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Lloyd Humberstone [68]L. Humberstone [4]
  1. Lloyd Humberstone, Equivalential Interpolation.
    By a consequence relation on a set L of formulas we understand a relation I — c p(L) x L satisfying the conditions called 'Overlap', 'Dilution', and 'Cut for Sets' at p.15 of [25]; we do not repeat the conditions here since we are simply fixing notation and the concept of a consequence relation is well known in any case. (The characterization in [25] amounts to that familiar from Tarski's work, except that there is no 'finitariness' restriction to the effect (...)
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  2. Lloyd Humberstone, On the BCI-Admissibility of an 'Abelian' Rule.
    Am(B m B). Specifically I was wondering whether for every BCI-provable formula A there is a B for which the inset formula was provable. If you want to read about this issue, which I..
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  3. Lloyd Humberstone, Weaker-to-Stronger Translational Embeddings in Modal Logic.
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  4. Lloyd Humberstone, How Not to Think About Modal Definability: A Modal Axiom From G. E. Hughes.
    In a 1990 paper, George Hughes axiomatized the logic determined by the class of all frames in which each point has a reflexive successor, and raised various questions along the way, one of which is answered incorrectly here by means of an interestingly fallacious argument.
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  5. Lloyd Humberstone, The Consequence Relation of Tautological Entailment is Maximally Relevant: Answering a Question of Graham Priest.
    Graham Priest has asked whether the consequence relation associated with the Anderson–Belnap system of Tautological Entailment,1 in the language with connectives ¬, ∧, ∨, and countably many propositional variables as tomic formulas, maximal amongst the substitution-invariant relevant consequence relations on this language. Here a consequence relation is said to be relevant just in case whenever for a set of formulas Γ and formula B, we have Γ B only if some propositional variable occurring in B occurs in at least one (...)
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  6. Lloyd Humberstone (forthcoming). Plural Logic, by Alex Oliver and Timothy Smiley. Australasian Journal of Philosophy:1-4.
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  7. Lloyd Humberstone (forthcoming). Sentence Connectives in Formal Logic. Stanford Encyclopedia of Philosophy.
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  8. Lloyd Humberstone (2013). Aggregation and Idempotence. Review of Symbolic Logic 6 (4):680-708.
    A 1-ary sentential context is aggregative (according to a consequence relation) if the result of putting the conjunction of two formulas into the context is a consequence (by that relation) of the results of putting first the one formula and then the other into that context. All 1-ary contexts are aggregative according to the consequence relation of classical propositional logic (though not, for example, according to the consequence relation of intuitionistic propositional logic), and here we explore the extent of this (...)
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  9. Lloyd Humberstone (2013). Inverse Images of Box Formulas in Modal Logic. Studia Logica 101 (5):1031-1060.
    We investigate, for several modal logics but concentrating on KT, KD45, S4 and S5, the set of formulas B for which ${\square B}$ is provably equivalent to ${\square A}$ for a selected formula A (such as p, a sentence letter). In the exceptional case in which a modal logic is closed under the (‘cancellation’) rule taking us from ${\square C \leftrightarrow \square D}$ to ${C \leftrightarrow D}$ , there is only one formula B, to within equivalence, in this inverse image, (...)
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  10. Lloyd Humberstone (2013). Logical Relations. Philosophical Perspectives 27 (1):175-230.
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  11. Lloyd Humberstone (2013). Replacement in Logic. Journal of Philosophical Logic 42 (1):49-89.
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  12. Lloyd Humberstone (2013). Zolin and Pizzi: Defining Necessity From Noncontingency. Erkenntnis 78 (6):1275-1302.
    The point of the present paper is to draw attention to some interesting similarities, as well as differences, between the approaches to the logic of noncontingency of Evgeni Zolin and of Claudio Pizzi. Though neither of them refers to the work of the other, each is concerned with the definability of a (normally behaving, though not in general truth-implying) notion of necessity in terms of noncontingency, standard boolean connectives and additional but non-modal expressive resources. The notion of definability involved is (...)
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  13. L. Humberstone & D. Makinson (2012). Intuitionistic Logic and Elementary Rules. Mind 120 (480):1035-1051.
    The interplay of introduction and elimination rules for propositional connectives is often seen as suggesting a distinguished role for intuitionistic logic. We prove three formal results concerning intuitionistic propositional logic that bear on that perspective, and discuss their significance. First, for a range of connectives including both negation and the falsum, there are no classically or intuitionistically correct introduction rules. Second, irrespective of the choice of negation or the falsum as a primitive connective, classical and intuitionistic consequence satisfy exactly the (...)
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  14. Lloyd Humberstone (2012). Minimally Congruential Contexts: Observations and Questions on Embedding E in K. Notre Dame Journal of Formal Logic 53 (4):581-598.
    Recently, an improvement in respect of simplicity was found by Rohan French over extant translations faithfully embedding the smallest congruential modal logic (E) in the smallest normal modal logic (K). After some preliminaries, we explore the possibility of further simplifying the translation, with various negative findings (but no positive solution). This line of inquiry leads, via a consideration of one candidate simpler translation whose status was left open earlier, to isolating the concept of a minimally congruential context. This amounts, roughly (...)
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  15. Lloyd Humberstone (2011). The Connectives. MIT Press.
    It will be an essential resource for philosophers, mathematicians, computer scientists, linguists, or any scholar who finds connectives, and the conceptual issues surrounding them, to be a source of interest.This landmark work offers both ...
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  16. Lloyd Humberstone (2011). Variation on a Trivialist Argument of Paul Kabay. Journal of Logic, Language and Information 20 (1):115-132.
    Impossible worlds are regarded with understandable suspicion by most philosophers. Here we are concerned with a modal argument which might seem to show that acknowledging their existence, or more particularly, the existence of some hypothetical (we do not say “possible”) world in which everything was the case, would have drastic effects, forcing us to conclude that everything is indeed the case—and not just in the hypothesized world in question. The argument is inspired by a metaphysical (rather than modal-logical) argument of (...)
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  17. Lloyd Humberstone (2010). Smiley's Distinction Between Rules of Inference and Rules of Proof. In T. J. Smiley, Jonathan Lear & Alex Oliver (eds.), The Force of Argument: Essays in Honor of Timothy Smiley. Routledge. 107--126.
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  18. Rohan French & Lloyd Humberstone (2009). Partial Confirmation of a Conjecture on the Boxdot Translation in Modal Logic. Australasian Journal of Logic 7:56-61.
    The purpose of the present note is to advertise an interesting conjecture concerning a well-known translation in modal logic, by confirming a (highly restricted) special case of the conjecture.
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  19. Lloyd Humberstone (2009). Collapsing Modalities. Notre Dame Journal of Formal Logic 50 (2):119-132.
    Sections 1 and 2 respectively raise and settle the question of whether, if an affirmative modality collapses (reduces to the null modality, that is) in a normal modal logic, then all modalities of the same length collapse in that logic, while Section 3 considers some special cases of an analogous phenomenon for congruential modal logics, closing with a general question about collapsing modalities in this broader range of logics.
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  20. Lloyd Humberstone (2009). Logical Pluralism. Australasian Journal of Philosophy 87 (1):162 – 168.
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  21. Lloyd Humberstone (2008). Béziau's Translation Paradox. Theoria 71 (2):138-181.
    Jean-Yves Béziau (‘Classical Negation can be Expressed by One of its Halves’, Logic Journal of the IGPL 7 (1999), 145–151) has given an especially clear example of a phenomenon he considers a sufficiently puzzling to call the ‘paradox of translation’: the existence of pairs of logics, one logic being strictly weaker than another and yet such that the stronger logic can be embedded within it under a faithful translation. We elaborate on Béziau’s example, which concerns classical negation, as well as (...)
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  22. Lloyd Humberstone (2008). Contrariety and Subcontrariety: The Anatomy of Negation (with Special Reference to an Example of J.-Y. Béziau). Theoria 71 (3):241-262.
    We discuss aspects of the logic of negation bearing on an issue raised by Jean-Yves Béziau, recalled in §1. Contrary- and subcontrary-forming operators are introduced in §2, which examines some of their logical behaviour, leading on naturally to a consideration in §3 of dual intuitionistic negation (as well as implication), and some further operators related to intuitionistic negation. In §4, a historical explanation is suggested as to why some of these negation-related connectives have attracted more attention than others. The remaining (...)
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  23. Lloyd Humberstone (2008). Can Every Modifier Be Treated as a Sentence Modifier? Philosophical Perspectives 22 (1):241-275.
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  24. Lloyd Humberstone (2008). Invitation to Autoepistemology. Theoria 68 (1):13-51.
    The phrase ‘autoepistemic logic’ was introduced in Moore [1985] to refer to a study inspired in large part by criticisms in Stalnaker [1980] of a particular nonmonotonic logic proposed by McDermott and Doyle.1 Very informative discussions for those who have not encountered this area are provided by Moore [1988] and the wide-ranging survey article Konolige [1994], and the scant remarks in the present introductory section do not pretend to serve in place of those treatments as summaries of the field. A (...)
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  25. Lloyd Humberstone (2008). Parts and Partitions. Theoria 66 (1):41-82.
    Our object is to study the interaction between mereology and David Lewis’ theory of subject-matters, elaborating his observation that not every subject matter is of the form: how things stand with such-and-such a part of the world. After an informal introduction to this point in Section 1, we turn to a formal treatment of the partial orderings arising in the two areas – the part-whole relation, on the one hand, and the relation of refinement amongst partitions of the set (...)
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  26. Lloyd Humberstone (2007). Investigations Into a Left-Structural Right-Substructural Sequent Calculus. Journal of Logic, Language and Information 16 (2):141-171.
    We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction (fission, ‘cotensor’, par) against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules (...)
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  27. Lloyd Humberstone (2007). Identical Twins, Deduction Theorems, and Pattern Functions: Exploring the Implicative BCsK Fragment of S. [REVIEW] Journal of Philosophical Logic 36 (5):435 - 487.
    We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we (...)
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  28. Lloyd Humberstone (2007). Modal Logic for Other-World Agnostics: Neutrality and Halldén Incompleteness. [REVIEW] Journal of Philosophical Logic 36 (1):1 - 32.
    The logic of 'elsewhere,' i.e., of a sentence operator interpretable as attaching to a formula to yield a formula true at a point in a Kripke model just in case the first formula is true at all other points in the model, has been applied in settings in which the points in question represent spatial positions (explaining the use of the word 'elsewhere'), as well as in the case in which they represent moments of time. This logic is applied here (...)
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  29. Lloyd Humberstone (2006). Identical Twins, Deduction Theorems, and Pattern Functions: Exploring the Implicative BCsK Fragment of S. [REVIEW] Journal of Philosophical Logic 35 (5):435 - 487.
    We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we (...)
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  30. Lloyd Humberstone (2006). Lloyd Humberstone. Aristotelian Society Supplementary Volume 80 (1):265–320.
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  31. Lloyd Humberstone (2006). Variations on a Theme of Curry. Notre Dame Journal of Formal Logic 47 (1):101-131.
    After an introduction to set the stage, we consider some variations on the reasoning behind Curry's Paradox arising against the background of classical propositional logic and of BCI logic and one of its extensions, in the latter case treating the "paradoxicality" as a matter of nonconservative extension rather than outright inconsistency. A question about the relation of this extension and a differently described (though possibly identical) logic intermediate between BCI and BCK is raised in a final section, which closes with (...)
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  32. Lloyd Humberstone & Herman Cappelen (2006). Sufficiency and Excess. Proceedings of the Aristotelian Society 106 (1):265-320.
    This paper assembles examples and considerations bearing on such questions as the following. Are statements to the effect that someone is too young (for instance) or that someone is old enough always to be understood in terms of someone's being too young or too old for such-and-such-for example, for them to join a particular organization? And when a 'such-and-such' has been specified, is it always at least tacitly modal in force-in the case just given, too young or old enough to (...)
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  33. Lloyd Humberstone, Beziau's Translation Paradox.
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  34. Lloyd Humberstone (2005). Geach's Categorial Grammar. Linguistics and Philosophy 28 (3):281 - 317.
    Geach’s rich paper ‘A Program for Syntax’ introduced many ideas into the arena of categorial grammar, not all of which have been given the attention they warrant in the thirty years since its first publication. Rather surprisingly, one of our findings (Section 3 below) is that the paper not only does not contain a statement of what has widely come to be known as “Geach’s Rule”, but in fact presents considerations which are inimical to the adoption of the rule in (...)
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  35. Lloyd Humberstone, Logical Discrimination.
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  36. Lloyd Humberstone (2005). Modality. In Frank Jackson & Michael Smith (eds.), The Oxford Handbook of Contemporary Philosophy. Oxford University Press.
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  37. Lloyd Humberstone (2005). For Want of an 'And': A Puzzle About Non-Conservative Extension. History and Philosophy of Logic 26 (3):229-266.
    Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free (...)
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  38. A. P. Hazen & Lloyd Humberstone (2004). Similarity Relations and the Preservation of Solidity. Journal of Logic, Language and Information 13 (1):25-46.
    The partitions of a given set stand in a well known one-to-onecorrespondence with the equivalence relations on that set. We askwhether anything analogous to partitions can be found which correspondin a like manner to the similarity relations (reflexive, symmetricrelations) on a set, and show that (what we call) decompositions – of acertain kind – play this role. A key ingredient in the discussion is akind of closure relation (analogous to the consequence relationsconsidered in formal logic) having nothing especially to do (...)
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  39. Lloyd Humberstone (2004). Archetypal Forms of Inference. Synthese 141 (1):45 - 76.
    A form (or pattern) of inference, let us say, explicitlysubsumes just such particular inferences as are instances of the form, and implicitly subsumes thoseinferences with a premiss and conclusion logically equivalent to the premiss and conclusion of an instanceof the form in question. (For simplicity we restrict attention to one-premiss inferences.) A form ofinference is archetypal if it implicitly subsumes every correct inference. A precise definition (Section 1)of these concepts relativizes them to logics, since different logics classify different inferences ascorrect, (...)
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  40. Lloyd Humberstone (2004). Special Issue: The Two-Dimensional Framework and its Applications: Metaphysics, Language, and Mind. Philosophical Studies 118:463-464.
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  41. Lloyd Humberstone (2004). Two-Dimensional Adventures. Philosophical Studies 118 (1-2):17--65.
    This paper recalls some applications of two-dimensional modal logic from the 1980s, including work on the logic of Actually and on a somewhat idealized version of the indicative/subjunctive distinction, as well as on absolute and relative necessity. There is some discussion of reactions this material has aroused in commentators since. We also survey related work by Leslie Tharp from roughly the same period.
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  42. Lloyd Humberstone, Yet Another Choice of Primitives Warning : Normal Modal Logics.
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  43. S. Chopra, A. G. Cohn, R. P. de Freitas, H. Field, A. Ghose, L. Goble, V. Halbach, L. Humberstone, N. Kamide & S. Kovac (2003). Benevides, MRF, 343 Berk, L., 323 Boėr, SE, 43 Calabrese, PG. Journal of Philosophical Logic 32 (669).
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  44. Lloyd Humberstone (2003). False Though Partly True – an Experiment in Logic. Journal of Philosophical Logic 32 (6):613-665.
    We explore in an experimental spirit the prospects for extending classical propositional logic with a new operator P intended to be interpreted when prefixed to a formula as saying that formula in question is at least partly true. The paradigm case of something which is, in the sense envisaged, false though still "partly" true is a conjunction one of whose conjuncts is false while the other is true. Ideally, we should like such a logic to extend classical logic - or (...)
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  45. Lloyd Humberstone (2003). Note on Contraries and Subcontraries. Noûs 37 (4):690–705.
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  46. Lloyd Humberstone (2003). A Strange Remark Attributed to Gödel. History and Philosophy of Logic 24 (1):39-44.
    We assemble material from the literature on matrix methodology for sentential logic?without claiming to present any new logical results?in order to show that Gödel once made (or at least, is quoted as having made) an uncharacteristically ill-considered remark in this area.
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  47. Lloyd Humberstone (2002). Implicational Converses. Logique Et Analyse 45:61-79.
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  48. Lloyd Humberstone (2002). The Modal Logic of Agreement and Noncontingency. Notre Dame Journal of Formal Logic 43 (2):95-127.
    The formula A (it is noncontingent whether A) is true at a point in a Kripke model just in case all points accessible to that point agree on the truth-value of A. We can think of -based modal logic as a special case of what we call the general modal logic of agreement, interpreted with the aid of models supporting a ternary relation, S, say, with OA (which we write instead of A to emphasize the generalization involved) true at a (...)
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  49. Vladimir Markin, Dmitry Zaitsev, Imaginary Logic, Lloyd Humberstone, Implicational Converses, Jose M. Mendez, Francisco Salto, Pedro Mendez, Roger Vergauwen & Ray Lam (2002). Tjeerd B. Jongeling, Teun Koetsier & Evert Wattel, a Logical Approach to Qualitative Reasoning With'several'... 15. Logique Et Analyse 45:1.
     
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  50. Lloyd Humberstone (2001). The Pleasures of Anticipation: Enriching Intuitionistic Logic. [REVIEW] Journal of Philosophical Logic 30 (5):395-438.
    We explore a relation we call 'anticipation' between formulas, where A anticipates B (according to some logic) just in case B is a consequence (according to that logic, presumed to support some distinguished implicational connective →) of the formula A → B. We are especially interested in the case in which the logic is intuitionistic (propositional) logic and are much concerned with an extension of that logic with a new connective, written as "a", governed by rules which guarantee that for (...)
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