How Russell's Problems of Philosophy solved the impasse between Rationalism and Empiricism and made Logic the Essence of Philosophy. Gregory Landini gregory-landini@uiowa.edu
The Principia Era There is an important era in which Russell s ideas in logic and philosophy were coming together to form a new research program for a scientific philosophy. I call this the Principia Era (1910-1917) and the fundamental books works in it are: Principia Mathematica (vol. 1 1910, vol 2 1912, vol 3 1913 The Problems of Philosophy (1911/12) The Theory of Knowledge (1912/13) Our Knowledge of the External World (1914) The fundamental articles are: On the nature of Cause Knowledge by Acquaintance and Knowledge by Description The Ultimate Constituents of Matter On Scientific Method in Philosophy The relation of sense-data to physics The Philosophy of Logical Atomism
Mathematics and the Metaphysicians 1901 One of the chief triumphs of modern mathematics consists in having discovered what mathematics really is
In these passages, and many others, we see that Russell maintains that the work of Cantor has revolutionized mathematics, and with the work of Weierstrass and Pieri etc.) order is what mathematics is all about. Order is what the Algebra of Relatives is all about. This fact is the key to understanding Russell s logicism. This algebra, Russell came quickly to see, is not quantification theory with relations (of first or higher level). It requires the comprehension of ever new relations.
What is Logic, in Russell s view? Logic is an abstract science of all of the kinds of structures that there are. Logic studies all structures, whether exemplified or not, by studying the ways relations order their fields. Logic involves comprehension. I call it cp Logic for it embraces/or emulates the comprehension of ever new simple-types of attributes in intension. cp Logic n is simple type theory of n-many types of attributes. cp Logic 2 is standard second order logic. Logic is not is it concerned with settling the question of what norms for proper human reasoning ought to be adopted. Logic is not built from logical particles or, and not and quantifiers. Logic is not the study of semantic consequence. The debates between intuitionistic, relevance, free, classical logics and the like are not debates about logic they are debates about what structures best guide reasoning with respect to a specific goal.
If cp Logic is the study of all the kinds of structures realized by relations, then why isn t chemistry and physics part of logic? Answer on behalf of Russell: Physics and chemistry studies what happens when particular kinds of properties and relations are exemplified in the events that compose nature. Events interact with other events and create regularities and patterns, but these interactions are not due to the ordering features of the structure alone. They are due to the interactions of the events in which they are exemplified.
Every structure can be studied (if only tediously) within simple type theory. Simple Type theoretical Logicism is the foundation of mathematics. - Principa Mathematica (PM) showed how the order-type structures can be captured (albeit tediously) within simple type theory of attributes in extension. - e.g., the progression of finite von-neuman ordinals violates simple-type theory. It has:, { }, { { }}, {, { }, { { }},.. Sx = x {x} but the general theory of the structure of a progression can be studied in simple-type theory. -e.g., PM showed that the structure of the order-types in general (finite and infinite ordinals) can be captured in simple type theory.
The point, is not to constrain the investigation of structures by requiring they be given simple type representations. Let mathematicians have their intuitions and practices. The point is that every mathematical study of structure has its foundations in intuitions of relational structures. All such intuitions are disguised forms of purely logical intuitions concerning relational structures. And this is always capturable within simple types. Our claim entails that where a given set theory departs from the study of structures (representable in simple types) it departs from mathematics itself. Obviously, on this view, set theory is not assumed to be part of mathematics.
what is Russell s Logicism? Russell-Logicism =df mathematical truths are cp Logic truths. Russell s embraces: A) Cantor s revolution in mathematics and B) Frege s revolution in logic. A1) Cantor was correct that cardinality consists in relational properties of similarity (i.e., one-to-one correspondence) A2) Cantor was correct that ordinality consists in relational properties of relational similarity (and well-ordering). B1) Frege was correct that impredicative comprehension is the foundation of a new informative Logic that is adequate to study all the kinds of relational structures (exemplified or otherwise) that there are. Russell s Logicism entails: 1) Mathematical necessity is logical necessity. 2) Mathematicians in practicing mathematics are studying relational structures. 3) Every relational structure can be studied in the cp Logic n of relations.
In the Problems of Philosophy (1912), Russell offered a non-kantian solution of the impasse between Rationalism and Empiricism over synthetic a priori truth. Acquaintance (a kind of experience ) with universals (and logical objects) provides the foundation of the synthetic a priori truths in mathematical logic, induction/probability, and even ethics. Russell s Theory of Knowledge (TK) book project of 1913 was to carry out this solution in detail by showing that acquaintance with logical objects and universals is the source of our knowledge of mathematical logic.
Russell s PRoblems of Philosophy Rationalist Empiricist
The 1913 TK offers a philosophy of mind based on Acquaintance with universals (properties and relations). The ontology is one of facts and universals. TK takes acquaintance as primitive and unanalyzable. As in Problems, truth is defined in terms of correspondence between judgment/belief facts and other facts. A belief consists in a fact of a subject in a belief relation which has multiple relata. For example, when Othello believes that Desdemona loves Cassio, there is a belief fact consisting of O related severally by belief to d and loves and c. This theory of synthetic a priori knowledge was abandoned in 1917.
In 1918, Russell embarked on a radically new philosophy of mind he called neutral monism. Its natural ally is Behaviorism and Russell embraced many behaviorist doctrines during its heyday in the 1920 s. Acquaintance is abandoned. Russell makes knowing an ability to appropriately to environmental stimuli. Russell s neutral monism maintains that physical events (transient particulars) are NEUTRAL between mind (continuants perduring in time obeying largely behaviorist laws) and matter (continuants perduring in time obeying largely the laws of the new relativistic physics.
When we understand a mathematical expression, that means that we can react to it in an appropriate manner When we come to algebra, and have to operate with x and y, there is a natural desire to know what x and y really are. To understand even the simplest formula in algebra, say (x + y) 2 = x 2 + 2xy + y 2, is to be able to react to two sets of symbols in virtue of the form which they express, and to perceive that the form is the same in both cases. This is a very elaborate business, and it is no wonder that boys and girls find algebra a bugbear. But there is no novelty in principle after the first elementary perceptions of form. And perceptions of form consist merely in reacting alike to stimuli which are alike in form Russell, Outline of Philosophy, p. 86)
This book, Analysis of Mind (1921), and especially and his book Outline of Philosophy (whose title in America was simply Philosophy ) clearly shows that Principia Era had ended. It is a book to be placed in contrast with Russell s Problems of Philosophy
Our focus is on the Principia Era and in particular on Problems of Philosophy. I shall offer quotes supporting the following as theses held by Russell during the Principia Era: Logic is synthetic a priori Philosophy is properly the science of the foundations of necessity. Ontology is understood as the study of what entities are logically necessary. Logical necessity is the only necessity. The ontology of logic is the only ontology. All synthetic a priori knowledge is knowledge of logic. There is synthetic a priori knowledge of probability (law of induction), mathematics, logic and perhaps ethics.
Russell, The Principles of Mathematics 1903
Russell, Necessity and Possibility (unpublished 1905/6)
As we see, Russell admits kinds of necessity, but only in the sense of degrees of universal generality, culminating in a full universally closed wffs of the language of logic. This pristine view of logical necessity was couched in Russell s logic of propositions (his 1905 substitutional theory ). Every fully general and true wff in the language of the substitutional theory is logically necessary (logically true). Unfortunately, the substitutional theory had formal difficulties with Cantor s power-theorem. It was abandoned in 1908 in favor of the formal language of Principia Mathematica. Nonetheless, Russell retains his view of degrees of necessity (i.e., degrees of generality) as restricted to first-order wffs of the language of quantification theory.
1918
Examples #1 (x is a unicorn) is possible with respect to x ( x)(x is a unicorn) #2 It is logially possible that ( x)(x is a unicorn). If unicorn were a primitive notion we d have: ( )( x)( x) But x is a unicorn perhaps means x is a horse with a horn projecting from its forehead i.e., ( y,z)( Hy & Fzx & Pyz). So we have: (, R, S)( x,y,z)( y & Rzx & Syz). The matter might become quite complicated. #3 It is biologically possible that ( x)(x is a unicorn). (, R, S)(, R, and S are biological attributes & ( x,y,z)( y & Rzx & Syz)). #1 Possibly Pegasus exists possibly E!(ixPx) ( )(E!( x x)). true #2 Necessarily Socrates = Socrates Necessarily (E!( xsx) [ xsx][x=x]) ( )(E!( x x) [ x x][x=x]). True Of course, in modern modal logic the connection between necessity and universal quantification (and between possibility and existential quantification) is captrued in a different way.
We now have found that for Russell: Necessity comes in degrees of universal generality culiminating in logical necessity = logical truth analytic = logically true Now surely Russell accepts the traditional view that p is known a priori if and only if p is necessary. But we shall see that Russell does not hold that p is known a priori p is logically true That is because there are degrees of necessity i.e., degrees of generality. However, Russell rejects metaphysical, mathematical, causal, etc., necessity.
Problems (chapter VII, p. 77) All pure mathematics is a priori, like logic. The fact is that, in simple mathematical judgements such as 'two and two are four', and also in many judgements of logic, we can know the general proposition without inferring it from instances, (p. 79). Problems (Chapter VIII, p. 82) Kant undoubtedly deserves credit for two things: first, for having perceived that we have a priori knowledge which is not purely 'analytic', i.e. such that the opposite would be self-contradictory, and secondly, for having made evident the philosophical importance of the theory of knowledge.
Problems (Chapter VII, p. 81) We have now seen that there are propositions known a priori, and that among them are the propositions of logic and pure mathematics, as well as the fundamental propositions of ethics. The question which must next occupy us is this: How is it possible that there should be such knowledge? And more particularly, how can there be knowledge of general propositions in cases where we have not examined all the instances, and indeed never can examine them all, because their number is infinite? These questions, which were first brought prominently forward by the German philosopher Kant (1724-1804), are very difficult, and historically very important.
Problems Chapter VIII, p. 84) The question which Kant put at the beginning of his philosophy, namely 'How is pure mathematics possible?' is an interesting and difficult one, to which every philosophy which is not purely sceptical must find some answer. The answer of the pure empiricists, that our mathematical knowledge is derived by induction from particular instances, we have already seen to be inadequate,
(Problems VII, p. 73) One of the great historic controversies in philosophy is the controversy between the two schools called respectively 'empiricists' and 'rationalists'. The empiricists, who are best represented by the British philosophers, Locke, Berkeley, and Hume, maintained that all our knowledge is derived from experience; the rationalists, who are represented by the Continental philosophers of the seventeenth century, especially Descartes and Leibniz, maintained that, in addition to what we know by experience, there are certain 'innate ideas' and 'innate principles', which we know independently of experience. It has now become possible to decide with some confidence as to the truth or falsehood of these opposing schools. It must be admitted, for the reasons already stated, that logical principles are known to us, and cannot be themselves proved by experience, since all proof presupposes them. In this, therefore, which was the most important point of the controversy, the rationalists were in the right.
On the other hand, even that part of our knowledge which is logically independent of experience (in the sense that experience cannot prove it) is yet elicited and caused by experience. It is on occasion of particular experiences that we become aware of the general laws which their connexions exemplify. It would certainly be absurd to suppose that there are innate principles in the sense that babies are born with a knowledge of everything which men know and which cannot be deduced from what is experienced. For this reason, the word 'innate' would not now be employed to describe our knowledge of logical principles. The phrase 'a priori' is less objectionable, and is more usual in modern writers. Thus, while admitting that all knowledge is elicited and caused by experience, we shall nevertheless hold that some knowledge is a priori, in the sense that the experience which makes us think of it does not suffice to prove it, but merely so directs our attention that we see its truth without requiring any proof from experience.
Problems (Chapter X, p. 103) Returning now to the problem of a priori knowledge, which we left unsolved when we began the consideration of universals, we find ourselves in a position to deal with it in a much more satisfactory manner than was possible before. All a priori knowledge deals exclusively with the relations of universals. This proposition is of great importance, and goes a long way towards solving our previous difficulties concerning a priori knowledge. Thus the statement 'two and two are four' deals exclusively with universals, and therefore may be known by anybody who is acquainted with the universals concerned and can perceive the relation between them which the statement asserts. It must be taken as a fact, discovered by reflecting upon our knowledge, that we have the power of sometimes perceiving such relations between universals, and therefore of sometimes knowing general a priori propositions such as those of arithmetic and logic. (p. 105).
Problems (Chapter X, p. 107) It will serve to make the point clearer if we contrast our genuine a priori judgement with an empirical generalization, such as 'all men are mortals'. Here as before, we can understand what the proposition means as soon as we understand the universals involved, namely man and mortal. Thus the difference between an a priori general proposition and an empirical generalization does not come in the meaning of the proposition; it comes in the nature of the evidence for it.
In the empirical case, the evidence consists in the particular instances. It is true that if physiology can prove, assuming the general laws that govern living bodies, that no living organism can last for ever, that gives a connexion between man and mortality which would enable us to assert our proposition without appealing to the special evidence of men dying. But that only means that our generalization has been subsumed under a wider generalization, for which the evidence is still of the same kind, though more extensive. The progress of science is constantly producing such subsumptions, and therefore giving a constantly wider inductive basis for scientific generalizations. But although this gives a greater degree of certainty, it does not give a different kind: the ultimate ground remains inductive, i.e. derived from instances, and not an a priori connexion of universals such as we in logic and arithmetic.
This possibility, of knowledge of general propositions of which no instance can be given, is often denied, because it is not perceived that the knowledge of such propositions only requires a knowledge of the relations of universals, and does not require any knowledge of instances of the universals in question (p. 108) We may now take a survey of the sources of our knowledge, as they have appeared in the course of our analysis. We have first to distinguish knowledge of things and knowledge of truths. In each there are two kinds, one immediate and one derivative. Our immediate knowledge of things, which we called acquaintance, consists of two sorts, according as the things known are particulars or universals. Among particulars, we have acquaintance with sense-data and (probably) with ourselves. Among universals, there seems to be no principle by which we can decide which can be known by acquaintance, but it is clear that among those that can be so known are sensible qualities, relations of space and time, similarity, and certain abstract logical universals.
According to Problems, the Principle of Induction is a logical truth, known a priori. Problems (Chapter VI, On Induction, p. 67) We may therefore repeat the two parts of our principle as regards the general law, thus: (a) The greater the number of cases in which a thing of the sort A has been found associated with a thing of the sort B, the more probable it is (if no cases of failure of association are known) that A is always associated with B; (b) Under the same circumstances, a sufficient number of cases of the association of A with B will make it nearly certain that A is always associated with B, and will make this general law approach certainty without limit.
Problems (Chapter XIV, p. 149) Principles such as the law of gravitation are proved, or rather are rendered highly probable, by a combination of experience with some wholly a priori principle, such as the principle of induction. Thus our intuitive knowledge, which is the source of all our other knowledge of truths, is of two sorts: pure empirical knowledge, which tells us of the existence and some of the properties of particular things with which we are acquainted, and pure a priori knowledge, which gives us connexions between universals, and enables us to draw inferences from the particular facts given in empirical knowledge. Our derivative knowledge always depends upon some pure a priori knowledge and usually also depends upon some pure empirical knowledge.
Problems (chapter VII, p, 73) In addition to the logical principles which enable us to prove from a given premiss that something is certainly true, there are other logical principles which enable us to prove, from a given premiss, that there is a greater or less probability that something is true. An example of such principles, perhaps the most important example is the inductive principle, which we considered in the preceding chapter. Problems (chapter XI, p. 117) The inductive principle has less self-evidence than some of the other principles of logic, such as 'what follows from a true premiss must be true'. Problems (Chapter XI ) we shall be driven back to the inductive principle, which we discussed in Chapter VI. But beyond that, there seems to be no further regress. The principle itself is constantly used in our reasoning, sometimes consciously, sometimes unconsciously; but there is no reasoning which, starting from some simpler self-evident principle, leads us to the principle of induction as its conclusion. And the same holds for other logical principles. Their truth is evident to us, and we employ them in constructing demonstrations; but they themselves, or at least some of them, are incapable of demonstration.
A question arises concerning a priori knowledge in Ethics. Is such knowledge, like the principle of induction, logical knowledge? Problems (Chapter X, p. 109) Our immediate knowledge of truths may be called intuitive knowledge, and the truths so known may be called self-evident truths. Among such truths are included those which merely state what is given in sense, and also certain abstract logical and arithmetical principles, and (though with less certainty) some ethical propositions. Problems (Chapter XI, p.112) It would seem, also, though this is more disputable, that there are some self-evident ethical principles, such as 'we ought to pursue what is good'.
Problems (chapter VI, p. 75): A priori knowledge is not all of the logical kind we have been hitherto considering. Perhaps the most important example of non-logical a priori knowledge is knowledge as to ethical value. I am not speaking of judgements as to what is useful or as to what is virtuous, for such judgements do require empirical premisses; I am speaking of judgements as to the intrinsic desirability of things. If something is useful, it must be useful because it secures some end; the end must, if we have gone far enough, be valuable on its own account, and not merely because it is useful for some further end. Thus all judgements as to what is useful depend upon judgements as to what has value on its own account.
We judge, for example, that happiness is more desirable than misery, knowledge than ignorance, goodwill than hatred, and so on. Such judgements must, in part at least, be immediate and a priori. Like our previous a priori judgements, they may be elicited by experience, and indeed they must be; for it seems not possible to judge whether anything is intrinsically valuable unless we have experienced something of the same kind. But it is fairly obvious that they cannot be proved by experience; for the fact that a thing exists or does not exist cannot prove either that it is good that it should exist or that it is bad. The pursuit of this subject belongs to ethics, where the impossibility of deducing what ought to be from what is has to be established. In the present connexion, it is only important to realize that knowledge as to what is intrinsically of value is a priori in the same sense in which logic is a priori, namely in the sense that the truth of such knowledge can be neither proved nor disproved by experience.
According to Russell, ethical knowledge such as we ought to pursue the good is (perhaps) a case of a priori knowledge. But there is a serious difficulty is singling out Ethics as a priori and yet not, as was the case of the principle of induction, not a matter of logic. The foundation (source) of all a priori knowledge is perceiving relations between universals with which we are acquainted. This source of a priori knowledge, Russell tells us, is the same in logic and mathematics as in ethics. But what is it to perceive relations between universals? The relations perceived must not be contingent features of the way the universals happen to be instantiated. They must be relations that hold independently of whether the universals are instantiated. In short, the relations between the universals must be necessary! If all a priori knowledge derives from the same source as our a priori knowledge of mathematics, then the necessity of these relations between universals is the same as necessity of logic itself.
What is Philosophy? Problems (Chapter XIV, p. 149) Philosophical knowledge, if what has been said above is true, does not differ essentially from scientific knowledge; there is no special source of wisdom which is open to philosophy but not to science, and the results obtained by philosophy are not radically different from those obtained from science. The essential characteristic of philosophy, which makes it a study distinct from science, is criticism. It examines critically the principles employed in science and in daily life; it searches out any inconsistencies there may be in these principles, and it only accepts them when, as the result of a critical inquiry, no reason for rejecting them has appeared.
What is Philosophy? Our Knowledge of the External World Lecture II Logic as the Essence of Philosophy The topics we discussed in our first lecture [Current Tendencies], and the topics we shall discuss later, all reduce themselves, in so far as they are genuinely philosophical to logic. This is not due to any accident, but to the fact that every philosophical problem, when it is subjected to the necessary analysis and purification, is found either not to be really philosophical at all, or else to be, in the sense in which we are using the word, logical.
Our Knowledge p. 28. The difference between a good word and a bad one is a difference in the particular characteristics of the particular things that exist in these worlds: it is not a sufficiently abstract difference to come within the province of philosophy. Love and hate, for example, are ethical opposites, but to philosophy the are closely analogous attitudes toward objects. The general form and structure of those attitudes towards objects which constitute mental phenomena is a problem for philosophy; but the difference between love and hate is not a difference of form or structure, and therefore belongs rather to the special science of psychology than to philosophy.
The above conclusion, of which we had an instance in the case of the inductive principle, is important since it affords a refutation of the older empiricists.. We must therefore admit that there is general knowledge, not derived from sense, and that some of this knowledge is not obtained by inference but is primitive. Such general knowledge is to be found in logic. Whether there is any such knowledge not derived from logic, I do not know. Logic, we may say, consists of two parts. The first part investigates what propositions there are what forms they may have. The second part consists of certain supremely general propositions, which assert that truth of all propositions of certin forms. merges into pure mathematics, whose propositions all turn out, on analysis, to be such general formal truths.
In Problems, Russell s conception of the nature of philosophy as a science as still being developed. We see that he has not yet reached his thesis, set out in his book Our Knowledge of the External World that logic is the essence of philosophy. Problems allows a priori knowledge grounded by perceiving necessary connections between universals. But there are no kinds of necessity. There are only degrees of generality culminating in the full generality that defines logical necessity. Russell came to see that logical necessity is the only kind of necessity and thus all a priori knowledge is knowledge of logically necessary relations between universals of which we are acquainted. Philosophy, is then the science that endeavors to provide the ontological ground of necessity, to discover what is, or what is not necessary, and to determine whether there are kinds of necessity (mathematical, causal/physical, apart from logical necessity.
Ontology is not, as Quine would have it, the study of what there is. Ontology, in Russell s view is the study of the ground of necessity. In Russell s view, Philosophy is the study of the ontology of necessity and logic is its essence. Why is logic its essence? Because logical necessity is the only necessity. Hence, the only ontology is the ontology of logical necessity. The task of the philosopher is to separate the logical/mathematical notions from the confused hybrids notions involved in (physical, chemical, biological, causal, psychology) that parade as if necessities. But we may feel that Russell s view is to narrow. So I would like to advocate a sympathetic extension: Philosophy =df the study of necessity. Whoever engages in arguing for (or against) a kind of necessity (logical, mathematical, physical, causal, biological, ethical, psychological, etc) is ipso facto a philosopher.
Philosophy is not a practice of criticism or critical thinking. Philosophy is not conceptual analysis. Philosophy has nothing to do with religion (but for the fact that religions make unsubstantiated claims of what is necessary and it is the job of philosophy to evaluate them). Who are the Philosophers? The philosophers include the well-known Plato, Aristotle, Descartes, Locke, Leibniz and Hume, Berkeley and Kant and Frege and Russell and Quine. But they also include many not currently recognized in the hall of fame, such as Newton, Huygens, Maxwell, Einstein, Heisenberg, Pasteur, and Darwin. (Darwin rejected essentialist dogmas concerning species; Newton, maintained that absolute space and univeral gravitation is physically necessary ; Einstein held that the invariance of the propagation of light in a vacuum is physicially necessary). We shall not hide from the implications of this definition though it is bound to stir controversy. Our definition may well exclude some professional philosophers both past and present. If you are not working on the foundations of some form of necessity, you are not doing philosophy though you may well be doing something very important in some other field.
Russell s view that logical necessity is the only necessity is a most liberating view. It is liberating because is frees us from the dogmatisms of culture, politics, religion and the like, which attempt to legitimate themselves by parading as necessities (essentialisms). If logical necessity is the only necessity, we are empowered to rather easily rid ourselves of these dogmatisms. They are not necessities because they are not truths about structures grounded in the way relations, whether exemplified or not, order their fields It is in this way that philosophy, as Russell so wonderfully put it in the Problems of Philosophy, liberates the mind from the prison produced by the metaphysical dogmatisms fashioned after self-interest. Let me end with some quotes:
Problems (Chapter The Value of Philosophy XV, p. 157) Apart from its utility in showing unsuspected possibilities, philosophy has a value, perhaps its chief value, through the greatness of the objects which it contemplates, and the freedom from narrow and personal aims resulting from this contemplation. The life of the instinctive man is shut up within the circle of his private interests In one way or another, if our life is to be great and free, we must escape this prison and this strife. One way of escape is by philosophic contemplation. Philosophic contemplation does not, in its widest survey, divide the universe into two hostile camps, friends and foes, helpful and hostile, good and bad, it views the whole impartially. Philosophic contemplation, when it is unalloyed, does not aim at proving that the rest of the universe is akin to man.
For this reason greatness of soul is not fostered by those philosophies which assimilate the universe to Man. By thus making a barrier between subject and object, such personal and private things become a prison to the intellect. The free intellect will see as God might see, without a here and now, without hopes and fears, without the trammels of customary beliefs and traditional prejudices, calmly, dispassionately, in the sole and exclusive desire of knowledge, knowledge as impersonal, as purely contemplative, as it is possible for man to attain.