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- Independence-friendly logic (IF logic; proposed by Jaakko Hintikka and in 1989) is an extension of classical first-order logic (FOL) by means of slashed quantifiers of the form and , where is a finite set of variables. The intended reading of is "there is a which is functionally independent from the variables in ". IF logic allows one to express more general patterns of dependence between variables than those which are implicit in first-order logic. This greater level of generality leads to an actual increase in expressive power; the set of can characterize the same classes of structures as existential second-order logic. For example, it can express branching quantifier sentences, such as the formula which expresses infinity in the empty signature; this cannot be done in FOL. Therefore, first-order logic cannot, in general, express this pattern of dependency, in which depends only on and , and depends only on and . IF logic is more general than branching quantifiers, for example in that it can express dependencies that are not transitive, such as in the quantifier prefix , which expresses that depends on , and depends on , but does not depend on . The introduction of IF logic was partly motivated by the attempt of extending the game semantics of first-order logic to games of imperfect information. Indeed, a semantics for IF sentences can be given in terms of these kinds of games (or, alternatively, by means of a translation procedure to existential second-order logic). A semantics for open formulas cannot be given in the form of a Tarskian semantics; an adequate semantics must specify what it means for a formula to be satisfied by a set of assignments of common variable domain (a team) rather than satisfaction by a single assignment. Such a team semantics was developed by Hodges. Independence-friendly logic is translation equivalent, at the level of sentences, with a number of other logical systems based on team semantics, such as dependence logic, dependence-friendly logic, exclusion logic and independence logic; with the exception of the latter, IF logic is known to be equiexpressive to these logics also at the level of open formulas. However, IF logic differs from all the above-mentioned systems in that it lacks locality: the meaning of an open formula cannot be described just in terms of the free variables of the formula; it is instead dependent on the context in which the formula occurs. Independence-friendly logic shares a number of metalogical properties with first-order logic, but there are some differences, including lack of closure under (classical, contradictory) negation and higher complexity for deciding the validity of formulas. Extended IF logic addresses the closure problem, but its game-theoretical semantics is more complicated, and such logic corresponds to a larger fragment of second-order logic, a proper subset of . Hintikka has argued that IF and extended IF logic should be used as a basis for the foundations of mathematics; this proposal has been met in some cases with skepticism. (en)
- Lógica de independência amigável (do inglês Independence-Friendly, Lógica IF), proposta por Jaakko Hintikka e Gabriel Sandu em 1989, objetiva ser uma alternativa mais natural e intuitiva à clássica lógica de primeira ordem (FOL). Lógica do SE é caracterizada por quantificadores ramificados. Esta é mais expressiva que FOL porque permite que sejam expressas relações independentes entre variáveis quantificadas. Por exemplo, a fórmula ∀a ∀b ∃c/b ∃d/a φ(a,b,c,d) ("x/y" deve ser lida como "x independente de y") não pode ser expressa em FOL. Isso é porque c depende apenas de a, e d depende apenas de b. Lógica de primeira ordem não pode expressar esses independências por qualquer reordenação linear de quantificadores. Em parte, a lógica do SE foi motivada pela semântica de jogos para jogos com Informação perfeita. A lógica IF é a tradução equivalente a lógica de segunda ordem existencial e também com dependência lógica de Väänänen e com a lógica de primeira ordem estendida com quantificadores de Henkin. Embora compartilhe várias propriedades meta lógicas com a lógica de primeira ordem, existem algumas diferenças, incluindo a falta de fechamento sob negação e uma complexidade superior para decidir a validade das fórmulas. A lógica IF expandida remete ao problema do fecho, mas sacrifica a semântica de jogos no processo, e isso pertence propriamente a um fragmento superior da lógica de segunda ordem. A proposição de Hintikka de que a lógica IF e sua versão estendida ser usada como fundações da matemática tem sido visto com ceticismo por outros matemáticos, incluindo Väänänen e Solomon Feferman. (pt)
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- Burgess, John P., "A Remark on Henkin Sentences and Their Contraries", Notre Dame Journal of Formal Logic 44 :185-188 . (en)
- Figueira, Santiago, Gorín, Daniel and Grimson, Rafael "On the Expressive Power of IF-Logic with Classical Negation", WoLLIC 2011 proceedings, pp. 135-145, ,http://www.glyc.dc.uba.ar/santiago/papers/iflso.pdf. (en)
- Eklund, Matti and Kolak, Daniel, "Is Hintikka’s Logic First Order?" Synthese, 131: 371-388 June 2002, https://link.springer.com/content/pdf/10.1023%2FA%3A1016184410627. (en)
- Hodges, Wilfrid , "Compositional semantics for a language of imperfect information". Journal of the IGPL 5: 539–563. (en)
- Cameron, Peter and Hodges, Wilfrid , "Some combinatorics of imperfect information". Journal of Symbolic Logic 66: 673-684. (en)
- Kontinen, Juha and Väänänen, Jouko, "On definability in dependence logic" , Journal of Logic, Language and Information 18 , 317-332. (en)
- Enderton, Herbert B., "Finite Partially-Ordered Quantifiers", Mathematical Logic Quarterly Volume 16, Issue 8 1970 Pages 393–397. (en)
- Hodges, Wilfrid, "Some Strange Quantifiers", in Lecture Notes in Computer Science 1261:51-65, Jan. 1997. (en)
- Mann, Allen L., Sandu, Gabriel and Sevenster, Merlijn Independence-Friendly Logic. A Game-Theoretic Approach, Cambridge University Press, . (en)
- Hintikka, Jaakko , "The Principles of Mathematics Revisited", Cambridge University Press, . (en)
- Janssen, Theo M. V., "Independent choices and the interpretation of IF logic." Journal of Logic, Language and Information, Volume 11 Issue 3, Summer 2002, pp. 367-387 http://dare.uva.nl/document/1273. (en)
- Hintikka, Jaakko, "Hyperclassical logic and its implications for logical theory", Bulletin of Symbolic Logic 8, 2002, 404-423http://www.math.ucla.edu/~asl/bsl/0803/0803-004.ps . (en)
- Sandu, Gabriel, "On the Logic of Informational Independence and Its Applications", Journal of Philosophical Logic Vol. 22, No. 1 , pp. 29-60. (en)
- Kolak, Daniel and Symons, John, "The Results are In: The Scope and Import of Hintikka’s Philosophy" in Daniel Kolak and John Symons, eds., Quantifiers, Questions, and Quantum Physics. Essays on the Philosophy of Jaakko Hintikka, Springer 2004, pp. 205-268 , . (en)
- Sandu, Gabriel, "If-Logic and Truth-definition", Journal of Philosophical Logic April 1998, Volume 27, Issue 2, pp 143–164. (en)
- Kolak, Daniel, On Hintikka, Belmont: Wadsworth 2001 . (en)
- Hintikka, Jaakko and Sandu, Gabriel , "Informational independence as a semantical phenomenon", in Logic, Methodology and Philosophy of Science VIII , North-Holland, Amsterdam, . (en)
- Hintikka, Jaakko and Sandu, Gabriel, "Game-theoretical semantics", in Handbook of logic and language, ed. J. van Benthem and A. ter Meulen, Elsevier 1996 Updated in the 2nd second edition of the book . (en)
- Feferman, Solomon, "What kind of logic is “Independence Friendly” logic?", in The Philosophy of Jaakko Hintikka ; Library of Living Philosophers vol. 30, Open Court , 453-469, http://math.stanford.edu/~feferman/papers/hintikka_iia.pdf. (en)
- Walkoe, Wilbur John Jr., "Finite Partially-Ordered Quantification", The Journal of Symbolic Logic Vol. 35, No. 4 , pp. 535-555. (en)
- Väänänen, Jouko, 2007, 'Dependence Logic -- A New Approach to Independence Friendly Logic', Cambridge University Press, , https://web.archive.org/web/20070921164444/http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521876599. (en)
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