# What are reasons why some symbols in mathematical logic are not standardized?

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?

• One reason is probably that completeness theorems tell us that the exact syntactic details often don't matter. – Michael Greinecker Jan 7 '14 at 17:22
• This isn't much of a mathematical question, is it now? It's more of a rant. – Asaf Karagila Jan 7 '14 at 17:24
• @Asaf - yes, it,is ... sorry. – Mauro ALLEGRANZA Jan 7 '14 at 17:25
• I think that something could be said about it, if the question was edited to be more like "What are some reasons why these things are not standardized" rather than sounding like a complaint. – Carl Mummert Jan 7 '14 at 17:26
• A lot of (modern) mathematics is not standardized to the extent one might like. Some basic questions that mathematicians will differ on: Is $0$ a natural number? What is $0^0$? Does normality of a topological space imply Hausdorffness? Rigour has little to do with standardisation of the terms/symbols employed; rather it is about the exactness and precision with which one employs these terms/symbols. While logicians may differ from each other on specific details, like all mathematicians they will ensure that their translations are faithful. – user642796 Jan 7 '14 at 18:43

## 5 Answers

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.

• I think you are right : less then 150 years from Frege to present time is not so a big amount of time. I'm wondering (I'm not prepared as an historian of math) how many years there were between the Renaissance "cossists" (the forerunnes of algebra) and the (reasonable) standardisation of modern algebraic symbolism ($+$, $-$, ...). – Mauro ALLEGRANZA Jan 8 '14 at 7:14
• have a look at en.wikipedia.org/wiki/History_of_mathematical_notation i think lot of modern notation goes back to Peano, but i may be wrong ( i think Peano used dot notation) – Willemien Jan 8 '14 at 13:10
• Thanks. I think that less than two centuries from Renaissance algebrists to Euler (that is quite "modern") can be an amount of time comaparable to that between Frege-Peano and today. I think that the pedagocical benefits of a standard notation (see for an example math.stackexchange.com/questions/630391/…) it is not sufficently percieved in math log due to the lack systematic teaching in school. With algebra and calculus, the needs of teaching them in internediate school forced the math community towards standardisation. – Mauro ALLEGRANZA Jan 8 '14 at 13:40

Why is so hard to find a standardisation regarding symbolism ... in Mathematical Logic ?

Well, if you compare the situation with (say) fifty to sixty years ago -- the time when the books I was looking at while I was a student were written -- I would have said that there has been quite a considerable standardisation in symbolism, at least in mainstream mathematical logic books/articles. And one reason for that, surely, has been the universal adoption of LaTeX, which makes it so easy to type \land [for 'logical and'] and always get '$$\land$$' (and not '&', or a dot, etc.) and to type \forall x and get '$$\forall x$$' (and not e.g. '$$(x)$$' or '$$\Pi x$$'). So, let's be duly grateful for all the standardisation there in fact now is!

True there is the annoying business with $$\to$$ vs $$\implies$$. This too is partly to down to LaTeX I guess, as \implies yields the second. Now 'A implies B' gets used in informal talk both as variant on 'if A then B' and as a variant of 'A logically entails B', i.e. as both what me might regiment as $$A \to B$$ and as $$A \vdash B$$ [or $$A \vDash B$$]. And low and behold, we find $$\implies$$ being confusingly used both ways [in the object language, or in the metalanguage]. Conservatism in symbolism is a Good Thing, so I think the use of $$\implies$$ is to be deprecated: I'd say, use $$\to$$ for an object language conditional, and the appropriate turnstile in in metalanguage. The exception might be in a sequent calculus. [Indeed, it was only when I started regularly visiting math.se that I really registered that the double arrow was used in non-sequent-calculi so widely outside the very narrow logic community I was most familiar with -- though that might just show I wasn't really paying attention!]

• \renewcommand{\land}{\mathbin{\&}} is as easy as pie. ;-) – egreg Jan 16 '14 at 11:40
• Of course it is! But mostly we go with the defaults, 'cos we are lazy/unimaginative/or (most likely) we have no special reason for changing them ... – Peter Smith Jan 16 '14 at 12:36
• How would you suggest saying $\varphi \rightarrow \psi$ if not "implies"? – goblin GONE Jan 16 '14 at 14:26
• @user18921 'If $\varphi$, then $\psi$'. – Peter Smith Jan 16 '14 at 14:33
• Any other alternatives? To me, the if-then phrase has strong connotations of entailment. Mathematicians always write: "Now if $x$ is strictly less than $0$, then the problem is trivial. So assume $x \geq 0$." – goblin GONE Jan 16 '14 at 14:39

To paraphrase Bill Thurston, mathematics is just a way to organize human thought. The purpose of mathematical notation is to help us understand each other.

One reason that different notation is used is that the same concept may need to be compared or contrasted to different things in different situations. For instance, I recently discussed the various division symbols with someone:

The fraction notation $\frac{a}{b}$ is most useful when simplifying equations by hand.

The long division notation is most useful when calculating exact values.

The slash / is great for computers as it works with a regular keyboard.

The 'obelus' ÷ is used on calculators because the other notations are dissimilar from +,-, and x.

I feel that trying to standardize division would be counterproductive.

Now, mathematical logic is different from arithmetic, but the same truths hold. Russell and Whitehead used the notation most helpful for pure symbolic calculation, but this may not be the best notation for writing on the board or writing a computer program.

TL;DR Mathematical notation is designed to express thought as clearly as possible, and strict standardization makes this difficult.

• I'm learning from the comments. One aspect of the issue regards, according to me, more "practical" aspects involving communication and theaching: from this side, I prefer an effort toward "standardisation" of basic notations, e.g. truth-functional connectives (like : $\rightarrow$) and metatheoric symbols (like : $\vdash$). From another side, I begin to understand the positive aspect of a richness in symbolism: may we say that there is an effect due to the power of abstraction inherent in the symbolism ? – Mauro ALLEGRANZA Jan 13 '14 at 21:34

The other answers don't seem to have addressed directly the point about implication $\rightarrow$ or $\implies$, so I will point out that (non-logician) mathematicians mostly use $\implies$ whereas logicians more often use $\rightarrow$. Possibly this is because logicians use it a lot more, so eventually one catches on that it is pointless to draw two lines when you can draw one. Since we are talking about two different scientific cultures, the difference in notation may be with us for the foreseeable future. For comparison, note that physicists often use different notation (than mathematicians are used to) for various mathematical concepts.

• I like this answer. I would hazard a guess that it would be very unusual to see a mathematician use anything other than $\Rightarrow$ for logical consequence. Having written quite a few logical derivations in the last couple of weeks, I also find myself abbreviating it to $\rightarrow$, but only for my own scribblings. I don't think logicians should use $\rightarrow$ for logical consequence, then at least they can agree with mathematicians, if not each other! – James Smith Jun 18 '16 at 13:28

With regards to the ambiguity between implication $\rightarrow$ and entailment $\Rightarrow,$ the problem of course is that most theories have a deduction theorem, which has the consequence that the following judgements are equivalent.

$$\varphi \Rightarrow \psi,\quad\quad \Rightarrow \varphi \rightarrow \psi$$

Furthermore, every formula $\alpha$ can be seen as shorthand for the judgement $\Rightarrow \alpha$. So in some sense (as much as it pains me), the following are equivalent.

$$\varphi \Rightarrow \psi,\quad\quad \varphi \rightarrow \psi$$

This is probably the cause of a lot of ambiguity, but also suggests that we needn't worry; to claim that these strings of symbols "true" amounts to roughly the same meaning, so the ambiguity isn't too problematic. (On the other hand, to claim that the first string of symbols is false is very different to claiming that the second string of symbols is false.)