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Next tasks

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  1. Put in refs on Talk:Randal Rauser#Canaanite genocide. — Charles Stewart (talk) 18:34, 1 February 2022 (UTC)
  2. Start article Lawverean set theory covering ETCS and Leinster's and link to it from William Lawvere; cf. Rethinking set theory (Chauvenet prize). -- Mar 2020
  3. Write history section for First-order logic -- Apr 2017
  4. Explain significance of Gaussian copula function in the financial meltdown -- Mar 2017
  5. Flesh out Roger Salmon (banker) and ensure that it is appropriately linked to -- Mar 2017
  6. Promote use of Wikipedia:WikiProject Logic/To do as way of coordinating work on logic tasks -- Mar 2017
  7. Nominate best articles in scope of Boolean algebra taskforce for good article status. Need to fix Boolean algebra (structure) -- May 2016
  8. Fix the mess that is Principle of bivalence -- May 2016
  9. Create Vending machine (process theory), cf. Process theory -- Jun 2009
  10. Linear logic - semantics section, Hopf algebra and [Category Theory for Linear Logicians http://www.site.uottawa.ca/~phil/papers/catsurv.web.pdf] -- Jul 2009
  11. Second-order logic: change talk of standard semantics to set-theoretic semantics, and contrast to the dependence of Henkin semantics on simply-typed lambda-calculus -- Apr 2009
  12. Another round of surgery on Paradoxes of material implication‎, see Talk:Paradoxes of material implication‎. -- Mar 2009
  13. Add discussion of fan theorem to bar induction, cf. Rathjen -- Mar 2009
  14. Alasdair Macintyre: fact check londonsocialisthistorians.org, look for other accts. Mar 2009
  15. Sort out intensional logic, start Imre Rusze (obit), ask User:Physis -- May 2009
  16. Add institute to Robert von Ostertag -- May 2009

And see Wikipedia:WikiProject Logic/To do

Article composition

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Böhm's theorem

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Constructivism (mathematics) and Mass problem

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  • Troelstra, 1992, Hist. of constructivisms in C20th: talks of 4 schools, success of Bishop's school 'winning'.
  • Refs
    • Terwijn, 2006, Constructive logic and the Medvedev lattice
    • Kolmogorov, 1932, 'On the interpretation of intuitionistic logic'
    • Yuri Medvedev, 1955, 'Degrees of difficulty of mass problems' (Doklady Akademii Nauk SSSR, 104/4:501--504),
    • Yuri Medvedev, 1962, 'Finite problems' (Doklady Akademii Nauk SSSR, 142:1015--1018)

Hilbert system

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Hilbert's finitism

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History of algebraic logic

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Laws of Thought

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  • Merge Boole's syllogistic into LoT. Points from Talk:Boole's syllogistic:
    1. Boole did not propose a separate syllogistic calculus, rather what he proposed was an interpretation of syllogistic into his algebra. There are a few accounts of the interpretation on the web, of which (Burris 2001) is perhaps the best.
    2. The current article is talking about how this interpretation proposes a resolution to the problem of existential import (see square of opposition), one incompatible with Aristotle assertions. One of the many problems with the article as it stands is that it isn't clear that that is what it is doing. I suppose no one else is going to fix it but me: I'll get around to it...
    3. Naturally this calculus is propositional logic and not predicate logic. The embedding of syllogistic into propositional logic shows that syllogistic corresponds to a very weak fragment of predicate logic.
  • Resources
    1. Boole, G. (1850). Gutenberg text of Laws of Thought
    2. Burris, Stanley (2000). The Laws of Boole's Thought.
    3. Burris, Stanley (2001). A Fragment of Boole's Algebraic Logic Suitable for Traditional Syllogistic Logic.
    4. Hailperin, T. (1976/1986). Boole's Logic and Probability. North Holland.
    5. Hailperin, T. (1981). Boole’s algebra isn’t Boolean algebra. Mathematics Magazine 54:172–184. Reprinted in A Boole Anthology, ed. by James Gasser. Synthese Library volume 291. Springer 2000.


Many-sorted logic

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Stoic logic

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  1. Farm out Stoic logic from history of logic
  2. Summarise Sect 3.5 of SEP:lvon-warsaw Lukasiewicz on Stoic logic as a system of rules, and section 3.3 on bivalence.
  3. Refs

Tasks for sometime: logic

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Core logic

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Formal axiology

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Ice (Dukaj novel)

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  • Literary section on logic (date=June 2009)
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Proof theory

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Semantics & MSfS

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Theories

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WP:WPLOG

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  1. Make list of logic articles of joint concern to maths & philosophy, reply to GregBard -- Feb 2009
  2. Start with: modal logic, predicate logic, soundness, completeness, Hilbert system
  3. What standards do we want for these articles? How about elementary formal reasoning, sufficient for detailed discussion of soundness theorem & deduction theorem, but not as much as needed for completeness theorem?


Tasks for sometime: &c

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Free software

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  1. Raph Levien: incorporate material on remailers (see talk) --- Mar 2007

Mathematics

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In order to show that the axioms of the class of algebras we consider capture exactly the collection of predicates we have in mind, a representation theorem is necessary. A representation theorem is a correspondence between an abstract algebra and its set-theoretical model. The first representation theorem is due to Cayley [Cay78] showing that every abstract group is isomorphic to a concrete group of permutations. A representation theorem for the algebra of all predicates was first proved by Lindenbaum and Tarski [Tar35]. They proved that a Boolean algebra is isomorphic to the collection of all subsets of some set if and only if it is complete and atomic. This general result restricts the class of Boolean algebras for which a concrete representation exists. It was Stone [Sto36] who first saw a connection between algebra and topology. He constructed from a Boolean algebra a set of points using prime ideals which can be made into a topological space in a natural way. Conversely, using a topology on a set of points he was able to construct a Boolean algebra. For certain topological spaces (later called Stone spaces) these constructions give an isomorphism. In a later paper [Sto37], Stone generalized this correspondence from Stone spaces to spectral spaces and from Boolean algebras to distributive lattices. Hofmann and Keimel [HK72] described the Stone representation theorem in a categorical framework showing a duality between the category of Boolean algebras and a sub-category of topological spaces. A representation theorem for Boolean algebras with operators has been considered by J'onsson and Tarski [JT51, JT52]. By means of an extension theorem they proved that operators on a Boolean algebra can be naturally extended to completely additive operators on a complete and atomic Boolean algebra.
Stone's representation theorem leaves open the problem of finding an abstract characterization of topological spaces. For every topological space, its lattice of open sets forms a frame. This fact leads Papert and Papert [PP58] to a representation theorem between spatial frames and sober spaces. Even further, Isbell [Isb72a] gives an adjunction between the category of topological spaces with continuous functions and the opposite category of frames with frame homomorphisms. This adjunction yields a duality between the category of sober spaces and the category of spatial frames.


Computer Science

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Gathering references

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