Translate $\sf CKKCpqCqrpr$ into classical notation

Question

From polish to classical notation

$\sf CKKCpqCqrpr$

I have $((p\to q)\land(q\to r)\land p)\to r~ $ for "$\sf CpqCqrpr $" but I don't understand what $\sf CKK$ means in the beginning.

Answer 1

No, the string of $\sf CpqCqrpr$ is not a well formed formulae .

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A string of the form $\sf CKKabcd$ translates to $\sf((a\land b)\land c)\to d$, as follows:

$$\sf{CKKabcd\\CK(a{\land}b)cd\\C((a{\land}b){\land}c)d\\((a{\land}b){\land}c){\to}d}$$

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So it is the entire statement of $\sf CKKCpqCqrpr$ that translates as $((p\to q)\land(q\to r)\land p)\to r~$ as follows:$$\sf{CKKCpqCqrpr\\CKK(p{\to}q)(q{\to}r)pr\\(((p{\to}q){\land}(q{\to}r)){\land}p){\to}r}$$

Answer 2

Graham Kemp has already explained how to translate the specific example you're looking at; let me try to address what I think is the source of your confusion.

I think the issue is with how you're interpreting concatenation - it sounds like you think concatenation is how we write conjunction in Polish notation, so that e.g. "$Cpqr$" means "$(p\rightarrow q)\wedge r$." This is not the case. Conjunction is denoted by "$K$" which behaves syntactically just like "$C$" does, so "$(p\rightarrow q)\wedge r$" in Polish notation would be written "$KCpqr$."

So what does concatenation mean? The answer is:

Nothing whatsoever!

An expression like "$Cpqr$" is not meaningful: it has two maximal well-formed components, namely "$Cpq$" and "$r$," but the whole thing isn't a complete expression yet. Similarly, the expression "$CpqCqrpr$" you consider in the OP is not meaningful on its own: it's just four well-formed formulas put next to each other, namely "$Cpq$," "$Cqr$," "$p$," and "$r$." Additional connectives are needed to combine these into a single expression, and this is exactly what the "$CKK$" at the front is doing.