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Modal deduction in second-order logic and set theory - I

J van Benthem
1997 Journal of Logic and Computation  
We investigate modal deduction through translation into standard logic and set theory.  ...  The employed techniques are mainly model-theoretic and set-theoretic, and they admit extensions to richer languages and modal deductive systems than that of basic modal logic.  ...  Weak second-order logic contains a set of axioms complete for first-order predicate logic, plus L 0 -substitution. Let us denote derivability in weak second-order logic with L 2 .  ... 
doi:10.1093/logcom/7.2.251 fatcat:f7yqrxrsabfttdwhvitcdqlc4u

Non-monotonic logic I

Drew McDermott, Jon Doyle
1980 Artificial Intelligence  
( The Inferential system used defines a first-order theory to be a set of axioms I I •~~~~.  ...  In order to proceed further , the tableau, must~~~~~~~~~~~~~~~~~~~~~~~~~~~ In case I., C is l abeled both 1 and 0. In case I I . , 'D is labeled both 1 and 0.  ... 
doi:10.1016/0004-3702(80)90012-0 fatcat:2np5actyufadji5i6glm4ehts4

The problem of logical omniscience, I

Robert Stalnaker
1991 Synthese  
beginning, I epistemic and doxastic logics -the logics of knowledge and belief have been modeled on modal logic -the logic of necessity and possibility.  ...  Some proponents of the sentence storage model argue that a belief set must meet certain minimal standards of logical coherence in order to count as a set of beliefs at all.  ... 
doi:10.1007/bf00413506 fatcat:uvinmfhk5rehfdlyeiypuvysom

Weakly Implicative (Fuzzy) Logics I: Basic Properties

Petr Cintula
2006 Archive for Mathematical Logic  
It contains the majority of logics studied in the literature under the name fuzzy logic. We present many general theorems for both classes, demonstrating their usefulness and importance.  ...  First we generalize the well-known class of implicative logics of Rasiowa and introduce the class of weakly implicative logics.  ...  Finally, I would like to thank to both of them and to Rostislav Horčík and Petr Hájek for the repeated reading of the drafts of this paper and helping me to improve it a lot in many aspects.  ... 
doi:10.1007/s00153-006-0011-5 fatcat:j276gfjzrvc7pnjyvff32trmqi

The Many Worlds of Modal λ-calculi: I. Curry-Howard for Necessity, Possibility and Time [article]

G. A. Kavvos
2016 arXiv   pre-print
We discuss logical aspects, modal λ-calculi, and their categorical semantics. The logics underlying the calculi surveyed are constructive versions of K, K4, S4, and LTL.  ...  We cover the prehistory of the subject and then concentrate on the developments that took place in the 1990s and early 2000s.  ...  of the proof theory of modal logic, and their associated computational and categorical interpretations.  ... 
arXiv:1605.08106v1 fatcat:q2hpdwpddvc43lxcaompezbm5u

Classical Logic I: First-Order Logic [chapter]

Wilfrid Hodges
2017 The Blackwell Guide to Philosophical Logic  
In the same spirit, there are several closely related languages called modal languages, and together they form modal logic. Likewise second-order logic, deontic logic and so forth.  ...  The contents of many undergraduate logic courses are first-order logic in this second sense.  ...  Modal logic takes on board the first example, epistemic logic the second. [See chapters 7 and 9.]  ... 
doi:10.1002/9781405164801.ch1 fatcat:3blbrl2zbrdfbifmo5qdwgtjiu

A theory of nonmonotonic rule systems I

W. Marek, A. Nerode, J. Remmel
1990 Annals of Mathematics and Artificial Intelligence  
Such systems include intuitionistic logic, classical logics, modal logics, and many others. Suppose that a nonempty set U is given.  ...  In the context of nonmonotone logic the intersection of all deductively closed sets containing I is a (non-deductively closed) set, called the set of secure consequences of I.  ... 
doi:10.1007/bf01531080 fatcat:zx35j3ewzneixmmpefdl5d7pia

Epistemic Closure and Epistemic Logic I: Relevant Alternatives and Subjunctivism

Wesley H. Holliday
2014 Journal of Philosophical Logic  
Finally, I consider the relation between closure and higher-order knowledge.  ...  By "modal decomposition" I obtain completeness and other results for two non-normal modal logics with respect to new semantics.  ...  Acknowledgements I wish to thank everyone who provided helpful feedback on earlier versions of this paper, especially Johan van Benthem and Krista Lawlor, as well as Davide Grossi, Peter Hawke, Tomohiro  ... 
doi:10.1007/s10992-013-9306-2 fatcat:npvw5qevevgc5igb5ciojjerji

Game logic and its applications I

Mamoru Kaneko, Takashi Nagashima
1996 Studia Logica: An International Journal for Symbolic Logic  
The limit case is an infinitary predicate extension of modal propositional logic KD~, and is of special interest in applications.  ...  In Part I, we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game.  ...  There still remain important problems in this direction from the viewpoint of both logic and game theory. These will be discussed in Kaneko [10] .  ... 
doi:10.1007/bf00370838 fatcat:rdsaefbgk5anje24ktqkbjzttq

Viral Vectors I: Major Viral Vectors

2000 Molecular Therapy  
In the second lecture we shall introduce the logical notions such as coequations and modalities and show, respectively, their appropriateness and completeness.  ...  I will present and contrast two different ways of presenting general classes of topological spaces that are amenable to analysis in subsystems of second-order arithmetic, the traditional setting for reverse  ... 
doi:10.1006/mthe.2000.0143 fatcat:t7k2ymu7rjeirdh3zhe4cxuxju

Modalities Over Actions, I. Model Theory [chapter]

L. Thome McCarty
1994 Principles of Knowledge Representation and Reasoning  
The combination of these two special cases is illustrated in [24] for a first-order language without a linear temporal order, and in [25] for the action language.  ...  In fact, the deontic modalities in the present paper are essentially the same as the deontic modalities in [16, 17] , but applied to the action language in [25].  ...  Notice, however, that Definition 3.1 is not restricted to first-order theories, and could apply equally well to second-order theories, such as the circumscription axiom.  ... 
doi:10.1016/b978-1-4832-1452-8.50136-6 fatcat:ddwxqyp5a5hkdebbdlql3ldcpe

Set theory for verification: I. From foundations to functions

Lawrence C. Paulson
1993 Journal of automated reasoning  
A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle.  ...  Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators.  ...  Philippe Noël's version of set theory, modified by Martin Coen, was the starting point of the present theory. Tobias Nipkow made great contributions to Isabelle, including the simplifier.  ... 
doi:10.1007/bf00881873 fatcat:e22sreqldfg4jkes3jhai6dike

MODALITY AND AXIOMATIC THEORIES OF TRUTH I: FRIEDMAN-SHEARD

JOHANNES STERN
2014 The Review of Symbolic Logic  
Second, we turn to semantic aspects and develop a modal semantics for the newly developed theory.  ...  In this first paper of our two part investigation we develop the general strategy and then apply it to the axiomatic theory of truth Friedman-Sheard.  ...  Already in the modal operator logic Q K , K in first-order logic plus the modal principles (N D ) and (B F ) we can prove the rigidity of syntax: LEMMA 5.2.  ... 
doi:10.1017/s1755020314000057 fatcat:nmejpurlmjfi7lwmpojw3c6bh4

A Foundation for Metareasoning Part I: The Proof Theory

G. Criscuolo
2002 Journal of Logic and Computation  
OM pairs allow us to generate deductively pairs composed of an object theory and a metatheory related via a so called reflection principle.  ...  This is done by imposing, via appropriate reflection rules, the relation we want to hold between the object theory and the metatheory. In this paper we concentrate on the proof theory of OM pairs.  ...  Examples of duality principles for propositional logic and modal logic can be found in [23] and [11] respectively.  ... 
doi:10.1093/logcom/12.1.167 fatcat:uj2e6ptcx5huvh62tic3y2xsqa

Pragmatic and dialogic interpretations of bi-intuitionism. Part I

Gianluigi Bellin, Massimiliano Carrara, Daniele Chiffi, Alessandro Menti
2014 Logic and Logical Philosophy  
We focus on the logical role of illocutionary forces and justification conditions in order to provide "intended interpretations" of logical systems that classify inferential uses in natural language and  ...  We consider a "polarized" version of bi-intuitionistic logic [5, 2, 6, 4 ] as a logic of assertions and hypotheses and show that it supports a "rich proof theory" and an interesting categorical interpretation  ...  López-Escobar [19] the first to notice that first order bi-intuitionistic logic is not a conservative extension of first order intuitionistic logic, 2 since in the standard theory of first order bi-intuitionistic  ... 
doi:10.12775/llp.2014.011 fatcat:gelonc7qlfdjvldyax3bmuaype
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