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I thought that ∧ is a standard symbol for logical conjunction used in textbooks. I understand that many typographically restricted environments, e.g. programming languages, will substitute it with: &, and, etc. I've noticed that books about Description Logic are using Π symbol for conjunction instead of ∧. Similarily with ∨ vs. ⊔. What is the reason?

Paul Jurczak
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    No "deep" reason... tradition and idiosyncrasies. – Mauro ALLEGRANZA Mar 17 '23 at 08:03
  • @MauroALLEGRANZA How old are Π and ⊔ symbols? Are they fairly new? – Paul Jurczak Mar 17 '23 at 08:06
  • I have never seen those notations in the context of mathematics. Mathematicians tend to use “$\land$” or would explicitly write out “and”. – Vivaan Daga Mar 17 '23 at 08:09
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    IMO $\sqcap$ usage is fairly new; see Description Logic. IMO its usage instead of $\land$ is due to the fact that it is used on concepts and it is more an "intersection" of sets (the extensions of the concepts) that a "conjunction" of statements. – Mauro ALLEGRANZA Mar 17 '23 at 09:28
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    Why would they be the same? Numeral systems have varied throughout humanity's history: https://en.wikipedia.org/wiki/List_of_numeral_systems Language varies among cultures. Different authors have different ideas about how to approach things. Also, sometimes logicians have thought a notation more clear than other notations. Additionally, there's the possibility and actuality of using letters as abbreviations for names as done in Polish notation, where 'K' is used for a conjunction operator. And hardly anyone chose to use Frege's notation. – Doug Spoonwood Mar 17 '23 at 15:02
  • @MauroALLEGRANZA Your comment is spot on. In the context of Description Logic, these symbols represent intersection and union of sets. This begs the question of why not just use set operators: ∩ and ∪? – Paul Jurczak Mar 17 '23 at 21:49
  • @DougSpoonwood Why would they be the same? So people can communicate efficiently. Imagine a world where everyone uses a different notation... – Paul Jurczak Mar 17 '23 at 21:53
  • Because Genesis 11:6-7. – MJD Mar 18 '23 at 14:27
  • I don't believe that efficient communication would get achieved by everyone using the same notation all the time. Also, it's rather clear that for some purposes, some notations are superior in some relevant respect. A simple example is that the algorithms to determine whether something is or is not a well-formed formula are simpler in Polish notation, because parentheses don't have to get parsed. Also, finding the main operator is easier in Polish notation than in infix notation. – Doug Spoonwood Mar 19 '23 at 03:20

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It is for the same reason as for any symbolic system: there are different communities that each developed their own language. Not everyone in the world speaks English, and indeed there is no one natural language spoken by everyone in the world.

With natural languages there are clear geographic reasons for this happening, but for later symbol systems used in math and science communities were formed in more abstract ways. So it is not as if every single individual just makes up their own language … that is unsustainable (I vaguely remember reading somewhere that linguistic anthropologists found that it takes something like 100 people to sustain a natural language … but please don’t hold me to that…) and of course would not work as a way to communicate with others. But it is also not true that everyone is part of the same one and single community.

The situation is similar to the lack of a single standard for which side of the road to drive on, or the fact that many countries are still not adopting the metric standards.

Also note that each community may have a different focus, perspective, or practical purpose for their symbol systems, even if the different communities talk about what is basically the same thing. In fact, depending on that context, perspective or purpose I myself may find one set of symbols preferable over another. For example, when I use logic to analyze arguments, I’ll use expressions like $P \land \neg Q$, for that can still be directly read and understood as the English ‘$P$ and not $Q$’, but when I use logic to analyze computer circuitry I’ll use $PQ’$, because that is easier and faster to algebraically work with. And people doing description logics may have their practical reasons for using a different notation yet.

Bram28
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  • The study of these various “logics” you mentioned are not really mathematical logic, where $\land$ is solely used. – Vivaan Daga Mar 18 '23 at 13:00
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    @Shinrin-Yoku Thanks for proving my point! Ok, so if what you are saying is right, the community of mathematical logic uses $P \land Q$ … but the community of circuitry logic uses $PQ$, in elementary logic you often see $P & Q$, in programming you may see $ P & & Q$, and apparently the community of description logicians use some symbol I don’t even know the MathJax notation for :P And from the post it is clear that the OP is looking at all those different logics – Bram28 Mar 18 '23 at 13:43
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    @Shinrin-Yoku There have been plenty of original results in the study of propositional calculi, which studies the deductive power or propositional systems, where P∧Q (or '(P∧Q)') wasn't the notation used. Not only was there Frege's notation and his work, there has also been a whole host of work in Polish notation, where 'Kpq' gets used instead. One such paper got entitled "Equational Logic" by A. N. Prior and C. A. Meredith. Pierce's work also come to think of it. – Doug Spoonwood Mar 19 '23 at 03:15
  • @Bram28 I was not aware how different Description Logic is from "Classical Logic" when I posted my question. Adding circuit logic notation and syntax of boolean expressions in programming languages is beyond the scope of my question. – Paul Jurczak Mar 19 '23 at 03:19
  • @PaulJurczak OK, that wasn't quite clear from your question as you do mention programming languages. Anyway, even within 'Classical Logic' you will find different symbols ... I have seen books use $P & Q$, $P \cdot Q$, and yes, some books do use $PQ$. DougSpoonwood also mentions some more historical notations ... which is line with my answer: you first get these 'local' communities that adopt a certain notation .. later on, some of them may fall by the wayside, but it is rare to ever get a single standard. That's just how language works in terms of creation and development. – Bram28 Mar 19 '23 at 13:35