Editing Principia Mathematica (section)

== Notation ==

{{main|Glossary of Principia Mathematica}}

One author<ref name="SEP"/> observes that "The notation in that work has been superseded by the subsequent development of logic during the 20th century, to the extent that the beginner has trouble reading PM at all"; while much of the symbolic content can be converted to modern notation, the original notation itself is "a subject of scholarly dispute", and some notation "embodies substantive logical doctrines so that it cannot simply be replaced by contemporary symbolism".<ref>{{cite book|url=https://plato.stanford.edu/archives/fall2016/entries/pm-notation/|title=The Stanford Encyclopedia of Philosophy|first=Bernard|last=Linsky|editor-first=Edward N.|editor-last=Zalta|year=2018|publisher=Metaphysics Research Lab, Stanford University|access-date=1 May 2018|via=Stanford Encyclopedia of Philosophy}}</ref>

[[Kurt Gödel]] was harshly critical of the notation: "What is missing, above all, is a precise statement of the syntax of the formalism. Syntactical considerations are omitted even in cases where they are necessary for the cogency of the proofs."<ref name="Gödel 1944" /> This is reflected in the example below of the symbols "''p''", "''q''", "''r''" and "⊃" that can be formed into the string "''p'' ⊃ ''q'' ⊃ ''r''". ''PM'' requires a ''definition'' of what this symbol-string means in terms of other symbols; in contemporary treatments the "formation rules" (syntactical rules leading to "well formed formulas") would have prevented the formation of this string.

'''Source of the notation''': Chapter I "Preliminary Explanations of Ideas and Notations" begins with the source of the elementary parts of the notation (the symbols =⊃≡−ΛVε and the system of dots):
:"The notation adopted in the present work is based upon that of [[Giuseppe Peano|Peano]], and the following explanations are to some extent modeled on those which he prefixes to his ''Formulario Mathematico'' [i.e., Peano 1889]. His use of dots as brackets is adopted, and so are many of his symbols" (''PM'' 1927:4).<ref>For comparison, see the translated portion of Peano 1889 in van Heijenoort 1967:81ff.</ref>
PM changed Peano's Ɔ to ⊃, and also adopted a few of Peano's later symbols, such as ℩ and ι, and Peano's practice of turning letters upside down.

''PM'' adopts the assertion sign "⊦" from Frege's 1879 ''[[Begriffsschrift]]'':<ref>This work can be found at van Heijenoort 1967:1ff.</ref>
:"(I)t may be read 'it is true that'"<ref>And see footnote, both at PM 1927:92</ref>
Thus to assert a proposition ''p'' ''PM'' writes:
:"⊦'''.''' ''p''." (''PM'' 1927:92)
(Observe that, as in the original, the left dot is square and of greater size than the full stop on the right.)
<!-- [[Modus ponens]]<ref>This is PM's very-first "primitive proposition" '''✱1.1''' Anything implied by a true proposition is true. Pp† [† The letters "Pp" stand for "primitive proposition"., as with Peano (''PM'' 1927:94).</ref> is also written in the contemporary manner, i.e., stacked formulae above the long bar; the detached conclusion below it. -->

Most of the rest of the notation in PM was invented by Whitehead.<ref>{{cite book|author=Bertrand Russell|title=My Philosophical Development|chapter=Chapter VII|year=1959}}</ref>

=== 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.

=== An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✱8–✱14.34) ===

These sections concern what is now known as [[predicate logic]], and predicate logic with identity (equality).

:* NB: As a result of criticism and advances, the second edition of ''PM'' (1927) replaces '''✱9''' with a new '''✱8''' (Appendix A). This new section eliminates the first edition's distinction between real and apparent variables, and it eliminates "the primitive idea 'assertion of a propositional function'.<ref>p. xiii of 1927 appearing in the 1962 paperback edition to '''✱56'''.</ref> To add to the complexity of the treatment, '''✱8''' introduces the notion of substituting a "matrix", and the [[Sheffer stroke]]:
:::* '''Matrix''': In contemporary usage, ''PM''{{'}}s ''matrix'' is (at least for [[propositional function]]s), a [[truth table]], i.e., ''all'' truth-values of a propositional or predicate function.
:::* '''Sheffer stroke''': Is the contemporary logical [[Sheffer stroke|NAND]] (NOT-AND), i.e., "incompatibility", meaning:
::::"Given two propositions ''p'' and ''q'', then ' ''p'' | ''q'' ' means "proposition ''p'' is incompatible with proposition ''q''", i.e., if both propositions ''p'' and ''q'' evaluate as true, then and only then ''p'' | ''q'' evaluates as false." After section '''✱8''' the Sheffer stroke sees no usage.

Section '''✱10: The existential and universal "operators"''': ''PM'' adds "(''x'')" to represent the contemporary symbolism "for all ''x'' " i.e., " ∀''x''", and it uses a backwards serifed E to represent "there exists an ''x''", i.e., "(Ǝx)", i.e., the contemporary "∃x". The typical notation would be similar to the following:
: "(''x'') '''.''' φ''x''" means "for all values of variable ''x'', function φ evaluates to true"
: "(Ǝ''x'') '''.''' φ''x''" means "for some value of variable ''x'', function φ evaluates to true"

Sections '''✱10, ✱11, ✱12: Properties of a variable extended to all individuals''': section '''✱10''' introduces the notion of "a property" of a "variable". ''PM'' gives the example: φ is a function that indicates "is a Greek", and ψ indicates "is a man", and χ indicates "is a mortal" these functions then apply to a variable ''x''. ''PM'' can now write, and evaluate:
: (''x'') '''.''' ψ''x''
The notation above means "for all ''x'', ''x'' is a man". Given a collection of individuals, one can evaluate the above formula for truth or falsity. For example, given the restricted collection of individuals { Socrates, Plato, Russell, Zeus } the above evaluates to "true" if we allow for Zeus to be a man. But it fails for:
: (''x'') '''.''' φ''x''
because Russell is not Greek. And it fails for
: (''x'') '''.''' χ''x''
because Zeus is not a mortal.

Equipped with this notation ''PM'' can create formulas to express the following: "If all Greeks are men and if all men are mortals then all Greeks are mortals". (''PM'' 1962:138)
:(''x'') '''.''' φ''x'' ⊃ ψ''x'' ''':'''(''x'')'''.''' ψ''x'' ⊃ χ''x'' ''':'''⊃''':''' (''x'') '''.''' φ''x'' ⊃ χ''x''

Another example: the formula:
:'''✱10.01'''. (Ǝ''x'')'''.''' φ''x'' '''.''' = '''.''' ~(''x'') '''.''' ~φ''x'' '''Df'''.

means "The symbols representing the assertion 'There exists at least one ''x'' that satisfies function φ' is defined by the symbols representing the assertion 'It's not true that, given all values of ''x'', there are no values of ''x'' satisfying φ'".

The symbolisms ⊃<sub>''x''</sub> and "≡<sub>''x''</sub>" appear at '''✱10.02''' and '''✱10.03'''. Both are abbreviations for universality (i.e., for all) that bind the variable ''x'' to the logical operator. Contemporary notation would have simply used parentheses outside of the equality ("=") sign:
:'''✱10.02''' φ''x'' ⊃<sub>''x''</sub> ψ''x'' '''.'''='''.''' (''x'')'''.''' φ''x'' ⊃ ψ''x'' '''Df'''
:: Contemporary notation: ∀''x''(φ(''x'') → ψ(''x'')) (or a variant)
:'''✱10.03''' φ''x'' ≡<sub>''x''</sub> ψ''x'' '''.'''='''.''' (''x'')'''.''' φ''x'' ≡ ψ''x'' '''Df'''
:: Contemporary notation: ∀''x''(φ(''x'') ↔︎ ψ(''x'')) (or a variant)

''PM'' attributes the first symbolism to Peano.

Section '''✱11''' applies this symbolism to two variables. Thus the following notations: ⊃<sub>''x''</sub>, ⊃<sub>''y''</sub>, ⊃<sub>''x, y''</sub> could all appear in a single formula.

Section '''✱12''' reintroduces the notion of "matrix" (contemporary [[truth table]]), the notion of logical types, and in particular the notions of ''first-order'' and ''second-order'' functions and propositions.

New symbolism "φ '''!''' ''x''" represents any value of a first-order function. If a circumflex "^" is placed over a variable, then this is an "individual" value of ''y'', meaning that "''ŷ''" indicates "individuals" (e.g., a row in a truth table); this distinction is necessary because of the matrix/extensional nature of propositional functions.

Now equipped with the matrix notion, ''PM'' can assert its controversial [[axiom of reducibility]]: a function of one or two variables (two being sufficient for ''PM''{{'}}s use) ''where all its values are given'' (i.e., in its matrix) is (logically) equivalent ("≡") to some "predicative" function of the same variables. The one-variable definition is given below as an illustration of the notation (''PM'' 1962:166–167):

'''✱12.1''' ⊢''':''' (Ǝ ''f'')''':''' φ''x'' '''.'''≡<sub>''x''</sub>'''.''' ''f'' '''!''' ''x'' '''Pp''';
:: '''Pp''' is a "Primitive proposition" ("Propositions assumed without proof") (''PM'' 1962:12, i.e., contemporary "axioms"), adding to the 7 defined in section '''✱1''' (starting with '''✱1.1''' [[modus ponens]]). These are to be distinguished from the "primitive ideas" that include the assertion sign "⊢", negation "~", logical OR "V", the notions of "elementary proposition" and "elementary propositional function"; these are as close as ''PM'' comes to rules of notational formation, i.e., [[syntax]].

This means: "We assert the truth of the following: There exists a function ''f'' with the property that: given all values of ''x'', their evaluations in function φ (i.e., resulting their matrix) is logically equivalent to some ''f'' evaluated at those same values of ''x''. (and vice versa, hence logical equivalence)". In other words: given a matrix determined by property φ applied to variable ''x'', there exists a function ''f'' that, when applied to the ''x'' is logically equivalent to the matrix. Or: every matrix φ''x'' can be represented by a function ''f'' applied to ''x'', and vice versa.

'''✱13: The identity operator "=" ''': This is a definition that uses the sign in two different ways, as noted by the quote from ''PM'':

:'''✱13.01'''. ''x'' = ''y'' '''.'''=''':''' (φ)''':''' φ '''!''' ''x'' '''.''' ⊃ '''.''' φ '''!''' ''y'' '''Df'''
means:
:"This definition states that ''x'' and ''y'' are to be called identical when every predicative function satisfied by ''x'' is also satisfied by ''y'' ... Note that the second sign of equality in the above definition is combined with "Df", and thus is not really the same symbol as the sign of equality which is defined".
The not-equals sign "≠" makes its appearance as a definition at '''✱13.02'''.

'''✱14: Descriptions''':
:"A ''description'' is a phrase of the form "the term ''y'' which satisfies φ''ŷ'', where φ''ŷ'' is some function satisfied by one and only one argument."<ref>The original typography employs an ''x'' with a circumflex rather than ''ŷ''; this continues below</ref>
From this ''PM'' employs two new symbols, a forward "E" and an inverted iota "℩". Here is an example:
:'''✱14.02'''. E '''!''' ( ℩''y'') (φ''y'') '''.'''=''':''' ( Ǝ''b'')''':'''φ''y'' '''.''' ≡<sub>''y''</sub> '''.''' ''y'' = ''b'' '''Df'''.
This has the meaning:
: "The ''y'' satisfying φ''ŷ'' exists," which holds when, and only when φ''ŷ'' is satisfied by one value of ''y'' and by no other value." (''PM'' 1967:173–174)

=== Introduction to the notation of the theory of classes and relations ===

The text leaps from section '''✱14''' directly to the foundational sections '''✱20 GENERAL THEORY OF CLASSES''' and '''✱21 GENERAL THEORY OF RELATIONS'''. "Relations" are what is known in contemporary [[set theory]] as sets of [[ordered pair]]s. Sections '''✱20''' and '''✱22''' introduce many of the symbols still in contemporary usage. These include the symbols "ε", "⊂", "∩", "∪", "–", "Λ", and "V": "ε" signifies "is an element of" (''PM'' 1962:188); "⊂" ('''✱22.01''') signifies "is contained in", "is a subset of"; "∩" ('''✱22.02''') signifies the intersection (logical product) of classes (sets); "∪" ('''✱22.03''') signifies the union (logical sum) of classes (sets); "–" ('''✱22.03''') signifies negation of a class (set); "Λ" signifies the null class; and "V" signifies the universal class or universe of discourse.

Small Greek letters (other than "ε", "ι", "π", "φ", "ψ", "χ", and "θ") represent classes (e.g., "α", "β", "γ", "δ", etc.) (''PM'' 1962:188):
: ''x'' ε α
:: "The use of single letter in place of symbols such as ''ẑ''(φ''z'') or ''ẑ''(φ '''!''' ''z'') is practically almost indispensable, since otherwise the notation rapidly becomes intolerably cumbrous. Thus ' ''x'' ε α' will mean ' ''x'' is a member of the class α'". (''PM'' 1962:188)
:α ∪ –α = V
::The union of a set and its inverse is the universal (completed) set.<ref>See the ten postulates of Huntington, in particular postulates IIa and IIb at ''PM'' 1962:205 and discussion at p. 206.</ref>
:α ∩ –α = Λ
::The intersection of a set and its inverse is the null (empty) set.

When applied to relations in section '''✱23 CALCULUS OF RELATIONS''', the symbols "⊂", "∩", "∪", and "–" acquire a dot: for example: "⊍", "∸".<ref>The "⊂" sign has a dot inside it, and the intersection sign "∩" has a dot above it; these are not available in the "Arial Unicode MS" font.</ref>

'''The notion, and notation, of "a class" (set)''': In the first edition ''PM'' asserts that no new primitive ideas are necessary to define what is meant by "a class", and only two new "primitive propositions" called the [[Axiom of reducibility|axioms of reducibility]] for classes and relations respectively (''PM'' 1962:25).<ref>Wiener 1914 "A simplification of the logic of relations" (van Heijenoort 1967:224ff) disposed of the second of these when he showed how to reduce the theory of relations to that of classes</ref> But before this notion can be defined, ''PM'' feels it necessary to create a peculiar notation "''ẑ''(φ''z'')" that it calls a "fictitious object". (''PM'' 1962:188)
: ⊢''':''' ''x'' ε ''ẑ''(φ''z'') '''.'''≡'''.''' (φ''x'')
:: "i.e., ' ''x'' is a member of the class determined by (φ''ẑ'')' is [logically] equivalent to ' ''x'' satisfies (φ''ẑ''),' or to '(φ''x'') is true.'". (''PM'' 1962:25)

At least ''PM'' can tell the reader how these fictitious objects behave, because "A class is wholly determinate when its membership is known, that is, there cannot be two different classes having the same membership" (''PM'' 1962:26). This is symbolised by the following equality (similar to '''✱13.01''' above:
: ''ẑ''(φ''z'') = ''ẑ''(ψ''z'') '''.''' ≡ ''':''' (''x'')''':''' φ''x'' '''.'''≡'''.''' ψ''x''
::"This last is the distinguishing characteristic of classes, and justifies us in treating ''ẑ''(ψ''z'') as the class determined by [the function] ψ''ẑ''." (''PM'' 1962:188)

Perhaps the above can be made clearer by the discussion of classes in ''Introduction to the Second Edition'', which disposes of the ''Axiom of Reducibility'' and replaces it with the notion: "All functions of functions are extensional" (''PM'' 1962:xxxix), i.e.,
: φ''x'' ≡<sub>''x''</sub> ψ''x'' '''.'''⊃'''.''' (''x'')''':''' ƒ(φ''ẑ'') ≡ ƒ(ψ''ẑ'') (''PM'' 1962:xxxix)

This has the reasonable meaning that "IF for all values of ''x'' the ''truth-values'' of the functions φ and ψ of ''x'' are [logically] equivalent, THEN the function ƒ of a given φ''ẑ'' and ƒ of ψ''ẑ'' are [logically] equivalent." ''PM'' asserts this is "obvious":
: "This is obvious, since φ can only occur in ƒ(φ''ẑ'') by the substitution of values of φ for ''p, q, r, ...'' in a [logical-] function, and, if φ''x'' ≡ ψ''x'', the substitution of φ''x'' for ''p'' in a [logical-] function gives the same truth-value to the truth-function as the substitution of ψ''x''. Consequently there is no longer any reason to distinguish between functions classes, for we have, in virtue of the above,
: φ''x'' ≡<sub>''x''</sub> ψ''x'' '''.'''⊃'''.''' (''x'')'''.''' φ''ẑ'' = '''.''' ψ''ẑ''".
Observe the change to the equality "=" sign on the right. ''PM'' goes on to state that will continue to hang onto the notation "''ẑ''(φ''z'')", but this is merely equivalent to φ''ẑ'', and this is a class. (all quotes: ''PM'' 1962:xxxix).