Leon Henkin. An extension of the Craig-Lyndon interpolation theorem. The journal of symbolic logic, vol. 28 no. 3 (for 1963, pub. 1964), pp. 201–216.

1965 ◽  
Vol 30 (1) ◽  
pp. 98-99
Author(s):  
M. A. Taitslin
Keyword(s):  
Interpolation Theorem ◽  
Symbolic Logic

scholarly journals Randomized feasible interpolation and monotone circuits with a local oracle

2018 ◽  
Vol 18 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Interpolation Theorem ◽  
Proof System ◽  
Symbolic Logic ◽  
Approximate Counting ◽  
Small Width ◽  
Randomized Protocols ◽  
Monotone Circuits

The feasible interpolation theorem for semantic derivations from [J. Krajíček, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic 62(2) (1997) 457–486] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two [Formula: see text] sets [Formula: see text] and [Formula: see text] a small communication protocol (a general dag-like protocol in the sense of Krajíček (1997) computing the Karchmer–Wigderson multi-function [Formula: see text] associated with the sets, and such a protocol further yields a small circuit separating [Formula: see text] from [Formula: see text]. When [Formula: see text] is closed upwards, the protocol computes the monotone Karchmer–Wigderson multi-function [Formula: see text] and the resulting circuit is monotone. Krajíček [Interpolation by a game, Math. Logic Quart. 44(4) (1998) 450–458] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity (e.g. to the cutting planes proof system CP). In this paper, we generalize the method to a still larger class of semantic derivations by allowing randomized protocols. We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for [Formula: see text] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of [Formula: see text], [Formula: see text] closed upwards, yields a small randomized protocol for [Formula: see text] and hence a small monotone CLO separating the two sets. This research is motivated by the open problem to establish a lower bound for proof system [Formula: see text] operating with clauses formed by linear Boolean functions over [Formula: see text]. The new randomized feasible interpolation applies to this proof system and also to (the semantic versions of) cutting planes CP, to small width resolution over CP of Krajíček [Discretely ordered modules as a first-order extension of the cutting planes proof system, J. Symbolic Logic 63(4) (1998) 1582–1596] (system R(CP)) and to random resolution RR of Buss, Kolodziejczyk and Thapen [Fragments of approximate counting, J. Symbolic Logic 79(2) (2014) 496–525]. The method does not yield yet lengths-of-proofs lower bounds; for this it is necessary to establish lower bounds for randomized protocols or for monotone CLOs.

2004 Spring Meeting of the Association for Symbolic Logic

2004 ◽  
Vol 10 (3) ◽  
pp. 438-446
Author(s):  
Michael Kremer
Keyword(s):  
Symbolic Logic ◽  
Spring Meeting

Association for Symbolic Logic

2008 ◽  
Vol 14 (4) ◽  
pp. 553-558
Keyword(s):  
Symbolic Logic

2008 Annual Meeting of the Association for Symbolic Logic

2008 ◽  
Vol 14 (3) ◽  
pp. 418-437
Author(s):  
Chris Laskowski
Keyword(s):  
Annual Meeting ◽  
Symbolic Logic

2008–2009 Winter Meeting of the Association for Symbolic Logic

2009 ◽  
Vol 15 (2) ◽  
pp. 237-245
Author(s):  
Ali Enayat
Keyword(s):  
Symbolic Logic

2010–2011 Winter Meeting of the Association for Symbolic Logic

2012 ◽  
Vol 18 (1) ◽  
pp. 142-149
Author(s):  
Michael Mislove
Keyword(s):  
Symbolic Logic

Algebra

2018 ◽  
Author(s):  
Andrea Henderson
Keyword(s):  
Euclidean Geometry ◽  
Symbolic Logic ◽  
Associative Networks ◽  
Aristotelian Logic ◽  
Modern Algebra ◽  
Symbolic Systems ◽  
The Difference ◽  
Founding Principle ◽  
Conventional Symbol

The difference between the transcendent Coleridgean symbol and the unreliable conventional symbol was of explicit concern in Victorian mathematics, where the former was aligned with Euclidean geometry and the latter with algebra. Rather than trying to bridge this divide, practitioners of modern algebra and the pioneers of symbolic logic made it the founding principle of their work. Regarding the content of claims as a matter of “indifference,” they concerned themselves solely with the formal interrelations of the symbolic systems devised to represent those claims. In its celebration of artificial algorithmic structures, symbolic logician Lewis Carroll’s Sylvie and Bruno dramatizes the power of this new formalist ideal not only to revitalize the moribund field of Aristotelian logic but also to redeem symbolism itself, conceived by Carroll and his mathematical, philosophical, and symbolist contemporaries as a set of harmonious associative networks rather than singular organic correspondences.

scholarly journals Frege and Wittgenstein: The Limits of the Analytic Style

Elements ◽  
2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Christopher Sheridan
Keyword(s):  
Twentieth Century ◽  
Logical Analysis ◽  
Linguistic Meaning ◽  
Bertrand Russell ◽  
Gottlob Frege ◽  
Symbolic Logic ◽  
Scientific Language ◽  
German Mathematician ◽  
Broad Application ◽  
Analytic Tradition

The analytic tradition in philosophy stems from the work of German mathematician and logician Gottlob Frege. Bertrand Russell brough Frege's program to render language-particularly scientific language-in formal logical terms to the forefront of philosophy in the early twentieth century. The quest to clarify language and parse out genuine philosophical problems remains a cornerstone of analytic philosophy, but investigative programs involving the broad application of formal symbolic logic to language have largely been abandoned due to the influence of Ludwig Wittgenstein's later work. This article identifies the key philosophical moves that must be performed successfully in order for Frege's "conceptual notation" and other similar systems to adequately capture syntax and semantics. These moves ultimately fail as a result of the nature of linguistic meaning. The shift away from formal logical analysis of language and the emergence of the current analytic style becomes clearer when this failure is examined critically.

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