Editing Principia Mathematica (section)

=== Construction ===

The theory of ''PM'' has both significant similarities, and similar differences, to a contemporary formal theory.{{clarify|date=October 2017|reason=What is the contemporary formal theory being referred to here?}} Kleene states that "this deduction of mathematics from logic was offered as intuitive axiomatics. The axioms were intended to be believed, or at least to be accepted as plausible hypotheses concerning the world".<ref>Quote from Kleene 1952:45. See discussion LOGICISM at pp. 43–46.</ref> Indeed, unlike a Formalist theory that manipulates symbols according to rules of grammar, ''PM'' introduces the notion of "truth-values", i.e., truth and falsity in the ''real-world'' sense, and the "assertion of truth" almost immediately as the fifth and sixth elements in the structure of the theory (''PM'' 1962:4–36):
# ''Variables''
# ''Uses of various letters''
# ''The fundamental functions of propositions'': "the Contradictory Function" symbolised by "~" and the "Logical Sum or Disjunctive Function" symbolised by "∨" being taken as primitive and logical implication ''defined'' (the following example also used to illustrate 9. ''Definition'' below) as<br />''p'' ⊃ ''q'' '''.'''='''.''' ~ ''p'' ∨ ''q'' '''Df'''. (''PM'' 1962:11)<br /> and logical product defined as<br /> ''p'' '''.''' ''q'' '''.'''='''.''' ~(~''p'' ∨ ~''q'') '''Df'''. (''PM'' 1962:12)
# ''Equivalence'': ''Logical'' equivalence, not arithmetic equivalence: "≡" given as a demonstration of how the symbols are used, i.e., "Thus ' ''p'' ≡ ''q'' ' stands for '( ''p'' ⊃ ''q'' ) '''.''' ( ''q'' ⊃ ''p'' )'." (''PM'' 1962:7). Notice that to ''discuss'' a notation ''PM'' identifies a "meta"-notation with "[space] ... [space]":<ref>In his section 8.5.4 ''Groping towards metalogic'' Grattan-Guinness 2000:454ff discusses the American logicians' critical reception of the second edition of ''PM''. For instance Sheffer "puzzled that ' ''In order to give an account of logic, we must presuppose and employ logic'' ' " (p. 452). And Bernstein ended his 1926 review with the comment that "This distinction between the propositional logic as a mathematical system and as a language must be made, if serious errors are to be avoided; this distinction the ''Principia'' does not make" (p. 454).</ref><br />Logical equivalence appears again as a ''definition'':<br />''p'' ≡ ''q'' '''.'''='''.''' ( ''p'' ⊃ ''q'' ) '''.''' ( ''q'' ⊃ ''p'' ) (''PM'' 1962:12),<br />Notice the appearance of parentheses. This ''grammatical'' usage is not specified and appears sporadically; parentheses do play an important role in symbol strings, however, e.g., the notation "(''x'')" for the contemporary "∀''x''".
# ''Truth-values'': "The 'Truth-value' of a proposition is ''truth'' if it is true, and ''falsehood'' if it is false" (this phrase is due to [[Gottlob Frege]]) (''PM'' 1962:7).
# ''Assertion-sign'': "'⊦'''.''' ''p'' may be read 'it is true that' ... thus '⊦''':''' ''p'' '''.'''⊃'''.''' ''q'' ' means 'it is true that ''p'' implies ''q'' ', whereas '⊦'''.''' ''p'' '''.'''⊃⊦'''.''' ''q'' ' means ' ''p'' is true; therefore ''q'' is true'. The first of these does not necessarily involve the truth either of ''p'' or of ''q'', while the second involves the truth of both" (''PM'' 1962:92).
# ''Inference'': ''PM''{{'}}s version of ''modus ponens''. "[If] '⊦'''.''' ''p'' ' and '⊦ (''p'' ⊃ ''q'')' have occurred, then '⊦ '''.''' ''q'' ' will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of '⊦'''.''' ''q'' ' [in other words, the symbols on the left disappear or can be erased]" (''PM'' 1962:9).
# ''The use of dots''
# ''Definitions'': These use the "=" sign with "Df" at the right end.
# ''Summary of preceding statements'': brief discussion of the primitive ideas "~ ''p''" and "''p'' ∨ ''q''" and "⊦" prefixed to a proposition.
# ''Primitive propositions'': the axioms or postulates. This was significantly modified in the second edition.
# ''Propositional functions'': The notion of "proposition" was significantly modified in the second edition, including the introduction of "atomic" propositions linked by logical signs to form "molecular" propositions, and the use of substitution of molecular propositions into atomic or molecular propositions to create new expressions.
# ''The range of values and total variation''
# ''Ambiguous assertion and the real variable'': This and the next two sections were modified or abandoned in the second edition. In particular, the distinction between the concepts defined in sections 15. ''Definition and the real variable'' and 16 ''Propositions connecting real and apparent variables'' was abandoned in the second edition.
# ''Formal implication and formal equivalence''
# ''Identity''
# ''Classes and relations''
# ''Various descriptive functions of relations''
# ''Plural descriptive functions''
# ''Unit classes''