Łukasiewicz developed his prefix system of logical notation (also called 'Polish' or 'normal Polish' notation) in the 1920s as an alternative to the infix notation which arose from mathematical notation, Russell & Whitehead's Principa Mathematica, and other sources. The main advantage of his system over infix or Russellian notation was its absence of parenthesis, required by the latter to avoid ambiguity in the scope of operators.
For instance, we represent the unwieldy infix \(\neg ((p\leftrightarrow (q\land r))\to(p\lor r))\) as NCEpKqrApr.
For contemporary people interested in using logical notation, the more pressing problem with Russellian notation is that it is difficult to represent using a computer, and, unless using a typesetting engine like \(\LaTeX{}\), it almost invariably looks bad. Outside of a formal context, it is extremely inconvenient to use Russellian notation on a computer, especially on the web – though FB Messenger and other platforms now support the necessary LaTeX rendering via software like MathJax, it's awkward to write. Łukasiewicz is better, but it's difficult for most people to remember the right symbols since they're derived from Polish. Hence a different schema, more closely related to Russellian notation.
In the table below, compare each operation's (named informally) representation in infix notation (in LaTeX mathmode and plain text) with classical Łukasiewicz notation and my proposed scheme.
| Operation | Infix (LaTeX) | Infix (plain) | Łukasiewicz | Proposal |
|---|---|---|---|---|
| Not | \(\neg p\) or \(\sim\! p\) | ¬p or ~p | Np | ¬p or ~p |
| Or | \(p\lor q\) | pvq | Apq | +pq |
| And | \(p\land q\) or \(p\cdot q\) &c. | p&q or p.q &c. | Kpq | &pq or *pq |
| Implies | \(p\to q\) or \(p\supset q\) | p→q | Cpq | ?pq |
| Iff | \(p\leftrightarrow q\) | p↔q | Epq | =pq |
| Operation | Infix (LaTeX) | Infix (plain) | Łukasiewicz | Proposal |
|---|---|---|---|---|
| All | \(\forall x(Fx)\) or \((x)(Fx)\) | (x)(Fx) or ∀x(Fx) | ΠxFx | *xFx |
| Some | \(\exists x(Fx)\) | ∃x(Fx) | ΣxFx | +xFx |
| Operation | Infix (LaTeX) | Infix (plain) | Łukasiewicz | Proposal |
|---|---|---|---|---|
| Strong | \(\Box p\) | □p | Lp | See below |
| Weak | \(\Diamond p\) | ◇p | Mp | See below |
Please note that my proposal is not that people should give up Russellian notation globally – in an academic context, for instance, or anywhere the notation should be pretty (here one has no option but to use LaTeX). It is rather that in informal contexts where the use of rendering software is unwarranted (e.g. email and social media discussion), Russellian notation is inappropriate, and we should use more user-friendly notation instead.
One of the advantages of the proposed notation is that it is consistent with the conventions for Russellian notation: p, q, r, ... are propositions; x, y, z, ... are object variables; &c. Majuscule romans are not taken up as logical operators.
Rationale for symbol choice
Not: ¬p or ~p: these are easy to type, especially '~'. One can even maintain the distinction between '~' for classical negation and '¬' for, e.g., intuitionistic negation.
Or: +pq: disjunction is logical sum, like addition is numerical sum.
And: &pq or *pq: the ampersand is a symbol for 'and' ('et'); conjunction is logical product, like multiplication is numerical product.
Implies: ?pq: this is maybe the strangest choice. It comes from the ternay ?: operator in programming: 'a?b:c' means roughly 'if a, then return b; else return c' (we just drop the third term). We can also think of ?pq thus: 'is p the case? then q is.'
Iff: =pq: here p and q have the same truth-value (in Fregean terms, they have the same extension – denote the same thing).
All: *xFx: universal quantification works like conjunction: *xFx iff *Fa*Fb*Fc... Some logicians write \(\bigwedge x(Fx)\)
Some: +xFx: existential quantification works like disjunction: +xFx iff +Fa+Fb+Fc... Some logicians write \(\bigvee x(Fx)\)
The modal operators
The standard notation for the modal operators is \(\Box p\ \Diamond p\), but there a few alternatives – Gödel even used N (for 'necessarily') for the strong operator. One can safely adopt Łukasiewicz's Lp and Mp, but this means they can't be used for predicates &c. I have seen []p <>p, approximating the box and diamond; these are an option, but aren't great either. A hesitant suggestion would be to drop the second character of the operator: [p <p. Here one also has the option of explicitly marking the end of the operator scope as 'syntactic sugar' (like removing the outermost parenthesis in Russellian notaion). E.g., LKpq becomes [*pq or [*pq]; MKpq becomes <*pq or <*pq>.
(This is even extendable to other modalities, where [ and < are taken for logical necessity and possibility – we can have {p or {p}, (p or (p), &c.)
