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P. D. Welch [31]Philip Welch [28]Philip D. Welch [8]P. Welch [5]
Patrick Welch [3]Patrick J. Welch [2]
  1.  48
    Ultimate Truth Vis- À- Vis Stable Truth.P. D. Welch - 2008 - Review of Symbolic Logic 1 (1):126-142.
    We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: -CA0 (...)
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  2.  20
    Some Observations on Truth Hierarchies.P. D. Welch - 2014 - Review of Symbolic Logic 7 (1):1-30.
  3. Possible-Worlds Semantics for Modal Notions Conceived as Predicates.Volker Halbach, Hannes Leitgeb & Philip Welch - 2003 - Journal of Philosophical Logic 32 (2):179-223.
    If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...)
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  4.  10
    Some Observations on Truth Hierarchies: A Correction.Philip D. Welch - forthcoming - Review of Symbolic Logic:1-4.
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  5.  62
    Games for Truth.P. D. Welch - 2009 - Bulletin of Symbolic Logic 15 (4):410-427.
    We represent truth sets for a variety of the well known semantic theories of truth as those sets consisting of all sentences for which a player has a winning strategy in an infinite two person game. The classifications of the games considered here are simple, those over the natural model of arithmetic being all within the arithmetical class of $\Sum_{3}^{0}$.
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  6.  75
    Revision Revisited.Leon Horsten, Graham E. Leigh, Hannes Leitgeb & Philip Welch - 2012 - Review of Symbolic Logic 5 (4):642-664.
    This article explores ways in which the Revision Theory of Truth can be expressed in the object language. In particular, we investigate the extent to which semantic deficiency, stable truth, and nearly stable truth can be so expressed, and we study different axiomatic systems for the Revision Theory of Truth.
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  7.  73
    Reflecting on Absolute Infinity.Philip Welch & Leon Horsten - 2016 - Journal of Philosophy 113 (2):89-111.
    This article is concerned with reflection principles in the context of Cantor’s conception of the set-theoretic universe. We argue that within such a conception reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity.
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  8.  8
    Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions.P. D. Welch - 2011 - Journal of Symbolic Logic 76 (2):418 - 436.
    We locate winning strategies for various ${\mathrm{\Sigma }}_{3}^{0}$ -games in the L-hierarchy in order to prove the following: Theorem 1. KP+Σ₂-Comprehension $\vdash \exists \alpha L_{\alpha}\ models"\Sigma _{2}-{\bf KP}+\Sigma _{3}^{0}-\text{Determinacy}."$ Alternatively: ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ "there is a β-model of ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17 em}}{\mathrm{\Sigma }}_{3}^{0}$ -Determinacy." The implication is not reversible. (The antecedent here may be replaced with ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\mathrm{\Pi }}_{3}^{1}\right)-{\mathrm{C}\mathrm{A}}_{0}:\text{\hspace{0.17em}}{\mathrm{\Pi }}_{3}^{1}$ instances of Comprehension with only ${\mathrm{\Pi }}_{3}^{1}$ -lightface definable parameters—or even weaker theories.) Theorem 2. KP +Δ₂-Comprehension +Σ₂-Replacement + ${\mathrm{\Sigma }}_{3}^{0}\phantom{\rule{0ex}{0ex}}$ -Determinacy. (Here AQI (...)
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  9. Necessities and Necessary Truths: A Prolegomenon to the Use of Modal Logic in the Analysis of Intensional Notions.V. Halbach & P. Welch - 2009 - Mind 118 (469):71-100.
    In philosophical logic necessity is usually conceived as a sentential operator rather than as a predicate. An intensional sentential operator does not allow one to express quantified statements such as 'There are necessary a posteriori propositions' or 'All laws of physics are necessary' in first-order logic in a straightforward way, while they are readily formalized if necessity is formalized by a predicate. Replacing the operator conception of necessity by the predicate conception, however, causes various problems and forces one to reject (...)
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  10. Godel's Disjunction: The Scope and Limits of Mathematical Knowledge.Leon Horsten & Philip Welch (eds.) - 2016 - Oxford University Press UK.
    To what extent can we hope to find answers to all mathematical questions? A famous theorem from Gödel entails that if our thinking capacities do not go beyond what an electronic computer is capable of, then there are indeed absolutely unsolvable mathematical problems. Thus it is of capital importance to find out whether human mathematicians can outstrip computers. Within this context, the contributions to this book critically examine positions about the scope and limits of human mathematical knowledge.
     
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  11.  21
    The Complexity of the Dependence Operator.P. D. Welch - 2015 - Journal of Philosophical Logic 44 (3):337-340.
    We show that Leitgeb’s dependence operator of Leitgeb is a \-operator and that this is best possible.
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  12.  77
    The Undecidability of Propositional Adaptive Logic.Leon Horsten & Philip Welch - 2007 - Synthese 158 (1):41-60.
    We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and (...)
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  13.  37
    On Gupta-Belnap Revision Theories of Truth, Kripkean Fixed Points, and the Next Stable Set.P. D. Welch - 2001 - Bulletin of Symbolic Logic 7 (3):345-360.
    We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their theory of circular definitions as those notions universally definable over the next stable set. We give a simplified account of varied revision sequences-as a generalised algorithmic theory of truth. This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.
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  14.  57
    On the Consistency Strength of the Inner Model Hypothesis.Sy-David Friedman, Philip Welch & W. Hugh Woodin - 2008 - Journal of Symbolic Logic 73 (2):391 - 400.
  15.  14
    Ramsey-Like Cardinals II.Victoria Gitman & P. D. Welch - 2011 - Journal of Symbolic Logic 76 (2):541-560.
  16.  41
    The Extent of Computation in Malament–Hogarth Spacetimes.P. D. Welch - 2008 - British Journal for the Philosophy of Science 59 (4):659-674.
    We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some relations on natural numbers that are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. (...)
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  17. Introduction.Salvatore Florio, Øystein Linnebo, Sean Walsh & Philip Welch - 2015 - Notre Dame Journal of Formal Logic 56 (1):1-2.
    Introduction to a special issue based on a summer school on set theory and high-order logic.
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  18.  12
    Recognizable Sets and Woodin Cardinals: Computation Beyond the Constructible Universe.Merlin Carl, Philipp Schlicht & Philip Welch - 2018 - Annals of Pure and Applied Logic 169 (4):312-332.
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  19.  41
    Set-Theoretic Absoluteness and the Revision Theory of Truth.Benedikt Löwe & Philip D. Welch - 2001 - Studia Logica 68 (1):21-41.
    We describe the solution of the Limit Rule Problem of Revision Theory and discuss the philosophical consequences of the fact that the truth set of Revision Theory is a complete 1/2 set.
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  20.  66
    Greatly Erdős Cardinals with Some Generalizations to the Chang and Ramsey Properties.I. Sharpe & P. D. Welch - 2011 - Annals of Pure and Applied Logic 162 (11):863-902.
    • We define a notion of order of indiscernibility type of a structure by analogy with Mitchell order on measures; we use this to define a hierarchy of strong axioms of infinity defined through normal filters, the α-weakly Erdős hierarchy. The filters in this hierarchy can be seen to be generated by sets of ordinals where these indiscernibility orders on structures dominate the canonical functions.• The limit axiom of this is that of greatly Erdős and we use it to calibrate (...)
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  21.  57
    Truth, Logical Validity and Determinateness: A Commentary on Field’s Saving Truth From Paradox.P. D. Welch - 2011 - Review of Symbolic Logic 4 (3):348-359.
    We consider notions of truth and logical validity defined in various recent constructions of Hartry Field. We try to explicate his notion of determinate truth by clarifying the path-dependent hierarchies of his determinateness operator.
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  22. Characterising Subsets of Ω1 Constructible From a Real.P. D. Welch - 1994 - Journal of Symbolic Logic 59 (4):1420 - 1432.
    A small large cardinal upper bound in V for proving when certain subsets of ω 1 (including the universally Baire subsets) are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that $\forall B \subseteq \omega_1 \lbrack B$ is universally Baire $\Leftrightarrow B \in L\lbrack r \rbrack$ for some real r] is preserved under set-sized forcing extensions if and only if there (...)
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  23.  10
    Coding the Universe.A. Beller, R. Jensen & P. Welch - 1982 - Cambridge University Press.
  24. On Elementary Embeddings From an Inner Model to the Universe.J. Vickers & P. D. Welch - 2001 - Journal of Symbolic Logic 66 (3):1090-1116.
    We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, (...)
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  25.  25
    On Revision Operators.P. D. Welch - 2003 - Journal of Symbolic Logic 68 (2):689-711.
    We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as “revision operations”. More particularly we: (i) characterise the revision theoretically definable subsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta’s theory of truth over arithmetic using \emph{fully varied revision} sequences yields a complete \Pi13 set of integers; (iii) the set of \emph{stably categorical} sentences using their revision operator ψ is similarly \Pi13 and which (...)
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  26.  41
    Global Reflection Principles.P. D. Welch - unknown
    Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia , Woodin’s Ω-Logic.
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  27.  11
    Hypermachines.Sy-David Friedman & P. D. Welch - 2011 - Journal of Symbolic Logic 76 (2):620 - 636.
    The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a "Σ₂-machine" in that it uses a Σ₂ Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σ n rule. Such machines either halt or enter an infinite loop by stage ζ(n) = df μζ(n)[∃Σ(n) > ζ(n) L ζ(n) ≺ Σn L Σ(n) ], again generalising precisely the ITTM case. The collection of (...)
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  28.  27
    Comparing Inductive and Circular Definitions: Parameters, Complexity and Games.Kai-Uwe Küdhnberger, Benedikt Löwe, Michael Möllerfeld & Philip Welch - 2005 - Studia Logica 81 (1):79 - 98.
    Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.
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  29.  20
    On the Possibility, or Otherwise, of Hypercomputation.Philip D. Welch - 2004 - British Journal for the Philosophy of Science 55 (4):739-746.
    We claim that a recent article of P. Cotogno ([2003]) in this journal is based on an incorrect argument concerning the non-computability of diagonal functions. The point is that whilst diagonal functions are not computable by any function of the class over which they diagonalise, there is no ?logical incomputability? in their being computed over a wider class. Hence this ?logical incomputability? regrettably cannot be used in his argument that no hypercomputation can compute the Halting problem. This seems to lead (...)
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  30. Field on Revenge.Agustin Rayo & Philip Welch - 2007 - In J. C. Beall (ed.), Revenge of the Liar: New Essays on the Paradox. Oxford University Press.
  31.  14
    Menachem Magidor. Representing Sets of Ordinals as Countable Unions of Sets in the Core Model. Transactions of the American Mathematical Society, Vol. 317 , Pp. 91–126. [REVIEW]Philip Welch - 1995 - Journal of Symbolic Logic 60 (2):701-704.
  32.  27
    Moral Psychology and the Problem of Moral Criteria.Patrick Welch - 2011 - Journal of Moral Education 40 (4):513-526.
    This article is intended as an initial investigation into the foundations of moral psychology. I primarily examine a recent work in moral education, Daniel Lapsley?s and Darcia Narvaez?s ?Character education?, whose authors seem to assume at points that criteria for discerning moral actions and moral traits can be derived apart from ethics or moral philosophy. This assumption, which appears to stem from misconceptions about both the virtues traditionally understood and the non-empirical nature of moral-philosophical theorising, is problematic: (1) it courts (...)
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  33.  13
    Discrete Transfinite Computation Models.Philip D. Welch - 2011 - In S. B. Cooper & Andrea Sorbi (eds.), Computability in Context: Computation and Logic in the Real World. World Scientific. pp. 375--414.
  34.  33
    Eventually Infinite Time Turing Machine Degrees: Infinite Time Decidable Reals.P. D. Welch - 2000 - Journal of Symbolic Logic 65 (3):1193-1203.
    We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated (...)
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  35.  11
    William J. Mitchell and John R. Steel. Fine Structure and Iteration Trees. Lecture Notes in Logic, No. 3. Springer-Verlag, Berlin, Heidelberg, New York, Etc., 1994, V + 130 Pp. [REVIEW]P. D. Welch - 1997 - Journal of Symbolic Logic 62 (4):1491-1493.
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  36.  19
    On Unfoldable Cardinals, Ω-Closed Cardinals, and the Beginning of the Inner Model Hierarchy.P. D. Welch - 2003 - Archive for Mathematical Logic 43 (4):443-458.
    Let κ be a cardinal, and let H κ be the class of sets of hereditary cardinality less than κ ; let τ (κ) > κ be the height of the smallest transitive admissible set containing every element of {κ}∪H κ . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H κ is as long as τ. (It is known that some weak (...)
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  37.  8
    Games and Ramsey-Like Cardinals.Dan Saattrup Nielsen & Philip Welch - 2019 - Journal of Symbolic Logic 84 (1):408-437.
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  38. 68 Solidarity.Patrick J. Welch & Stuart D. Yoak - 2009 - In Jan Peil & Irene van Staveren (eds.), Handbook of Economics and Ethics. Edward Elgar.
  39.  34
    On Successors of Jónsson Cardinals.J. Vickers & P. D. Welch - 2000 - Archive for Mathematical Logic 39 (6):465-473.
    We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming $O^{Sword}$ does not exist. Namely, if $\kappa$ is a Jónsson cardinal then $\kappa^+ = \kappa^{+K}$ , provided that there is no non-trivial elementary embedding $j:K \longrightarrow K$ . There are a number of related results in ZFC concerning $\cal{P}(\kappa)$ in V and inner models, for $\kappa$ a Jónsson or singular cardinal.
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  40.  28
    Countable Unions of Simple Sets in the Core Model.P. D. Welch - 1996 - Journal of Symbolic Logic 61 (1):293-312.
    We follow [8] in asking when a set of ordinals $X \subseteq \alpha$ is a countable union of sets in K, the core model. We show that, analogously to L, and X closed under the canonical Σ 1 Skolem function for K α can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.
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  41.  17
    Ethics: Universal, but Neither Empirical nor a Priori: A Response to Lapsley and Narvaez.Patrick Welch - 2011 - Journal of Moral Education 40 (4):533-533.
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  42.  11
    Rethinking Revision.P. Welch - 2019 - Journal of Philosophical Logic 48 (1):137-154.
    We sketch a broadening of the Gupta-Belnap notion of a circular or revision theoretic definition into that of a more generalized form incorporating ideas of Kleene’s generalized or higher type recursion. This thereby connects the philosophically motivated, and derived, notion of a circular definition with an older form of definition by recursion using functionals, that is functions of functions, as oracles. We note that Gupta and Belnap’s notion of ‘categorical in L’ can be formulated in at least one of these (...)
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  43.  5
    Coding the Universe.A. Beller, R. Jensen & P. Welch - 1985 - Journal of Symbolic Logic 50 (4):1081-1081.
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  44.  30
    Bounded Martin's Maximum, Weak Erdős Cardinals, and $\Psi\Sb {AC}$.David Asperó & Philip Welch - 2002 - Journal of Symbolic Logic 67 (3):1141-1152.
  45.  1
    Bounded Martin's Maximum, Weak Erdӧs Cardinals, and Ψ Ac.David Asperó & Philip D. Welch - 2002 - Journal of Symbolic Logic 67 (3):1141-1152.
  46.  11
    On Elementary Embeddings From an Inner Model to the Universe.J. Vickers & P. D. Welch - 2001 - Journal of Symbolic Logic 66 (3):1090-1116.
    We consider the following question of Kunen: Does Con imply Con? We use core model theory to investigate consequences of the existence of such a j : M $\rightarrow$ V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the (...)
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  47.  25
    Some Descriptive Set Theory and Core Models.P. D. Welch - 1988 - Annals of Pure and Applied Logic 39 (3):273-290.
  48.  5
    The Ramified Analytical Hierarchy Using Extended Logics.Philip D. Welch - 2018 - Bulletin of Symbolic Logic 24 (3):306-318.
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  49.  12
    Determinacy in the Difference Hierarchy of Co-Analytic Sets.P. D. Welch - 1996 - Annals of Pure and Applied Logic 80 (1):69-108.
  50.  30
    Large Cardinals, Inner Models, and Determinacy: An Introductory Overview.P. D. Welch - 2015 - Notre Dame Journal of Formal Logic 56 (1):213-242.
    The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines of development.
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