Three centuries ago the philosopher, logician, mathematician, scientist and engineer Gottfried Wilhelm Leibniz conceived, as a young man, a grand project, so far ahead of its time that no-one has yet come close to realizing it. The principal elements his project were:
He devoted considerable energy to progressing these ideas. But his ideas were pie in the sky, there was no hope of success.
Since then many important developments have taken place in philosophy, logic, mathematics and information technology, and most of the now known impediments to the realization of something close to his dream have now been surmounted.
Today it is natural to think that computers solve the mechanical side of the project, and that the rest of the project is just software of various kinds. Though Leibniz did work on the hardware, in his day mechanical calculators, his main interest was in what we would now call software, the algorithms and the data upon which they operated, and his approach to this was philosophical and logical.
This is at once an information technology project and philosophical research. To engineer a ``cognitive agent'', to build something which has or can acquire knowledge, and can reason from that knowledge to solve problems, can only be done in the context of a suitable philosophical framework. At its most abstract, architectural design for cognitive artifacts is philosophy.
The kind of philosophy required to underpin such a project is not piecemeal philosophical analysis, it is a systematic philosophical synthesis. It is the aim of this book to undertake such a synthesis, in such a way that its relationship with the engineering of a certain kind of intelligent artifact is made clear. The engineering enterprise I call here ``the project''. The philosophy is intended to underpin that project and to constitute its earliest most abstract stages.
It is therefore intended to present by stages together,
In this introduction I shall survey the structure of the exposition and sketch the principal themes.
It is in the nature of this project that it combines materials from and seeks to interest devotees of diverse disciplines. The various themes discussed demand varied background, much of which I cannot hope to supply. Though the philosophical the logical and the technological aspects are intimately intertwined, it is hoped that readers with a particular competence and interest in one aspect will find most of the material of interest to him intelligible without a complete mastery of the other aspects of the presentation.
The collection of those works of Aristotle which concerned logic is known as the organon. This word comes from the Greek word for a tool.
It has this name because logic was conceived by Aristotle and by later scholars as providing a tool, rather than as a purely academic pursuit. The principal application of that tool was to be demonstrative science, the derivation of necessary truths in the various sciences from the first principles of those sciences.
The emphasis on logic as a tool might well have been unimportant through most of the history of logic, since until recently Aristotle's organon has dominated the field. However, though well intentioned, Aristotle's formal logic is inadequate for any non-trivial scientific application, and the study of logic has remained the province of Philosophers.
With the advent of modern logic, beginning in the second half of the nineteenth century with mathematicians and philosophers such as Boole, Frege and Pierce, there was spawned a new discipline of mathematics, mathematical logic, which was primarily meta-theoretical in character. Academic interest in logic now spans multiple disciplines, of which the most important are philosophy, mathematics and computer science. There are various ways in which these disciplines make use of logic as a tool, but its use as a tool in the manner envisaged by Aristotle, as the means whereby conclusions are drawn from various first principles, is rare.
This book belongs to a line of philosophical works (notably, Aristotle, Leibniz and Carnap) of which the aim is to contribute to the means for the application of formal deductive methods in the establishment and application of knowledge. This work is subordinate to that purpose.
The realization of that objective depends upon a philosophical context which cannot be taken for granted, or even taken from a shelf and dusted off. Over the last half-century philosophy has moved in other directions, and in order to do so has undermined the philosophical basis for the most recent manifestation of the project in the philosophy of Rudolf Carnap. For a resumption of the project, new philosophical foundations are required.
The philosophical innovation required in support of the project consists substantially in the re-establishment of ideas which have fallen into disrepute. This includes specific ideas such as the analytic/synthetic dichotomy, and entire philosophical perspectives or systems such as positivism. Their reestablishment, at least as viable alternatives to received opinion, will not be realized by a detailed refutation of the arguments which displaced them.
In logic over the last 150 years and in Computing over the last 50 years there have been very many new developments of a kind which one might expect to give philosophers pause for thought. But new ideas in philosophy itself are very rare, most of the twists and turns in philosophical fashion are revivals of old ideas in new clothes. It is in the nature of philosophical progress that it often appears through millennia of debate as ideas are proposed, developed, disputed, rejected and perhaps ignored for a while before rising again re-engineered for a new philosophical climate.
If we seek afresh to understand and to advance such ideas, tracing their development through history may be helpful in getting or in conveying an understanding of their contemporary manifestation.
Several historical threads serve I hope to illuminate aspects of the present proposal. The philosophers whose ideas are touched upon in these sketches have spent a lifetime developing the ideas. The purpose of the sketches is to make the ideas presented here clearer by connecting them with their historical roots.
The first of these historical threads sketches the analytic/synthetic dichotomy and a variety of related dichotomies and concepts. This is closely connected with the development of ideas of logical truth, and of truth conditional semantics.
Alongside such questions relating to the establishment of meaning, there is the question of truth and how it may be established or evaluated. Doubts about our ability to establish truths are at their most severe in scepticism which part of the historical background to and continuous with positivism, which combines elements of scepticism with a strict attitude towards rigour in science. Our positivism departs from its predecessors substantially in ways which can be illuminated by consideration of those sceptical roots, and which make it natural to think of as a graduated positive scepticism. This is closely connected with the question of rigour, and with the tension between rigour and progress, in science, mathematics and philosophy.
Though there appears to be at any point in time a trade-off between rigour and productivity (which is perhaps easiest to see in mathematics), at times advances are made which allow both to advance at once, and a new balance to be achieved. This happened in the nineteenth century for mathematics, a period of consolidation in which standards of rigour in mathematical analysis were transformed. The last stages in this transformation involved the invention of modern methods in logic, and established the possibility of the formal derivation of mathematics. The greater precision in locating the most abstract subject matters of mathematics, through the agency of axiomatic set theory, resulting in achievements of high standards of rigour which were sustained through a century of continuous mathematical innovation. These higher standards depended only peripherally on the new mathematical logic. Axiomatic set theory provided sufficient additional clarity to the definition of mathematical concepts, that standards of rigour were able to advance without the adoption of formal derivation as the standard for mathematical proof. It sufficed for a mathematical to convince his peers that a formal derivation would be possible.
In the second half of the century, the application of digital computers to the support of formal notations and deductive systems has created a new domain in which for the first time formal notations and languages are extensively used. Stored program computers demand and support the use of such formal languages. Communicating unambiguously with computers becomes a prime motivation for the use of formal notations, which provide not only the motivation but also the kinds of support which facilitate the use of formal languages.
If we look for the most fundamental ideas in philosophy we find a inextricably interrelated complex of ideas belonging to several distinct areas of philosophy.
Philosophically this work is primarily epistemological, as one might expect, and the philosophical system presented gives central place to epistemology.
Epistemology is approached in what might perhaps be described as an instrumental manner. Two aspects of epistemology receive no attention. The first is the meaning of the word ``know'', and hence for some philosophers the question of what knowledge em, is not considered. Second is any aspect of how knowledge is acquired or applied which is peculiarly human.
Instead of considering what knowledge is we are concerned with how knowledge might be represented and applied.
The epistemology is foundational both in relation to logical and empirical knowledge. The distinction between these two, the logical and empirical, is a keystone of the project and the philosophy. The project realizes an general analytic method a principal feature of which is the systematic and thorough separation of these two kinds of knowledge. Logical knowledge (under a broad conception of logical truth as analyticity) is represented formally and established by deductively sound methods. Empirical knowledge is represented formally using abstract logical models. The relationship between abstract formal theories and the systems which they model is ultimately informal.
The first stage of this, in Chapter 2, is to give the next stage of detail in describing the project, the aims and the architecture. This is presented by starting with an account of Leibniz's project, tracing the history of the ideas from the through to the century, and then offering a revised conception of such a project for the .
The philosophical side of this history ends in debacle. The last major philosophical proponent of a significant fragment of the Leibnizian enterprise was Rudolf Carnap in whose philosophy the formalization of science had a central place. In mid century, fundamental philosophical ideas with a key place in Carnap's philosophy were repudiated by W.V.Quine. In particular, the analytic/synthetic distinction, subject to continuous critique by Quine since his first exposure to Carnap's philosophy, was given a full blooded and uncompromising repudiation in Quine's influential ``Two Dogmas of Empiricism''. Though not aligning himself with Quine on the analytic/synthetic dichotomy Saul Kripke then teased apart the triumvirate of concepts which had been identified by Carnap (analyticity, necessity and the a priori), allowing Kripke to inaugurate a new kind of metaphysics, and set analytic philosophy on a new track fundamentally at odds with the philosophy of Carnap. Thus Kripke contributed to the subsequent widely held view that the philosophy of Rudolf Carnap (sometimes known as Logical Positivism) had been decisively refuted as a result of technical advances by two of the most highly competent and respected philosopher-logicians of the period.
The philosophical framework I offer here is closer to the philosophy of Carnap than to that of any other philosopher, and it is therefore necessary in Chapter 3 to repair some of the damage done to this point of view. Chapter 3 considers the status of these three dichotomies, the distinctions between analytic and synthetic, between necessity and contingency, and that between the a priori and the a posteriori. Though not always with this vocabulary, similar distinctions have been talked of throughout almost the entire history of western philosophy. Through this history one can see both a gradual refinement of these concepts and also reversals. I therefore lightly trace this history showing how Carnap's understanding of these dichotomies was reached, and then review and respond to some of the modern criticisms which are still held by many to be decisive against Carnap. In this chapter I argue that the refutation of Carnap on these matters is one more demonstration of the contrast between the rigour of mathematics and that of philosophy. An illustration, I might say, of the irrationality of philosophy.
In Chapter 4 I then take the concept of analyticity as established, and the question of its significance is examined in greater detail. In philosophy the twentieth century was called the age of analysis, and the principal kind of philosophy progressed by academics was called analytic philosophy. In the philosophy of Rudolf Carnap and the logical positivists the connection between the concept of analyticity and analytic philosophy was simple. Insofar as philosophy was concerned with establishing the truth of propositions (in the manner in which mathematicians establish results by proving theorems, or science establishes physical laws by observation and experiment) the results of the kind of philosophical analysis envisaged by Carnap would be analytic, though in his hands such results play a secondary role to the articulation of methods and the definition of languages or concepts suitable for science. For none of the many other conceptions of philosophical analysis which appear in the century was there such a simple connection between analyticity and analysis. In this chapter we look at the relevance of analytic truth to various kinds of analysis, both philosophical and scientific. This is done using a kind of philosophical thought experiment. Suppose that we had an oracle (man or machine) which could tell us of any conjecture whether or not it was an analytic truth? What impact would that have on the various kinds of analysis under consideration?
With the concept of analyticity and hence of deductive soundness in place it is time to further refine my characterization of the project, as being concerned, firstly, with deductively sound computation and ultimately with useful approximations to the terminator. These terms are explained in Chapter 5, and a variety of contemporary research trends are compared with the research thus envisaged. Since the beginning of mathematical logic many different conceptions of have evolved of proof and its relation to computation. The approach envisaged here is clarified in the context of a general discussion of these different conceptions of proof.
Carnap's philosophy had been intended to provide a way forward for philosophy to the achievement of standards of rigour comparable to those of mathematics (an ideal which has been held by many philosophers over the last 2500 years), but his program had been defeated, illustrating just those defects that he sought to remedy. To reinstate this idea I sketch another historical thread in Chapter 6. This is a history of rigour in mathematics and in philosophy. It is a history also of those philosophers who have perceived the rational deficit in philosophy, or in the search for knowledge more generally, of the sceptics of ancient Greece, and of the more modern tradition of positivism consisting of a kind of mitigated or constructive scepticism in which high standards of rigour are articulated for both a priori and a posteriori sciences.
The epistemologically conservative aspects of positivism give rise in this philosophy and architecture to the notion of ``epistemic retreat''. This involves an admission of general doubt, but the acceptance of degrees of doubt, and hence a partial ordering of conjectures indicating in a relative way how well they have been established, and what level of confidence they may be viewed.
Carnap's pluralism, a willingness to accept the use of any well-defined language on a pragmatic basis, gives rise to the problem of ``language planning'', addressing for example the problems arising from the use of multiple different languages for different aspects of the same problem. In Chapter 8 the architecture is further developed by addressing these matters.
Carnap built on the purely mathematical foundational ideas of Frege and Russell, but sought to apply the new logical methods to the empirical sciences. He believed that this required innovation on his part to admit languages suitable for talking about the material world rather than purely about mathematics, and in making this transition he also moved from a purely universalistic conception of logic (in which one language sufficed) to a pluralistic conception of the language of science. His contribution to this pluralistic world was by way of meta-theory, he spoke about how languages might be defined in their syntax, semantics and proof rules, taking this to be a proper philosophical contribution to the methodological advancement of science. The proliferation of languages thus envisaged would demand some kind of activity which he called ``language planning'', but did little on.
The project I envisage embraces the pluralism of Carnap, and therefore depends upon an architecture which admits multiple languages and permits large scale applications involving more than one language. This connects with other initiatives in computing, notably the idea of an ``Extensible Markup Language'' (XML) and the many related ideas which have built up around it, including the idea of a ``semantic web''.
So far as applications to empirical science are concerned the project we outline does not adopt the approach of Carnap. Instead we envisage that the project provides or empirical science (and ultimately of various engineering enterprises which depend upon it) by regarding these as working exclusively with abstract models of the physical world, and take the connection between such models and the concrete world to be beyond the scope of these formal methods.
Instead of asserting the truth of an empirical theory we instead evaluate the contexts in which it provides a useful model of aspects of the real world, and evaluate the model in different application domains in terms of reliability and fidelity or precision. Carnap's notion of language planning (on which he himself said little), is one domain in which our project might best be compared with the W3C Semantic Web initiative.
In chapter 9 we now table an architectural proposal, in the form of a set of key requirements and a set of architectural features intended to realize those requirements, and a rationale for the belief that they do indeed realize them.
With the architectural proposal in place we return in chapter 10 to the philosophy.
This first involves gathering together a coherent and rounded philosophy sufficient to underpin the proposed architecture, an important element of this is foundational. The second part concerns the methods supported by the architecture and their scope of applicability.
Metaphysical Positivism does not answer all philosophical problems, but it does influence what might be considered a worthwhile philosophical problem for further investigation. Chapter 11 looks at some of these.
Where does this architecture take us, why should it be implemented?
Chapter is concerned with the outer reaches of the significance of the project, beyond the confines of philosophy or of academia.
Far be it for me to say how you, reader should proceed. However, here are some ideas, and some observations on how I have tried to write it which might be helpful.
I rarely myself read a book linearly from cover to cover. The book nominally addresses a very broad range of potential readers, many of whom will be interested in only some aspects of its subject matter. In writing it I have therefore tried to make it possible for readers to reach those parts which matter most to them without having to struggle through too great a jungle of detail which might seem to them peripheral.
To this end I have tried to begin and end each chapter with summary material which for some readers might suffice, and to include references as specific as possible in the text to prior materials upon which an understanding may depend.
I have felt it desirable, in order to make as clear as possible the ideas which I present, to make use of stories about the history of various aspects of the subject matters. Often the work of philosophers who have spent a lifetime producing an important body of original work will be spoken of in a few sentences which cannot be a fair account of their work, even in some special corner.
In order to avoid misrepresentation I have used wherever possible the device of enunciating a position which, whether it was ever held by any philosopher or not, is useful in making a point. One or more philosophers may be named as having inspired this position, without going into a detailed examination (which I am rarely best equipped to undertake) of how closely it does correspond to the positions they in fact held.
This is a method not unrelated to that of the philosopher Saul Kripke in his examination of certain ideas suggested to him by the writings of Ludwig Wittgenstein. The method may be adopted of connecting a philosophical problem which is thought to be of interest in its own right with the history of the subject. Alternatively, for those whose interest is primarily historical and exegetical, the process of rational reconstruction may begin in this way, with a definite model of some aspect of a philosophers work, which may cast light by evaluation of its similarities and differences with the textual sources, and which may be refined in the light of such comparisons into progressively more complex and subtle models less readily seen as diverging from the target of analysis.
In this work it is the former motivation which concerns us exclusively. The second, as a kind of analysis is of interest from a meta-theoretic point of view, but is not here practiced in anger.
Too rigorous an attempt to distill historical illustrations into hypothetical positions not directly attributed would however be unduly cumbersome. This mode of presentation is reserved for the most important and substantial points, and much background is presented as if historical fact, but should nevertheless be thought of in a similar manner, as so simple an account as could at best be true in spirit, only to be dissipated on closer inspection.
Roger Bishop Jones 2012-09-23