## Notre Dame Journal of Formal Logic

### A Syntactic Embedding of Predicate Logic into Second-Order Propositional Logic

#### Abstract

We give a syntactic translation from first-order intuitionistic predicate logic into second-order intuitionistic propositional logic IPC2. The translation covers the full set of logical connectives ∧, ∨, →, ⊥, ∀, and ∃, extending our previous work, which studied the significantly simpler case of the universal-implicational fragment of predicate logic. As corollaries of our approach, we obtain simple proofs of nondefinability of ∃ from the propositional connectives and nondefinability of ∀ from ∃ in the second-order intuitionistic propositional logic. We also show that the ∀-free fragment of IPC2 is undecidable.

#### Article information

Source
Notre Dame J. Formal Logic Volume 51, Number 4 (2010), 457-473.

Dates
First available in Project Euclid: 29 September 2010

http://projecteuclid.org/euclid.ndjfl/1285765799

Digital Object Identifier
doi:10.1215/00294527-2010-029

Mathematical Reviews number (MathSciNet)
MR2741837

Keywords
propositional quantification IPC2

#### Citation

Sørensen, Morten H.; Urzyczyn, Paweł. A Syntactic Embedding of Predicate Logic into Second-Order Propositional Logic. Notre Dame Journal of Formal Logic 51 (2010), no. 4, 457--473. doi:10.1215/00294527-2010-029. http://projecteuclid.org/euclid.ndjfl/1285765799.

#### References

• [1] Arts, T., and W. Dekkers, "Embedding first order predicate logic in second order propositional logic", Technical Report 93-02, Katholieke Universiteit Nijmegen, 1993.
• [2] Fujita, K., and A. Schubert, Existential type systems with no types in terms,'' pp. 112--26 in Typed Lambda Calculi and Applications, edited by P.-L. Curien, vol. 5608 of Lecture Notes in Computer Science, Springer, Berlin, 2009.
• [3] Fujita, K., Galois embedding from polymorphic types into existential types,'' pp. 194--208 in Typed Lambda Calculi and Applications, edited by P. Urzyczyn, vol. 3461 of Lecture Notes in Computer Science, Springer, Berlin, 2005.
• [4] Gabbay, D. M., "On 2nd order intuitionistic propositional calculus with full comprehension", Archiv für mathematische Logik und Grundlagenforschung, vol. 16 (1974), pp. 177--86.
• [5] Gabbay, D. M., Semantical Investigations in Heyting's Intuitionistic Logic, vol. 148 of Synthese Library, D. Reidel Publishing Co., Dordrecht, 1981.
• [6] de Groote, P., "On the strong normalisation of intuitionistic natural deduction with permutation-conversions", Information and Computation, vol. 178 (2002), pp. 441--64.
• [7] Löb, M. H., "Embedding first order predicate logic in fragments of intuitionistic logic", The Journal of Symbolic Logic, vol. 41 (1976), pp. 705--18.
• [8] Matthes, R., "Non-strictly positive fixed points for classical natural deduction", Annals of Pure and Applied Logic, vol. 133 (2005), pp. 205--30.
• [9] Nakazawa, K., M. Tatsuta, Y. Kameyama, and H. Nakano, "Undecidability of type-checking in domain-free typed lambda-calculi with existence", pp. 478--92 in Computer Science Logic. Proceedings of the 22nd International Workshop (CSL 2008, Bertinoro), edited by M. Kaminski and S. Martini, vol. 5213 of Lecture Notes in Computer Science, Springer, Berlin, 2008.
• [10] Nakazawa, K., and M. Tatsuta, "Strong normalization of classical natural deduction with disjunctions", Annals of Pure and Applied Logic, vol. 153 (2008), pp. 21--37.
• [11] Pitts, A. M., "On an interpretation of second-order quantification in first-order intuitionistic propositional logic", The Journal of Symbolic Logic, vol. 57 (1992), pp. 33--52.
• [12] Połacik, T., "Pitts' quantifiers are not topological quantification", Notre Dame Journal of Formal Logic, vol. 39 (1998), pp. 531--44.
• [13] Sobolev, S. K., "On the intuitionistic propositional calculus with quantifiers", Matematicheskie Zametki, vol. 22, (1977), pp. 69--76.
• [14] Sørensen, M. H., and P. Urzyczyn, Lectures on the Curry-Howard Isomorphism, vol. 149 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 2006.
• [15] Statman, R., "Intuitionistic propositional logic is polynomial-space complete", Theoretical Computer Science, vol. 9 (1979), pp. 67--72.
• [16] Tatsuta, M., "Second-order permutative conversions with Prawitz's strong validity", Progress in Informatics, vol. 2 (2005), pp. 41--56.
• [17] Tatsuta, M., "Second-order system without implication nor disjunction", Bulletin of Symbolic Logic, vol. 15 (2009), pp. 262--3.
• [18] Tatsuta, M., K. Fujita, R. Hasegawa, and H. Nakano, "Inhabitation of existential types is decidable in negation-product fragment", Proceedings of Second International Workshop on Classical Logic and Computation (CLC2008, Reykjavik), 2008.
• [19] Tatsuta, M., "Simple saturated sets for disjunction and second-order existential quantification", pp. 366--80 in Typed Lambda Calculi and Applications, edited by S. Ronchi Della Rocca, vol. 4583 of Lecture Notes in Computer Science, Springer, Berlin, 2007.
• [20] Urzyczyn, P., "Inhabitation in typed lambda-calculi (a syntactic approach)", pp. 373--89 in Typed Lambda Calculi and Applications (Nancy, 1997), edited by P. de Groote and J. R. Hindley, vol. 1210 of Lecture Notes in Computer Science, Springer, Berlin, 1997.
• [21] Wojdyga, A., Short proofs of strong normalization,'' pp. 613--23 in Mathematical Foundations of Computer Science 2008, edited by E. Ochmański and J. Tyszkiewicz, vol. 5162 of Lecture Notes in Computer Science, Springer, Berlin, 2008.
• [22] Zdanowski, K., "On second order intuitionistic propositional logic without a universal quantifier", The Journal of Symbolic Logic, vol. 74 (2009), pp. 157--67.