In the same manner that Intuitionist PC and can be mapped onto S4 standard PC and S5 are related. [Zeman, 1973, p231] [Hughes and Cresswell, 1996, p225]
Finding other basis for PC seems to have been a popular sport historically. Not included below is a "tour de force" of Jean Porte. "Un systèm logistique très faible pour le calcul propositionannel classique." in Comptes rendus hebdomadaires des séances de l' Académie des Sciences (Paris), vol. 254 (1962), pp 2500-2502 which manages with ONE axiom, and 136 one-premise rules.
PC is the Positive Propositional Logic plus
Principia: ~ and + primitive, the rest added by definition.
[Russell and Whitehead, 1910, Volume 1, p100 (First edition), (p96 second edition)] The axiom numbers are Russell and Whitehead's. Almost nobody else ever used them. It is more usual to see the four axioms referred to as A1-A4, although some authors use A1-A5, with *1.5 below being A4.
Note that the original also has
But this was shown redundant by Paul Bernays, "Axiomatische Untersuchungen des Aussagenkalküls der Principia Mathematica", Mathematiche Zeitschrift, Vol 25, 1926, pp 305-320, Bernay's states that Haskel Curry also had a proof, using ideas from one originally due to Peirce. (Paul Bernays, in his review of Peter Nidditch's article "A Note on the Redundant Axiom of Principia Mathematica". His review appeared in The Journal of Symbolic Logic, Vol. 36, No. 2 (Jun., 1971), pp. 332-333)
The second edition of Principia changed from being "+" and "~" based, to being Sheffer stroke ("|") based, with the definitions:
Added to chapter 1.
PC is the Positive Propositional Logic plus
PC is the Positive Propositional Logic plus
PC is the Positive Propositional Logic plus
The axioms of Łukasiewicz:
Rosser's Axioms
[Zeman 1973, p 26] [Rosser, 1953, Page 55]
Tarski, Bernays, and Wajsberg basis 1 (F primitive):
Wajsberg himself credits these to Tarski and Bernays. [Wajsberg, 1937, pages 154-157] (Page 285 of the English translation: [McCall, 1967]) I belive that Quine adds Wajsberg to the list because Wajsberg proved a lot of the results for this axiomatic basis. -JH
The Peirce axiom can be replaced by the rule: From |- (p>q) > p infer p. [Wajsberg, 1937] (Page 285 of the English translation: [McCall, 1967])
Tarski, Bernays, and Wajsberg basis 2 (~ primitive):
Wajsberg himself credits these to Tarski and Bernays. [Wajsberg, 1937, pages 154-157] (Page 285 of the English translation by S. McCall and P.Woodruff appearing in: [McCall, 1967], pages ) I belive that Quine adds Wajsberg to the list because Wajsberg proved a lot of the results for this axiomatic basis. -JH
Wajsberg basis 1 (~ primitive):
[Wajsberg, 1937] (Page 319 of the English translation S. McCall and P.Woodruff appearing in: [McCall, 1967])
Wajsberg basis 2 (~ primitive):
[Wajsberg, 1939] (Page 319 of the English translation by S. McCall appearing in [McCall, 1967])
(: Almost everything... :)
Standard PC has some alternatives...
Editorial comment:
[I personally don't see why any these alternatives shouldn't be a basis for many of the modal logic systems instead of PC. Of course, if the systems were build on this framework they wouldn't be the same systems. But I think that they would still be interesting systems. In fact, the intuitionist versions of S4 and S5 have actually been investigated in the literature. -JH
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