists y (\phi(x,y))$. Sometimes parentheses are suppressed if there is no confusion.) – Hanul Jeon May 22 '20 at 19:51
  • There isn't, but I personally like the first formulation $∀a∃b(P(a,b))$, and I hope it become official. – Ethan May 22 '20 at 20:30

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