[Math] What are reasons why some symbols in mathematical logic are not standardized

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Why is so hard to find a standardisation regarding symbolism and/or terminology in Mathematical Logic ?

We see again and again students asking if e.g. $\rightarrow$ and $\implies$ means the same thing : somebody answer : "yes", somebody answer : "no".

The same thing happens with "tautology" and "validity", with "logical consequence" and "logically implies", and so on …

Why is this problem still with us, and what can we do about it?

Best Answer

I think there is more than one cause of it. My ideas:

  • Symbolic logic is still a reasonably new field. (Different than you may think, symbolic logic didn't start with the old Greeks but with Frege's Begriffsschrift in 1879, not even 150 years ago, and don't even try to follow his notation.)
  • Some philosophers thought that they knew everything about logic already and didn't even study it and thus were never confronted with the standard notation.
  • Some logicians needed other kinds of implication (relevant, strict, material), negation (minimal, subminimal, constructive), or entailment (standard, fuzzy, quasi-, degree) for their own logic and created their own new symbols for it.
  • Some logicians were comparing different logics and decided to use a different set of connectives to not get utterly confused.
  • A couple started their own notation because they were not satisfied with the old one: Polish notation, dot notation, compressed dot notation, Lambda notation...

And maybe some wanted to confuse everybody :)

To add a bit:

Even with truth tables you see some publications where $0$ stands for true and others where $1$ stands for true. And that is just with two valued logic. If you are lucky you have a book that uses $T$ and $F$ or $\top$ and $\bot$. In either case the $T$ or $\top$ stands for true, and the $F$ or $\bot$ for false. But even so, be warned: always check the meaning first.

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