| 1. |
- Blanck, Rasmus, 1982
(författare)
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Hierarchical incompleteness results for arithmetically definable extensions of fragments of arithmetic
- 2021
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Ingår i: The Review of Symbolic Logic. - 1755-0203 .- 1755-0211. ; 14:3, s. 624-644
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Tidskriftsartikel (refereegranskat)abstract
- There has been a recent interest in hierarchical generalisations of classic incompleteness results. This paper provides evidence that such generalisations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “T is Σn-ill” is a canonical example of a Σn+1formula that is Πn+1-conservative over T.
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| 2. |
- Cantwell, John
(författare)
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An Expressivist Analysis Of The Indicative Conditional With A Restrictor Semantics
- 2021
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Ingår i: The Review of Symbolic Logic. - : Cambridge University Press (CUP). - 1755-0203 .- 1755-0211. ; 14:2, s. 487-530
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Tidskriftsartikel (refereegranskat)abstract
- A globally expressivist analysis of the indicative conditional based on the Ramsey Test is presented. The analysis is a form of 'global' expressivism in that it supplies acceptance and rejection conditions for all the sentence forming connectives of propositional logic (negation, disjunction, etc.) and so allows the conditional to embed in arbitrarily complex sentences (thus avoiding the Frege-Geach problem). The expressivist framework is semantically characterized in a restrictor semantics due to Vann McGee, and is completely axiomatized in a logic dubbed ICL ('Indicative Conditional Logic'). The expressivist framework extends the AGM (after Alchourron, Gardenfors, Makinson) framework for belief revision and so provides a categorical ('yes'-'no') epistemology for conditionals that complements McGee's probabilistic framework while drawing on the same semantics. The result is an account of the semantics and acceptability conditions of the indicative conditional that fits well with the linguistic data (as pooled by linguists and from psychological experiments) while integrating both expressivist and semanticist perspectives.
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| 3. |
- Goranko, Valentin, et al.
(författare)
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LOGICS FOR PROPOSITIONAL DETERMINACY AND INDEPENDENCE
- 2018
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Ingår i: The Review of Symbolic Logic. - 1755-0203 .- 1755-0211. ; 11:3, s. 470-506
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Tidskriftsartikel (refereegranskat)abstract
- We introduce and study formal logics for reasoning about propositional determinacy and independence. These relate naturally with the philosophical concept of supervenience, which can also be regarded as a generalisation of logical consequence. Propositional Dependence Logic D, and Propositional Independence Logic I are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics L_D and L_I, based on Kripke semantics, and propose them as alternatives for D and I, respectively. We analyse and compare the relative expressive powers of these four logics and also discuss how they relate to the natural language use and meaning of the concepts of determinacy and independence. We argue that L_D and L_I naturally resolve a range of interpretational problems that arise in D and I. We also obtain sound and complete axiomatizations for L_D and L_I and relate them with the recently studied inquisitive logics and their semantics.
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| 4. |
- Hansson, Sven Ove
(författare)
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Outcome level analysis of belief contraction
- 2013
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Ingår i: The Review of Symbolic Logic. - 1755-0203 .- 1755-0211. ; 6:2, s. 183-204
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Tidskriftsartikel (refereegranskat)abstract
- The outcome set of a belief change operator is the set of outcomes that can be obtained with it. Axiomatic characterizations are reported for the outcome sets of the standard AGM contraction operators and eight types of base-generated contraction. These results throw new light on the properties of some of these operators.
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| 5. |
- Kindvall Gorbow, Paul, 1981, et al.
(författare)
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The Copernican Multiverse of Sets
- 2022
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Ingår i: The Review of Symbolic Logic. - 1755-0203 .- 1755-0211. ; 15:4, s. 1033-1069
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Tidskriftsartikel (refereegranskat)abstract
- We develop an untyped framework for the multiverse of set theory. ZF is extended with semantically motivated axioms utilizing the new symbols Uni(U) and Mod(U), expressing that U is a universe and that is true in the universe U, respectively. Here ranges over the augmented language, leading to liar-style phenomena that are analysed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philo- sophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally inter- preting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consis- tency strength is at most slightly above that of the base theory ZF, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins' multiverse theory. © 2021 Cambridge University Press. All rights reserved.
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| 6. |
- Pagin, Peter, 1953-
(författare)
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COMPOSITIONALITY, COMPUTABILITY, AND COMPLEXITY
- 2021
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Ingår i: The Review of Symbolic Logic. - : Cambridge University Press (CUP). - 1755-0203 .- 1755-0211. ; 14:3, s. 551-591
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Tidskriftsartikel (refereegranskat)abstract
- This paper starts from the observation that the standard arguments for compositionality are really arguments for the computability of semantics. Since computability does not entail compositionality, the question of what justifies compositionality recurs. The paper then elaborates on the idea of recursive semantics as corresponding to computable semantics. It is then shown by means of time complexity theory and with the use of term rewriting as systems of semantic computation, that syntactically unrestricted, noncompositional recursive semantics leads to computational explosion (factorial complexity). Hence, with combinatorially unrestricted syntax, semantics with tractable time complexity is compositional.
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| 7. |
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| 8. |
- Segerberg, Krister
(författare)
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Some completeness theorems in the dynamic doxastic logic of iterated belief revision
- 2010
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Ingår i: The Review of Symbolic Logic. - 1755-0203. ; 3:2, s. 228-246
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Tidskriftsartikel (refereegranskat)abstract
- The success of the AGM paradigm-the theory of belief change initiated by Alchourron, Gardenfors, and Makinson-is remarkable, as even a quick look at the literature it has generated will testify.(1) But it is also remarkable, at least in hindsight, how limited was the original effort. For example, the theory concerns the beliefs of just one agent; all incoming information is accepted; belief change is uniquely determined by the new information; there is no provision for nested beliefs. And perhaps most surprising: there is no analysis of iterated change. In this paper it is that last restriction that is at issue. Our medium of study is dynamic doxastic logic (DDL). The success of the AGM paradigm The particular contribution of the paper is detailed completeness proofs for three dynamic doxastic logics of iterated belief revision. The problem of extending the AGM paradigm to include iterated change has been discussed for years, but systematic discussions have appeared only recently (see Segerberg, 2007 and forthcoming, but also van Benthem, 2007; Rott, 2006; Zvesper, 2007).(2)
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| 9. |
- Westerståhl, Dag, 1946
(författare)
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Decomposing generalized quantifiers
- 2008
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Ingår i: The Review of Symbolic Logic. - 1755-0203. ; 1:3, s. 355-371
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Tidskriftsartikel (refereegranskat)abstract
- This note explains the circumstances under which a type <1> quantifier can be decomposed into a type <1,1> quantifier and a set, by fixing the first argument of the former to the latter. The motivation comes from the semantics of Noun Phrases (also called Determiner Phrases) in natural languages, but in this article, I focus on the logical facts. However, my examples are taken among quantifiers appearing in natural languages, and at the end, I sketch two more principled linguistic applications.
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