Polish notation is a mathematical notation system that allows you to eliminate parentheses without ambiguity. It’s called “Polish” because the name of its Polish creator, Jan Łukasiewicz, was too difficult for people to pronounce.
A motivating example: Suppose somebody says “p and q implies r”. There are two possible interpretations of this: “(p and q) implies r” and “p and (q implies r)”. The usual way to disambiguate these two is to simply add in parentheses like I just did. Another way is to set an order-of-operations convention, like that “and” always applies before “implies”. This is what’s used in basic algebra, and what allows you to write 2 + 2 ⋅ 4 without any fear that you’ll be interpreted as meaning (2 + 2) ⋅ 4.
Łukasiewicz’s method was to make all binary connectives into prefixes. So “A and B” because “and A B”, “P implies Q” becomes “implies P Q”, and so on. In this system, “(p and q) implies r” translates to “implies and p q r”, and “p and (q implies r)” translates to “and p implies q r”. Since the two expressions are different, there’s no need for parentheses! And in general, no ambiguity ever arises from lack of parentheses when using Polish notation.
If this is your first time encountering Polish notation, your first reaction might be to groan and develop a slight headache. But there’s something delightfully puzzling about reading an expression written in Polish notation and trying to understand what it means. Try figuring out what this means: “implies and not p or q s r”. Algebra can be written in Polish notation just as easily, removing the need for both parentheses AND order-of-operations. “2 + 2 = 4” becomes “+ 2 2 = 4”, or even better, “= + 2 2 4”.
Other binary connectives can be treated in Polish notation as well, creating gems like: “If and you’re happy you know it clap your hands!” “When life is what happens you’re busy making plans.” “And keep calm carry on.” “Therefore I think, I am.” (This last one is by of the author the Meditations). Hopefully you agree with me that these sentences have a nice ring to them, though the meaning is somewhat obscured.
But putting connectives in front of the two things being connected is not unheard of. Some examples in English: “ever since”, “because”, “nonwithstanding”, “whenever”, “when”, “until”, “unless”. Each of these connects two sentences, and yet can appear in front of both. When we hear a sentence like “Whenever he cheated on a test the professor caught him”, we don’t have any trouble parsing it. (And presumably you had no trouble parsing that entire last sentence either!) One could imagine growing up in a society where “and” and “or” are treated the same way as “ever since” and “until”, and perhaps in this society Polish notation would seem much more natural!
Slightly related to sentential connectives are verbs, which connect subjects and objects. English places its verbs squarely between the subject and the object, as does Chinese, French, and Spanish. But in fact the most common ordering is subject-object-verb! 45% of languages, including Hindi, Japanese, Korean, Latin, and Ancient Greek, use this pattern. So for instance, instead of “She burned her hand”, one would say “she her hand burned”. This is potentially weirder to English-speakers than Polish notation; it’s reverse Polish notation!
9% of languages use Polish notation for verbs (the verb-subject-object pattern). These include Biblical Hebrew, Arabic, Irish, and Filipino. In such languages, it would be grammatical to say “Loves she him” but not “She loves him”. (3% of languages are VOS – loves him she – 1% are OVS – him loves she – and just a handful are OSV – him she loves).
Let’s return to English. Binary prepositions like “until” appear out front, but they also swap the order of the two things that they connect. For instance, “Until you do your homework, you cannot go outside” is the same as “You cannot go outside until you do your homework”, not “You do your homework until you cannot go outside”, which sounds a bit more sinister.
I came up with some examples of sentences with several layers of these binary prepositions to see if the same type of confusion as we get when examining Polish notation for “and” or “implies” sets in here, and oh boy does it.
Single connective
Since when the Americans dropped the bomb the war ended, some claimed it was justified.
Two connectives, unlayered
Since when the Americans dropped the bomb the war ended, when some claimed it was an atrocity others argued it was justified.
Still pretty readable, no? Now let’s layer the connectives.
One layer
Whenever he was late she would weep.
She would weep whenever he was late.
Two layers
Since whenever he was late she would weep, he hurried over.
He hurried over, since she would weep whenever he was late.
Three layers
Because since whenever he was late she would weep he hurried over, he left his wallet at home.
He left his wallet at home, because he hurried over since she would weep whenever he was late.
Four layers
Because because since whenever he was late she would weep he hurried over he left his wallet at home, when he was pulled over the officer didn’t give him a ticket.
The officer didn’t give him a ticket when he was pulled over, because he left his wallet at home because he hurried over since she would weep whenever he was late.
Five layers
When he heard because because since whenever he was late she would weep he hurried over he left his wallet at home, when he was pulled over the officer didn’t give the man a ticket, the mayor was outraged at the lawlessness.
The mayor was outraged at the lawlessness when he heard the officer didn’t give the man a ticket when he was pulled over because he left his wallet at home because he hurried over since she would weep whenever he was late.
Read that last one out loud to a friend and see if they believes you that it makes grammatical sense! With each new layer, things become more and more… Polish. That is, indecipherable. (Incidentally, Polish is SVO just like English). Part of the problem is that when we have multiple layers like this, phrases that are semantically connected can become more and more distant in the sentence. It reminds me of my favorite garden-path sentence pattern:
The mouse the cat the dog chased ate was digested.
(The mouse that (the cat that the dog chased) ate) was digested.
The mouse (that the cat (that the dog chased) ate) was digested.
The phrases that are meant to be connected, like “the mouse” and “was digested” are sandwiched on either side of the sentence, and can be made arbitrarily distant by the addition of more “that the X verbed” clauses.
Does anybody know of any languages where “and” comes before the two conjuncts? What about “or”? English does this with “if”, so it might not be too much of a stretch.
Regarding the question at the end, maybe it doesn’t show up since the symbol is just left out and to be derived from context.
Imagine a paper boy, listing what stories or news he has to sell to you, or a guy at a market asking you what you want.
“Fresh melons, fresh lemons…?”