3. Fundamental Dichotomies

[In this chapter, as in all chapters which are primarily historical, it is particularly important not to slip into too purely chronological an account. All the life is in the particular themes which the narrative is intended to illuminate, and it is the development of these themes which must be at all times in the foreground. I do not know how to achieve this. Part of the difficulty in producing the story is that the detailed content of these themes has to be well understood in order to find a good way of presenting them, but one feels that the required level of understanding can only be extracted from the history. ]

We saw in the last chapter how the most recent successor to Leibniz's project, in the philosophy of Rudolf Carnap, was derailed by the rejection of some of the most fundamental tenets of Carnap's philosophy.

It is conceivable that the project could be revived on a different basis, for neither Quine nor Kripke, despite their devastating critiques, actually abandoned the idea of deductive reason or the use of formal deductive systems. If I was myself convinced of the soundness of those criticisms then I might follow that course, but I am not. It is best instead to answer them.

In this chapter I therefore focus on the concept of analyticity and its relation to those of necessity and of the a priori. Before addressing specifically the criticisms of Carnap's position by Quine and Kripke, I propose to sketch the history of the evolution of these concepts over the past two and a half millennia. I will present this as falling in two principal stages, first the development of predecessors of these fundamental concepts and then the subsequent refinement of our understanding of the concept. The point of transition I suggest, is with Hume, though this is more an expository device than a dogmatic claim. The idea that there exists any such definite point of transition is tenuous, but the supposition helps to give structure to the narrative.

It is with Hume, I suggest, that we find a first account of the dichotomy which is not easily seen to be in some respects defective. This is perhaps as much to do with Hume not seeing himself as defining the distinction, but rather as drawing attention to it and placing it in a central place in his philosophy.

Before entering into the history I will sketch the technical content of the narrative, to provide a framework in which the evolving detail can be placed. This is done using the ideas of Hume.

3.1 Hume's Fork

David Hume was a philosopher of the Scottish Enlightenment. The enlightenment was a period of ascendency in the place of reason in the discussion of human affairs, when science had secured its independence from the authority of church and state and had a new confidence in its powers derived substantially from the successes of Newtonian physics.

Hume looked upon the philosophical writings of his contemporaries and found in them two principal kinds, an ``easy'' kind which appealed to the sentiments of the reader, and a ``hard'' kind which trawled deeper and appealed to reason. This latter kind, ``commonly called'' metaphysics, was preferred by Hume, but found nevertheless, by him, to be lacking, infested with religious fears and prejudices. Hume's feelings about these aspects of philosophy were not vague misgivings. He had a specific epistemological criterion which he saw these philosophical doctrines as violating.

Hume's project involves an enquiry into the nature of human reason for the purpose of eliminating those parts of metaphysics which go beyond the limits of knowledge, and establishing a new metaphysics on a solid foundation limited to those matters which fall within the scope of human understanding.

David Hume wrote his philosophical magnum opus, A Treatise on Human Nature [Hum39] as a young man. He was disappointed to find his work largely ignored and otherwise misunderstood, and thought perhaps that his presentation had been at fault. To improve matters he wrote a much shorter work more tightly focussed on the core messages which he thought of greatest importance, which he called An Enquiry into Human Understanding [Hum48].

In a central place both logically and physically in this more concise account of his philosophy he says:

``ALL the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact.''

We shall see that Hume is here identifying a single dichotomy which corresponds to all three of the distinctions which here concern us. In his next two paragraphs he expands in turn on the kinds he has thus introduced.

3.1.1 Relations of Ideas

``Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypotenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.''

Hume is distinctive here among empiricist philosophers in having a broad conception of the a priori (though he does not use that term here), allowing notable the whole of mathematics. In this he may be contrasted for example with Locke who allowed only certain rather trivial logical truths to be knowable a priori. Nevertheless, Hume's conception of the a priori remains narrow by comparison with the rationalists, and in particular, as Hume will later emphasize, excludes metaphysics.

3.1.2 Matters of Fact

``Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind.''

The evolution of following three dichotomies are the subject matter, though we will find other related dichotomies which feature in the history.

The terms which I will use to speak of them, in this chapter are:

As I shall use these terms these are divisions of different kinds of entity, by different means.

The first is a division of sentences, understood in sufficient context to have a definite meaning, and is a division dependent upon that meaning.

The second is a division of propositions, which may be understood for present purposes as meanings of sentences in context. The division is made according to whether the proposition expressed must under all circumstances have the same truth value, or whether its truth value varies according to circumstance. In this we are concerned with two particular notions of necessity, those of logical and of metaphysical necessity, the latter being sometimes taken to be broader than the former. A part of the role of Hume's fork in positivist philosophy is to banish metaphysical necessity insofar as this goes beyond logical necessity.

The third is for our purposes also a division of propositions, on a different basis. It concerns the status of claims or of supposed knowledge of propositions. It is expected that such a claim must in some way be justified if we are to accept it, and that the kind of justification required depends upon the proposition to be justified. The justification is a priori if it makes no reference to observations about the state of the world.

3.1.3 The Place of The Fork in Hume's Philosophy

The mere statement of the fork (which we shall see, is not original in Hume) is of lesser significance than the role which it plays in Hume's philosophy, which serves to clarify the distinction at stake and draw out its significance.

Hume's philosophy, like Descartes' comes in two parts of which the first is sceptical in character, and the second constructive. In both cases the sceptical part clears the ground for a new approach to philosophy which is then adopted in the constructive phase.

For our present purposes we are concerned principally with the first sceptical phase of Hume's philosophy, because of the delineation of the scope of deductive reason, and hence of the analytic/synthetic dichotomy which is found in Hume's sceptical arguments. This delineation is baldly stated in Hume's first description of the distinction between ``relations between ideas'' and ``matters of fact'', for there Hume tells us that no matter of fact is demonstrable.

This bald statement would by itself have little persuasive force if it were not followed up with more detail, even though ultimately this detail does not so much underpin the distinction as depend upon it.

Hume's further discussion begins with the consideration of what matters of fact can be known `beyond the present testimony of our senses or the records of our memory'. The inference beyond this immediate data is invariable causal, we infer from the sensory impressions or memories to the supposed causes of those impressions. But these are not logical inferences, causal necessity is for Hume no necessity at all (even less the inference from effect to cause). Hume's central thesis that matters of fact are not demonstrable is in this way reduced first to the logical independence of cause and effect, and then to the distinction between deductive (and hence sound) inference and inductive inference (whereby we infer causal regularities and their consequences).

Given that Hume considers all inferences from senses to be based on induction, and sees no validity in causal inference, it follows that from information provided directly to us by the senses nothing further can be deduced which is not simply a restatement, selection or summary of the information itself. Further enlightenment from this sceptical doctrine is primarily the application of this principle to various kinds of knowledge. In the process Hume does a certain amount of

3.2 A Contemporary Perspective

In tracing the history of the analytic/synthetic and related distinctions I hope to show how their various historical manifestations relate, and to present a perspective from which the various developments may be evaluated perhaps as plain progress, perhaps as advances in certain respects, perhaps as involving or constituting regress.

Such evaluations can only be made from a particular point of view, and in this section that point of view is outlined, and related, in the first instance to Hume's fork.

3.3 Before Hume

The distinctive place of Hume in the history of ``the fork'' which bears his name is not a mark of his priority in making the distinction, but rather of an important point in its development, a clear identification of exactly the right distinction (I now suggest) of the central place which he gave it in his philosophy, and of the evidence he gives (in what is are now regarded as Hume's various sceptical doctrines) of a sound intuitive grasp of the scope and limitations of logical truth and deductive inference.

To underpin the significance of Hume's distinction, and to see that, notwithstanding its apparent simplicity, it is by no means simple to arrive at, I now look at some of the most important stages in the earlier development of the ideas.


The philosophers I will consider here are Socrates, Plato, Aristotle, Descartes, Locke and Leibniz.

Most of the elements which we find in Hume's fork can be found in the earliest of these philosophers.

The relevant ideas which these philosophers discussed include Plato's world of ideals and of appearances, the notions of essential and accidental predication, and of necessary and contingent truth, and Aristotle's logic and the idea of demonstrative truth. Leibniz is important for his conception of the scope of logic and the possibility of arithmetisation and mechanisation. His intended method for these depends upon a conceptual atomism which was to exert significant influence on the philosophy of Russell and of the Early Wittgenstein, and thus indirectly upon Carnap's conception of logical truth.

The distinction between the logical and the empirical influenced two major tendencies in modern philosophy, namely rationalism and empiricism of which Leibniz represented the first and Hume the second. The distinctive feature of these tendencies has been respectively the overestimation and underestimation of the scope of deduction, championed by metaphysically and scientifically inclined philosophers (or philosophically inclined scientists). The dialogue between these has therefore served to refine the distinction between the logical and the empirical. These two tendencies may be seen to have been anticipated by Plato and Aristotle, of whom Plato seems like the rationalist, and Aristotle is closer to empiricism. Curiously the connections with Leibniz and Hume are crossed over, it is Plato the rationalist who seems to provide the clearer anticipation of Hume's fork, and it is Aristotle the less rationalistic of the two who provides the logic on which Leibniz's ideas were built.

Hume's principle description of his fork is in terms of subject matter, the distinction between knowledge of ideas and knowledge of the world. In Plato we see something quite similar, he distinguishes between the eternally stable world of Platonic ideals, of which we can have certain knowledge obtained by reason, and a world of appearances, in constant flux, and of whose fleeting nature we can at best have tentative opion based on the unreliable testimony of our senses. Not only does the distinction of subject matter match, but the key characteristics of these domains, that in the one we have solid precise, reliable knowledge, and in the other, nothing of the kind, are agreed between them.

For an understanding of the philosophy of Socrates and Plato it may be helpful to consider first the pre-Socratic philosophers.

3.3.1 The Pre-Socratics

The history of the analytic/synthetic dichotomy is connected with that of deductive inference, which is generally held to have begun with Greek mathematics.

It might be argued that the ability to undertake elementary deductions is an essential part of competence in a descriptive language. For to know the meaning of concepts, one must also know at least the more obvious cases of conceptual inclusion. One cannot be said to know the meaning of the word ``mammal'' if one does not know that all mammals are animals, and hence that any general characteristic of animals is possessed by mammals. The ability to draw inferences purely based on an understanding of language does not however entail the capability to discriminate between such deductive inferences and inferences which are based not merely on meanings but also on supposed facts, so no clear grasp of what inferences are deductive need be involved. Furthermore, competence in drawing conclusions may by entirely unwitting, there may be not awareness of the idea of inference.

It is in Ionia3.1in the $6^{th}$ century BC that mathematics was transformed into a theoretical discipline, a key feature of which is the practice of proving general mathematical laws (rather than simply recording them for general use). At this time philosophy, the love of knowledge, encompassed mathematics and science as well as what we might now call philosophy. The use of reason in mathematics was very successful, and Greek mathematics grew into a substantial body of rigorously derived, knowledge of which much is known to us through the thirteen books of the Elements of Euclid (c300 BC). In Euclid's geometry, developed deductively using an explicitly documented deductive axiomatic method, Greek mathematics achieved a standard of rigour which was not to be surpassed for two thousand years.

The success of deduction in mathematics was not reflected in other aspects of the work of the pre-Socratic philosophers. It is distinctive of Greek philosophy, beginning with the Ionians, that religion and myth was not a dominant influence, and that philosophers sought knowledge by rational means, rather than deferring to any kind of authority. Ionian philosophers, often described as cosmologists, sought unifying principles, as a way of understanding the diversity of the world around them. Different philosophers adopted different hypothesis about the ultimate constituents of the universe. Thales held that the world originated from water, Anaximenes from air. Empedocles held that all matter was formed from air, water, earth and fire. By contrast with mathematics, Greek philosophy became a great diversity of irreconcilable doctrines.

The philosophical search for unity and stability underlying apparent diversity and flux yielded no universally accepted result. This is nowhere more conspicuous than in the philosophies of Heraclitus and Paremenides, the first asserting an external flux in which nothing remained stable and the second that nothing changes. The failure of reason to resolve differences in this domain of enquiry was underlined by the arguments of Zeno of Elea, who devised many paradoxical arguments3.2to reduce to absurdity the possibility of change, in support of the philosophy of Parmenides. In this way it is made conspicuous that apparently the same method, reason, works very well for mathematics but fails miserably in metaphysics.

3.3.2 Two Kinds of Stability

We can understand the search for the pre-Socratic philosophers for the unity and stability underlying the diversity of the world by analogy either with modern science or with mathematics.

In the case of science we seek physical laws which govern the changes which take place in the world. The world changes, but the laws are immutable. In modern science the preferred way of formulating scientific laws is using mathematics. The scientific hypothesis is that the relevant aspects of the world correspond more or less exactly with the structure of some mathematical model.

The unchanging fundamental truths may therefore be sought either in empirical scientific laws or in mathematics.

These two alternatives connect with tendencies in modern philosophy, empiricism and rationalism, and to aspects of the two great philosophical systems of classical Greece, those of Plato and Aristotle. The systems of Plato and Aristotle in different ways anticipate Hume's fork, and in the rationalist and empiricist tendencies we can see a dialectic on the scope of deduction through which the distinction evolved towards Hume's formulation of the fork.

3.3.3 Plato's Theory of Ideals

Plato sought to place philosophy on as rigorous a footing as mathematics, and he did this with a synthesis of the ideas of Heraclitus and Parmenides which recognized two distinct domains of rational discourse. Mathematics succeeded because it reasoned deductively about abstract ideas

In Plato Hume's fork is anticipated as a distinction between two worlds, the world of ideal forms and the world of appearances. Thought of in this way the distinction is about subject matter, Distinction

the world of forms and world of appearances Distinction

the a priori (knowledge) and a posteriori (opinion) Distinction

essential and accidental predication Connection

definition and essence

3.3.4 Aristotle's Logic and Metaphysics

3.3.5 Rationalist Philosophy

3.3.6 British Empiricism Before Hume

Hume was one of a line of British philosophers who emphasized the role of experience in the acquisition of knowledge. Important figures in this tradition were Bacon, Hobbes and Locke.

3.3.7 Leibniz

Leibniz conceived of the project to which this book is devoted. We are concerned here with just one aspect of his work, which is his contribution to the ideas leading to Hume's fork.

Through most of the intellectual history right up to the $20^{th}$ century Aristotle is a dominating influence on thinking about all aspects of Hume's fork. Some of the influence of Aristotle is negative, his subject-predicate analysis of propositions and his syllogistic conception of logic were both to narrow and adherence to Aristotle's ideas may have inhibited the development of logic sufficient for our purposes.

Leibniz conceived of the idea that reason might be automated as a young man, and pursued this project for the rest of his life. His insight was specifically about how the Aristotelian syllogism could be automated through arithmetisation, thus anticipating a method which was to become famous in logic when applied by Gödel in his incompleteness results.

Within this context the logical and metaphysical ideas which underpinned and provided technical substance to his project are important.

Like Aristotle, Leibniz did have the concepts of Necessary and Contingent truth, he also had the concepts of a priori and a posteriori truth which were closely connected, as in Hume. There is also a connection with semantics, similar to that in Aristotle, through their role in a priori proof and hence in the establishment of necessary truths. To this is added a precise description of a mechanizable process of analysis whereby necessary truths might be established, which differs greatly in character from any previous work in logic.

Though Leibniz's work in this area (by contrast with his work on the calculus) exerted little influence in his time, but proved an inspiration for some of the leading figures in the development of logic in the $19^{th}$ and $20^{th}$ centuries, including Gottlob Frege and Bertrand Russell. His influence is conspicuous in Russell's philosophy of logical atomism[Rus18] and Wittgenstein's Tractatus Logico-Philosophicus [Wit61].

Leibniz was a rationalist philosopher and his conception of the division which concerns us sometimes sounds radical. He holds for example that, at least for God, all truths are necessary. But he nevertheless does distinguish between necessary and contingent truths and, for us mere mortals the dividing line between these two falls more or less where Hume will put it. Mathematics is necessary, science is contingent. He also closely connects, as will Hume, necessity and a priority.

In relation to the analytic/synthetic distinction, Leibniz still uses these concepts as they are used in classical Greece, to describe two different approaches to logical proof, contemporary usage of ``synthetic'' comes from Kant. Leibniz does have something like a semantic and a proof theoretic characterization of necessary and a priori truth. On the proof theoretic side it is of interest not just that Leibniz considers necessary truths to be those susceptible of a priori proof, thus implicitly distinguishing the method of verification from the manner of discovery. On the semantic side we have an explanation of necessity and of proof via the analysis of definitions.

Leibniz tells us that truth is a matter of conceptual containment, the containment of the concept of the predicate in the concept of the subject. This kind of containment we would now describe as extensional and would not accept as a characterization of logical or necessary truth, but rather of truth in general, including contingent truths. There is another kind of conceptual containment which applies only to necessary truths, and that may variously described as intensional, essential or risking circularity, as necessary or analytic. This is the containment of meanings and descriptions of the role of definitions in the process of proof are consistent with the view that logical truths correspond to containment of meanings rather than containment of extensions.

This distinction does not feature in Leibniz, his distinction between necessary and contingent propositions for us mortals is made in two other ways. The first is by appeal to the omniscience of God and the limitations of man. Men are not capable of the kind of complete comprehension of meaning which God exhibits. Propositions which are contingent for us are those whose analysis is beyond the limits of our knowledge or analytic capability, but will nevertheless, if true, be knowable by analysis for God. The second way of making the distinction is through the principle of ``sufficient reason''. Logical necessities can be shown to be true by reduction to the law of identity of which self-predication is an instance. The procedure is to analyze the predicate and subject by expanding their definitions and discarding components present in the subject but not in the predicate, until the subject and predicate turn out to be identical. The remaining necessities arise from ``the principle of sufficient reason'', which tells us that nothing happens without sufficient reason, and more specifically that the world is the way it is because from the logically possible alternatives available to God, he has chosen the best.

We seem to have here two kinds of regression from the position occupied by Aristotle. We see a reduction in the clarity of the concept of demonstrative truth, by introducing the idea that the definitions of concepts may be beyond our ken, and intelligible only to God. This replaces the distinction between essence and accident, which is closer to the idea that in one case meaning suffices, and in the other observation will prove necessary.

Leibniz's ideas on mechanisation of the syllogism flow from a conceptual atomism. The word ``term'' is used in Aristotelian logic for the subject and predicate of a proposition in subject/predicate form, both of which may be thought of in more modern language as expressing concepts. Leibniz's atomism is the idea that there are simple concepts from which all other concepts are formed in specific ways, together with the retention from Aristotle of the idea that all propositions have subject/predicate form.

In this is it important to distinguish the simple concepts from simple expressions which may be used to express them. Leibniz atomism here is not about syntax, it is about those things which the syntax expresses, the underlying concepts. A complex concept may nevertheless have a simple name, a complex concept is one which is defined in terms of other concepts, a simple concept is one which has no such definition. The atomistic thesis is that there are simple concepts, and that all complex concepts have a definition directly or indirectly in terms of simple concepts.

More specifically, a complex concept can always be analyzed as a finite conjunction of literals (in todays terminology) where a literal is either a simple concept or the negation of a simple concept.

Leibniz saw that if every distinct simple concept were represented by a unique prime number, then a conjunction of concepts could be represented by the product of primes. The fundamental theorem of arithmetic tells us that any such a product can be factorized to retrieve the primes corresponding to the simple constituents. He proposed to represent arbitrary complex concepts by a pair of such products, i.e. a pair of numbers, of which the first number is the product of the primes representing the concepts which occur positively in the definition of the complex concept, and the second number is the product of the primes representing the concepts which occur negatively in the complex. The same concept cannot appear both positively and negatively, or else the definition is inconsistent, so these two numbers will be relatively prime (having no common factors). If Leibniz were correct then once the simple concepts have been identified and given unique prime numbers, all complex concepts are representable as pairs of co-prime numbers.

It is then possible to describe (quite simply) calculations which determine the truth of any proposition of the four forms which Aristotle uses in his syllogistic, (when these propositional schemes, which include variables instead of definite concepts in the subject and predicate places, are instantiated with specific concepts).

3.4 After Hume

Hume's fork abolishes a certain conception of metaphysics, making a difficulty in establishing any other.

What it leaves is something that looks rather like science, falling into two parts. One part contains only matters devoid of empirical content, albeit including the whole of mathematics. The other contains opinions of the empirical world, obtained by guesswork based on sensory impressions.

Not many philosophers, or scientists find this a very attractive picture. Much philosophical reaction to this seeks to refute the general scepticism in Hume's position, but the refutation of his empirical scepticism is more of benefit to science than to philosophy. The significance to philosophy of Hume's fork is more acute in relation to those special kinds of knowledge, beloved particularly to Plato, which are the truest subject matter of philosophy, and were later to be known as metaphysics. From this point of view positivism, first amply exemplified in Hume, is an anti-philosophical philosophy, in which philosophy gives up its own true ground yielding knowledge to science.

It is understandable that many philosophers will react specifically against this central feature of positivism, and in this section we will consider some of these reactions.

These we consider in three aspects.

  1. Firstly there arises in the ongoing dialectic, further refinement of our knowledge of exactly where the fundamental line is drawn.
  2. Secondly there are challenges to and reaffirmation of the idea that a single dichotomy is involved.
  3. Finally, coupled with the previous two there are various ways of reviving and of dismissing the possibility of some kind of metaphysics.

Alongside these philosophical themes there are two technical problems which are gradually addressed. In order for the dichotomies to be definite it is necessary for languages to have definite meaning. One way to achieve this is to define new languages, and for them, give a definite semantics and deductive system. The ability to do this appears on the scene very late, little more than a century ago. The history both before and after that point of transition of ideas about semantics and proof, is part of our present concern.

The clarity which I find in Hume's description of his fork is like a moment of lucidity in life of confusion. Not until Rudolf Carnap do we find a return to and a positive refinement of that distinction.

3.4.1 A Broad Sketch of the Development

It would be easy in a sketch of the subsequent history of Hume's fork to loose the central issues in the detail of the very considerable developments in logic and philosophy since then.

To help make draw these out the headlines are sketched here before a little more detail is supplied.

Kant supplied the first challenge to Hume's conception of a single dichotomy. Kant rescued a limited conception of metaphysics by separating the a priori from analytic truth (which is contrasted with synthetic truth), and declaring that our knowledge of arithmetic, of space and time were synthetic a priori. It is not clear here whether Kant's conception of analyticity differed materially from Hume's relations between ideas, or whether the concept was the same but Kant disagreed about its extension.

3.4.2 Kant

The first challenge we consider came from Kant, who was awoken from his dogmatic slumbers by Hume, and was moved to reject the unity of the triple dichotomy.

Kant was the first philosopher to make prominent use of the notion of analyticity. He has a dual conception, in both following Leibniz in ideas going back to Aristotle. The first is apparently semantic, that an analytic sentence is one in which the subject contains the predicate. This is in Leibniz, but Leibniz's conception of conceptual containment is extensional, and therefore corresponds to truth not analyticity. The other is ``proof theoretic'' (insofar as one can talk of proof theory at this time), a sentence is analytic if it follows from the law of non contradiction. This latter corresponds to Hume's ``intuitively or demonstratively certain'' which in turn is from Aristotle (the ``intuitive'' part corresponds to first principles which must be essential truths in which again the predicate is contained in the subject).

What is distinctive about Kant's philosophy is not his definition of analyticity, which adds nothing to what is found in Hume and Leibniz, but his view on what things fall under the definition. He has reverted to something closer to the earlier empiricist Locke, for whom the corresponding notion is confined to trivia.

Notably Kant excludes two groups of a priori truths from analyticity, those of arithmetic and of geometry.

3.4.3 Bolzano

With Bolzano we see a first approach to the definition of concepts relevant to Hume's fork which advance significantly beyond Aristotle.

The techniques used by Bolzano are a considerable advance, and the conception of logical truth he promulgated is quite close to the idea of logical truth which has dominated mathematical logic ever since. However, this is a narrower concept than the ones we have considered so far, and depends upon the classification of concepts into logical and non-logical.

Though the technical apparatus which Bolzano deploys is a great advance, the concepts he defines with this apparatus take us further away from a notion of logical or analytic truth which is properly complementary to the notion of empirical or synthetic truth.

3.4.4 Frege

Frege's logical and philosophical work was in part aimed at overturning Kant's critique of Hume, in particular Kant's refusal to accept that mathematics is analytic. To achieve this aim Frege invented a new kind of formal logical system of which the two most important features were the abandonment of the Aristotelian emphasis on the subject/predicate form of propositions. Instead of predicates Frege worked more generally with functions, among which predicates are functions whose results are always truth values. Crucially, logical sentences now have arbitrary complexity, and the universal quantifier is introduced to operate over an arbitrarily complex propositional function.

For Frege the notion of logical and analytic truths are separated. Logical truth similar to Bolzano's in being defined through a separation of concepts into logical and non-logical, but analytic truth in Frege represents the most precise formulation of Hume's truths of reason to that date.

In the analysis of Frege's contribution the notion of free logic becomes significant, (though this is not a term he uses). This is so because in his Begriffsschrift or concept notation those aspects of deductive logic which do not involve ontology, or questions of what exits, are satisfactorily addressed. However, when he later comes to apply essentially the same logical system to the development of arithmetic it is necessary to incorporate axioms which suffice to establish an ontology sufficient for mathematics, and at this stage Frege introduces principles which are not logically consistent.

3.4.5 Russell

Russell began his work on mathematical logic rather later than Frege, having been born in 1872, not long before the publication in 1879 of Frege's Begriffsschrift. Furthermore, because Frege's work was ignored, Russell did not become aware of it until he was already well progressed with his own approach to the logicisation of mathematics.

Prior to this work Russell had already studied and written his own account of the work of Leibniz, and Russell's philosophy is significantly influenced by Leibniz. Russell perceived Leibniz as having been handicapped by too closely following the logic or Aristotle, most particularly his retaining the idea that all propositions have subject/predicate form. Russell's logical ideas build on the work of Pierce and Schröder on the relational calculus. The relational calculus provided a more general form for propositions. A sentence might in effect have multiple subjects of which jointly some predicate is asserted (predicates involving multiple subjects are called relations).

The divergence from Aristotle in this matter was independently progressed by Frege and by Pierce and Schröder.

3.4.6 Wittgenstein

While studying engineering at Manchester, Ludwig Wittgenstein became interested in philosophy through an interest in the logical foundations of mathematics. On advice from Frege, Wittgenstein went to Cambridge to study under Russell at a time when Russell was deeply involved in the completion of Principia Mathematica[WR13].

3.4.7 Tarski

3.4.8 Carnap

3.4.9 Quine

Having studied Russell's Principia Mathematica even as an undergraduate,

3.4.10 Kripke

The following table gives a chronology.

Table 3.1: Development of the Analytic/Synthetic distinction
585 BC Thales Mathematics
428-348 BC Plato The Theory of Forms
384-322 BC Aristotle Essential v. Accidental
    Necessary v. Contingent
    Demonstrative v. Dialectical
1632-1704 Locke on trifling propositions
1646-1716 Leibniz  
1711-1776 Hume relations between ideas v. matters of fact
1724-1804 Kant the synthetic, a priori
1781-1848 Bolzano the method of variations
1845-1918 Cantor set theory
1848-1825 Frege Begriffsschrift, sinn and bedeutung
1848-1825 Russell the theory of types
1871-1953 Zermelo the cumulative hierarchy
1889-1951 Wittgenstein logical truths as tautologies
1901-1983 Tarski definition of truth
1891-1970 Carnap logical syntax, the method of intensions and extensions
1908-2000 Quine against the analytic/synthetic distinction and modal logics, holism
1940- Kripke separating analyticity, necessity and a priority via rigid designators

Roger Bishop Jones 2012-09-23