Knowing and Understanding
Relations between meaning and truth,
meaning and necessary truth,
meaning and synthetic necessary truth
-- -- --
The thesis is now freely available in machine-readable PDF and TXT formats.
However, proof-reading of the typed transcript is still in progress.
Please report any errors.
Information from Oxford (Bodleian Library) Web site:
Abstract and Keywords
Note on angle trisection
(Original image files and derived PDF and text files.)
Detailed table of contents and links to downloadable chapters
(Free) online book version coming
(Draft) Historical note
Update history (since installation in 2007)
Information from Oxford Web site
Oxford University Research Archive (ORA)
Bodleian Libraries, University of Oxford
Digital Origin: Digitized other analog Type of Award: DPhil Level of Award: Doctoral Awarding Institution: University of Oxford About The AuthorsA. Sloman Search for more by this author on ORA site
website http://www.cs.bham.ac.uk/~axs/ institution University of Oxford faculty Faculty of Literae Humaniores oxford College St Antony's College (Balliol College 1957-60, St Antony's 1960-62)
Citable link to the Oxford page:
Scanned image files were made available at:
which links to:
This was the first Oxford DPhil Thesis to be digitised for the archive.
The scanned image PDF files were downloaded from Oxford to Birmingham in 2007:
Suitable OCR technology was not available, so the PDF files contained only images of the pages.
Humanly transcribed versions of the files were added (details below) in April/May 2016, thanks to the efforts of Luc Beaudoin, who identified Hi-Tech Transcription Services as suitable for the task, then helped with proof-reading and getting the files to work in Libreoffice. (Proof-reading still in progress 14 May 2016).May 2016:
The aim of the thesis is to show that there are some synthetic necessary truths, and that synthetic apriori knowledge is possible. This is really a pretext for an investigation into the general connection between meaning and truth, or between understanding and knowing, which, as pointed out in the preface, is really the first stage in a more general enquiry concerning meaning. (Not all kinds of meaning are concerned with truth.) After the preliminaries (chapter one), in which the problem is stated and some methodological remarks made, the investigation proceeds in two stages. First there is a detailed inquiry into the manner in which the meanings or functions of words occurring in a statement help to determine the conditions in which that statement would be true (or false). This prepares the way for the second stage, which is an inquiry concerning the connection between meaning and necessary truth (between understanding and knowing apriori). The first stage occupies Part Two of the thesis, the second stage Part Three. In all this, only a restricted class of statements is discussed, namely those which contain nothing but logical words and descriptive words, such as "Not all round tables are scarlet" and "Every three-sided figure is three-angled". (The reasons for not discussing proper names and other singular definite referring expressions are given in Appendix I.)
Meaning (Philosophy), Vagueness, Truth, Immanuel Kant, Gottlob Frege, Synthetic necessary truth, Synthetic apriori knowledge, Logic, Geometry, Arithmetic, Functions vs rogators.
Many years later I learnt that it is possible to trisect an arbitrary angle
using origami geometry, and, perhaps more interestingly, it is possible using
the "neusis" construction, which is a simple extension to Euclidean geometry
that allows a straight edge to be marked in two places and moved around subject
to constraints. The neusis construction and its use was known to Archimedes.
For more details and a demonstration of its use see:
The "original" PDF files are (bulky) scanned images of thesis pages, scanned in 2007 by the Oxford library from a carbon copy version of the typed thesis deposited in 1962.
The re-typed files with machine-searchable text are much smaller. They will later be merged to form a single online book.
The first three files were transcribed using OCR package tesseract, which required a huge amount of post editing. That produced .TXT files from which the first three new PDF files were derived.
In 2016, thanks to help from Luc Beaudoin, who found a suitable company to do the work (HiTech), the remaining chapters (Chapter 2 on) and appendices, etc., were re-typed by humans, producing MS Word files. These required some post editing but most of the typing seems to have been accurate. Please report any remaining flaws.
For post-editing, the word files were converted to ODT format, using Libreoffice. After checking and fixing, three derived files were produced from each word file using Libreoffice: .DOCX, .PDF, and .TXT. Only the PDF and TXT files are linked below. The ODT files will later be used to merge all the separate parts into one document.
|Front Matter Part||Pages||
|New PDF files||New TXT files|
|Title and Abstract||Pages i-ix||PDF (2.1MB)||New PDF (125KB)||TXT|
Table of Contents
|Pages i-v||PDF(716KB)||New PDF(74KB)||TXT|
|Part or Chapter Title||Start Page||Original PDF (scanned)||New PDF files  ||New TXT files|
|Chapter one: Introduction||. . 1||PDF 3.7 MB,||
PDF of TXT
|..... 1.A. The problems||. . 1|
|..... 1.B. Methodological remarks||. . 5|
|..... 1.C. The programme||. 13|
|PART TWO: MEANING AND TRUTH|
|Chapter two: Propositions and meanings||. 18||PDF 8.7 MB||NEW Ch2 PDF (197KB)||Ch2 TXT|
|..... 2.A. Criteria of identity||. 18|
|..... 2.B. General facts about language||. 24|
|..... 2.C. Universals and strict criteria||. 38|
|..... 2.D. The independence of universals||. 50|
|Chapter three: Semantic rules||. 63||PDF 7.2 MB||NEW Ch3 PDF (242KB)||Ch3 TXT|
|..... Introduction||. 63|
|..... 3.A. F-words||. 64|
|..... 3.B. Logical syntheses||. 70|
|..... 3.C. How properties explain||. 83|
|..... 3.D. Non-logical syntheses||. 93|
|..... 3.E. Concluding remarks and qualifications||102|
|Chapter four: Semantic rules and living languages||107||PDF 3.9 MB||NEW Ch4 PDF (102KB)||Ch4 TXT|
|..... 4.A. Indefiniteness||107|
|..... 4.B. Ordinary language works||117|
|..... 4.C. Purely verbal rules||125|
|Chapter five: Logical form and logical truth||129||PDF 12.2 MB||NEW Ch5 PDF (241KB)||Ch5 TXT|
|..... 5.A. Logic and syntax||130|
|..... 5.B. Logical techniques||144|
|..... 5.C. Logical Truth||166|
|..... 5.D. Some generalisations||176|
|..... 5.E. Conclusions and qualifications||181|
|-||PART THREE: MEANING AND NECESSARY TRUTH|
|Chapter six: Analytic propositions||194||PDF 12.5 MB||NEW Ch6 PDF (238KB)||Ch6 TXT|
|..... 6.A. Introduction||194|
|..... 6.B. Some unsatisfactory accounts of the distinction||199|
|..... 6.C. Identifying relations between meanings||217|
|..... 6.D. Indefiniteness of meaning||229|
|..... 6.E. Knowledge of analytic truth||236|
|..... 6.F. Concluding remarks||249|
|Chapter seven: Kinds of necessary truth||260||PDF13 MB||NEW Ch7 PDF (297KB)||Ch7 TXT|
|The features of an analytic proposition in virtue of which it is true ensure that it would be true in all possible states of affairs, so we can say that it could not possibly be false, that it must be true, that it is necessarily true, and so on. All these truth-guaranteeing features are topic-neutral and can be described in purely logical terms, such as that the proposition is made up of certain logical words in a certain order, with non-logical words whose meanings stand in certain identifying relations. This chapter will be concerned with the question whether there is any other way in which a proposition can be necessarily true. In order to give this question a clear sense I must explain what is meant by "necessary", that is, give an account of the way in which the necessary-contingent distinction is to be applied. I shall start off by talking about the meaning of "possible". The next section will attempt to explain the meaning of "necessary". The rest of the chapter will be concerned to describe and distinguish kinds of necessary truths, and ways in which a proposition may be known to be true independently of observation of contingent facts.|
|..... 7.A. Possibility||261|
|..... 7.B. Necessity||272|
|..... 7.C. Synthetic necessary connections||283|
|..... 7.D. Informal proofs||294|
|..... 7.E. Additional remarks||319|
|Chapter eight: Concluding summary||329||PDF 1 MB||NEW Ch8 PDF (35KB)||Ch8 TXT|
|335||PDF 9.7 MB||NEW Appendices PDF (40KB)||NEW appendices TXT|
|..... Bibliography||389||PDF 374.4 KB||NEW Bibliography PDF (40KB)||NEW Bibliography TXT|
.....p 335 .. Appendix I. Singular referring expressions
.....p 340 .. Appendix II. Confusions of formal logicians
This appendix presents arguments against the view that a natural language must include a formal system, and that logic is just a matter of syntax. One of the key points, also made by Frege, is that semantics cannot emerge from syntax alone: we also need to take account of the functions of the symbols used, not just their form.
.....p 357 .. Appendix III. Implicit knowledge
This appendix gives examples of several kinds of implicit knowledge, including allowing for the deployment of implicit knowledge to be unreliable sometimes (Compare Chomsky's Competence/Performance distinction, 1965). The ability to do logic and mathematics, as well as many other kinds of things, depends on the use of implicit knowledge, which can be very difficult to make explicit. (At that point I knew nothing about the young science of AI which was beginning to provide new techniques for articulating implicit knowledge.)
.....p 372 .. Appendix IV. Philosophical analysis
The ideas about implicit knowledge in Appendix III are used in Appendix IV to explain some of the puzzling features of the activity of conceptual analysis (disagreeing with R.M. Hare's explanation). This leads to further discussion of the nature of philosophical analysis and the claim that it cannot be concerned merely with properties of concepts: it must also be concerned with the world those concepts are used to describe, which may support different sets of concepts.
.....p 381 .. Appendix V. Further examples
.....p 386 .. Appendix VI. Apriori knowledge
When it seems that the new versions are sufficiently free from transcription errors they will be combined into a single new document that will be made available online here free of charge, probably in two formats: html and pdf. I hope this will be completed some time in 2016.
If you would like to have a full book-length pdf or plain text version (the latter without diagrams) before then, let me know and I'll generate one. But it is likely to go out of date as proof-reading continues.
My first degree was in mathematics and physics (Cape Town, 1957) after which I went to Oxford planning to become a research mathematician. In Oxford I became friendly with several philosophy graduate students and attended their seminars and some philosophy lectures. I soon realised that the philosophers I encountered had a view of the nature of mathematics that was deeply mistaken, and did not fit my experience of doing mathematics, including discovering and proving, or disproving, conjectures. In particular, it seemed to be commonly thought that the rejection of Euclid's parallel postulate on the basis of work by Einstein and Eddington demonstrated that Euclidean geometry was empirical. However this ignored the fact that a great deal of Euclidean geometry was also common to its alternatives. And work of Imre Lakatos (mentioned briefly in the thesis (Chapter 7), later published as Proofs and Refutations) showed that it was important to distinguish the non-empirical characteristics of mathematical discoveries from a claim that mathematicians are infallible.
After a year or two registered as a mathematics student (details forgotten) I switched from Mathematics to Logic (supervised for a while by Hao Wang) and then later switched to Philosophy and became a philosophy research student. David Pears was named my supervisor, though I continued to attend lectures by Hao Wang, and also by Michael Dummett, John Lemmon, and others. My "moral tutor" at Balliol College was Richard Hare, who helped me to broaden my (miniscule) philosophical education, including introducing me to his version of Kantian meta-ethics.
For one term my college arranged for me to be supervised by Michael Dummett, but that was a time when all his energies were spent on trying to help refugees who were being obstructed by immigration authorities, and phone calls continually disrupted our meetings. Insofar as we did communicate I tended to disagree with his anti-realism. (A few years later he decided that the political situation was beyond repair and re-focused on academic work.) I have never been able to understand how highly intelligent people can take religion seriously: and he was an example. But I did not challenge him on that. (I did challenge Hare, whose response was that God did not need to exist for people to believe in God: for him that seemed to be a kind of moral stance rather than a factual belief.)
During my five years in Oxford, mostly after the second year, I attended lectures and seminars by Gilbert Ryle, John Austin, Peter Strawson, William Kneale, Martha Kneale, Friedrich Waismann, Michael Hinton, Anthony Quinton, Geoffrey Warnock, Mary Warnock, Paul Grice, Elizabeth Anscombe, Philippa Foot, Sybil Wolfram, Stuart Hampshire, among others, as well as lectures by visiting philosophers, including Karl Popper, Carl Hempel, Hilary Putnam, Georg Kreisel, John Mackie, and John Wisdom, some of whom kindly gave me some of their time. (This list may be extended, as memories return.)
I did not find that the quality of what I learnt from well known philosophers was always correlated with their reputation.
I owe a great deal to many philosophy graduate students who kindly spent time arguing with and educating me, several of whom later achieved great distinction, including Ian Hacking whom I met first at a Philosophy graduate conference, and John and Dagmar Searle, whom I met on a skiing holiday arranged for Oxford students, and in seminars. They even kindly gave me a tutorial.
After officially converting to philosophy, I read enough of Kant's Critique of Pure Reason to be convinced that he understood better than most contemporary philosophers what mathematical discovery was, and how it provided knowledge that was different from analytical truths, which could be established purely on the basis of logic and definitions, and empirical truths, that could only be discovered on the basis of observation and experiment and were liable to refutation by some newly observed phenomenon. So I set out to explain why he was right to describe mathematical knowledge as synthetic, not analytic, non-empirical, and non-contingent (i.e. mathematical truths are necessary truths, and mathematical falsehoods are necessarily false, e.g. "There is a largest prime number").
The thesis was finally submitted in 1962, and accepted. The examiners were Elizabeth Anscombe and Geoffrey Warnock. In those days Oxford University was too arrogant to involve external examiners. Theses were either accepted or rejected: only later did the practice of acceptance subject to revision evolve. I am sure my thesis would have benefitted from that!
The defense of Kant built on Frege's distinction between Sense (Sinn) and Reference (Bedeutung) but showed how those concepts needed to be refined and used with great care.
At the time, although I knew about Turing machines I had had no experience with computers or programming, and did not realise how important programming ideas and developments in Artificial Intelligence were going to be for philosophy (unfortunately still largely ignored by teachers of philosophy, except for those who believe or discuss greatly exaggerated claims about what machines will soon be able to do). Without realising it, I was anticipating developments in AI by systematically interpreting Frege's notion of Sense in terms of semantic procedures, e.g. procedures for identifying referents or for establishing truth values. However, since those procedures typically required interrogating some part of the world they were not like Frege's functions, which simply associate an argument (input) with a value (output). The semantic procedures associate an argument and some portion of the world with the value. The value of "the tallest person in my office" at a particular time might be a particular individual, but if the situation in the office had been different it could have been a different individual.
So unlike Frege's functions (modelled on functions in mathematics) these semantic procedures have to ask the world some questions in order to determine a value, and I therefore called them "rogators" (from the Latin for "ask"). These rogators produce results that depend on contents of limited portions of the world, so this idea is very different from the idea of "possible world semantics" which became popular later. (It seems that Barbara Vetter has developed some closely related ideas independently: e.g. Barbara Vetter, 2013, 'Can' without possible worlds: semantics for anti-Humeans, Imprint Philosophers 13, 16, Aug, 2013.)
This defence of Kant required a theory of compositional semantics that allowed the semantic content of complex linguistic structures to depend not only on the semantic contents of parts and the syntactical relationships used, but also on relevant parts of the world, the parts to which the associated procedures had to be applied in order to determine referents and truth values.
Another unwitting anticipation of a theme in AI is the emphasis on what would now be called meta-cognitive abilities: the ability not only to apply procedures, but also to reflect on the process of application and in some cases discover that the result of applying the procedure can be determined without applying it. This generalises to abilities to reflect on aspects of perception, learning, and reasoning and notice structural relationships that could go unnoticed in machines or organisms without the required meta-cognitive architecture. There has been work on meta-cognition in AI (including workshops and publications), but I think there is not yet a well understood and implemented specification for a meta-cognitive architecture capable of making the kinds of mathematical discovery that Kant drew attention to, for example, the ability to discover not merely what the result of applying a certain procedure is but also that that procedure cannot produce any other result no matter what portion of the world it is applied to. A Kantian example in the thesis is that no three planar surfaces can bound a finite region of space.
In some cases the discovered impossibility arises out of aspects of the logical/mathematical structure of the procedure. In other cases it is because of the structure of the portion or aspect of the world to which it is applied. Related meta-cognitive processes could lead to discoveries about what alternative results a procedure could produce if some feature of a situation were varied. E.g. "A is outside B" happens to be true, but if might be false if the location of A or B or both were changed. However, if A is incompressible and is much larger than B then it can be concluded that if neither A nor B changes shape or size then no rearrangement could make "A is outside B" false.
From this viewpoint discoveries in logic are a special case of a broader class
of non-empirical, mathematical, discoveries about possible values for various
procedures applied in specified situations. The thesis explored a subset of
examples but later work covers a much broader variety, which turned out to be
related to a generalisation of J.J.Gibson's theory of perception of affordances.
Unfortunately research in AI has recently become dominated by the assumption that intelligent agents constantly seek and make use of empirically based statistical regularities: this approach cannot shed light on discovery of non-statistical, structure-based regularities. The AI work that does attempt to model non-empirical (e.g. logical, mathematical) reasoning mostly assumes that that can be done by building machines that are presented with sets of axioms and rules of inference and abilities to determine which formulae are or are not derivable from the axioms by the rules. This ignores the possibility that there might be a prior form of discovery leading to the axioms and rules. This thesis suggests that that could be based on abilities to examine procedures and work out their constraints and powers, meta-cognitive abilities that go far beyond abilities to simply follow pre-specified rules and constraints. The vast majority of computers that can do the latter cannot (yet) do the former, which is not generally recognized as one of the aims of AI. In other words, AI is diminished by not yet being sufficiently Kantian.
The thesis is, in part, an attempt to spell out some of the requirements for satisfactory answers to Kant's questions. There is still work to be done to meet those requirements.
A decade after finishing the thesis I started trying to use ideas from
programming and AI to demonstrate why Kant was right about mathematics, but the
project turned out to be extremely difficult because of the mis-match between
the procedural operations provided by computers and the sorts of procedures
required for biological perceptual and reasoning mechanisms. Part of the problem
that I think is still unsolved is to specify what the requirements for
appropriate biological information processing mechanisms are. Work on that is
now a key part of the Turing-inspired Meta-Morphogenesis project, attempting to
understand the diversity of biological information processing mechanisms and how
they evolved. It seems that evolution itself "discovered" and made productive
use of many important mathematical aspects of the world, long before there were
any animals that were able to extend those mathematical discoveries.
This web page was originally loosely based on the corresponding ORA web page, from which the originally scanned chapters were downloaded. These pdf files contained only images of the pages, with no machine readable textual information. From time to time, this web site was expanded and/or reorganized, first in order to provide more information about the thesis, and, later on to provide transcribed versions of the PDF files. All chapters of the thesis have now been transcribed, along with bibliography and appendices. So the text is searchable and can be copied and pasted, e.g. into notes.
26 May 2016
Updated and reorganised, yet again.
9 May 2016
Added links to subsequent related publications, and historical note.
I previously wrote: "If anyone is able to automate the conversion to text of the remaining chapters I shall be very grateful!". As of April 2016, this has been done by manual typing, by an agency in India. Links to the newly provided PDF files are included for Chapters 3 to 8, Appendices and bibliography, alongside links to the original scanned images (also PDF).
There are still likely to be transcription errors. Please email me if you find any.
At a later stage, a single document combining all the new text will be available, possibly in PDF and HTML formats.
Updated 7 May 2016:
Found and fixed some problems in derived .TXT files.
Also re-formatted this page, to make things clearer, I hope.
Updated 5 May 2016:
Added Chapter 8, Appendices, and Bibliography.
Added diagrams to Chapter 7 and corrected more typos.
Updated 3 May 2016:
More corrections in transcribed chapters, namely
Chapters 2, 3, 4, 5 and 6
Updated 28 Apr 2016:
I had previously produced text versions of the Abstract, Preface, Table of contents and Chapter 1, from which searchable PDF had been derived. Now (April 2016) thanks to much help from a former PhD student, Luc Beaudoin (http://cogzest.com/), the chapters that had not previously been transcribed were re-typed by Hitech because OCR technology is not yet able to cope with the very fuzzy scanned carbon copies of the original thesis. Luc Beaudoin also helped with subsequent proof-reading, and a host of minor problems (still in progress).
For more convenient printing, the text files were then converted to searchable PDF.
Updated 11 Feb 2014:
Split part 2 (preface, contents and chapter 1) into 2a (preface+contents) and 2b (Ch 1) (above).
Updated 10 Feb 2014:
Used OCR (tesseract) to create searchable, selectable plain text versions
of the Abstract, preface, table of contents and Chapter 1 (above).
Updated 9 Feb 2014:
Fixed link to Oxford Research Archive. (They change from time to time, unfortunately.)
Updated 8 Jan 2008:
Added more information about the contents of Appendices III and IV,
including links to some of my more recent work on those topics.
Updated 10 Jun 2007:
Slight reformatting, and added full table of contents copied from the PDF version.