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This essay’s content is rendered by the titles of the successive sections. 1. Effective solvability versus intuitive solvability. — 2. Decidability, i.e. effective solvability, in predicate logic. The speedup phenomenon — 3. Contributions of the second-order logic to the problems of solvability — 4. The infinite progress of science in the light of Turing’s idea of the oracle. The term “oracle” is a technical counterpart of the notion of mathematical intuition.

A more detailed summary can be obtained through juxtaposing the textboxes labelled with letters A...F. Conclusion: in the progress of science an essential role is played by the feedback between intellectual intuitions (intuitive solvability) and algorithmic procedures (effective solvability).

complexity of the spanning tree congestion problem, Algorithmica 64 (2012) 85-111. doi:10.1007/s00453-011-9565-7 [5] H.L. Bodlaender and A.M.C.A. Koster, Combinatorial optimization on graphs of bounded treewidth, Comput. J. 51 (2008) 255-269. doi:10.1093/comjnl/bxm037 [6] B. Courcelle, The monadic second-order logic of graphs III: tree-decompositions, minors and complexity issues, Theor. Inform. Appl. 26 (1992) 257-286. [7] T. Hasunuma, Completely independent spanning trees in the underlying graph of a line digraph, Discrete Math. 234 (2001) 149-157. doi:10.1016/S0012

Applied Mathematics, 9 (1984), pp. 27-39. [5] B. Courcelle, The monadic second-order logic of graphs. I. Recognizable sets of finite graphs , Inform. and Comp. 85(1) (1990) 64-75. [6] A.P. De Villiers, Edge criticality in secure graph domination , Ph.D. Dissertation Stellenbosch: Stellenbosch University, (2014). [7] M.R. Garey, and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness , Freeman, New York, (1979). [8] M.C. Golumbic, and C.F. Goss, Perfect elimination and chordal bipartite graphs , Journal of Graph Theory, 2 (1978

circuits on two-dimensional manifolds, Ann. of Math. 23 (1921) 144-168. doi:10.2307/1968030 [6] M. Cohn and A. Lempel, Cycle decomposition by disjoint transpositions, J. Combin. Theory Ser. A 13 (1972) 83-89. doi:10.1016/0097-3165(72)90010-6 [7] B. Courcelle, Circle graphs and monadic second-order logic, J. Appl. Log. 6 (2008) 416-442. doi:10.1016/j.jal.2007.05.001 [8] W.H. Cunningham, Decomposition of directed graphs, SIAM J. Alg. Disc. Meth. 3 (1982) 214-228. doi:10.1137/0603021 [9] J. Daligault, D. Gonçalves and M. Rao, Diamond-free circle graphs are Helly circle

. Sen, Analogues of cliques for (m, n)-colored mixed graphs, Graphs Combin. 33 (2017) 735 - 750. doi: 10.1007/s00373-017-1807-2 [5] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer-Verlag, London, 2008). [6] R.A. Brualdi and J.J. Quinn Massey, Incidence and strong edge colorings of graphs, Discrete Math. 122 (1993) 51-58. doi: 10.1016/0012-365X(93)90286-3 [7] B. Courcelle, The monadic second order logic of graphs VI: on several representa- tions of graphs by relational structures., Discrete Appl. Math. 54 (1994) 117-129. doi: 10.1016/0166-218X(94)90019-1 [8] W

. Courcelle, The monadic second-order logic of graphs. I. Recognizable sets of fi nite graphs , Inform. and Comput. 85 (1990) 12–75. doi:10.1016/0890-5401(90)90043-H [9] A.P. De Villiers, Edge Criticality in Secure Graph Domination, Ph.D. Dissertation (Stellenbosch University, 2014). [10] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979). [11] S. Guha and S. Khuller, Approximation algorithms for connected dominating sets , Lecture Notes in Comput. Sci. 1136 (1996) 179–193. doi:10

Consequence”, [in:] Lou Goble (ed.), The Blackwell Guide to Philosophical Logic. Oxford: Blackwell Publishers, 2001, 115-35. 15. Bonnay, D., “Logicality and Invariance”, Bulletin of Symbolic Logic 14.1 (2008): 29-68. 16. Bonnay, D. and D. Westerståhl, “Consequence mining: constants versus consequence relations”, Journal of Philosophical Logic 41.4 (2012): 671-709. 17. Boolos, G., “On Second-order Logic”, Journal of Philosophy 72 (1975): 509-27. 18. Brown D.J. and R. Suszko, “Abstract Logics”, Dissertationes Mathematicae 102 (1973): 7-41. 19. Carnielli, W. A., M. E. Coniglio

property wisdom’ is derivative of the more ordinary ‘Socrates is wise’. Hale’s Fregean approach to ontology motivates an abundant con- ception of properties: a property is anything that could be referred to by a meaningful predicate, no matter how heterogenous. How- ever, due to the linguistic constraints, it is not so abundant as to allow for properties of infinite complexity. Any property has to be initely speciiable. Chapter 8 discusses an important consequence for the semantics of second-order logic: “we should interpret second- order variables as ranging over

-solvability much depends on such a device as the due repertoire of grammatical categories. This is exemplified with such systems as Frege’s second-order logic, Rullsell’s theory of types, etc. As for the share of rhetoric in the output of “Studies”, there is no need to encourage new activities. For it is enough to skip the titles of the hitherto edited volumes in order to notice the considerable share of logical theory of argumentation. As examples of so conceived rhetorical issues, one may consider the volumes: 29, 32, 36, 55, 65. Section 2 suggests an approach to the issue of

conjunction and negation they are close to having logical constants. This is far from quantified modal logic or proofs in second-order logic, but it is a basic conception of how language could be structured from essen- tially pre-linguistic elements. The following notions should be remembered here: conditions of satisfaction, pre-linguistic forms of representation, as op- posed to merely expression (several animals do have these) and conventions, i.e. standard procedures. This is a powerful apparatus, because intentional acts performed using a convention introduce a crucial