Alonzo Church Research Articles

Singular concepts

Alonzo Church proposed a powerful and elegant theory of sequences of functions and their arguments as surrogates for Russellian singular propositions and singular concepts. Church’s proposed theory accords with his Alternative (0), the strictest of his three competing criteria for strict synonymy. The currently popular objection to strict criteria like (0) on the basis of the Russell–Myhill antinomy is here rebutted. Russell–Myhill is not a problem specifically for Alternative (0); it is a refutation of unrestrained concept comprehension. Unrestricted comprehension is also inconsistent with facts about sets of properties. Criteria more lax than (0) are philosophically inadequate. In particular, the rival conception of propositions as classes of possible worlds is subject to a fatal philosophical collapse. It follows on that conception, given that each of us is fallible, that everyone believes everything. It is shown, however, that Church’s proposed theory is vulnerable under (0) to a version of Russell’s notorious Gray’s Elegy objection. Some amendments to Church’s proposal are proffered, including an amendment, first proposed in the author’s Frege’s Puzzle (1986), that addresses Russell’s objection. Church’s response (personal correspondence) is considered.

From the Editor-in-Chief

Welcome everybody to the third issue of 2024! Some of us are already in holiday, some are still at work; surely, many of us will meet in the forthcoming conferences, mostly our flagship conference (LICS). In the meanwhile, I invite you to read the present issue of SIGLOG News, whose contents are the following: -The SIGLOG Chair introduces this year edition of LICS and reports on the current situation of SIGLOG. -We present the outcome of the 2024 Alonzo Church Award. This prize was established in 2015 by SIGLOG, by the European Association for Theoretical Computer Science (EATCS), by the European Association for Computer Science Logic (EACSL), and by the Kurt Goedel Society (KGS). In 2022 the KGS was dissolved and, since then, the award has been sponsored by EACSL, EATCS and SIGLOG. The award is for an outstanding contribution represented by a paper or by a small group of papers published within the past 25 years. This year's recipients are Thomas Ehrhard and Laurent Regnier. -Then, you can read a new semantics column, edited by Prakash Panangaden and written by Samson Abramsky and Luca Reggio, on Game comonads ; these offer a categorical view of a number of model-comparison games central to model theory and provide a new bridge between categorical methods developed for semantics, and the combinatorial and algorithmic methods of resource-sensitive model theory. -To conclude, we wrap up this number with the latest issue of SIGLOG Monthly, prepared by Elli Anastasiadi.

SIGLOG Monthly 251: July 2024

The 2024 Alonzo Church Award for Outstanding Contributions to Logic and Computation is presented jointly to Thomas Ehrhard and Laurent Regnier for giving a logical and computational account of differentiation, bringing Taylor expansion to the Curry-Howard correspondence, which had a major impact on programming language semantics.

WAND/SET THEORIES: A REALIZATION OF CONWAY’S MATHEMATICIANS’ LIBERATION MOVEMENT, WITH AN APPLICATION TO CHURCH’S SET THEORY WITH A UNIVERSAL SET

Abstract Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive way of fleshing out this idea is synonymous with a ZF-like theory. This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974).

From the Editor-in-Chief

Happy 2024 to all! SIGLOG News starts with the current issue its 11th volume; differently from the normal issues, here you will find a retrospective of the first 10 volumes of this journal, including, among other things, the list of all the columns published so far and a few statistics on them. Then, you will find the call for nominations for the Alonzo Church Award, for which we hope to receive many outstanding proposals from our community. Finally, we wrap up this number with the latest issue of SIGLOG Monthly, prepared by David Purser and Elli Anastasiadi.

Applying Theory to Practice

Abstract By making use of three IBM case studies involving the author and colleagues, this paper is about applying theory to practice. In the first case study, the system builders (or practitioners) initiated the interaction. This interaction led to the following problem. Assume that there is a set of objects, each with multiple attributes, and there is a numerical score assigned to each attribute of each object. In the spirit of real-valued logics, there is a scoring function (such as the min or the average), and a ranking of the objects is obtained by applying the scoring function to the scores of each object’s attributes The problem is to find the top $k$ objects, while minimizing the number of database accesses. An algorithm is given that is optimal in an extremely strong sense: not just in the worst case or the average case, but (up to a constant factor) in every case! Even though the algorithm is only 8 lines long (!), the paper containing the algorithm won the 2014 Gödel Prize, the top prize for a paper in theoretical computer science. The interaction in the second case study was initiated by theoreticians, who wanted to lay the foundations for ‘data exchange’, in which data is converted from one format to another. Although this problem may sound mundane, the issues that arise are fascinating, and this work made data exchange a new subfield, with special sessions in every major database conference. This work won the 2020 Alonzo Church Award, the highest prize for research in logic and computation. The third case study, specifically on real-valued (or ‘fuzzy’) logic, arose as part of a large ‘Logical Neural Nets’ (LNN) project at IBM. The inputs to, say, an ‘and’ gate could each be any numbers in the interval [0,1]. The system builders of LNN wanted a sound and complete axiomatization for real-valued logic, so that they could arrive at truth values given other truth values whenever possible. This recent work provides a sound and complete axiomatization for a large class of real-valued logics, including the most common ones. It also allows weights, where the importance of some subformulas can be greater than that of other subformulas. This paper is aimed at both theoreticians and system builders, to show them the mutual benefits of working together. This is via the three case studies mentioned above: two initiated by the system builders, and one by the theoreticians. The moral for the theoreticians is to show by example how to apply theory to practice, and why applying theory to practice can lead to better theory. The moral for the system builders is the value of theory, and the value of involving theoreticians. This paper is written in a very informal style. In fact, it is based closely on a talk on ‘Applying theory to practice’ that the author has presented a number of times.

Knowledge Representation and The Semantic Web: An Historical Overview of Influences on Emerging Tools

Abstract: suite of standards known as the Semantic Web is transforming the Internet into a semantic graph rather than a graph of hypertext links. This paper will describe how various ideas and initiatives in artificial intelligence knowledge representation influenced its design. We begin with the seminal work by Alan Turing and Alonzo Church that led to the definition of Turing Machines, enabled digital computing, and provided the mathematical theory of computation, which has been one of the determining factors for Artificial Intelligence knowledge representation. We then provide a brief history of artificial intelligence knowledge representation, starting with groundbreaking researchers, such as Newell and Simon, then to the first "AI boom" driven primarily by rule-based expert systems, followed by major initiatives such as Cyc and the DARPA Knowledge Sharing Initiative. We will discuss how innovations from these initiatives affected standards that, in turn, led to the suite of standards known as the Semantic Web. We conclude with a brief overview of the most important issues currently facing those who wish to see widespread adoption of Semantic Web technology in the industry.

SIGLOG Monthly 233

An annual award, called the Alonzo Church Award for Outstanding Contributions to Logic and Computation, was established in 2015 by the ACM Special Interest Group for Logic and Computation (SIGLOG), the European Association for Theoretical Computer Science (EATCS), the European Association for Computer Science Logic (EACSL), and the Kurt Goedel Society (KGS). The award is for an outstanding contribution represented by a paper or by a small group of papers published within the past 25 years. This time span allows the lasting impact and depth of the contribution to have been established. The award can be given to an individual, or to a group of individuals who have collaborated on the research. For the rules governing this award, see https://siglog.org/alonzo-church-award/, https://www.eatcs.org/index.php/church-award/, and https://www.eacsl.org/alonzo-church-award/.

2022 Alonzo Church Award Announcement

The ACM Special Interest Group on Logic (SIGLOG), the European Association for Theoretical Computer Science (EATCS), the European Association for Computer Science Logic (EACSL), and the Kurt Goedel Society (KGS) are pleased to announce that has been selected as the winner of the 2022 Alonzo Church Award for Outstanding Contributions to Logic and Computation for his fundamental work on developing the theory and applications of Kleene Algebra with Tests, an equational system for reasoning about iterative programs, published in: Kleene Algebra with Tests. ACM Transactions on Programming Languages and Systems 19(3): 427-443 (1997).

From the Editor

We announce the winner of the 2022 Alonzo Church Award for Outstanding Contributions to Logic and Computation and present the Call for Nominations for the next edition of the Award. In the Automata Column, edited by Mahesh Viswanathan, Andreas Pavlogiannis overviews the CFL/Dyck reachability problem on graphs. Mahsa Shirmohammadi presents a tutorial on stochastic games in Ranko Lazić's column on Verification. As usual, we wrap up with the latest issue of SIGLOG Monthly, prepared by David Purser. Finally, I would like to announce that, from January 2023, SIGLOG News will be edited by Daniele Gorla from the Sapienza University of Rome. I am very grateful to Daniele for agreeing to take over and wish him every success in this role! It has been my great pleasure and honour to edit SIGLOG News since its foundation in 2014. I thank all the contributors and column editors for the enjoyable and fruitful co-operation and, most of all, for all their expert articles and introductions!

Phenomenology, anti‐realism, and the knowability paradox

AbstractHusserl endorsesideal verificationism, the claim that there is a necessary correlation between truth and the ideal possibility of experience. This puts him in the company of semantic anti‐realists like Dummett, Tennant, and Wright who endorse theknowability thesisthat all truths are knowable. Unfortunately, there is a simple, seductive, and troubling argument due to Alonzo Church and Frederic Fitch that the knowability thesis collapses into theomniscience thesisthat all truths are known. Phenomenologists should be worried. I assess the damage by surveying responses that may be open to Husserl. In particular, I explore whether Husserl ought to have adopted intuitionistic logic and motivate a restriction of ideal verificationism on phenomenological grounds.

LEVEL THEORY, PART 3: A BOOLEAN ALGEBRA OF SETS ARRANGED IN WELL-ORDERED LEVELS

AbstractOn a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural extension of BLT is definitionally equivalent with ZF.

Conjoining Meanings: Semantics without Truth Values

<i>Conjoining Meanings: Semantics without Truth Values</i>

The 2021 Alonzo Church award for outstanding contributions to logic and computation

An annual award, called the Alonzo Church Award for Outstanding Contributions to Logic and Computation, was established in 2015 by the ACM Special Interest Group for Logic and Computation (SIGLOG), the European Association for Theoretical Computer Science (EATCS), the European Association for Computer Science Logic (EACSL), and the Kurt Gödel Society (KGS). The award is for an outstanding contribution represented by a paper or by a small group of papers published within the past 25 years. This time span allows the lasting impact and depth of the contribution to have been established. The award can be given to an individual, or to a group of individuals who have collaborated on the research. For the rules governing this award, see https://siglog.org/alonzo-church-award/, https://www.eatcs.org/index.php/church-award/, and https://eacsl.org/. The 2020 Alonzo Church Award was given jointly to Ronald Fagin, Phokion G. Kolaitis, Renée J. Miller, Lucian Popa, and Wang Chiew Tan for their ground-breaking work on laying the logical foundations for data exchange. Lists containing this and all previous winners can be found through the links above.

The 2020 Alonzo Church award for outstanding contributions to logic and computation

An annual award, called the Alonzo Church Award for Outstanding Contributions to Logic and Computation, was established in 2015 by the ACM Special Interest Group for Logic and Computation (SIGLOG), the European Association for Theoretical Computer Science (EATCS), the European Association for Computer Science Logic (EACSL), and the Kurt Gödel Society (KGS). The award is for an outstanding contribution represented by a paper or by a small group of papers published within the past 25 years. This time span allows the lasting impact and depth of the contribution to have been established. The award can be given to an individual, or to a group of individuals who have collaborated on the research. For the rules governing this award, see: https://siglog.acm.org/alonzo-church-award-for-outstanding-contributions-to-logic-and-computation-2020/, https://www.eatcs.org/index.php/church-award/, and https://eacsl.org/. The 2019 Alonzo Church Award was given jointly to Murdoch J. Gabbay and Andrew M. Pitts for their ground-breaking work introducing the theory of nominal representations, see: http://siglog.org/winners-of-the-2019-alonzo-church-award/. Previous awardees are listed at https://www.eatcs.org/index.php/church-award/.

The Church-Turing thesis

In its original form, the Church-Turing thesis concerned computation as Alan Turing and Alonzo Church used the term in 1936--- human computation.

Cognition and recognition

Abstract Expressions are synonymous if they have the same semantic content. Complex expressions are synonymously isomorphic in Alonzo Church’s sense if one is obtainable from the other by a sequence of alphabetic changes of bound variables or replacements of component expressions by syntactically simple synonyms. Synonymous isomorphism provides a very strict criterion for synonymy of sentences. Several eminent philosophers of language hold that synonymous isomorphism is not strict enough. These philosophers hold that ‘Greeks prefer Greeks’ and ‘Greeks prefer Hellenes’ express different propositions even if they are synonymously isomorphic. They hold that the very recurrence (multiple occurrence) of ‘Greeks’ contributes to the proposition expressed something that indicates the very recurrence in question. Kit Fine argues that this thesis, which he labels semantic relationism calls for a radically new conception of semantics. I have argued that the relevant phenomenon is wholly pragmatic, entirely non-semantic. Here I supplement the case with a new argument. No cognition without recognition—or almost none. With this observation, standard Millianism has sufficient resources to confront Frege’s puzzle and related problems without injecting pragmatic phenomena where they do not belong.

Kalmár's Argument Against the Plausibility of Church's Thesis

In his famous paper, An Unsolvable Problem of Elementary Number Theory, Alonzo Church (1936) identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. Only a few papers have been written against it. One of these is László Kalmár's An Argument Against the Plausibility of Church's Thesis from 1959. The aim of this paper is to present Kalmár's argument and to fill in missing details based on his general philosophical thoughts on mathematics.

The 2017 Alonzo Church award

SIGLOG is delighted to announce that the 2017 Church Award goes to 6 people: Samson Abramsky, Martin Hyland, Radha Jagadeesan, Pasquale Malacaria, Hanno Nickau and Luke Ong for [Quoting from the official citation] "providing a fully-abstract semantics for higher-order computation through the introduction of games models, thereby fundamentally revolutionising the field of programming language semantics, and for the applied impact of these models."

‘In Defence of Sententialism’

Propositional attitude sentences, such as (1) Pierre believes that snow is white, have proved to be formidably difficult to account for in a semantic theory. It is generally agreed that the that-clause ‘that snow is white’ purports to refer to the proposition that snow is white, but no agreement has been reached on what this proposition is. Sententialism is a semantic theory which tries to undermine the very enterprise of understanding what proposition is referred to in (1): according to sententialists, in (1) reference is made to the sentence ‘Snow is white’. Sententialism is generally considered doomed. The two main reasons why are the famous translation argument, first suggested by Alonzo Church, and a problem raised by Stephen Schiffer. The purpose of this paper is to provide a unified solution to both criticisms. What I take to be the key ingredient sententialists may exploit is an observation that concerns the nature of languages and quotations: since quotation marks display the quoted material, if you are a speaker of the language the quoted material belongs to, you usually cannot but understand what is quoted. Moreover, I show that sententialists may appeal to that very observation also in order to answer another problem, pointed out by Kent Bach. I conclude that there are good reasons for resisting the temptation of introducing propositions in order to account for propositional attitude sentences.