Editing Principia Mathematica (section)

=== 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)