F. P. Ramsey. On a problem of formal logic. Proceedings of the London Mathematical Society, 30:264285, 1930.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....abstract binary clause of the form p(x) p(y) in unf a (P ) there exists a level mapping f such that p( t) is rigid with respect to f and = f(p(x) f(p(y) then P terminates for all initial queries described by G . We present here a proof of this proposition based on Ramsey s Theorem [5]. Theorem: Ramsey s Theorem] Let = #a, b# a, b IN and a b , be a finite set of colors and let A##L be a function associating the elements of with colors from L. Then, there is a color f#L and an infinite set IN such that F(#a, b#) f for each a, b for which a ....

F.P Ramsey. On a problem of formal logic. Proc. London Math. Society, 30:264-- 286, 1930.

....to the van der Waerden Theorem. The Schur Theorem states that for every positive integer k there is an integer m such that every partition of f1; mg into k blocks contains a block that is not sum free. Similarly, the Ramsey Theorem (which gave name to this whole area in combinatorics) Ram28] concerns the existence of monochromatic cliques in edge colored graphs, and the Hales Jewett theorem [HJ63] concerns the existence of monochromatic lines in colored cubes . Each of these results gives rise to a particular function defined on pairs or triples of integers and determining the ....

F.P. Ramsey. On a problem of formal logic. Proceedings of London Mathematical Society, 30:264--286, 1928.

....THEOREM 3.5. The typechecking problem for projectionfree non recursive QL queries without tag variables, regular input DTDs, and regular output DTDs, is decidable. It remains open whether Theorem 3.5 holds without the projection free restriction. The proof of Theorem 3. 5 uses Ramsey s Theorem [14, 23] and requires developing some technical machinery. We dedicate the remainder of the section to this development. Assume we are given some projection free non recursive QL query q. By the definition of projection free, q is equivalent to sonhe query q obtained from qt by replacing in each nested ....

....Ri. So, if Ri was not empty the projection results in one tuple the empty tuple. jr. It follo;vs that T still has the modulo property, vhich contradicts the minimaliW of T. To conclude, we state Ramsey s Theorem, and the corollary used above. THEOREM 3.13. Ramsey s Theorem) ld] see also [23], pp.7 9) For all natural numbers k, m, w there exists a finite number R(k,m,w) such that for every set Y of elements with IYI R(k, m, w) and every coloring of the family of all the subsets of Y of size k with w colors, Y contains a subset X C Y of size IXI =m where all the subsets of X of size ....

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F. P. Ramsey, On a problem of formal logic. Proceedings of the London Mathematical Society, 30(2):264 286, 1929.

....an odd cycle on 2k 1 vertices, then (G) k, G) 2, whereas the stability number of the edge intersection of G and G is 2k 1. Furthermore, there exist graphs on n vertices for which both the stability and the clique number are not larger than 2 log n= log 2 [2] The Ramsey number R(k; [5] is the smallest integer such that every graph on R(k; vertices has either stability number at least k or clique number at least . Proposition 2 Let G be the edge intersection of G 1 and G 2 . Then (G) R( G 1 ) 1; G 2 ) 1) 1: 2) Proof. Let G; G 1 ; G 2 be a counterexample, and ....

F.P. Ramsey, \On a Problem of Formal Logic", Proceedings of the London Mathematics Society 30 (1930) 264-286. 3

....# #, where # # is the Hindman space, i.e. the space of all increasing infinite sequences of pairwise disjoint nonempty finite sets. This yields a common generalization of a theorem of Taylor (cf. Theorem 0.4) and a theorem of Prmel and Voigt (cf. Theorem 0. 7) INTRODUCTION Ramsey s Theorem [Ra30] is an important extension of the pigeonhole principle: If ##### 0 # . # P k 1 is a partition of # into finitely many pieces, then for some i # k, P i is infinite. THEOREM 0.1. Ramsey) Let l ## ### ### ### l = P 0 # . # P k 1 is a ############ #### l into finitely many pieces, there ....

RAMSEY, F. P.: On a problem of formal logic, Proceedings of the London Mathematical Society (2), vol. 30 (1930), pp. 264-286

....DMS 9970329 and AFOSR grant F49620 01 1 0264. Draft 1.0 (November 9, 2001) 2 1 Introduction All graphs considered in this paper are finite and simple. We follow [12] for our terminology. In particular, the complement of a graph G will be denoted by G. We begin with a classical result of Ramsey [9]. Ramsey s Theorem. There exists a function r(n) defined on the set of positive integers, such that every graph on at least r(n) vertices must contain either K n or K n as an induced subgraph. In graph theory, there are many results that are similar to Ramsey s theorem and they are known as ....

F.P. Ramsey. On a problem for formal logic, Proc. London Math. Soc., 30 (1930), 264--286.

....f j Y ) is of pattern P , and Y is conditionally monochromatic for F as K colored set, i.e. for Z 2 [Y ] j the value F (Z) depends only on the K pattern of (Z; f j Z ) By iterated applications of Theorem 2. 2 in [AH78] we can derive the following generalization of the classical Ramsey theorem [Ram28]: Theorem 7. KV87] For arbitrary integers e; M , a system of colors K, and a K pattern P , there is a K pattern Q such that Q , P ) e M . With every nite vocabulary R in which the maximum arity is n we can associate a system of colors K such that every nite structure B = B; R B ; ....

Ramsey, F.P.: On a problem in formal logic. Proc. London Math. Soc. 30(1928). pp. 264-286.

....Theorem 3.5. The typechecking problem for projectionfree non recursive QL queries without tag variables, regular input DTDs, and regular output DTDs, is decidable. It remains open whether Theorem 3.5 holds without the projection free restriction. The proof of Theorem 3. 5 uses Ramsey s Theorem [14, 23] and requires developing some technical machinery. We dedicate the remainder of the section to this development. Assume we are given some projection free non recursive QL query q 0 . By the definition of projection free, q 0 is equivalent to some query q obtained from q 0 by replacing in ....

....if R i was not empty the projection results in one tuple the empty tuple. j l . It follows that T 0 still has the modulo property, which contradicts the minimality of T . To conclude, we state Ramsey s Theorem, and the corollary used above. Theorem 3.13. Ramsey s Theorem) 14] see also [23], pp.7 9) For all natural numbers k; m;w there exists a finite number R(k; m;w) such that for every set Y of elements with jY j R(k; m;w) and every coloring of the family of all the subsets of Y of size k with w colors, Y contains a subset X ae Y of size jXj = m where all the subsets of X of size ....

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F. P. Ramsey, On a problem of formal logic. Proceedings of the London Mathematical Society, 30(2):264--286, 1929.

....through Frederickson and Lynch s [7] paper on a problem in distributed computing and Moran, Snir and Manber s [13] work on decision trees, and many other papers. For a set S and an integer p jSj, let [S] p denote the set of subsets A S with jAj = p. We use the following theorem due to Ramsey [16]. For information on Ramsey Theory see [9] 5 Theorem 3.1 (Ramsey) For any p; m and c, there is a number R(p; m; c) such that the following holds. Let S be a set of size at least R(p; m; c) For any coloring of [S] p with at most c colors, there is a T S with jT j = m such that all of [T ....

F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc., 2nd Ser., 30 (1930), pp. 264--286.

....for example, the Bernays Sch onfinkel Ramsey class, which consists of all relational first order sentences with quantifier prefix of the form # # # # . The satisfiability problem for this class without equality was shown to be decidable by Bernays and Sch onfinkel [4] moreover, Ramsey [50] extended this result 1 to the case with equality. Lewis [42] showed that the satisfiability problem for this class without equality is NEXPTIME complete; it is easy to see that the same holds true for the case with equality. The first decidability result for FO 2 was obtained by Scott [53] ....

F. P. Ramsey, On a problem in formal logic, Proceedings of the London Mathematical Society, vol. 30 (1928), pp. 264--268.

....is obvious since the element = t) of S is a simple term of order 1. Thus, we may assume p to be in nite. First consider the case that p = Q i2 p i is an in nite product of nite N free pomsets p i . Let i : p i ) A standard application (cf. PP99, Thm. II.3. 2] of Ramsey s Theorem [Ram30] cf. also [Cam94] yields the existence of a strictly increasing sequence of positive integers n i for i 2 and a linked pair ( 2 S 2 such that 12 = 0 1 n 0 and = n i 1 n i 2 n i 1 for i 2 . Hence p 2 L ( Since ( is a simple term of order 1 ....

F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264-286, 1930.

....lattices In this section, we will characterize those classes of nite distributive lattices whose elementary, monadic, monadic chain, and monadic antichain theory are decidable. Before we can actually prove this characterization, we need Ramsey s theorem and Lemma 5.1. Ramsey s Theorem [Ram30] Let c; r be positive integers. Then there is a positive integer R r (c) such that for any mapping d of the two elements subsets of R r (c) into c there exists an r elements subset A R r (c) such that we have d(B) d(C) for any two elements subsets B and C of A. Lemma 5.1 Let (L; be a ....

F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264-286, 1930.

....whenever M is width bounded. Therefore, it is necessary to show that the automaton A has only finitely many reachable states. This is achieved by bounding the length of the monoid elements occurring in reachable states. This proof is based on a well known result by Ramsey: Ramsey s Theorem [Ram30] Let k; c; r be positive integers. Then there is a positive integer R(k; c; r) such that for any mapping d of the k elements subsets of [R(k; c; r) into [c] there exists an r elements subset A [R(k; c; r) such that d(B) d(C) for any k elements subsets B and C of A. First we use Ramsey s ....

F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264-- 286, 1930.

....If t is nite, the lemma is obvious since the element s = t) of S is a simple term of order 1. Thus, we may assume t to be in nite. First consider the case that t = Q i2 t i is an in nite product of nite N free pomsets t i . Let s i : t i ) A standard application of Ramsey s Theorem [19] (cf. also [17] yields the existence of positive integers n i for i 2 and a linked pair (s; e) 2 S 2 such that s = s 0 s 1 s n0 and e = s n i 1 s n i 2 s n i 1 for i 2 . Hence t 2 L (s; e) Since (s; e) is a simple term of order 1 w(t) we showed the lemma for in nite ....

F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264-286, 1930.

....is obvious since the element = t) of S is a simple term of order 1. Thus, we may assume p to be infinite. First consider the case that p = Q i2 p i is an infinite product of finite N free pomsets p i . Let i : p i ) A standard application (cf. PP99, Thm. II.3. 2] of Ramsey s Theorem [Ram30] cf. also [Cam94] yields the existence of a strictly increasing sequence of positive integers n i for i 2 and a linked pair ( 2 S 2 such that = 0 1 n0 and = n i 1 n i 2 n i 1 for i 2 . Hence p 2 L ( Since ( is a simple term of order 1 ....

F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264--286, 1930.

....(of compactness) that every infinite binary tree has an infinite branch. They showed that neither of these two systems is a subtheory of the other, over the base system RCA 0 . We continue in this line by comparing BCT # 0 1 with the assertions of the various cases of Ramsey s Theorem. See (Ramsey 1930). # Slaman was partially supported by NSF Grant DMS 91 06714 and SERC Visiting Fellowship Research Grant ( Leeds Recursion Theory Year 1993 94 ) No. GR H 91213. The authors would like to thank S. Simpson for bringing the Brown and Simpson question to their attention. 1 2 Michael E. Mytilinaios ....

Ramsey, F. P. (1930). On a problem in formal logic. Proc. London Math.

....Show: by assumption (2) the first player can indeed avoid those positions, and by assumption (1) doing so he must win. Bibliographic Remarks For an introduction to inductive definitions, see Aczel 1977. The least fixed point result goes back to Tarski 1955. Ramsey s Theorem B. 8 is from Ramsey 1928. Determinacy of open games is due to Gale and Stewart 1953. The strongest determinacy result that is provable in ZF is due to Martin: it says that Borel games (where one of the players has a winning set that is Borel in the topology described) are determined. The Axiom of Determinacy states that ....

Ramsey, F.P. 1928. On a problem in formal logic. Proceedings of the London Mathematical Society 30:264--286.

....Observations 1 3. Remark 1. It is possible to extend these arguments and obtain a proof of theorem 5.2 in general. For details cf [Th 89] Remark 2. Erd os and Mills [EM 81] gave upper bounds for the Paris Harrington function for coloring pairs with a fixed number of colors, the Ramsey case [Ra 30] The above results cover the canonical min homogeneous case for pairs and k tuples in general. 13 x 6 Outlook and problems Here we concentrated on the levels of the Grzegorczyk Wainer hierarchy up to ffi . Of course one could and did go beyond. S 87] gives an account on the finite ....

Ramsey, F. P., On a problem of formal logic. (1930), Proceedings of the London Mathematical Society, 30, 264-286.

....2 does not imply B# 3 . In contrast, J. Hirst showed that RCA 0 RT 2 # does imply B# 3 , and we include a proof of a slightly strengthened version of this result. It follows that RT 2 # is strictly stronger than RT 2 2 over RCA 0 . 1. Introduction Ramsey s theorem was discovered by Ramsey [1930] and used by him to solve a decision problem in logic. Subsequently it has been an important tool in logic and combinatorics. Definition 1.1. i. X] n = Y # X : Y = n . ii. A k coloring C of [X] n is a function from [X] n into a set of size k. iii. A set H # X is homogeneous ....

RAMSEY'S THEOREM 71 Ramsey, F. P. [1930]. On a problem in formal logic, Proc. London Math. Soc. (3) 30: 264--286.

....= 2n 5, for n 5, and sat(n; K 3 ; 3) 3n 15, for n 10. 3 Ramsey Type Games Let natural numbers n and s be given. Two players R and B colour, in turn, any uncoloured edge of Kn red and blue correspondingly. The player who rst creates a monochromatic K s is the winner. 2 By Ramsey Theorem [24] if n = n(s) is suciently large a draw is impossible. Furthermore, then the rst player, say R, has a winning strategy. Suppose not, i.e. B has a winning strategy, but then R can make any rst move which won t do to him any harm and then use the strategy of the second player a contradiction. ....

F. P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264-286, 1930.

....automata with a more general acceptance condition, the Muller condition [Mu63] and uses the fact that for deterministic Muller automata the complementation step is obvious. Both approaches involve nontrivial arguments: In the first case, a combinatorial result is applied (Ramsey s Theorem B [Ra29]) and in the second a very intricate automaton construction is required. In the present paper, we expose a third proof strategy which so far did not attract much attention in the literature but has some advantages. Instead of reducing the nondeterminism of Buchi automata to determinism with a ....

....is possible via the input v. It is easy to verify that A is a congruence with finitely many (regular) equivalence classes. Denoting the A class of the word u by [u] one observes that an language [u] Delta [v] is either contained in L(A) or disjoint from L(A) Invoking Ramsey s Theorem B ([Ra29]) one shows that any word can be cut into a sequence u 0 u 1 : of finite words where all u i for i 1 belong to a fixed A class. Applying this decomposition to the words outside L(A) one sees that A n L(A) is representable as a union of sets [u] Delta [v] taking those pairs ....

F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1929), 264-286.

....0; m 0, where is quantifier free. This is the well known Bernays Schonfinkel prefix class [BS28] which we denote 9 8 FO. We use similar notation for prefix classes such as 9 FO and 8 FO, with the obvious meaning. The decidability of finite satisfiability of 9 8 FO sentences was shown in [Ram30]. The decidability follows from a straightforward observation: if a sentence 9x 1 : 9x k 8y 1 : 8ym has a model, then it has a model with max(1; k) elements. The complexity of the decision procedure was investigated in [Lew80] and it was proven that the problem is complete in nexptime. ....

F. Ramsey. On a problem in formal logic. Proc. of the London Math. Society, 2nd Series(30):264--286, 1930.

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F. Ramsey, On a Problem of Formal Logic, Proceedings of the London Mathematics Society 30 (1930), 264-286.

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F. P. Ramsey. On a problem in formal logic. Proc. London Math. Soc., 30:264--286, 1930.

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F. Ramsey, On a Problem of Formal Logic, Proceedings of the London Mathematics Society 30 (1930), 264-286.

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F. P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264--286, 1930. 191

..... 47 6.3 The discrete cube . 52 7 Open Problems 55 1 Introduction The philosophy of modern Ramsey Theory states that large systems necessarily contain large, highly structured sub systems. The classical Ramsey coloring theorem [49, 27] is a prime example of this principle: Here large refers to the cardinality of a set, and highly structured means being monochromatic. Another classical theorem, which can be viewed as a Ramsey type phenomenon, is Dvoretzky s theorem on almost spherical sections of convex bodies. This ....

F. P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 48:122-160, 1930.

....Computer Science of the last two decades not contained in these. 1 Introduction Ramsey type theorems have roots in di#erent branches of mathematics, and the theory developed from it influenced such diverse areas as number theory, ergodic theory, or theoretical computer science. Ramsey [211] stated his theorem in a general setting and applied it to formal logic. The finite version says: For all t, n, k N there exists R N so that, for m R, if the k tuples of a set M of cardinality m are t colored, then there exists M # M of cardinality n with all the k tuples of M # ....

.... observation: If the assertion Algorithm A solves correctly an order invariant problem in t steps can be formally expressed by a universal formula of first order predicate calculus using the predicates , P 1 , P k , then the following formulation of Ramsey s theorem, due to Ramsey [211] can be used. For each j, k, m,n there is a number N(j, k, m,n) such that the following holds: let F be a universal formula of size j in first order predicate calculus, with predicates P 1 , P k , and n variables. If this formula can be satisfied by a model of size N(j, k, m,n) then it ....

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F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264--286.

....function that approximates the size of the largest clique of G, to(G) within a factor of n 0, then P NP. There are upper bounds on 0; but the best value is unknown. Our main result is the observation that there is a fundamental connection between these results and certain Ramsey like results [7,10,13]. This seems to be surprising. Roughly the connection shows that if certain approximation problems are NP hard, then either co NP has feasible proofs or certain Ramsey results are true. It is, of course, believed that co NP does not have polynomial size proofs. However, by feasible proofs we ....

....state the known results about hard approximation problems. Further, it makes the connection with Ramsey theory easy to establish. The notion of an NP hard pair is related to an old concept from recursion theory of effectively inseparable sets [14] 2. Definitions We need a Ramsey like function [13]. It is essentially the functional inverse of the usual one. For a graph G, define R(G) to be the size of the largest monochromatic clique that must appear in any 2 coloring of the edges of G. Recall that a monochromatic clique is one that has all its edges colored in the same way. Clearly, ....

F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (193) 264-286.

....1 vertices. The graphs constructed in the proof have size np, 1) Theorems 1 arid 2 imply that lbr every n A arid I In particular, p(1) r= 1) hence results on Ramsey numbers can be used to bound the largest increase from 7 ( t. o 7 ( A special case of) the well known theorem of Ramsey [18] asserts that any v vertex graph G contains a logarithmic size clique or independent set: max (C) C) 5 log v. S) Erd6s [8] showed that up to a constant factor this bound is tight. For every v 2 there is a v vertex graph O containing neither a clique nor an independent set of size 2 log ....

F. P. Ramsey. On a problem of formal logic. Proceedings of the London Mathematical Society, 30(2):264 286, 1929.

....Remarks 22 9. References 22 A through D 22 E through G 25 H through L 29 M through R 32 S through Z 35 2 THE ELECTRONIC JOURNAL OF COMBINATORICS (2001) DS1.8 1. Scope and Notation There is a vast literature on Ramsey type problems starting in 1930 with the original paper of Ramsey [Ram]. Graham, Rothschild and Spencer in their book [GRS] present an exciting development of Ramsey Theory. The subject has grown amazingly, in particular with regard to asymptotic bounds for various types of Ramsey numbers (see the survey paper [GrRo] but the progress on evaluating the basic numbers ....

F.P. Ramsey, On a Problem of Formal Logic, Proceedings of the London Mathematical Society, 30 (1930) 264-286.

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F. P. Ramsey. On a problem of formal logic. Proceedings of the London Mathematical Society, 30:264285, 1930.

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F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264-286.

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Ramsey F.P. (1930) On a Problem of Formal Logic, Proc. London Math. Society, 30, 264-286

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F.P. Ramsey. On a problem of formal logic, Proceedings of London Mathematical Society, 30:264-286, 1928.

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F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264-286.

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F. P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264--286, 1930.

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F. P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264--286, 1930.

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F.P. Ramsey. On a problem of formal logic, Proceedings of London Mathematical Society, 30:264--286, 1928.

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F.P. Ramsey. On a problem of formal logic, Proceedings of London Mathematical Society, 30:264--286, 1928.

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F. P. Ramsey. On a problem of formal logic. Proceedings of the London Mathematical Society, 30:264-285, 1930.

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F. P. Ramsey. On a problem of formal logic. Proceedings of the London Mathematical Society, 30:264285, 1930.

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F. P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society, 30(2):264--286, 1929

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F. Ramsey. On a Problem of Formal Logic. Proceedings of the London Mathematical Society, 30:264-286, 1930.

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F. P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society, 30(2):264--286, 1929. 37

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F. P. Ramsey, On a problem of formal logic. Proceedings of the London Mathematical Society, 30(2):264-286, 1929.

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F.D. Ramsey, 1929, On a problem of formal logic, Proc. of the London Math. Soc. 30, 338--384.

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F. D. Ramsey, On a problem of formal logic, Proc. of the London Math. Soc. 30, (1929) 338--384.

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-46, 2000. #23# F. P. Ramsey, On a problem of formal logic. Proceedings of the London Mathematical Society,

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F.P. Ramsey (1928), On a problem of formal logic. Proc. London Math. Society 30 (1928) pp. 264-286.

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