In mathematics and logic expressions often get written with the operation in infix notation... we write a+b, a^b, a v b. As logicians and mathematicians know this means that we either have to establish an order of operations when we have more than operation involved, such as Exponents, Multiplication, Division, Addition, Subtraction for arithmetic, or use parentheses, or have ambiguous expressions. Since at least the time of Lukasiewicz, mathematical and logical scholars have known that we can stop excusing our dear Aunt Sally. In Lukasiewicz notation instead of writing (5+2)*4, we can simply write *+5 2 4. In reverse Lukasiewicz notation we can write 5 2+4*. I highly recommend that you read the link there, and translate some common logical or mathematical expressions into reverse Lukasiewicz notation. Since these notational systems establish that we can write mathematical and logical expressions in prefix notation (Lukasiewicz notation), infix notation, and postfix notation (reverse Lukasiewicz notation), can we write mathematical and logical expressions in a way such that the operations involved can come *anywhere* in the expression *without* having a fixed order as declined languages do?
For variables or constants, suppose we write a unique subscript below each of them. For an n-ary operation suppose we write n sequenced subscripts which correspond to the subscripts in the order of variables or constants it operates on. E. G. if we write xa yb+ab, then we indicate that the operation "+" operates on x and y. We can then immediately write xa+abyb as well as +abxayb. So, suppose we have a function F such that a b cF=b c*a+. Then we could write 4c8dFcba6a7e+de2b. So, do you believe that this gives us an order free notation for the operations in a logical or mathematical expression? Please answer this question and write a response down on paper before reading my further comment.
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As the example illustrates the order in which F and + get "computed" isn't clear, so the method as described doesn't work. One might try and patch it up by saying that we need to use finite ordinals for the subscripts, and that each individual sequence of ordinals deals only with immediate successors and predecessors (i. e. we can only use sequences like 23 or 32, but not 42 or 24). Does this create an order and parenthesis free notation?
ReplyDeleteI would say no, it hasn't. It has created an order-free notation for the explicit operations which have subscripts like +ab (read ab as a subscript here) above in the computations we first have to look for the operations with their subscripts. But all the subscripted symbols for n-ary functions where n>=2, themselves come as ordered operations from the function to the subscripted symbol to the constant or variable which matches it (it's a composition of operations). However, the subscripts still have to come in a definite order. So, although one may say that the need for order or parentheses has gotten pushed back a step, it still exists in the expression.