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  • 1
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    DUALLING: A CRITIQUE OF AN ARGUMENT OF POPPER AND MILLER (1986)
    Oxford : Periodicals Archive Online (PAO)
    The British journal for the philosophy of science. 37 (1986) 220 
    Details
    ISSN: 0007-0882
    Topics: Natural Sciences in General , Philosophy
    Notes: DISCUSSIONS
    URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pao:&rft_dat=xri:pao:article:4064-1986-037-00-000024
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  • 2
    Electronic Resource
    Electronic Resource
    Positive modal logic (1995)
    Springer
    Studia logica 55 (1995), S. 301-317 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply ∧, ∨, □, ◊. The postulates (and theorems) are all deducibility statements ϕ ⊢ ψ. The only postulates that might not be obvious are $$\diamondsuit \varphi \wedge \square \psi \vdash \diamondsuit (\varphi \wedge \psi )\square (\varphi \vee \psi ) \vdash \square \varphi \vee \diamondsuit \psi $$ . It is shown thatK + is complete with respect to the usual Kripke-style semantics. The proof is by way of a Henkin-style construction, with “possible worlds” being taken to be prime theories. The construction has the somewhat unusual feature of using at an intermediate stage disjoint pairs consisting of a theory and a “counter-theory”, the counter-theory replacing the role of negation in the standard construction. Extension to other modal logics is discussed, as well as a representation theorem for the corresponding modal algebras. We also discuss proof-theoretic arguments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01061239
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  • 3
    Electronic Resource
    Electronic Resource
    Partiality and Its Dual (2000)
    Springer
    Studia logica 66 (2000), S. 5-40 
    Details
    ISSN: 1572-8730
    Keywords: consequence ; four-valued ; paraconsistent ; partial ; three-valued ; constructible falsity ; relevance logic ; intuitionistic logic
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including Łukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity".
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026740726955
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  • 4
    Electronic Resource
    Electronic Resource
    A Kripke-style semantics for R-Mingle using a binary accessibility relation (1976)
    Springer
    Studia logica 35 (1976), S. 163-172 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02120878
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  • 5
    Electronic Resource
    Electronic Resource
    Quantification and RM (1976)
    Springer
    Studia logica 35 (1976), S. 315-322 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02282493
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  • 6
    Electronic Resource
    Electronic Resource
    A theorem in 3-valued model theory with connections to number theory, type theory, and relevant logic (1979)
    Springer
    Studia logica 38 (1979), S. 149-169 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract Given classical (2 valued) structures $$\mathfrak{A}$$ and $$\mathfrak{A}'$$ and a homomorphism h of $$\mathfrak{A}$$ onto $$\mathfrak{A}'$$ , it is shown how to construct a (non-degenerate) “3-valued counterpart” $$3\mathfrak{A}'$$ of $$\mathfrak{A}'$$ . Classical sentences that are true in $$\mathfrak{A}$$ are non-false in $$3\mathfrak{A}'$$ . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00370439
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  • 7
    Electronic Resource
    Electronic Resource
    Relevant Robinson's arithmetic (1979)
    Springer
    Studia logica 38 (1979), S. 407-418 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in relevant arithmetic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00370478
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  • 8
    Electronic Resource
    Electronic Resource
    Recent work at the interface of logic, combinatorics, and computer science (1992)
    Springer
    Annals of mathematics and artificial intelligence 6 (1992), S. 1-15 
    Details
    ISSN: 1573-7470
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01531020
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