Editing Logicism (section)

== Neo-logicism<!--'Neo-Fregeanism', 'Neo-Fregeanism', 'Neo-logicism', 'Neo-Logicism', 'Neologicism', 'Scottish School (philosophy of mathematics)', 'Stanford–Edmonton School', 'Stanford-Edmonton School', 'Abstractionist Platonism', and 'Modal neo-logicism' redirect here--> ==
'''Neo-logicism'''<!--boldface per WP:R#PLA--> describes a range of views considered by their proponents to be successors of the original logicist program.<ref>Bernard Linsky and [[Edward N. Zalta]], [http://mally.stanford.edu/Papers/neologicism2.pdf "What is Neologicism?"], ''The Bulletin of Symbolic Logic'', '''12'''(1) (2006): 60–99.</ref> More narrowly, neo-logicism may be seen as the attempt to salvage some or all elements of [[Gottlob Frege#Work as a logician|Frege's]] programme through the use of a modified version of Frege's system in the ''Grundgesetze'' (which may be seen as a kind of [[second-order logic]]). 

For instance, one might replace [[Basic Law V]] (analogous to the [[axiom schema of unrestricted comprehension]] in [[naive set theory]]) with some 'safer' axiom so as to prevent the derivation of the known paradoxes.  The most cited candidate to replace BLV is [[Hume's principle]], the contextual definition of '#'  given by '#''F'' = #''G'' [[if and only if]] there is a [[bijection]] between ''F'' and ''G'''.<ref>[http://seis.bris.ac.uk/~plxol/Courses/PHIL30067/Syllabus.htm PHIL 30067: Logicism and Neo-Logicism] {{webarchive|url=https://web.archive.org/web/20110717200246/http://seis.bris.ac.uk/~plxol/Courses/PHIL30067/Syllabus.htm|date=2011-07-17}}.</ref> This kind of neo-logicism is often referred to as '''neo-Fregeanism'''<!--boldface per WP:R#PLA-->.<ref name=SEP>{{cite SEP |url-id=logicism |title=Logicism and Neologicism}}</ref> Proponents of neo-Fregeanism include [[Crispin Wright]] and [[Bob Hale (philosopher)|Bob Hale]], sometimes also called the '''Scottish School'''<!--boldface per WP:R#PLA--> or '''abstractionist Platonism'''<!--boldface per WP:R#PLA-->,<ref>Bob Hale and Crispin Wright (2002), "Benacerraf's dilemma revisited", ''European Journal of Philosophy'' '''10'''(1):101–129, esp. "6. Objections and Qualifications".</ref> who espouse a form of [[epistemic]] [[foundationalism]].<ref name="st-andrews">[http://www.st-andrews.ac.uk/~mr30/papers/EbertRossbergPurpose.pdf st-andrews.ac.uk]. {{webarchive|url=https://web.archive.org/web/20061224165534/http://www.st-andrews.ac.uk/~mr30/papers/EbertRossbergPurpose.pdf|date=2006-12-24}}.</ref>

Other major proponents of neo-logicism include Bernard Linsky and [[Edward N. Zalta]], sometimes called the '''Stanford–Edmonton School''',<!--boldface per WP:R#PLA--> [[abstract structuralism]] or '''modal neo-logicism''',<!--boldface per WP:R#PLA--> who espouse a form of [[axiomatic metaphysics]].<ref name=st-andrews/><ref name=SEP/> Modal neo-logicism derives the [[Peano axioms]] within [[Second-order logic|second-order]] [[Modal logic|modal]] [[Abstract object theory|object theory]].<ref>[[Edward N. Zalta]], "Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege's ''Grundgesetze'' in Object Theory", ''Journal of Philosophical Logic'', '''28'''(6) (1999): 619–660.</ref><ref>[[Edward N. Zalta]], "Neo-Logicism? An Ontological Reduction of Mathematics to Metaphysics", ''Erkenntnis'', '''53'''(1–2) (2000), 219–265.</ref>

Another quasi-neo-logicist approach has been suggested by M. Randall Holmes. In this kind of amendment to the ''Grundgesetze'', BLV remains intact, save for a restriction to stratifiable formulae in the manner of Quine's [[New Foundations|NF]] and related systems. Essentially all of the ''Grundgesetze'' then 'goes through'. The resulting system has the same consistency strength as [[Ronald Jensen|Jensen]]'s NFU + [[J. Barkley Rosser|Rosser]]'s Axiom of Counting.<ref>M. Randall Holmes, [https://randall-holmes.github.io/Gottlob/fregenote.pdf "Repairing Frege’s Logic"], August 5, 2018.</ref>