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What are the rules of omission of parentheses of formulas in mathematical logic ?

in my text , first order logic mathematical logic by angelo margaris ed 1990 dover , the paretheses is omitted for example in page 49 ,

$S$ $\rightarrow$ $.$ $P$ $\rightarrow$ $Q$ $:$$\rightarrow$$:$$S$$\rightarrow$$P$ $.$$\rightarrow$ $.$ $S$$\rightarrow$ $Q$

when the paretheses should be ?!!

how can i translate this " ." points to the right parentheses ??

what are the rules of these translation in general ?

Bram28
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FNH
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2 Answers2

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"What are the rules of omission of parentheses?" The answer is: different rules in different texts!

But what is really being asked is something different, i.e. "What are the rules for using dots instead of parentheses?" [and I have edited the title question to fit]. Again, though, the answer is that there are different rules in different texts. Church's Mathematical Logic (1956) is a canonical text here, though Church's rules are slightly different from Principia's. There's a slightly simpler set of rules, as I recall, in Robbin's Mathematical Logic. (1969) Anyway, the shared basic idea for using dots is revealed in the rules for getting rid of them again in favour of brackets:

A dot/dots to the immediate left of an implication sign indicates a right bracket, and its matching left bracket would appear as far left as we can go before encountering a higher number of dots.

Symmetrically for a dot/dots to the immediate right of an implication sign: this indicates a left bracket, and its matching right bracket would appear as far right as we can go before encountering a higher number of dots.

And we are to eliminate single dots before double dots, etc. Phew!

That sounds messy, but in practice isn't at all bad. So consider ...

$$S \to . P \to Q :\to:S \to P .\to. S \to Q$$

$$S \to (P \to Q) :\to:S \to P .\to. S \to Q$$

$$S \to (P \to Q) :\to:(S \to P) \to. S \to Q$$

$$S \to (P \to Q) :\to:(S \to P) \to (S \to Q)$$

$$(S \to (P \to Q)) \to:(S \to P) \to (S \to Q)$$

$$(S \to (P \to Q)) \to ((S \to P) \to (S \to Q))$$

You might wonder, indeed, about the virtues of this kind of bracketing system! But in fact, if you stick to single dots and mix with the use of parentheses it can be surprisingly readable. But I can't immediately think of any book (first) published in the last thirty years that uses the dotty system: it has fallen into disuse.

Community
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Peter Smith
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  • i think these note will help me to understand the notation in the text , my text is not first pubkished in the last 30 years . it's first order mathematical logic by angelo margaris , 1990 ed dover publication . it has a few exercises and the explnation is not excellent , but i started studying the subject from it for 2 reason , first to give me some knowledge about the subject which make me able to study from more advanced text , seconed , because i couldn't decide the best text i can use so i chose arbitary one to study from it !, - i will continue my talk in the next comment- – FNH Feb 23 '13 at 19:45
  • so if you recommend a excellent text - which doesn't use dotty system of course ! - which i can study the subject from it , that will be great , i heared about many many many texts and that made me confused about the one which i should use ! – FNH Feb 23 '13 at 19:55
  • Check out the Teach Yourself Logic guide at http://logicmatters.net/students/tyl/ – Peter Smith Feb 23 '13 at 22:13
  • thanx , can you tell me where can i find more about dotty system ? – FNH Feb 23 '13 at 22:16
  • Interesting; I'd never encountered this convention. I wonder whether it's related to the convention that "$\lambda x.$" in the lambda calculus (and by extension any variable binder in computer-scientist notation) will gobble up all the text to the right of it until it hits an unmatched right parenthesis. – hmakholm left over Monica Feb 25 '13 at 16:53
  • @HenningMakholm I suppose it might be the same sort of idea. Especially as Church had a big hand in both ... – Peter Smith Feb 25 '13 at 16:56
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I think this should rather read $$ S.\to.P\to Q:\to:S\to P.\to.S\to Q$$ If I remember correctly, the number of dots in such a notation marks the precedence, hence this is $ S.\to.P\to Q$ and $S\to P.\to.S\to Q$ joined by a $\to$ and similarly, $ S.\to.P\to Q$ is $S$ and $P\to Q$ joined by a $\to$ and $S\to P.\to.S\to Q$ is $S\to P$ and $S\to Q$ joined by a $\to$. Thus in "readable" form all this becomes $$ \bigl(S\to(P\to Q)\bigr)\to\bigl((S\to P)\to(S\to Q)\bigr)$$ Note that the dot-count is reflected here by the size of parentheses.

Hagen von Eitzen
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