# Importance of Logical Notation

Does better notation lead to ease of abstraction and shorter proofs? I ask because I tried translating the following from Euclid’s Elements into my own idiosyncratic notation: Prime numbers are more than any assigned multitude of prime numbers. This can be translated as follows:

1. A=X is a set
2. B=X is finite
3. C=X contains only prime numbers
4. D=Y is prime
5. E=Y has the property of not being contained in X
6. ∀X∃Y(ABC→DE) Note, (6) should be translated as thus: For all X there exists Y such that if X is a set, is finite, and contains only prime numbers, then Y is prime and Y has the property of not being contained in X. All you would have to do prove (6) is to show that its negation is unsatisfiable. Or if the negation is satisfied, it would lead to a contradiction. Note, my idiosyncratic notation was inspired by Boole’s use of multiplication for conjunction and addition for disjunction.
• Wrt "better notation" in general, the impressive development of modern mathematics in Early Modern Europe is due to the invention of algebraic symbolism and its later modification to be applied to calculus. Commented May 29 at 9:46
• Specificlly regarding your attempt, you have to use predicate logic. Commented May 29 at 9:47
• @AUTISTINC "Prime numbers are more than any assigned multitude of prime numbers." If that's a quote from Euclid, could you please give a reference which also shows the context ; thanks. Commented May 29 at 10:00
• @JoWehler - the meaning is simply: assume that there are n prime numbers: p1, p2, ..., pn; then there is a prime p(n+1) that is different from (and strictly greater than) all the previous ones. Commented May 29 at 10:07
• @AUTISINC Thanks, now I see what Euclid means. He means: For each finite number of primes there is a bigger prime. It is not difficult to understand the meaning of this statement. And I do not see that what you call your "idiosyncratic notation" is better. Commented May 29 at 10:17

The matter of mathematical style is not necessarily trivial. That SEP article quotes a certain Granger[68] such that:

These different ways of grasping a concept, of integrating it in an operative system and of associating to it some intuitive implications—of which one will have to delimit the exact extent—constitute what we call aspects of style. It is evident that the structural content of the notion is not here affected, that the concept qua mathematical object subsists identically through these effects of style. It is however not always so and we will encounter stylistic positions which demand true conceptual variations. What changes always, in any case, is the orientation of the concept towards this or that usage, this or that extension. Thus, style plays a role that is perhaps essential both with respect to the dialectic of the internal development of mathematics and to that of its relation to worlds of more concrete objects.

And so for example there is the Lvov-Warsaw school of logic to consider:

Łukasiewicz invented a parenthesis-free logical notation. ... The parenthesis-free notation is unambiguous in the sense that any finite sequence of symbols for connectives and variables is interpretable in a unique way. This implies that any wff coded in Polish notation has only one translation into the standard symbolism. The main advantage of Polish notation is its economy, because it avoids special punctuation devices such as brackets or dots. When Łukasiewicz met Turing in 1949, the latter remarked that Polish notation was much better for computers, because formulas with function-symbols in front could be better elaborated by mechanical devices.

And then:

The parenthesis-free symbolism was closely associated with some ideas of Polish logicians concerning the good properties of formal systems. Of course, any correct logical system should be consistent and, if possible, syntactically and semantically complete. It should also be based on independent sets of primitive terms and axioms. The Warsaw School of Logic strongly emphasized the last property, often considered secondary. Thus, the dependence of primitive terms or axioms was regarded as an essential defect. Moreover, some additional structural properties of logical systems were recommended: (a) a system with fewer primitive concepts is better; (b) a system with fewer axioms is better; (c) if we define the length of an axiom system as the number of symbols occurring in all of its axioms, the shortest axiom system is the best; (d) a system with fewer different symbols is better; (e) if we define an organic theorem as one which has no other theorem inside it (for example, the formula CpCqq is not an organic theorem), organic axioms are better than non-organic ones. Thus, the ideal axiom system consists of a sole organic axiom of the shortest possible length, provided that it is consistent.

The related article on Łukasiewicz specifically says:

On this basis, and using an extremely compressed linear notation for proofs which is at the opposite extreme of Frege’s space-occupying proofs, Łukasiewicz proves around 140 theorems in a mere 19 pages. ... They strove to find axioms sets satisfying a number of normative criteria: axioms should be as few as possible, as short as possible, independent, with as few primitives as possible. Undoubtedly there was a competitive element to the search for ever better axiom systems, in particular in the attempt to find single axioms for various systems, and the exercise has been smiled upon or even belittled as a mere “sport”, but the Polish preoccupation with improving axiom systems was a search for logical perfection, an illustration of what Jan Woleński has termed “logic for logic’s sake”. At one time it was thought, not without some justification, that only Poles could compete. When Tarski once congratulated the American logician Emil Post on being the only non-Pole to make fundamental contributions to propositional logic, Post replied that he had been born in Augustów and his mother came from Białystok. Later, Łukasiewicz was to find in the Irish mathematician Carew Meredith a worthy non-Pole who could outdo even the Poles in the brevity of his axioms (see Meredith 1953).