Extra Credit: Polish Notation
We have specified the syntax of PL by setting out the symbols of the language and by providing formation rules. Our choice of symbols is just that, a matter of choice. Other logic texts use other symbols, and it's important to know that. Some texts us a dot for conjunction. Others use an arrow for the conditional and a double pointed arrow for the biconditional.
We noted that except for the tilde, our operators are infix operators - they go between wffs to form new wffs. The tilde is a prefix operator. It goes in front of (pre) wffs. There is an alternative notation that uses only prefix operators, and it also dispenses with parentheses or other grouping indicators. It is called Polish Notation because it was developed by logicians in Poland.
Let's begin with the symbols for the operators in Polish Notation:
| English | PL | Polish Notation |
| not | ~ | N |
| and | & | K |
| or | v | A |
| if ... then | ⊃ | C |
| if and only if | ≡ | E |
As you can see, the symbols for the operators are completely different from those of PL. The choice of letters is based on the German and Polish words for the operators. "N" is for "Negation," "K" is for "Konjunction," "A" is for "Alternation," "C" is for "Condition," and "E" is for "Equivalenz."
Another difference between Polish Notation and ours is that sentence letters in Polish Notation are lower case letters, to distinguish them from the operators. The final difference is that there are no parentheses in Polish Notation.
How does it work? Polish notation is completely prefix. Let's start with some simple examples:
| PL | Polish Notation |
| ~A | Na |
| (A & B) | Kab |
| (A v B) | Aab |
| (A ⊃ B) | Cab |
| (A ≡ B) | Eab |
What happens when you have more than one operator? For
example, how do you write the negation of a conditional in Polish notation?
You simply make sure that the main operator is on the left. Here are some
examples:
| PL | Polish Notation |
| ~ (A ⊃ B) | NCab |
| (A & B) v (C ≡ D) | AKabEcd |
| ~((A ≡ B) ≡ ~(A ⊃ ~B)) | NEEabNCaNb |
Have you ever heard of reverse Polish Notation? Many calculators have it. You input the numbers on which you wish to perform an operation. Then you enter the operation. For example, to add 2 and 3, you enter 2, 3, and then the addition key. It's reverse Polish because the operator is postfix, while in Polish Notation it is prefix.
One nice feature of Polish Notation is that it uses only letters from our natural language alphabet. No special characters are used.
The Polish Notation Exercise is available for your logical pleasure - much better than watching television, but perhaps not better than this.
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