Mark Colyvan, an Australian philosopher of mathematics, published a paper in 2004 which examined the philosophical and logical assumptions of Cox’s theorem (assumptions usually left implicit by its proponents), and argued that these are inappropriate for many (perhaps even most) domains with uncertainty.

M. Colyvan [2004]: The philosophical significance of Cox’s theorem. International Journal of Approximate Reasoning, 37: 71-85.

Colyvan’s work complements Glenn Shafer’s attack on the theorem, which noted that it assumes that belief should be represented by a real-valued function.

G. A. Shafer [2004]: Comments on “Constructing a logic of plausible inference: a guide to Cox’s theorem” by Kevin S. Van Horn. International Journal of Approximate Reasoning, 35: 97-105.

Although these papers are several years old, I mention them here for the record -  and because I still encounter invocations of Cox’s Theorem.

IME, most statisticians, like most economists, have little historical sense. This absence means they will not appreciate a nice irony: the person responsible for axiomatizing classical probability theory – Andrei Kolmogorov – is also one of the people responsible for axiomatizing intuitionistic logic, a version of classical logic which dispenses with the law of the excluded middle. One such axiomatization is called BHK Logic (for Brouwer, Heyting and Kolmogorov) in recognition.

“Ah,” said Rowe, “there’s a difference between a man and a machine when it comes to praying.”   “Aye. The machine would do it better. It wouldn’t pray for things it oughtn’t pray for, and its thoughts wouldn’t wander.”

“Y-e-e-s. But the computer saying the words wouldn’t be the same . . .”

“Oh, I don’t know. If the words ‘O Lord, bless the Queen and her Ministers‘ are going to produce any tangible effects on the Government, it can’t matter who or what says them, can it?”

“Y-e-e-s, I see that. But if a man says the words he means them.”

“So does the computer. Or at any rate, it would take a damned complicated computer to say the words without meaning them. I mean, what do we mean by ‘mean’? If we want to know whether a man or a computer means ‘O Lord, bless the Queen and her Ministers,’ we look to see whether it’s grinning insincerely or ironically as it says the words. We try to find out whether it belongs to the Communist Party. We observe whether it simultaneously passes notes about lunch or fornication. If it passes all the tests of this sort, what other tests are there for telling if it means what it says? All the computers in my department, at any rate, would pray with great sincerity and single-mindedness. They’re devout wee things, computers.” (pages 109-110).

Reference:

Michael Frayn [1995/1965]: The Tin Men (London, UK: Penguin, originally published by William Collins, 1965)

In July 2005, inspired by a talk on formation flying by unmanned aircraft by Sandor Veres at the Liverpool Agents in Space Symposium, I wrote down some rules of thumb I have been using informally for determining whether an agent-based modeling (ABM) approach is appropriate for a particular application domain.  Appropriateness is assessed by answering the following questions:

1. Are there multiple entities in the domain, or can the domain be represented as if there are?
2. Do the entities have access to potentially different information sources or do they have potentially different beliefs? For example, differences may be due to geographic, temporal, legal, resource or conceptual constraints on the information available to the entities.
3. Do the entities have potentially different goals or objectives? This will typically be the case if the entities are owned or instructed by different people or organizations.
4. Do the entities have potentially different preferences (or utilities) over their goals or objectives ?
5. Are the relationships between the entities likely to change over time?
6. Does a system representing the domain have multiple threads of control?

If the answers are YES to Question 1 and also YES to any other question, then an agent-based approach is appropriate. If the answer to Question 1 is NO, or if the answers are YES to Question 1 but NO to all other questions, then a traditional object-based approach is more appropriate.

Traditional object-oriented systems involve static relationships between non-autonomous entities sharing the same beliefs, preferences and goals, and in a system with a single thread of control.

Brian Arthur has an article about automated and intelligent machine-to-machine communications creating a second digital economy underlying the first physical one, in the latest issue of The McKinsey Quarterly here.

I want to argue that something deep is going on with information technology, something that goes well beyond the use of computers, social media, and commerce on the Internet. Business processes that once took place among human beings are now being executed electronically. They are taking place in an unseen domain that is strictly digital. On the surface, this shift doesn’t seem particularly consequential—it’s almost something we take for granted. But I believe it is causing a revolution no less important and dramatic than that of the railroads. It is quietly creating a second economy, a digital one.

. . . .

We do have sophisticated machines, but in the place of personal automation (robots) we have a collective automation. Underneath the physical economy, with its physical people and physical tasks, lies a second economy that is automatic and neurally intelligent, with no upper limit to its buildout. The prosperity we enjoy and the difficulties with jobs would not have surprised Keynes, but the means of achieving that prosperity would have.

This second economy that is silently forming—vast, interconnected, and extraordinarily productive—is creating for us a new economic world. How we will fare in this world, how we will adapt to it, how we will profit from it and share its benefits, is very much up to us.”

Reference:

W. Brian Arthur [2011]:  The Second Economy.  The McKinsey Quarterly, October 2011.

An index to posts about the Matherati is here.

May began life as an applied mathematician and theoretical physicist (in the Sydney University of Harry Messel), then applied his models to food webs in ecology, and now finds the same types of network and lattice models useful for understanding inter-dependencies in networks of banks.  Although, as he said in his talk, these models are very simplified, to the point of being toy models, they still have the power to demonstrate unexpected outcomes:  For example, that actions which are individually rational may not be desirable from the perspective of a system containing those individuals.  (It is one of the profound differences between Computer Science and Economics, that such an outcome would be unlikely to be surprising to most computer scientists, yet seems to be so to mainstream Economists, imbued with a belief in metaphysical carpal entities.)

From the final section of Haldane and May (2011):

The analytic model outlined earlier demonstrates that the topology of the financial sector’s balance sheet has fundamental implications for the state and dynamics of systemic risk. From a public policy perspective, two topological features are key.

First, diversity across the financial system. In the run-up to the crisis, and in the pursuit of diversification, banks’ balance sheets and risk management systems became increasingly homogenous. For example, banks became increasingly reliant on wholesale funding on the liabilities side of the balance sheet; in structured credit on the assets side of their balance sheet; andmanaged the resulting risks using the same value-at-risk models. This desire for diversificationwas individually rational from a risk perspective. But it came at the expense of lower diversity across the system as whole, thereby increasing systemic risk.Homogeneity bred fragility (N. Beale and colleagues, manuscript in preparation).

In regulating the financial system, little effort has as yet been put into assessing the system-wide characteristics of the network, such as the diversity of its aggregate balance sheet and risk management models. Even less effort has been put into providing regulatory incentives to promote diversity of balance sheet structures, business models and risk management systems. In rebuilding and maintaining the financial system, this systemic diversity objective should probably be given much greater prominence by the regulatory community.

Second, modularity within the financial system. The structure of many non-financial networks is explicitly and intentionally modular.  This includes the design of personal computers and the world wide web and the management of forests and utility grids. Modular configurations prevent contagion infecting the whole network in the event of nodal failure. By limiting the potential for cascades, modularity protects the systemic resilience of both natural and constructed networks.

The same principles apply in banking. That is why there is an ongoing debate on the merits of splitting banks, either to limit their size (to curtail the strength of cascades following failure) or to limit their activities (to curtail the potential for cross-contamination within firms). The recently proposed Volcker rule in the United States, quarantining risky hedge fund, private equity and proprietary trading activity from other areas of banking business, is one example of modularity in practice. In the United Kingdom, the new government have recently set up a Royal Commission to investigate the case for encouraging modularity and diversity in banking ecosystems, as a means of buttressing systemic resilience.

It took a generation for ecological models to adapt. The same is likely to be true of banking and finance.”

It would be interesting to consider network models which are more realistic than these toy versions, for instance, with nodes representing banks with goals, preferences and beliefs.

References:

F. Caccioli, M. Marsili and P. Vivo [2009]: Eroding market stability by proliferation of financial instruments. The European Physical Journal B, 71: 467–479.

Andrew Haldane and Robert May [2011]: Systemic risk in banking ecosystems. Nature, 469:  351-355.

Robert May, Simon Levin and George Sugihara [2008]: Complex systems: ecology for bankers. Nature, 451, 893–895.

This page lists the people I have currently written about or mentioned, in alpha order:

Alexander d’Arblay,  John Aris, John Atkinson, John Bennett,  Christophe Bertrand,  Matthew Piers Watt Boulton, Joan Burchardt,   Nicolas Fatio de Duillier, Michael Dummett, Martin Gardner,   Kurt Godel,   Charles Hamblin,   Thomas Harriott,  Martin Harvey, Robert May, Robin Milner, Isaac Newton,  Mervyn Pragnell,   Malcolm Rennie, Dennis Ritchie, Ibn Sina, Adam Spencer, Alan Turing, Alexander Yessenin-Volpin.

Liking cladistics as I do, I thought it useful to list all the potential purposes of models and modeling.   The only discussion that considers this topic that I know is a brief discussion by game theorist Ariel Rubinstein in an appendix to a book on modeling rational behaviour (Rubinstein 1998).  Rubinstein considers several alternative purposes for economic modeling, but ignores many others.   My list is as follows (to be expanded and annotated in due course):

• 1. To better understand some real phenomena or existing system.   This is perhaps the most commonly perceived purpose of modeling, in the sciences and the social sciences.
• 2. To predict (some properties of) some real phenomena or existing system.  A model aiming to predict some domain may be successful without aiding our understanding  of the domain at all.  Isaac Newton’s model of the motion of planets, for example, was predictive but not explanatory.   I understand that physicist David Deutsch argues that predictive ability is not an end of scientific modeling but a means, since it is how we assess and compare alternative models of the same phenomena.    This is wrong on both counts:  prediction IS an end of much modeling activity (especially in business strategy and public policy domains), and it not the only means we use to assess models.  Indeed, for many modeling activities, calibration and prediction are problematic, and so predictive capability may not even be  possible as a form of model assessment.
• 3. To manage or control (some properties of) some real phenomena or existing system.
• 4. To better understand a model of some real phenomena or existing system.  Arguably, most of economic theorizing and modeling falls into this category, and Rubinstein’s preferred purpose is this type.   Macro-economic models, if they are calibrated at all, are calibrated against artificial, human-defined, variables such as employment, GDP and inflation, variables which may themselves bear a tenuous and dynamic relationship to any underlying economic reality.   Micro-economic models, if they are calibrated at all, are often calibrated with stylized facts, abstractions and simplifications of reality which economists have come to regard as representative of the domain in question.    In other words, economic models are not not usually calibrated against reality directly, but against other models of reality.  Similarly, large parts of contemporary mathematical physics (such as string theory and brane theory) have no access to any physical phenomena other than via the mathematical model itself:  our only means of apprehension of vibrating strings in inaccessible dimensions beyond the four we live in, for instance, is through the mathematics of string theory.    In this light, it seems nonsense to talk about the effectiveness, reasonable or otherwise, of mathematics in modeling reality, since how we could tell?
• 5. To predict (some properties of) a model of some real phenomena or existing system.
• 6. To better understand, predict or manage some intended (not-yet-existing) artificial system, so to guide its design and development.   Understanding a system that does  not yet exist is qualitatively different to understanding an existing domain or system, because the possibility of calibration is often absent and because the model may act to define the limits and possibilities of subsequent design actions on the artificial system.  The use of speech act theory (a model of natural human language) for the design of artificial machine-to-machine languages, or the use of economic game theory (a mathematical model of a stylized conceptual model of particular micro-economic realities) for the design of online auction sites are examples here.   The modeling activity can even be performative, helping to create the reality it may purport to describe, as in the case of the Black-Scholes model of options pricing.
• 7. To provide a locus for discussion between relevant stakeholders in some business or public policy domain.  Most large-scale business planning models have this purpose within companies, particularly when multiple partners are involved.  Likewise, models of major public policy issues, such as epidemics, have this function.  In many complex domains, such as those in public health, models provide a means to tame and domesticate the complexity of the domain.  This helps stakeholders to jointly consider concepts, data, dynamics, policy options, and assessment of potential consequences of policy options,  all of which may need to be socially constructed. 
• 8. To provide a means for identification, articulation and potentially resolution of trade-offs and their consequences in some business or public policy domain.   This is the case, for example, with models of public health risk assessment of chemicals or new products by environmental protection agencies, and models of epidemics deployed by government health authorities.
• 9. To enable rigorous and justified thinking about the assumptions and their relationships to one another in modeling some domain.   Business planning models usually serve this purpose.   They may be used to inform actions, both to eliminate or mitigate negative consequences and to enhance positive consequences, as in retroflexive decision making.
• 10. To enable a means of assessment of managerial competencies of the people undertaking the modeling activity. Investors in start-ups know that the business plans of the company founders are likely to be out of date very quickly.  The function of such business plans is not to model reality accurately, but to force rigorous thinking about the domain, and to provide a means by which potential investors can challenge the assumptions and thinking of management as way of probing the managerial competence of those managers.    Business planning can thus be seen to be a form of epideictic argument, where arguments are assessed on their form rather than their content, as I have argued here.
• 11. As a means of play, to enable the exercise of human intelligence, ingenuity and creativity, in developing and exploring the properties of models themselves.  This purpose is true of that human activity known as doing pure mathematics, and perhaps of most of that academic activity known as doing mathematical economics.   As I have argued before, mathematical economics is closer to theology than to the modeling undertaken in the natural sciences. I see nothing wrong with this being a purpose of modeling, although it would be nice if academic economists were honest enough to admit that their use of public funds was primarily in pursuit of private pleasures, and any wider social benefits from their modeling activities were incidental.
  

POSTSCRIPT (Added 2011-06-17):  I have just seen Joshua Epstein’s 2008 discussion of the purposes of modeling in science and social science.   Epstein lists 17 reasons to build explicit models (in his words, although I have added the label “0″ to his first reason):

0. Prediction
1. Explain (very different from predict)
2. Guide data collection
3. Illuminate core dynamics
4. Suggest dynamical analogies
5. Discover new questions
6. Promote a scientific habit of mind
7. Bound (bracket) outcomes to plausible ranges
8. Illuminate core uncertainties
9. Offer crisis options in near-real time. [Presumably, Epstein means "crisis-response options" here.]
10. Demonstrate tradeoffe/ suggest efficiencies
11. Challenge the robustness of prevailing theory through peturbations
12. Expose prevailing wisdom as imcompatible with available data
13. Train practitioners
14. Discipline the policy dialog
15. Educate the general public
16. Reveal the apparently simple (complex) to be complex (simple).

These are at a lower level than my list, and I believe some of his items are the consequences of purposes rather than purposes themselves, at least for honest modelers (eg, #11, #12, #16).

References:

Joshua M Epstein [2008]: Why model? Keynote address to the Second World Congress on Social Simulation, George Mason University, USA.  Available here (PDF).

Robert E Marks [2007]:  Validating simulation models: a general framework and four applied examples. Computational Economics, 30 (3): 265-290.

David F Midgley, Robert E Marks and D Kunchamwar [2007]:  The building and assurance of agent-based models: an example and challenge to the field. Journal of Business Research, 60 (8): 884-893.

Robert Rosen [1985]: Anticipatory Systems. Pergamon Press.

Ariel Rubinstein [1998]: Modeling Bounded Rationality. Cambridge, MA, USA: MIT Press.  Zeuthen Lecture Book Series.

Ariel Rubinstein [2006]: Dilemmas of an economic theorist. Econometrica, 74 (4): 865-883.

Charles Leonard Hamblin (1922-1985) was an Australian philosopher and one of Australia’s first computer scientists. His main early contributions to computing, which date from the mid 1950s, were the development and application of reverse polish notation and the zero-address store. He was also the developer of one of the first computer languages, GEORGE. Since his death, his ideas have become influential in the design of computer interaction protocols, and are expected to shape the next generation of e-commerce and machine-communication systems.

Hamblin was born in 1922 and attended North Sydney Boys’ High School and Geelong Grammar. He then took degrees in Arts (Philosophy and Mathematics) and in Science (Physics), followed by an MA in Philosophy (First Class Honours) at Melbourne University, with his studies interrupted by work as a radar officer in the RAAF during World War II. Following the war, he gained a PhD at the London School of Economics, University of London on the topic, “Language and the Theory of Information”, apparently under Karl Popper (Hamblin 1957a). Hamblin’s thesis presented a critique of Shannon’s theory of information from a semantic perspective, and developed a possible-worlds semantics for question-response exchanges. Between 1955 and his death in 1985 he was a Lecturer and Professor in the School of Philosophy at New South Wales University of Technology (NSWUT), which later became the University of New South Wales (UNSW).

In 1956, the University purchased a DEUCE computer manufactured by the English Electric Company (EEC), an early British computer manufacturer, and Hamblin, with his radar background, became involved in working with this machine. This was the second academic computer in Australia, after that of the University of Sydney. Hamblin soon became aware of the problems of (a) computing mathematical formulae containing brackets, and (b) the memory overhead in having dealing with memory stores each of which had its own name. One solution to the first problem was Jan Lukasiewicz’s Polish notation, which enables a writer of mathematical notation to instruct a reader the order in which to execute the operations (e.g. addition, multiplication, etc) without using brackets. Polish notation achieves this by having an operator (+, *, etc) precede the operands to which it applies, e.g., +ab, instead of the usual, a+b. Hamblin, with his training in formal logic, knew of Lukasiewicz’s work.

However, this does not solve the second problem. Hamblin realized that placing the operation symbol to the right of the operands (i.e., reversing the polish notation, as in ab+) would enable the machine to make use of a store which did not require an address – the current operation would always be conducted on the most-recent operands inserted and still remaining in the store. This store came to be called a stack, or last-in, first-out (LIFO) store. He implemented these ideas in a programming language for the DEUCE machine, a language he called GEORGE, for General Order Generator. This work was undertaken at a time when there were only a handful of programming languages, and indeed still some resistance to the idea of non-assemblor languages (due to their greater memory requirements). Hamblin’s work on the DEUCE machine at UNSW overlapped with that of Gordon Bell and Bob Brigham, who wrote a symbolic assembler and run-time system called SODA (or Symbolic Optimum DEUCE Assembly Program) (Brigham and Bell 1959). GEORGE used the SODA runtime library.

Hamblin presented his work at the first Australian conference on computing, which was held at the Weapon Research Establishment in Salisbury, South Australia, in June 1957 (Hamblin 1957b). Employees of the English Electric Company were present at this conference, and took his ideas back to England. As a consequence, Hamblin’s architecture was implemented in the company’s next machine, which came to be called the KDF9. The architecture of this machine even used Hamblin’s terminology. This machine was announced in 1960 and delivered (i.e. made available commercially) in 1963. Hamblin published his ideas in 1957 (Hamblin 1957b, 1957c) and 1962 (Hamblin 1962). An earlier paper (Burks et al. 1954) presented the same ideas in a more general notational framework, and that paper was briefly reviewed in the Journal of Symbolic Logic in 1955 (Nelson 1955). Hamblin may have seen the Burks paper, or (with greater probability) the JSL review, although neither of these articles is cited in his 1962 Computer Journal paper which presents RPN (Hamblin 1962). (When accessed on 2010-07-20, the catalogue of the Library of the University of New South Wales indicated that the UNSW Library did not currently carry the journal in which the Burks paper was published, Mathematical Tables and Other Aids to Computation; of course, the Library may have carried this journal in the past.)

Another computer, the American Burroughs B5000, announced in 1961 and delivered in 1963, also used a zero-address architecture, and also enabled reverse polish notation to be used for programming. R. S. Barton, one of the designers of the B5000, has written that he developed RPN independently of Hamblin, sometime in 1958 while reading a textbook on symbolic logic, and before he was aware of Hamblin’s work (Barton 1970). A decade after Hamblin first published his ideas, engineers at Hewlett-Packard (HP) developed a personal calculator, the 9100A Desktop Calculator, which used RPN. This calculator, the first in a long line by HP, was released in 1968, and it popularized RPN among the scientific and engineering communities; note, however, that early advertisements for the 9100A did not mention RPN.

Even if Hamblin’s work on RPN was not the first to be published that applied Polish Notation to a computational domain, people at the  time thought it was, as evidenced by the refereed publication of his 1962 paper in the Computer Journal, and Barton’s comments published in 1970.   Hamblin’s contribution to computer science was also recognized with an obituary in the Australian Computer Journal (Allen 1985) and in an influential history of British computing (Lavington 1980). In addition, GEORGE is listed in Bill Kinnersley’s comprehensive directory of computer languages, The Language List. In the 1960s, Hamblin also worked on implementing Tarski’s decision method for real closed fields (Tarski 1951), the first order theory of real numbers with addition and multiplication, and hired two programmers to assist in this project, Malcolm Newey and Vaughan Pratt. However, only in 1974 was it shown by Fischer and Rabin (1974) that the running time of this problem had an exponential lower bound.

Although usually not credited, Hamblin was the originator of two other ideas which subsequently became important in Artificial Intelligence. Firstly, Hamblin appears to have been the first person to define a formal measure of plausibility, distinct from that of probability, in a paper published in 1959. Alternative formalisms for uncertainty have come to play a very important role in Artificial Intelligence, particularly in the design of knowledge-based systems, due to the failure of the standard Kolmogorov axioms of probability to adequately account for all forms of uncertainty and for its manipulation. One person particularly taken by Hamblin’s work in this area was the British economist, George Shackle, who in the 1940s and 1950s had developed a theory of decision-making under uncertainty based on the potential surprise of rival uncertain beliefs, and focusing on the best-case and worst-case outcomes of alternative decision-options (see pp. 97 – 100 of Shackle 1969). (Shackle’s theory, based on his real-world experience of government economic policy making and business investment decisions, differed from the Maximum Expected Utility theory of Leonard J. Savage which has unfortunately come to dominate mainstream economics.)

Secondly, Hamblin was the first person to propose an axiomatic account of time based on intervals, rather than points. This was in a paper published in 1969. An interval calculus for time was later proposed by James Allen (1984), and has been influential in AI, both as a basis for reasoning about time, and, when extended to multiple dimensions, as a basis for reasoning about space (Anger and Rodriguez 1991).

From the 1960s, Hamblin returned to work in philosophy, particularly the philosophy of argumentation, and wrote two very influential books. One of these, Fallacies, published in 1970, is a study of the classical logical fallacies, such as begging the question, which Hamblin examined by means of formal dialogue games. These are games between speakers who utter statements according to strict rules, and they were first studied by Aristotle. Being rule-governed, these games have gained the attention of computer scientists, and, from about 1989, they have been applied in a number of areas, including: natural language processing; human-machine interaction; the design of complex software; and for dialogues between autonomous software agents (McBurney and Parsons 2009). Interaction and communication protocols based on formal dialogue games are likely to form the basis for the next generation of e-commerce systems and systems supporting high-level machine-to-machine communications. Another of Hamblin’s books, Imperatives, published posthumously in 1987, has also been influential in recent work in computer science, in modeling and implementing delegation of tasks between software agents (Atkinson et al. 2008, Reed and Norman 2007, Norman and Reed 2010).

Hamblin was fluent in several languages, including ancient Greek and Latin. He was one of three fellow-students from his time at Geelong Grammar to become professors of philosophy (the others being David Armstrong and Michael Scriven). At the time of his death, he was apparently attempting to set text of Wittgenstein to music.

Charles Hamblin was a pioneer computer scientist and a prominent philosopher, whose influence on the subject is still being felt. His contributions to applied and theoretical computing show the deep links which Computer Science has had, and continues to have, with philosophy and logic.

References:

In addition to the works cited in the text above, I have also listed all of Hamblin’s publications known to me.

J. F. Allen [1984]: Towards a general theory of action and time. Artificial Intelligence, 23(2): 123-154.

M. W. Allen [1985]: Charles Hamblin (1922-1985). The Australian Computer Journal, 17(4): 194-195.

F. D. Anger and R. V. Rodriguez [1991]: Time, tense, and relativity revisited. In: B. Bouchon-Meunier, R. R. Yager and L. A. Zadeh (Editors): Uncertainty in Knowledge Bases: Proceedings of the Third International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 1990), pp. 286-295. Heidelberg, Germany: Springer.

K. Atkinson, R. Girle, P. McBurney and S. Parsons [2008]: Command dialogues. In: I. Rahwan and P. Moraitis (Editors): Proceedings of the Fifth International Workshop in Argumentation in Multi-Agent Systems (ArgMAS 2008). AAMAS 2008, Lisbon, Portugal.

R. S. Barton [1970]: Ideas for computer systems organization: a personal survey. pp. 7-16 of: J. S. Jou (Editor): Software Engineering: Volume 1: Proceedings of the Third Symposium on Computer and Information Sciences held in Miami Beach, Florida, December 1969. New York, NY, USA: Academic Press.

R. C. Brigham and C. G. Bell [1959]: A Translation Routine for the DEUCE Computer. The Computer Journal, 2 (2): 76-84.

A. W. Burks, D. W. Warren and J. B. Wright [1954]: An analysis of a logical machine using paranthesis-free notation. Mathematical Tables and Other Aids to Computation, 8 (46): 53-57.

M. J. Fischer and M. O. Rabin [1974]: Super-exponential complexity of Pressburger arithmetic. Complexity of Computation, AMS-SIAM Proceedings. 7: 27-41.

R. J. Gillings and C. L. Hamblin [1964]: Babylonian reciprocal tables on UTECOM. Technology, 9 (2): 41-42, August 1964. An expanded version appeared in Australian Journal of Science, 27, 1964.

C. L. Hamblin [1957a]: Language and the Theory of Information. PhD Thesis, Logic and Scientific Method Programme, University of London, London, UK. Submitted October 1956, awarded 1957.

C. L. Hamblin [1957b]: An addressless coding scheme based on mathematical notation. Proceedings of the First Australian Conference on Computing and Data Processing, Salisbury, South Australia: Weapons Research Establishment, June 1957.

C. L. Hamblin [1957c]: Computer Languages. The Australian Journal of Science, 20: 135-139. Reprinted in The Australian Computer Journal, 17(4): 195-198 (November 1985).

C. L. Hamblin [1957d]: Review of: W. R. Ashby: Introduction to Cybernetics. Australasian Journal of Philosophy, 35.

C. L. Hamblin [1958a]: Questions. Australasian Journal of Philosophy, 36(3): 159-168.

C. L. Hamblin [1958b]: Review of: Time and Modality, by A. N. Prior. Australasian Journal of Philosophy, 36: 232-234.

C. L. Hamblin [1958c]: Surprises, innovations and probabilities. Proceedings of the ANU Symposium on Surprise, Canberra, July 1958.

C. L. Hamblin [1958d]: Review of: Formal Analysis of Normative Systems, by A. R. Anderson. Australasian Journal of Philosophy, 36.

C. L. Hamblin [1958e]: GEORGE Programming Manual. Duplicated, 1958. Revised and enlarged, 1959.

C. L. Hamblin [1959]: The Modal “Probably”. Mind, New Series, 68: 234-240.

C. L. Hamblin [1962]: Translation to and from Polish notation. Computer Journal, 5: 210-213.

C. L. Hamblin [1963]: Questions aren’t statements. Philosophy of Science, 30(1): 62-63.

C. L. Hamblin [1964a]: Has probability any foundations? Proceedings of the Symposium on Probability of the Statistical Society of New South Wales, May 1964. Reproduced in Science Yearbook, University of New South Wales, Sydney, 1964.

C. L. Hamblin [1964b]: Review of: Communication: A Logical Model, by D. Harrah. Australasian Journal of Philosophy, 42.

C. L. Hamblin [1964c]: Review of: Analysis of Questions, by N. D. Belnap.Australasian Journal of Philosophy, 42.

C. L. Hamblin [1965]: Review of: A Preface to the Logic of Science, by P. Alexander. The British Journal for the Philosophy of Science, 15(60): 360-362.

C. L. Hamblin [1966]: Elementary Formal Logic, a Programmed Course. (Sydney: Hicks Smith). Republished by Methuen, in London, UK, 1967. Also translated into Swedish by J. Mannerheim, under the title: Element”ar Logik, ein programmerad kurs. (Stockholm: Laromedelsf”orlagen, 1970).

C. L. Hamblin [1967a]: One-valued logic. Philosophical Quarterly, 17: 38-45.

C. L. Hamblin [1967b]: Questions, logic of. Encyclopedia of Philosophy. (New York: Collier Macmillan).

C. L. Hamblin [1967c]: An algorithm for polynomial operations. Computer Journal, 10.

C. L. Hamblin [1967d]: Review of: New Approaches to the Logical Theory of Interrogatives, by L. Aqvist. Australasian Journal of Philosophy, 44.

C. L. Hamblin [1969]: Starting and stopping. The Monist, 53: 410-425.

C. L. Hamblin [1970a]: Fallacies. London, UK: Methuen.

C. L. Hamblin [1970b]: The effect of when it’s said. Theoria, 36: 249-264.

C. L. Hamblin [1971a]: Mathematical models of dialogue. Theoria, 37: 130-155.

C. L. Hamblin [1971b]: Instants and intervals. Studium Generale, 24: 127-134.

C. L. Hamblin [1972a]: You and I. Analysis, 33: 1-4.

C. L. Hamblin [1972b]: Quandaries and the logic of rules. Journal of Philosophical Logic, 1: 74-85.

C. L. Hamblin [1973a]: Questions in Montague English. Foundations of Language, 10: 41-53.

C. L. Hamblin [1973b]: A felicitous fragment of the predicate calculus. Notre Dame Journal of Formal Logic. 14: 433-446.

C. L. Hamblin [1974]: La logica dell’iniziare e del cessare. Italian translation by C. Pizzi of an unpublished article: The logic of starting and stopping. Pages 295-317 in: C. Pizzi (Editor): La Logica del Tempo. Torino: Bringhieri.

C. L. Hamblin [1975a]: Creswell’s colleague TLM. Nous, 9(2): 205-210.

C. L. Hamblin [1975b]: Saccherian arguments and the self-application of logic. Australasian Journal of Philosophy, 53: 157-160.

C. L. Hamblin [1976]: An improved “Pons Asinorum”? Journal of the History of Philosophy, 14: 131-136.

C. L. Hamblin [1984]: Languages of Asia and the Pacific: A Phrasebook for Travellers and Students. (North Ryde, NSW: Angus and Robertson).

C. L. Hamblin [1987]: Imperatives. Oxford, UK: Basil Blackwell.

C. L. Hamblin and P. J. Staines [1992]: An extraordinarily simple theory of the syllogism. Logique et Analyse, 35: 81.

S. H. Lavington [1980]: Early British Computers: The Story of Vintage Computers and the People who Built Them. Manchester, UK: Manchester University Press.

P. McBurney and S. Parsons [2009]: Dialogue games for agent argumentation. Chapter 13 in: I. Rahwan and G. Simari (Editors): Argumentation in Artificial Intelligence. Berlin, Germany: Springer, pp. 261-280.

R. J. Nelson [1955]: Review of: “An analysis of a logical machine using paranthesis-free notation” by Arthur W. Burks, Don. W. Warren and Jesse B. Wright, The Journal of Symbolic Logic, 20 (1): 70-71.

T. J. Norman and C. Reed [2010]: A logic of delegation. Artificial Intelligence, 174 (1): 51-71.

T. Pearcey [1994]: Australian Computing: The Second Generation. Published in: J. M. Bennett, R. Broomham, P. M. Murton, T. Pearcey and R. W. Rutledge (Editors): Computing in Australia: The Development of a Profession. Australian Computer Society.

C. A. Reed. and T. J. Norman [2007]: A formal characterisation of Hamblin’s action-state semantics. Journal of Philosophical Logic, 36 (4): 415-448.

G. L. S. Shackle [1969]: Decision Order and Time in Human Affairs. Cambridge, UK: Cambridge University Press. Second Edition.

A. Tarski [1951]: A Decision Method for Elementary Algebra and Geometry. Berkeley, CA, USA: University of California Press.

R. A. Vowels [1978]: Introduction to PL/I, Algorithms and Structured Programming. Melbourne, 1978.

G. Williams [1985]: A shy blend of logic, maths and languages. (Obituary of Charles Hamblin). Sydney Morning Herald, 1985.

In compiling this biography, I am grateful for information and support received from: Gordon Bell, Jim Crosswhite, David Hitchcock, Jim Mackenzie, Vaughan Pratt, Michael Scriven, Phillip Staines, Robin Vowels, Doug Walton, and the family of the late Charles Hamblin. The views I express here are, of course, solely my own.

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