Editing Principia Mathematica (section)

=== An introduction to the notation of "Section A Mathematical Logic" (formulas ✱1–✱5.71) ===

''PM''{{'}}s dots<ref>The original typography is a square of a heavier weight than the conventional full stop.</ref> are used in a manner similar to parentheses. Each dot (or multiple dot) represents either a left or right parenthesis or the logical symbol ∧. More than one dot indicates the "depth" of the parentheses, for example, "'''.'''", "''':'''" or "''':.'''", "'''::'''". However the position of the matching right or left parenthesis is not indicated explicitly in the notation but has to be deduced from some rules that are complex and at times ambiguous. Moreover, when the dots stand for a logical symbol ∧ its left and right operands have to be deduced using similar rules. First one has to decide based on context whether the dots stand for a left or right parenthesis or a logical symbol. Then one has to decide how far the other corresponding parenthesis is: here one carries on until one meets either a larger number of dots, or the same number of dots next that have equal or greater "force", or the end of the line. Dots next to the signs ⊃, ≡,∨, =Df have greater force than dots next to (''x''), (∃''x'') and so on, which have greater force than dots indicating a logical product ∧.

Example 1. The line
:✱'''3.4'''. ⊢ ''':''' p '''.''' q '''.''' ⊃ '''.''' p ⊃ q
corresponds to
:⊢ ((p ∧ q) ⊃ (p ⊃ q)).
The two dots standing together immediately following the assertion-sign indicate that what is asserted is the entire line: since there are two of them, their scope is greater than that of any of the single dots to their right. They are replaced by a left parenthesis standing where the dots are and a right parenthesis at the end of the formula, thus:
:⊢ (p '''.''' q '''.''' ⊃ '''.''' p ⊃ q).
(In practice, these outermost parentheses, which enclose an entire formula, are usually suppressed.) The first of the single dots, standing between two propositional variables, represents conjunction. It belongs to the third group and has the narrowest scope. Here it is replaced by the modern symbol for conjunction "∧", thus
:⊢ (p ∧ q '''.''' ⊃ '''.''' p ⊃ q).
The two remaining single dots pick out the main connective of the whole formula. They illustrate the utility of the dot notation in picking out those connectives which are relatively more important than the ones which surround them. The one to the left of the "⊃" is replaced by a pair of parentheses, the right one goes where the dot is and the left one goes as far to the left as it can without crossing a group of dots of greater force, in this case the two dots which follow the assertion-sign, thus
:⊢ ((p ∧ q) ⊃ '''.''' p ⊃ q)
The dot to the right of the "⊃" is replaced by a left parenthesis which goes where the dot is and a right parenthesis which goes as far to the right as it can without going beyond the scope already established by a group of dots of greater force (in this case the two dots which followed the assertion-sign). So the right parenthesis which replaces the dot to the right of the "⊃" is placed in front of the right parenthesis which replaced the two dots following the assertion-sign, thus
:⊢ ((p ∧ q) ⊃ (p ⊃ q)).

Example 2, with double, triple, and quadruple dots:
:'''✱9.521'''. ⊢ : : (∃x). φx . ⊃ . q : ⊃ : . (∃x). φx . v . r : ⊃ . q v r
stands for
:((((∃x)(φx)) ⊃ (q)) ⊃ ((((∃x) (φx)) v (r)) ⊃ (q v r)))

Example 3, with a double dot indicating a logical symbol (from volume 1, page 10):
:''p''⊃''q'':''q''⊃''r''.⊃.''p''⊃''r''
stands for
:(''p''⊃''q'') ∧ ((''q''⊃''r'')⊃(''p''⊃''r''))
where the double dot represents the logical symbol ∧ and can be viewed as having the higher priority as a non-logical single dot.

Later in section '''✱14''', brackets "[ ]" appear, and in sections '''✱20''' and following, braces "{ }" appear. Whether these symbols have specific meanings or are just for visual clarification is unclear. Unfortunately the single dot (but also "''':'''", "''':.'''", "'''::'''", etc.) is also used to symbolise "logical product" (contemporary logical AND often symbolised by "&" or "∧").

Logical implication is represented by Peano's "Ɔ" simplified to "⊃", logical negation is symbolised by an elongated tilde, i.e., "~" (contemporary "~" or "¬"), the logical OR by "v". The symbol "=" together with "Df" is used to indicate "is defined as", whereas in sections '''✱13''' and following, "=" is defined as (mathematically) "identical with", i.e., contemporary mathematical "equality" (cf. discussion in section '''✱13'''). Logical equivalence is represented by "≡" (contemporary "if and only if"); "elementary" propositional functions are written in the customary way, e.g., "''f''(''p'')", but later the function sign appears directly before the variable without parenthesis e.g., "φ''x''", "χ''x''", etc.

Example, ''PM'' introduces the definition of "logical product" as follows:
:'''✱3.01'''. ''p'' '''.''' ''q'' '''.'''='''.''' ~(~''p'' v ~''q'') '''Df'''.
:: where "''p'' '''.''' ''q''" is the logical product of ''p'' and ''q''.
:'''✱3.02'''. ''p'' ⊃ ''q'' ⊃ ''r'' '''.'''='''.''' ''p'' ⊃ ''q'' '''.''' ''q'' ⊃ ''r'' '''Df'''.
:: This definition serves merely to abbreviate proofs.

'''Translation of the formulas into contemporary symbols''': Various authors use alternate symbols, so no definitive translation can be given. However, because of criticisms such as that of [[Kurt Gödel]] below, the best contemporary treatments will be very precise with respect to the "formation rules" (the syntax) of the formulas.

The first formula might be converted into modern symbolism as follows:<ref>The first example comes from plato.stanford.edu (loc.cit.).</ref>
: (''p'' & ''q'') =<sub>df</sub> (~(~''p'' v ~''q''))
alternately
: (''p'' & ''q'') =<sub>df</sub> (¬(¬''p'' v ¬''q''))
alternately
: (''p'' ∧ ''q'') =<sub>df</sub> (¬(¬''p'' v ¬''q''))
etc.

The second formula might be converted as follows:
: (''p'' → ''q'' → ''r'') =<sub>df</sub> (''p'' → ''q'') & (''q'' → ''r'')
But note that this is not (logically) equivalent to (''p'' → (''q'' → ''r'')) nor to ((''p'' → ''q'') → ''r''), and these two are not logically equivalent either.