Editing Principia Mathematica (section)

=== Primitive propositions ===

The ''first'' edition (see discussion relative to the second edition, below) begins with a definition of the sign "⊃"

'''✱1.01'''. ''p'' ⊃ ''q'' '''.'''='''.''' ~ ''p'' ∨ ''q''. '''Df'''.

'''✱1.1'''. Anything implied by a true elementary proposition is true. '''Pp''' modus ponens

('''✱1.11''' was abandoned in the second edition.)

'''✱1.2'''. ⊦''':''' ''p'' ∨ ''p'' '''.'''⊃'''.''' ''p''. '''Pp''' principle of tautology

'''✱1.3'''. ⊦''':''' ''q'' '''.'''⊃'''.''' ''p'' ∨ ''q''. '''Pp''' principle of addition

'''✱1.4'''. ⊦''':''' ''p'' ∨ ''q'' '''.'''⊃'''.''' ''q'' ∨ ''p''. '''Pp''' principle of permutation

'''✱1.5'''. ⊦''':''' ''p'' ∨ ( ''q'' ∨ ''r'' ) '''.'''⊃'''.''' ''q'' ∨ ( ''p'' ∨ ''r'' ). '''Pp''' associative principle

'''✱1.6'''. ⊦''':.''' ''q'' ⊃ ''r'' '''.'''⊃''':''' ''p'' ∨ ''q'' '''.'''⊃'''.''' ''p'' ∨ ''r''. '''Pp''' principle of summation

'''✱1.7'''. If ''p'' is an elementary proposition, ~''p'' is an elementary proposition. '''Pp'''

'''✱1.71'''. If ''p'' and ''q'' are elementary propositions, ''p'' ∨ ''q'' is an elementary proposition. '''Pp'''

'''✱1.72'''. If φ''p'' and ψ''p'' are elementary propositional functions which take elementary propositions as arguments, φ''p'' ∨ ψ''p'' is an elementary proposition. '''Pp'''

Together with the "Introduction to the Second Edition", the second edition's Appendix A abandons the entire section '''✱9'''. This includes six primitive propositions '''✱9''' through '''✱9.15''' together with the Axioms of reducibility.

The revised theory is made difficult by the introduction of the [[Sheffer stroke]] ("|") to symbolise "incompatibility" (i.e., if both elementary propositions ''p'' and ''q'' are true, their "stroke" ''p'' | ''q'' is false), the contemporary logical [[Sheffer stroke|NAND]] (not-AND). In the revised theory, the Introduction presents the notion of "atomic proposition", a "datum" that "belongs to the philosophical part of logic". These have no parts that are propositions and do not contain the notions "all" or "some". For example: "this is red", or "this is earlier than that". Such things can exist ''ad finitum'', i.e., even an "infinite enumeration" of them to replace "generality" (i.e., the notion of "for all").<ref>This idea is due to Wittgenstein's ''Tractatus''. See the discussion at ''PM'' 1962:xiv–xv)</ref> ''PM'' then "advance[s] to molecular propositions" that are all linked by "the stroke". Definitions give equivalences for "~", "∨", "⊃", and "'''.'''".

The new introduction defines "elementary propositions" as atomic and molecular positions together. It then replaces all the primitive propositions '''✱1.2''' to '''✱1.72''' with a single primitive proposition framed in terms of the stroke:
: "If ''p'', ''q'', ''r'' are elementary propositions, given ''p'' and ''p''|(''q''|''r''), we can infer ''r''. This is a primitive proposition."

The new introduction keeps the notation for "there exists" (now recast as "sometimes true") and "for all" (recast as "always true"). Appendix A strengthens the notion of "matrix" or "predicative function" (a "primitive idea", ''PM'' 1962:164) and presents four new Primitive propositions as '''✱8.1–✱8.13'''.

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'''✱88'''. Multiplicative axiom

'''✱120'''. Axiom of infinity