### Books on Polish Logic

•Philonous
Forum Veteran Usergroup: Members Joined: Apr 21, 2011 Total Topics: 13 Total Posts: 677 |
Posted Apr 30, 2013 - 2:25 AM:
Subject: Books on Polish Logic I am reading through some papers in journals on logic, and one of them uses a notation developed by Jan Lukasiewicz. One of the papers is by A.N. Prior and he uses Polish Notation in logic. For example: ~x = Nx x & q = Kxq x v q = Axq x--->y = Cxy Lx = Necessarily x Mx = Possibly x I'm wondering if anyone knows of a logic text book that uses Polish notation to help me get to understand them a little better, just to see how it all works out with the mechanics. |

•andrewk
Inexhaustibly Curious Usergroup: Moderators Joined: Oct 13, 2011 Location: Sydney, Australia Total Topics: 47 Total Posts: 5725 Last Blog: Standing on one leg. Number One in a series of Adult Amusements |
Posted Apr 30, 2013 - 5:06 AM:
The wikipedia article on it gives a pretty good explanation. http://en.wikipedia.org/wiki/Polish_notation It's just a different notation by the way. The logic is exactly the same. I think what's in that article is all there is to it. It's a prefix notation rather than the usual infix notation, meaning that the application of binary operators is written as: <Operator> <1st Operand> <2nd Operand> rather than the infix <1st Operand><Operator> <2nd Operand> If you've ever used a Hewlett Packard calculator, you'll be familiar with Reverse Polish Notation, which is postfix: <1st Operand> <2nd Operand><Operator> It looks like it also uses different symbols for the operators: N for not, K for And, A for Or etc. The main ones are listed in the wiki article. One advantage of Polish notation is that you don't need any brackets or rules of precedence of operators. The disadvantage is that it's harder to read. |

•Willemien
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Posted Apr 30, 2013 - 7:20 AM:
Welcome to the world of Polish notation Andrewk is right in that it is just a different notation, but there are other advantages as well, but I do disagree that it is harder to read, you just need to get used to it. (and of course I can name some psychological reasons for it) I guess the book you are refering to is "Formal Logic" by A.N Prior, another book in Polish Notation is "Modal Logic" by Zeman freely downloadable at: http://www.clas.ufl.edu/users/jzeman/modallogic/ Zeman also names other advantages of Polish notation ( Ćukasiewicz parenthesis-free notation) : " The researcher himself gains an obvious advantage in his choice of this notation, for almost all of the characters he will use in setting down his results are to be found on a standard typewriter keyboard. But the notation gives him another more important advantage. It lends itself exceptionally well to the use of the rules of inference commonly used with propositional calculi : substitution for variables and detachment. " (Zeman refers to condenced detachment a strong way to generate theorems, see http://en.wikipedia.org/wiki/Condensed_detachment sadly it only works with polish notation) Other advantages of Polish notation are: - formula's in Polish notation are shorter,(around half the size, and how more complicated the formula , the bigger the difference) - the main connective is always easy to find (it is just the first character) Problems are off course - it is sadly not used so much anymore. (although I am writing a computer program that works with it, Polish notation is just so much easier for computers) - formulas look rather alien at first, but that is because it is another language. Some hints on how to learn to use it. - Don't translate polish formula's to infix formula's, just because you can, it is better to treat polish formula's as a seperate logic language, follow its flow and treat it atfirst as a seperate logic. A problem here is also that formula's grow in this translation. CCpCqrCCpqCpr (13 characters) translates to (P -> (Q -> R)) -> ((P -> Q) -> (P ->R)) (23 characters, counting -> as one character) ENApqKNpNq( 10 characters) translates to ~(P v Q) <-> (~P & ~Q) (14 characters) CCCCCpqCrostCCtpCrp (single axiom C-o logic, Prior page 303) 19 characters (((((P -> Q) -> (R -> _|_) -> S) -> T) -> ((T -> P) -> (R -> P)))) 35 characters - learn to recognize well formed formulas and subformula's (there is a simple way see attached txt file) GOOD LUCK |

•andrewk
Inexhaustibly Curious Usergroup: Moderators Joined: Oct 13, 2011 Location: Sydney, Australia Total Topics: 47 Total Posts: 5725 Last Blog: Standing on one leg. Number One in a series of Adult Amusements |
Posted Apr 30, 2013 - 8:12 AM:
Willemien, I find those Polish formulas quite easy to understand reading from right to left, but very difficult to understand reading from left to right. Do you think that's a feature of the language, or just a bias induced in me by the languages I'm used to? Or is it perhaps just a feature of the particular formulae you wrote? Can you write a formula that's easier to read from left to right? Thinking about it, I think what I'm doing when reading from right to left is that, whenever the next symbol I read is not an operand, I |

•Willemien
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Posted Apr 30, 2013 - 11:40 AM:
Hi Andrew I think you try to hard to translate from Polish to Infix, |

•andrewk
Inexhaustibly Curious Usergroup: Moderators Joined: Oct 13, 2011 Location: Sydney, Australia Total Topics: 47 Total Posts: 5725 Last Blog: Standing on one leg. Number One in a series of Adult Amusements |
Posted Apr 30, 2013 - 7:52 PM:
Thanks Willemien. Your examples show nicely how one can keep track of things working from left to right. Your 'deduct 1' corresponds to what I think of as 'pushing up the stack' in my computer paradigm, and 'add 1' corresponds to 'pop off the stack'. When we go from left to right we push one and two-place connectives up into the stack as well as variables and zero-connectives. This is nice and methodical. On reflection though, I still find it easier to work from right to left, because then we only need to push variables and zero-connectives into the stack. One and two-place connectives are used as soon as they are encountered. This means the stack usually doesn't get as high when working from right to left. For example, taking your example of CCpCqrCCpqCpr: Working right to left (the yellow highlighted C is a character that I erroneously omitted in the first version of this post. This version is fixed). We can read off the subformulas from this by just taking the bottom (leftmost) element of the stack at each step. There are 13 steps, one for each symbol in the formula, which gives us 13 subformulas. Now working from left to right. Going this way we push any variables or zero-connectives onto the stack and we execute the lowest (leftmost) one-connective or two-connective on the stack as soon as it has the required operand(s) below it. This has 19 steps, but it would reduce to 16 if we combine the execution of a connective with the preceding step where that preceding step is a 'read-and-push'. But that is still more steps than working from right to left. The stack gets bigger working in this direction, because we have to push connectives into it as well as variables. Working from right to left a connective is used and disposed of as soon as it is read. I think that may be why reverse polish (postfix) is so commonly used. Reverse polish is just normal polish, written in reverse order. With reverse polish we work from left to right, as we are accustomed to, but obtain the efficiency benefits of the first of the above examples. It would be interesting to see if there is any formula that generates a higher stack and/or a lower number of steps in evaluation, working from right to left than left to right. I can't think of one. Edited by andrewk on May 1, 2013 - 8:06 PM. Reason: Fixed error & restored inadvertent deletion |

•Philonous
Forum Veteran Usergroup: Members Joined: Apr 21, 2011 Total Topics: 13 Total Posts: 677 |
Posted May 1, 2013 - 12:50 AM:
Willemien wrote: I guess the book you are refering to is "Formal Logic" by A.N Prior, another book in Polish Notation is "Modal Logic" by Zeman freely downloadable at: http://www.clas.ufl.edu/users/jzeman/modallogic/ Thank you. The paper I was reading was A.N. Prior, and he did use the Polish Notation so it makes sense to check out his book on it. And I do also recall coming across the very book you linked, I just forgot who the author was. Thanks for that link and I shall use it. I recently came across this notion of polish notation, which is simpler to present on a computer screen, at least for the person typing it out. I was also interested in the works of Lukasiewicz. What he did was create a three value logic system and from this we can come up with multi valued logics, and one of the ways that they do this is done by some Matrix system. Does anyone know of a book that presents this Matrix system to come up with two valued, three valued, or more valued logical systems? |

•Willemien
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Posted May 1, 2013 - 9:17 AM:
to Philonous what you are talking about is called many valued logic, http://en.wikipedia.org/wiki/Many-valued_logic http://plato.stanford.edu/entries/logic-manyvalued/ I am not well versed in them, it is al semantic model theory to me so maybe I should read more about it. I did some study in lattice theory (a related field i guess, but am not sure in how far they coincide or not) Maybe you can advice me on a good book about it. (ps i just bought a second hand copy of "Aristotle's Syllogistic: From the Standpoint of Modern Formal Logic" Jan Lukasiewicz; Hardcover; £15.00 , my first book by him) also Aristo in forums.philosophyforums.com...out-mathematics-60793.html is busy with multivalued logic let me know what you come up with to Andrew you are doing something completely different than me, I am splitting a formula up in subformula's (looking at the structure of the formula) you are evaluating formula's (simplifying them to a single (truth) value) The difference goes a bit back to my question "What is the concequent of CCpCqrCCpqCpr?" With your method you cannot even answer the question (which answer should be a formula) all your method does is evaluating the complete formula (the method doesn't even know when it has evaluated the consequent) Calculators are doing that as well they just generate a value (number), nothing else. for calculaters that is right , for logic it isn't. That doesn't mean that your method is wrong, both approuches have their values (i need to write an routine that does exactly what you are doing soonish so thanks for your help ) but it is a different part of logic. |

•ciceronianus
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Posted May 1, 2013 - 12:12 PM:
For a moment, I thought this thread might be a "Polish joke." I'm just saying. |

•andrewk
Inexhaustibly Curious Usergroup: Moderators Joined: Oct 13, 2011 Location: Sydney, Australia Total Topics: 47 Total Posts: 5725 Last Blog: Standing on one leg. Number One in a series of Adult Amusements |
Posted May 1, 2013 - 6:03 PM:
Well I've made a complete mess of this Willemien! In trying to reply to your post #8 I inadvertently edited my above post #6, so my reply has now over-written that one, to which you were replying. I shall try to restore it, but it'll be a bit tricky because the table formatting doesn't copy properly. I apologise for the inconvenience. Normal service will be resumed shortly. ETA: Fixed now. Crikey these WSN tables are unreliable b**rs to try and edit, quote or copy. Most of the time they just don't work at all, despite my trying using several different browsers and operating systems! I gave up in the end and just used hyphens to space out the columns. Edited by andrewk on May 1, 2013 - 8:10 PM |

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