Propositions and Reasoning in Frege and Russell

© Gary Kemp; penultimate draft; final draft in the Pacific Philosophical Quarterly 1998 79/3: 218-35. PROPOSITIONS AND REASONING IN RUSSELL AND FREGE By Gary Kemp For Frege, and for Russell during the crucial years immediately following his disavowal of idealism, the essential subject-matter of logical theory was not language but the propositions expressed by language (I shall use ‘propositions’ for either Fregean thoughts or Russellian propositions). The point has both a metaphysical and an epistemological side. The proposition is the real bearer of truth and falsity, and its existence and identity are independent not only of language but of the mind. Still, it admits of being apprehended, of being ‘grasped’ by the mind. Thus Frege pronounced that knowledge or cognition is ‘embodied in judgement’, which is nothing but the acceptance of a proposition as true.1 Accordingly, to draw an inference - to reason - is to move from judgement to judgement; it is to accept a proposition as true ‘on the basis of’ other propositions accepted as true.2 It follows that however psychologically dependent any actual creatures may be upon language for their cognitive prowess, thought or reasoning is not essentially something linguistically embodied; its essential condition is just the apprehension of, and the taking of attitudes towards, propositions.3 I shall call this thesis that of the Primacy of the Proposition. Frege was ever more explicitly commited to it. Russell held it during the aforementioned period, except that unlike Frege he was inclined to think of the constituents of propositions as being potential items of apprehension independently of the mind’s engagement with them as parts of whole propositions (and even characterized this so-called acquaintance as constituting a form of knowledge - in this case of ‘things’ rather than truths). Thus Russell, despite having hit upon the technique of defining certain sorts of expressions by defining their sentential contexts, did not accept anything like Frege’s context principle, understood as a bit of epistemology.4 But for my purposes that difference is immaterial. So is Russell’s having rejected Frege’s sense-reference distinction, and so is his having ceased, around 1910, to believe in propositions. For he continued to believe in their parts, and allocated to them what was substantially the same role in judgement: whereas he had thought of judgement as a binary relation between the judger and an independently existing and unified proposition, he now thought of it as a many-termed relation between the judger and what on the old scheme were the components of the proposition, which achieve propositional unity, if ever, only if they should peradventure be combined in some actual judgement (unless the judgement is true, in which case they are combined in the corresponding fact). On either view, the constituents stand before the mind in the act of judging, and on neither view does language enter into the conception of what it is to judge or think. My primary aim in what follows is to show that the thesis of the Primacy of the Proposition is not compatible with the formal treatments of inference expounded in Frege’s or 1 © Gary Kemp; penultimate draft; final draft in the Pacific Philosophical Quarterly 1998 79/3: 218-35. Russell’s logical works. A further purpose is to cast doubt upon it more generally. I I begin with Frege. The thesis of the Primacy of the Proposition figured ever more prominently in his life-long crusade for logical rigour in mathematics. As noted, Frege conceives an inference as that sort of judgement which is made on the basis of other propositions already accepted as true. Consider, for example, modus ponens: if I have already accepted the propositions that p and that if p then q, then, without further ado, I may judge that q (on pain of a well-known vicious regress, we cannot suppose that such an inference also requires the premise that q does follow from those premises). Now a perfectly presented, ‘systematic science’ is a true and complete account of its subject which, by explaining their truth, explicitly displays the real epistemic grounds for all its assertions. It thus consists confining ourselves to the a priori sciences - of statements expressing a set of true propositions which collectively embody a certain structure: Every proposition is either an axiom - a proposition neither requiring nor admitting of proof - or a theorem derived from axioms by a gap-free chain of purely logical inferences (since definitions are mere abbreviations which add no content, they can be ignored).5 Frege’s commitment to this ideal can be made vivid by briefly considering his central criticism of Hilbert’s Grundlagen der Geometrie (some terminological anachronism will have to be risked here for the sake of concision).6 Hilbert - or more accurately his apologist Korselt - was espousing an incipient formalism or deductivism, whereby mathematical theories are investigated strictly for their logical properties - their customary interpretations, for this purpose, set aside. Some of this was expressed in terms of what for Frege were such alarming ideas as that axioms ‘implicitly define’ the concepts occurring in them, but what we would now describe as a set of axioms determining a set of models. Now a number of related issues, both historical and conceptual, will have to be kept at bay here; but the crucial point is that Frege cannot accept the idea that an axiom or theorem is something which can be given an interpretation - as happens, for example, when Hilbert declares the consistency of Euclidean Geometry by specifying a model with a domain of real numbers. In Frege’s view, since Hilbert’s deductions employ schematic (uninterpreted) predicate letters, the formulae in which they appear do not express propositions, hence do not have truth-values, and hence do not themselves represent genuine advances in knowledge. Since a theorem is simply a true proposition, a formula which admits of more than one interpretation expresses no proposition in particular, and hence no theorem. Thus, if a theory is a system of theorems concerning a certain subject-matter, it makes no sense to speak of its being susceptible to different interpretations. Instead, the idea that an axiomatized theory should be considered purely ‘formally’ - 2 © Gary Kemp; penultimate draft; final draft in the Pacific Philosophical Quarterly 1998 79/3: 218-35. that is, independently of the concepts expressed by its predicates - should not be explained in terms of signs and their interpretations, but in terms of second-order quantification. In particular, each theorem must be thought of as a second-order, universally quantified conditional whose antecedent is the conjunction of what Hilbert called the ‘axioms’ of the theory, and whose consequent is what Hilbert called the ‘theorem’ (since this will not itself express a proposition, Frege calls it a ‘pseudo-sentence’)7. For example, where Hilbert had derived ‘(x1, ... xn)i’ from a set of axioms {A0, ... An} - with ‘i’ appearing in the axioms and, strictly speaking, serving as a mere schematic predicate - the thing ought to appear as ‘(1 ... n)(x1 ... xn)(A0 & ... & An  i’ - with ‘i’ now a bound variable. The results are not facts of geometry at all. They are simply truths of logic, to the effect that any properties and relations possessing the second-order properties and relations specified in the antecedents of those conditionals must also possess those specified in their consequents. The application to Euclidean Geometry then depends upon the premise that such relations as  is between  and  and  lies upon  do possess the requisite second-order properties and stand in the requisite second-order relations (this argument is independent of Frege’s view that that premise is supplied by our intuitive acceptance of the genuine Euclidean axioms). What is striking is that Frege should have regarded this issue as critical - that, in response to Hilbert, he should have seized upon this. The point, of course, was not that Hilbert’s Grundlagen was of no mathematical importance.8 Frege’s concern, as ever, was that even the best mathematicians were labouring under various false conceptions of the nature of mathematical knowledge, and hence of mathematics itself. In Frege’s view, Hilbert and many others could not adequately address foundational questions because they had yet to recognize the character of truly systematic science. That can only be a philosophical question, and in Frege’s view the answer is strongly constrained by a proper account of judgement: since knowledge is generated by judgement (as Frege conceives it), and a theory is something which actually embodies our knowledge of something, there is simply no such thing as a theory which is not fully interpreted, no such thing as reinterpreting a theory.9 In particular, then, there is no such thing as a kind of reasoning which is merely structural or formal - if that means that it does not consist in advancing from judgement to judgement, from complete proposition to complete proposition.10 What might seem to be mere pettifogging literalism is in Frege’s eyes philosophically inescapable. Let us ask, then, how well Frege manages to obey his principles in his own mathematical work - the relevant work being Begriffsschrift and Grundgesetze. The commitment to the Primacy of the Proposition, in Frege’s work (as well as in Russell’s) is closely allied to a rather heroic conception of the science of logic itself, which Jean van Hejinoort has aptly called the ‘Universalist’ conception. Everyone will agree that the mission of logic is in some sense to formulate universally applicable standards of reasoning. But for Frege, this mission is not to be carried out by devising a truth-preserving calculus, a calculus 3 © Gary Kemp; penultimate draft; final draft in the Pacific Philosophical Quarterly 1998 79/3: 218-35. ratiocinatur. In the years immediately subsequent to the publication of Begriffsschrift, Frege repeatedly explained that unlike Boole’s or Schröder’s, his logic is represented in a fully interpreted language, not as a mere formalism adaptable for the purposes of the moment. Thus in the footnote to §17 of Grundgesetze, Frege says that in a concept-script, ‘we always proceed directly from one asserted proposition to another’; likewise the heading of §32 is that ‘Every Sentence of Begriffsschrift Expresses a Thought’.11 In claiming this, he is not saying that every formula, once its non-logical symbols are assigned interpretations, expresses a thought; he is saying that every logical formula expresses a thought in its own right. Accordingly, he characterizes the theorems of logic, not as formulae or schemata true on all interpretations, but simply as the ‘most general laws’: whereas the special sciences represent the facts concerning some particular domain of reality - some particular range of objects and concepts - logic represents the most general and abstract facts, ones which hold for all objects and concepts whatsoever (Russell advances a similar view in The Principles of Mathematics12 ). Logic thus serves as the prescriptive ‘laws of thought’, not as a universally applicable method of ensuring validity, but as a set of universally constraining facts. It follows that there is no external perspective from which the deduction of logical laws from others can be thought of as being justified. Every judgement needed in the establishment of a theorem of logic must itself be expressed as a line in the proof, not as part of a separate argument to the effect that some particular mode of inference is correct.13 So Frege presumably believed he had obeyed the strictures he derives from his theory of judgement. And beginning with those in Begriffsschrift, his explanations of generality suggest that he had taken some pains to ensure that he could. Of course he recognizes the necessity, in carrying out inferences involving generality, of dropping quantifiers in order to expose the truth-functional complexity required for the application of inference rules such as modus ponens. But here, as well as repeatedly elsewhere, he explains his ‘Roman letters’ which correspond syntactically to the free variables of more recent formulations of logic - as actually ‘conferring generality’ upon the content of a proposition. Indeed in Begriffsschrift he describes ‘Fx’ merely as an abbreviation for ‘xFx’ (of course I am pretending that Frege employed a notation like those now standard).14 Elsewhere, ‘xFx’ is explained, not as the truth of ‘Fx’ on all assignments of ‘x’, but as the subsumption of the concept denoted by ‘F’ under the second-order concept denoted by the quantifier; it follows that ‘Fx’ itself involves a tacit quantifier. By the theory of Sense and Reference, then, the general proposition is composed of senses corresponding to those entities.15 In Grundgesetze (§8), the matter is turned around: Roman letters are introduced first as conferring generality; it is then pointed out that expressing generality merely by means of Roman letters will not enable one adequately to demarcate the scope of a generalization. In particular, it will not enable one to distinguish the ‘generality of a negation’ from the ‘negation of a generality’: if ‘Fx’ is a way of writing that everything is F, then still we need some way of distinguishing the negation of 4 © Gary Kemp; penultimate draft; final draft in the Pacific Philosophical Quarterly 1998 79/3: 218-35. that generalization from the assertion that everything is not-F. The notation of quantifiers and bound variables is then introduced for the purpose, and everything suggests that the point is strictly one of symbolism. Roman and Gothic letters (free and bound variables) are said equally to ‘indicate’ rather than ‘denote’, by which Frege means that by flagging the relevant argument-places into which object or function-denoting names may be inserted, they indicate precisely what generalization is being expressed (§17). When Frege acknowledges the necessity of dropping quantifiers in order to carry out the Barbara inference, he says merely that this is the necessity of ‘writing’ the premises in a way that admits the application of modus ponens, again implying that the change is only notational. If one glances over the theorems of Grundgesetze, one might quickly gain the impression that this conception has been realized. The theorems do contain Roman letters, but if one thinks of each theorem as being preceded by implicit quantifiers whose scope extends over the whole proposition, one for each distinct Roman letter (including ones which indicate functions), then the theorem will indeed express the desired general proposition. Even the letters appearing in truth-functional theorems are Roman letters, but that is as it should be: for example, if ‘p  (q  p)’ is to express the envisaged general fact, then we must think of the letters occurring in it as universally quantified variables ranging over all objects; it only makes this more vivid if we express the principle as ‘(p,q)[p  (q  p)]’. Since for Frege sentences are names of truth-values and truth-functions are literally functions on truth-values, any instantiation to sentences expresses a truth.16 The trouble is that procedures of inference are obstructed by this way of regarding theorems. Suppose, for example, we wished to prove (the proposition expressed by) ‘xy[Gy  (Fx  Gy)]’, following the stricter regimen of Begriffsschrift. (Since this is really a generalization about concepts, strictly speaking we should add quantifiers binding the predicate-letters as well, but there is no need to add that complication here.) We substitute from the relevant truth-functional axiom to formulae involving Roman letters, then append the appropriate universal quantifiers, yielding the theorem: (1) (2) (3) (p,q)[p  (q  p)] Gy  (Fx  Gy) xy[Gy  (Fx  Gy)] Since, according to Frege’s pronouncements, (2) and (3) express the same proposition, it all hangs on the inference from (1), a generalization, to (2), another. Clearly the inference cannot be one of Universal Instantiation. For if it were, then, in deriving (2), we must substitute names of truth-values, that is sentences, for ‘p’ and ‘q’, in which case the result would be something truth-functionally complex, not something synonymous with (3). (By ‘truth-functionally complex’ I mean a proposition which is the immediate result of combining 5 © Gary Kemp; penultimate draft; final draft in the Pacific Philosophical Quarterly 1998 79/3: 218-35. one or more propositions with a truth-functional connective.) That is, if (2) is truthfunctionally complex then ‘Gy’ there either expresses some particular ascription of a particular concept to a particular object, or is itself a universal generalization. Since what we are after is certainly not a statement about any particular object or concept, it cannot be the former. But if it’s the latter, then - even ignoring the question of the meaning of the predicate-letter - we cannot maintain that the scope of ‘y’ in (2) is the whole proposition, which we must, if (2) is to express the desired generalization. For if Roman letters express generality, then ‘Gy’ and ‘Gz’ express the same proposition; hence for (2) we could just as well have written ‘Gy  (Fx  Gz)’, which is no help towards (3) (the same reasoning repeated for the predicate variables makes things all the worse). Of course, the normal route nowadays would go from a schematically understood ‘p  (q  p)’ to (2) by substitution, to (3) by Universal Generalization. That is not available to Frege, but it might be suggested that the inference from (1) to (2), where (2) is reckoned synonymous with (3), is after all sanctioned by Frege’s principle of substitution, whereby any Roman letter may be replaced by any expression of the same grammatical category, even if the substitutends themselves contain Roman letters. In fact that is exactly what Frege does. And, ignoring the fact that Frege also regards lines such as (2) as truth-functionally complex, Frege’s procedure is valid, in the sense described above: every line so derived, understood as bound by appropriate quantifiers, is true. But this technical fact does not address the philosophical issue: it does not explain in what manner the inference may be regarded essentially as an operation upon propositions. Substitution as conceived nowadays is a syntactical operation upon logical schemata, not a generalization about objects, concepts, or propositions; it is justified semantically by showing that the validity of the original schema ensures the truth of the universal closure of every substitution instance. Now the substitution procedure in Frege’s logic - explicitly formulated only in Grundgesetze - would be correct from that point of view, but it ought to be puzzling from his. For according to that point of view, the validity of substitution (together with universal generalization) represents a fact about particular languages, not a logical principle in its own right. Frege calls it a ‘simple inference’; but, if conceived as a step from one proposition to another, it is by no means generally simple, and - in contrast with his careful discussions elsewhere of modus ponens and universal instantiation - Frege offers no explanation of it that elucidates its character as a step from one proposition to another.17 At most, there is one variety of substitution that can readily be explained as an operation on propositions. Suppose for example that in a firstorder generalization ‘(p)’ we wish to substitute the expression ‘Fx  Gx’ for ‘p’ (where the former expresses a generalization). This cannot be explained as the instantiation of a generalization, but it might be explained as a reduction of its scope: for it is clear that if something holds for all objects - hence for all truth-values - then it must hold for all those which are values of functions of a given internal structure (though for Frege, this talk of the 6 internal structure of a function would be problematic). But this does nothing to address the problem of cross-referring variables exemplified, at its most elementary, by the inference above from (1) to (2); the simple variety of substitution-instance just described may add syntactical structure within an argument-place, but not add structure which extends across argument-places. Put more intuitively, (1) indicates a truth-functional relationship between two propositions; (3) does not. The grammatical category of ‘Fx’, as it occurs here, is not that of ‘p’ (irreducibly polyadic examples will make this more vivid). The problem is general: given the way that his epistemological imperatives force Frege to think of inference, and of truth-functional theorems - as generalizations about the meanings of complete sentences - it is not at all clear what it is to derive an instance of a truth-functional theorem whose linguistic expression will comprise formulae with variables referring across that truth-functional complexity, bound by (possibly tacit) quantifiers whose scope is the whole proposition.18 The syntactical procedure of substitution-and-universalgeneralization tells us how to obtain sentences which express the required generalizations, but we have as yet no explanation of how this might be made consistent with the Primacy of the Proposition, and certainly none of how it might connect with or be justified by what, on the Universalist conception of logic, could rightly be called logical truth (remember that proofs, on the Universalist conception, cannot be thought of as being backed by a global argument for the soundness of the calculus). I shall return to this, including the question of in what way the thesis of the Primacy of the Proposition demands the Universalist conception. Now the only sign that Frege recognized this difficulty is that, after having said all that I have related concerning the role of Roman letters, he effectively takes it all back towards the end of §17 of Grundgesetze. He now says that combinations of signs containing Roman letters are not names, but expressions which become names when the Roman letters are replaced by names. Hence sentences containing Roman letters - including many theorems of Grundgesetze - are not names of truth-values, and do not express complete propositions.19 Perhaps Frege saw that this was inconsistent with his epistemological strictures, but unlike his frank worries over various related issues such as his worries about Hilbert - and unlike his careful explanations of other, simpler sorts of inference - he never puzzled over it in his writings. II Russell, however, puzzled over it openly and at length. His divagations range widely over the years 1903 to around 1924, and they are highly ramified at each point. Indeed, the Russell I shall consider is not one who existed at any one time. I shall think of Russell as having combined the binary theory of judgement assumed in The Principles of Mathematics (1903), with the account of generality in terms of propositional functions propounded in Principia Mathematica (1910). We shall thus avoid the tangles surrounding the Principles 7 theory of denoting concepts, and the incipient multiple relation theory of judgement of Principia (expounded in more detail, and with various adjustments, in succeeding works).20 I think this will be harmless: nothing I say depends upon features of the binary theory not shared by the multiple-relation theory, and those familiar with the theory of denoting concepts will know that those logical homunculi cannot provide any genuine relief from the sorts of difficulties I shall discuss. I shall, however, stress the epistemological ground common to Russell’s theories of judgement: that any constituent of a judgement must be something with which we are acquainted: that in order to judge that p, the constituents of p must be real mind-independent entities which in some sense stand before the mind.21 Russell retained this view throughout his ever-diminishing estimates of what we may assume ourselves to be acquainted with. Notoriously, Russell never explained how the logical theory of ‘On Denoting’ or Principia was to combine with the principle of acquaintance in the explanation of general judgements, but something like the following seems inevitable. The crucial notion is that of the propositional function. A propositional function x_ (such as _x is wise) is a function whose values are propositions: a for example is the value of x_ for the argument a. To assert that everything is , according to Russell, is to assert the truth of all values of x_. What, then, are the constituents of the proposition that everything is ? It cannot simply be composed of all values of x_, for that would require acquaintance with every object, and the proposition would be of implausibly enormous, if not infinite, complexity. Instead, Russell thinks of a universal generalization as ‘involving’ the propositional function itself. Whereas the proposition a does not itself contain the propositional function x_, the universal generalization does contain it (equivalently: is about it); it says that all its values are true. The conclusion seems inevitable then that for Russell, much as for Frege, the universal quantifier signifies a property of propositional functions, one which forms a constituent of general propositions. Unlike Frege, however, Russell not only sees clearly that quantifiers must be dropped for purposes of logical deduction, he sees from the start that the significance of doing so is not merely symbolic, and he struggles to come to terms with it philosophically. The issue is stated most directly in ‘Mathematical Logic as based on the Theory of Types’ (1908). Of a universal generalization, as just explained, Russell says that it is a ‘determinate proposition’ (in Principia, that it is ‘in no sense ambiguous’). By this Russell means that it does not contain a variable; its linguistic expression contains the letter ‘x’, but this is merely an ‘apparent’ variable, needed, as for Frege, to indicate precisely what is being said to hold in all cases. In carrying out deductions, however, we must operate with what Russell calls ‘real variables’. In particular, where we drop the quantifier and write ‘x’ as a line in a deduction, what we are doing is asserting a propositional function. In this case what we assert does contain a variable; to assert a propositional function is not to assert all its values, but to assert 8 ‘any value’ of it.22 Now in spite of that way of putting the point, and in spite of the slightly misleading first paragraph of Russell’s initial discussion of propositional functions in Principia, Russell does not mean that in such a case the propositional function is itself the object of judgement. Instead, to assert x is to assert something which is just like a, except that it contains a real variable where the latter contains the object a. x is thus not the propositional function but an undetermined value of it, and to assert it amounts to an ‘ambiguous assertion’. Thus Russell writes: When we assert something containing a real variable x, we cannot strictly be said to be asserting a proposition, for we only obtain a definite proposition by assigning a value to the variable. This could be taken in either of two ways. It could mean that there is a certain something containing a real variable x, and which we assert, but that this thing is not to be called a ‘proposition’. Or it could mean that there is no thing in particular that is the object of assertion (or judgement); rather we make an assertion of an arbitrary member of a certain class of propositions, namely the values of the asserted propositional function, in such a way that the truth of what we assert does not depend upon which member of the class happens to be the arbitrary one. On the first alternative there seems to be no good reason not to call the thing asserted a proposition; even if for some reason truth or falsehood can attach directly only to the values of the function, still it has at least derivative truth-conditions, and it is a thing which can be asserted or judged. On the second alternative there is only mystery, which I suspect it would be unremunerative to enter into at any length. Suffice it to say that where judgement is conceived as a relation between mind and a nonlinguistic object which is apprehended by the mind, and assertion as the linguistic expression of judgement, it makes no sense to say that assertions may take place when there is no particular object - especially not when it is quite clear that the conditions of success (the truth-conditions) of such an assertion are exactly specified, as in the present case they certainly are (it should be conceded, however, that the notion of ambiguous assertion is relatively unmysterious so long as sentences are understood to be the objects of assertion). To be driven towards the first alternative, however, is to be driven back to a problem which had bedevilled Russell unmercifully in the Principles. To assert, as a universal generalization, xRx (that everything bears R to itself) is clearly different from asserting, as a universal generalization, that xRy (that everything bears R to everything). Both propositions can be obtained by extracting the relation R from a proposition aRb, and replacing a and b with variables. But if we are discussing propositions rather than sentences, and these 9 propositions contain variables, then the variables x and y must be nonlinguistic objects, and must furthermore be distinct. But how could they be? After all, ‘xRy’ and ‘yRx’ (understood as generalizations) express the same proposition, as do ‘xRx’ and ‘yRy’. If variables enjoy an extralinguistic existence, then each can be signified by any letter, ‘x’ or ‘y’. As Russell had put it in the Principles, the (unrestricted) variable x (or y, or ...) must be ‘any term’, any object whatsoever; equivalently, something which ‘denotes’ but in an ‘impartial distributive manner’.23 It is hardly comprehensible how there could be two such entities - distinct, yet each such as to be any term whatsoever. Moreover, such entities would have to be not only distinct in themselves but distinguishable in thought, such that their appearance before the mind would enable one to distinguish xRx from xRy (without thereby distinguishing xRx from yRy).24 These near absurdities may have contributed to Russell’s having introduced, in the second edition of Principia, a method of quantificational inference which dispenses with real variables. All sentences containing unbound variables, Russell now says, are to be understood as governed by tacit universal quantifiers, one for each variable. Quantificational inference is now to be carried out, not by means of instantiation and generalization, but by means of a new inference rule: from (x)x and (x)(x  x) we may now infer (x)x directly. He then sketches a method for proving general propositions using the new rule. Whether Russell was motivated as I suggest is clouded by his later uncertainty over the status of logic. For example by 1937 - ten years after the second edition of Principia - Russell was saying that logic is ‘much more linguistic then I believed it to be at the time when I wrote the Principles’, and was willing to consider that the logical constants are nothing but symbols, not things expressed or denoted by symbols. He may simply have called it progress to reduce the number of primitive ideas in logic, which, in the first edition, did explicitly include ambiguous assertion. In any case the emendation announced in the second edition is clearly insufficient to make good on the idea of deduction as the movement from whole proposition to whole proposition. For substitution survives in the new system unchanged, and indeed now bears more of the burden; as it emerged earlier in connection with Frege, the real stumbling-block is substitution, not quantificational inference generally. Technically speaking, the discussion of substitution in Principia establishes almost nothing. Philosophically it is not without suggestive nuance, but from any point of view it is clearly inadequate to rest, as Russell does, with saying that to invoke it in deducing fresh theorems is nothing more than to ‘notice that they are instances’ of previous theorems, for the requisite notion of ‘instance’ is left unexplained. Without a precise statement of it there is no fact of the matter as to whether or not even the elementary portion of the system is consistent. For his part, Frege formulated a substitution rule explicitly and, from the point of view of logic as a formalism subject to interpretations, correctly in Grundgesetze. But neither Russell nor Frege had made any effort to come to terms philosophically with its centrality in logic 10 (though it is conceivable that Russell’s strange remark to the effect that substitution is somehow unstateable was philosophically motivated, not mere technical indolence). Again, the only evident path open to a Russell or a Frege, trying not to conceive of logic as being about language, would be to devise a principle of substitution as a generalization about structural relationships amongst propositions. No doubt the reasoning behind such a substitution principle can be plausibly sketched.25 What we need can be explained in terms of the logic of Principia as follows (the simplifying restriction here to oneplace propositional functions is inessential). We want to derive instances of truth-functional formulae with variables referring across their truth-functional structure. Thus suppose that F is some truth-function, a proposition F(p1...pn) is a truth functional theorem, and we want to derive (1 ...n)(x1 ... xn)[F(1x1 ....nxn)]. We can argue: For any choice of functions 1x_ ...nx_, every instance F[1a1 ... nan] of the truth-functional theorem is true, since the a’s are propositions in each case. But then every value of the propositional function F[1x_1 ... nx_n] is true, so by Russell’s definition of generality we may add the quantifiers to bind the ‘x’s, yielding the proposition (x1 ... xn)[F(1x1 ....nxn). But that was any choice of functions 1x_ ...nx_; hence by the same argument we may add the quantifiers to bind the ‘’s, which gives us the desired substitution-instance. That much is straightforward, and seems to generalize about propositions and propositional functions - rather than the symbols which express them - in the requisite way without mentioning propositions containing variables. But it is to no avail. In the first place it gives every appearance of using real variables, such as ‘a proposition F(p1...pn)’, thus assuming the unwanted thing in the attempt to eliminate it. Second, its premises will presuppose what look too much like matters of fact. That’s all right, as well as inevitable, for the philosophical, meta-theoretic enterprise known as logical theory, but not for logic itself. To speak of ‘a proposition F(p1...pn)’ , or ‘any instance F(a...)’ is like speaking of ‘any set {a, ...}’ - insupportable until we have a reassuringly explicit theory of the supposed entities, from which such loose expressions may be said to borrow their significance. The advent of such a theory would only make it clear that we were building logic atop a substantive theory of abstract entities. The needed generalization would have to presuppose a theory of the structure and constitution of propositions, something largely analogous to set theory except intensional rather than extensional. But, whereas Frege, and although less clearly Russell, could distinguish in a principled way between the laws of deduction (the truths of logic) and the theory of the composition of propositions (logical theory), the latter will now be essential to the content of the former. In expounding such a theory, one’s hand may be steadier in imposing some of the restrictions necessary for consistency, but even the most obvious restrictions are not thereby any less substantive; in principle, all the terrors of set theory and more will now be endemic to elementary logic itself, hence implied by whatever is needed for elementary reasoning (even in Russell the principles of type theory are not themselves 11 propositions of logic).26 That would be a lot even for Russell to swallow, and if you believe as Frege did that the basic principles of logic cannot coherently be thought of as being subject to doubt then it is out of the question (Russell too held this view in the Principles, but had definitely hedged it by the time of Principia). Of course, it might be pointed out that even where logic comprises the employment of formal systems and semantical generalizations about them, the validity of substitution must be established. Normally this is achieved indirectly by means of a soundness proof, which indeed one might regard as part of the argument for anything derived within the formal system. But the informal reasoning in such a proof is itself elementary - availing itself, at most, of finitary mathematical induction - and makes explicit reference to nothing more worrisome than linguistic expressions (whose structure is something rigorously stipulated, not out there to be discovered). It is at least arguable that the codification of inference does not thereby rest upon something more doubtful than what is being codified. Such is far from being the case if the codification rests upon a theory of propositions. III Formal first-order logics exist which employ no free variables, such that any interpretation of their predicates ensures that every provable formula is a true sentence. Quine’s system in Mathematical Logic is one such system; the example Tarski employs for a truth-definition of a first-order language is another, and indeed the formalism of the second edition of Principia provides materials for a third. But what must be appreciated is that the possibility of such systems provides no comfort to the thesis of the primacy of the proposition. To make this vivid, consider that decision procedures exist for the validity of a wide range of polyadic quantificational schemata. Choosing some such procedure (or set thereof), and taking the widest manageable class of such schemata as one’s object, one could declare one’s logic to consist of nothing but a specification of its formulae and its semantics, along with a single inference rule: every formula which is identified as valid by the decision procedure is a theorem. No one - and certainly no believer in the primacy of the proposition would say that a proof thereby delivered actually displays the justification for a given assertion. Clearly the justification of any such theorem would be contained in the argument for the reliability of the decision procedure. Yet the case is substantially the same, if less vividly, for systems like those just mentioned due to Quine and Tarski. Confidence in the logic is supplied not as it were from within, as Frege and Russell envisaged, but from without, by metalogical argument. The thesis of the primacy of the proposition requires that all reasoning consist in operating with complete propositions: perhaps the operations themselves cannot at all events be identified with propositions, but it would surely be satisfactory if they could be reduced, say, to nothing more momentous than modus ponens.27 Needless to say, for Quine or Tarski, avoiding open sentences as theorems does not bring a 12 requirement in its train that metalogical or metalinguistic argumentation should avoid such natural language analogues of free variables as ‘any predicate’ or ‘any sentential function’. But for Russell or Frege there must, as it were, be a level at which the buck stops: there must be a way of giving the full justification for any assertion in terms of complete propositions, all on the same level. If formalizing is like ascending a ladder, then we have to be able to draw it up behind us, if not kick it away. I have argued that the only evident way of achieving this by supplementing logic with an explicitly propositional principle of substitution - would be implausibly problematic: it carries too much weight to be passed off as an inference rule on a level with modus ponens, and it insinuates too much into logic to appear as an axiom.28 But that is to say that anyone who accepts the thesis of the Primacy of the Proposition, and thinks of the syntax of Begriffsschrift or Principia as embodying essentially the correct account of logical structure, must face the same difficulty. Some will see the positive wisdom of the Tractatus view of logic in this. The rest of us might settle with the idea that Wittgenstein the younger was right at least as far as saying that reasoning is in some sense calculating or operating with language; for in that case it would not seem ruled out that reasoning should be at least partly schematic.29 Perhaps, consistently with that, the language could be mental; what is more fundamentally missing from the propositional view of Frege or Russell might be more generally expressed as a failure to give proper place to the idea that thought is representational. University of Glasgow 13 Notes 1.PHW p. 144 (NS p. 155); for Frege’s characterizations of judgement, see PHW p. 2, p. 139, p. 185, p. 198 (NS p. 2, p. 151, p. 201, p. 214); CP pp. 164-165, p. 356 (KS pp. 150-1, pp.346-7). 2.PHW p. 3, p. 175, p. 204 (NS p. 3, p. 190, p. 220); GG §14, PMC p. 22 III/3 (WB p. 35 ix/4). 3.See PHW pp. 269-70 (NS pp. 288-9), ‘On the Scientific Justification of a Conceptual Notation’, CN pp. 83-4 (the original of this piece is reprinted in Frege 1964b); cf. PHW p. 259 (NS p. 279). 4.In particular, Frege held that in order to think a thought containing a given part, there need not be a kind of mental apprehension of that part (or of the referent of that part) which can be isolated from the thinking of whole thoughts containing it. See Foundations §§58-60. 5.See the beginning of ‘Logic in Mathematics’, PHW pp. 204-205 (NS pp. 220-1). 6.CP pp. 306-333 (KS pp. 293-317), PHW pp. 247-250 (NS pp. 266-70). 7.In the English translations Frege’s ‘Satz’ is ‘proposition’ (hence ‘uneigentlicher Satz’ is ‘pseudo-proposition’). For obvious reasons, I have avoided this. 8.At CP p. 318 (KS p. 304) Frege admits that Hilbert’s schematic derivations may at least ‘serve as the occasion’ for constructing proper proofs. Hilbert might have been happy to accept that his axiomatization was a work in pure logic; what Frege does not appreciate is that Hilbert, or at least the Hilbert of the 1920's, would nevertheless continue to speak of interpretations where Frege speaks of second-order logic. 9.CP p. 316 (KS p. 302). 10.Of course one can use words such as ‘theory’ as one likes, and the modern idea of a formal theory as precisely a syntactical apparatus subject to varying interpretations is not inconsistent with Frege’s epistemological strictures, so long as one distinguishes clearly between assertions of the theory (under a given interpretation), and metalinguistic assertions about the theory. Clearly Hilbert and Korselt at this time (1903-1906) were well on the way towards that conception, but it had not yet been realized fully (Hilbert’s proof theory was not to emerge completely before the 1920's). Frege himself was on the one hand making fun of those who would advance a ‘theory’ which doesn’t actually mean anything, and saying on the other that if symbols are the objects of a theory then the truths of that theory ought to be discoverable by means of a chemical analysis of (the tokens of) those symbols. 14 11.Other characterizations of inference, especially in his unpublished writings, reinforce this point. See especially PHW pp. 260-1 (NS 280-1). 12.Ch. 1-2, esp. §17, in which Russell observes that basic ‘principles of deduction’ cannot be subject to doubt, on pain of being unable to justify any argument. On this theme see Van Heijenoort, Ricketts, Hylton (pp. 198-204), Kemp (1995). Partly in order to maintain that the Axiom of Reducibility is a logical axiom, Russell eventually dropped the idea that axioms of logic need have any special epistemic distinction. 13.Of course, inference rules cannot themselves be cited as lines of proofs. Frege recognizes this, but he sees also that the choice of inference rules is partly open, hence that there is no one principle that must ever remain ineffable. 14.B §1. 15.PHW pp. 254-5 (NS 274-5). 16.Those letters can be instantiated to any name of an object, but Frege’s definitions ensure that p will be inferrable only when p is in fact the true, as opposed not only to the false but also as opposed to other objects. 17.A further issue is that substitution transmits only validity, and not truth generally. This fact, indeed, is difficult for Frege’s philosophy of logic to assimilate, because a substitutioninference requires that the premise be recognized not merely as true but as logically true. I believe that this will be problematic for Frege, because of the special relationship he envisages between the notions of truth and judgement. But the issue is too complex to enter into here. 18.In fact the same sort of problem arises within truth-functional logic itself: understood as generalizations about truth-values, truth-functional theorems are no longer truth-functionally complex. But these could plausibly be made amenable to inference by adopting a generalized version of modus ponens. 19.Frege does sometimes characterize the use of Roman letters as representing the idea of reasoning about an arbitrary object - e.g., ‘Suppose n is any perfect square; then ...n...’. The nature of the case perhaps makes this inevitable, but this idea finds no reflection in Frege’s philosophical explanations of generality or inference. I have thus suppressed this, since Russell takes up the point explicitly and at length. 20.Principia pp. 43-44, Problems pp. 125-130, Theory of Knowledge pp. 105-118. Also see Candlish. 21.More exactly: if, after adopting the multiple-relation theory, Russell could no longer hold precisely that ‘[every] proposition which we can understand must be composed wholly of 15 constituents with which we are acquainted’ (Problems p. 58), he still maintained, in effect, that ‘judges that’_s can be true of A only if A is acquainted with the meanings of the parts of s (where s is fully analysed, in which case s is significant only if all its parts do stand for something). Cf. Problems pp. 125-130. It may also be noted that, in Russell’s Logical Atomism, sense-data are in the requisite way independent of the mind. See Problems p. 41, and ‘The Relation of Sense-Data to Physics’, reprinted in Mysticism and Logic, pp. 108-131. 22.This account leans upon the account of Russell on generality in Hylton pp. 296-297, though departing from it slightly for the sake of simplicity. 23.Principles 56-60, 86, 88. 24.For greater illumination of this theme see Hylton, pp. 212-218 and pp. 255-6. In a wonderful letter to Jourdain, Frege suggests that this sort of idea is so egregiously incoherent that the expression ‘variable’ ought to be banned outright, since people will continue to talk as if they could be anything but mere letters (PMC VIII/13, pp. 81-84 (WB XXI/13, pp. 12933)). See also PHW pp. 234-8 (NS pp. 252-7); at p. 238 (NS p. 257) Frege makes the point about the indistinguishability of x and y, if variables were nonlinguistic entities. 25.This spells out in slightly more detail the brief discussion in §4 of Section II of the Introduction to the Second Edition of Principia (which is indeed Russell’s most explicit discussion of substitution; it deals only with the instantiation of truth-functional theorems). 26.It should be repeated that, having assumed the Axioms of Reducibility and Choice, Russell denied that logical axioms must be self-evident or peculiarly obvious (see note 12). But, to cut a long story short, it is not as if Russell could expect everyone to agree that those are principles of logic, in advance of showing that, along with principles which are so agreed upon, they suffice to derive mathematics. Russell never strayed so far from plausibility as to suppose that all reasoning - not just that needed for mathematics - assumes among its premises a substantive theory which explicitly quantifies over sets or propositions. It should also be acknowledged that Russell’s truth-functional theorems already generalize about propositions (for the truth-functions are propositional functions which take propositions as arguments); for that purpose, however, no account is needed of their structure. 27.Though erroneously, Frege supposed that Begriffsschrift employs only modus ponens as an inference rule. The situation is clouded because of Frege’s inconsistency in his explanations of Roman Letters (free variables), but if we think of the strictly first-order part of the system as a formalism in the modern sense then it employs universal generalization and substitution (Universal Instantiation is an axiom). 28.One might also worry that if propositional substitution must be an explicit axiom rather than an inference rule, then, where S is the relation of one proposition’s being a substitution instance of another, the principle would have to be of the form ‘if S(p,q) then if p is true then 16 so is q’ - in which case a truth-predicate appears in logic itself, hence consistency only at the price of yet more complexity. 29. For a recent account of inference which copes with this, see Lance. Jim Edwards is also working on a paper which suggests that, if we weaken the constraint on inference from preservation of truth to preservation of Tarski-truth, then although natural deduction systems cannot satisfy the Universalist Conception of logic, Axiomatic systems such as Frege’s might. 17