.....SIX KINDS OF PROPOSITION - Sections 2-8
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2. PROPOSITIONS
Most philosophers understand "propositions" to be the meanings conveyed by sentences such as The traffic lights have just changed to red, "Red" means "stop", Red is a colour, Nothing is both blue and red all over, Either you are at home or you are not at home, All fish have gills, Some fish live in holes, Sabrina avait un crayon jaune, There are about 20 million people living in Australia, 400 x 50,000 = 20,000,000, The shortest distance between two points is a straight line, If Harry plays the Socceroos will win, It will rain in Timbuctu on 31st March 2066, Had it rained the cabbages would have grown, If it rains 2+2=4, I have a terrible headache, Lucy has a terrible headache, I believe what she said, Jack will consider marrying Jill, Allah is the one true god, Frodo is a Hobbit, The King of Australia has a green beard, Smokeless aeroplanes don't exist, The only thing people have in common is that they are different. Such propositions are said to be true if what they say is factual or correct. These are simple examples, but they are sufficient to illustrate that there are different kinds of proposition, apparently requiring different techniques for deciding whether they are true or false (if, indeed, any such technique can be found).
Not all theories of verbal communication, meaning and truth call for propositions. Although of course there are sentences and utterances, according to these theories there are no universals corresponding to "the meanings" of verbal expressions. This viewpoint appears to have its origins in the belief that if propositions fall short of the eternal ideal - if a given sentence-form is not associated more or less unchangingly with a certain meaning - there's nothing useful left for them to do. It seems to me, on the contrary, that most propositions are short-lived, dynamic creatures whose being derives from the interaction of people with their environment, and which are generally well served by whatever ingenious turn of phrase kindles the pleasure of the moment. If it makes sense to speak of (at most) the same sentence communicating the same information on more than one occasion, then it makes sense to speak of meaning and of propositions. And to abandon propositions in this sense is to deny the possibility of language.
Accordingly I consider a proposition (or loosely a statement) to be a declarative sentence which (i) acquires a sufficient meaning to perform a particular communicative function and (ii) conserves this meaning throughout the performance of that function. But, in taking it for granted that any number of sentences (forms or tokens) may represent the same proposition, I intend the term "proposition" to refer primarily to its functional meaning, content or utility, and only passively to whatever apparatus conveys this meaning.
The notion of performance of a function involves two distinct ideas. In the first place it implies that propositions take part in active, purpose-oriented processes, including extended language processes such as "arguments", a term which I employ loosely to refer to collections of propositions (only) linked by connectives which may be open to a logical interpretation. Traditionally, a valid argument is one that exhibits (as a minimum requirement) the structure of a tautology, and it might be considered a primary purpose of arguments, in a stricter sense, to approach this type of structure. In the second place the notion of performance of a function is associated with that of assertion - the idea that the work of a proposition consists in advancing a specific fact or connection of meanings, regardless of tacit background assumptions or "presuppositions". Often the brunt of this work is borne by a discrete component of the sentence, typically the predicate. In what follows I shall lean on the view that there is a limited, indeed a proper, sense of falsification which consists in denying only what is asserted and never what is merely presupposed (though the difference is not always clear). On this account, a "genuine" proposition purports to be true, in just one of the several senses of truth, and if not true it is either false - implying that there is a true proposition which contradicts what is asserted - or else incompetent - implying that the underlying conditions required for truth or falsity are somehow flawed, by reason of which the assertion relinquishes its argumentive potential. (This is the underlying form of each of the "hexagonal" schemes outlined in §9.)
Purportment is not a psychological notion: a proposition purports to be true even when an utterance of it is a deliberate lie: indeed we should not call an expression a lie if it did not purport to be true - for how, then, could it mislead us? It is just that truth (of some type) is, so to speak, the default value of every proposition (of that type) - an observation whose chief importance is that it apparently establishes an inviolable link between truth and meaningfulness. For if every (declarative) proposition contains the possibility of being true, then there's a lot going for the claim that expressions which lack that possibility are meaningless, and therefore not genuine propositions at all. (Amongst the repurcussions of this view are that it puts verificationism on a firmer footing than falsificationism.) The "default value" idea receives further comment in §9.
The purportment (predication etc) of a simple proposition thus determines, among other properties, its type, though there are invariably non-assertoric elements present whose "type" differs from that assumed by the proposition as a whole. I include all such inactive elements under the "presupposition" label, and in §9 I submit a classification of basic propositions which broaches the idea that certain of their presuppositional aspects conform to a hierarchy of types, which may be invoked to examine the propositional integrity of any expression.
Whether or not this concept of purportment is sustainable, it's clear that most propositions are each endowed with just one way of being true (one kind of meaning), and that although they might fail to be true for any of a wide variety of reasons nevertheless they generally carry expectations as to how they might legitimately be falsified: that is, expectations as to the kind of meaning conveyed by the negation of the proposition. And since this kind of meaning is typically the same as that required for verifiability, there exist classes of typical propositions whose evaluation depends on only a single species of criterion and which, therefore, are logically unrelated to any of the typical propositions evaluated by other species of criterion. Such propositions conserve their type both under negation and (where applicable) under quantificational manipulation. These represent the basic classes which I contend number at least six. In respect of every basic proposition, therefore, its being held true in the characteristic sense does not imply that it is true in any other sense, nor can the proposition be "reduced" to any other kind.
Of putatively non-basic propositions, however, there appear to be some of mixed type and arguably some which can only be tested for truth in one way and for falsehood in another, or which may change type under quantification. Moreover some assertoric sentences in ordinary language situations appear to discharge two or more functions concurrently (or one or another of them indistinctly). While these mongrel types of expression are quite acceptable in ordinary discourse (for examples see §s 6 and 9), philosophically they are the result and the cause of intolerable confusion, and, were it not for the fact that they often deploy the various types of concepts in a pragmatically coordinated manner, I'd be tempted to dismiss them at once. It must also be acknowledged, however, that many apparently unobjectionable arguments combine different types of proposition, and since any argument can be represented as a complex proposition it might seem that there must exist viable mongrel propositions. This matter receives further consideration in §s 7 and 9, where I suggest that many such arguments are underpinned by a single, type-determining orientation which permits an unequivocal evaluation; otherwise the components of the argument are irrelevant to one another and the argument as a whole is effectively vacuous.
In spite of this setting, it cannot be presumed that all meaningful assertions have determinable truth values. But in keeping with the conventional idea that to assert a proposition, from a position of integrity, is to submit a truth of some sort, this discussion will be more or less confined to the range of expressions that seem to me to possess acceptable and clear criteria of truth determinability. Nonetheless these same criteria find application in a variety of expressions - including probability statements and counterfactual and subjunctive conditionals - which I think properly belong in the class of propositions even though they are commonly supposed to be neither true nor false but only inductively strong or weak. In practice the test of truth for most conditionals is similar to that for most direct statements. The fact is, many conditional statements (including subjunctives and counterfactuals) are supported by much more evidence than many direct statements, and this suggests that any distinctions as to truth status formulated on logical or grammatical grounds are almost certainly trivial. Much the same applies to probability statements.
Our present concern, however, is with certainties rather than probabilities.
A proposition might be considered certain or necessary if it seems absurd to deny that it correctly depicts a particular process or state of affairs; or if a process or state of affairs that it depicts cannot be conceived to be other than it is; or if the proposition couches a well established regularity of nature or an instance of such regularity; or if its symbolic or semantic relationships reiterate established conventions of usage; or if regardless of symbology or meaning its structure appears to reflect unchallengeable canons of reason; or if its structural elements are arranged in a prescribed pattern or comply with an agreed rule of calculation; or perhaps for less credible reasons, such as that it expounds a self-evident duty or voices the will of an absolute authority. While there may be some who consider such reasons respectable enough to explain the necessity of the propositions, it seems to me that, on the contrary, none of these cases supplies an intrinsic answer to the question why we feel certain in just this case while in a variety of others we remain doubtful. Nor shall I directly address that question here. My aim in this regard is only to suggest (where necessary) that, although the categories I shall describe may be distinguished by the unique quality and durability of their limits, in fact they embrace no indubitable concepts and therefore provide no foundation for the common assumption that human enterprise is constrained by rigid built-in principles.
3. CONSTITUTIVE PROPOSITIONS
Propositions, then, may be classified according to the kinds of criteria used for deciding their truth values. A commonly held belief, derived from Leibniz and Hume and championed by the logical positivists, is that every proposition belongs to one of only two such classes: analytic or synthetic. The evaluation of analytic propositions is said to depend only on their logical structure (overt or implicit) while that of synthetic propositions appeals to the results of empirical investigation. It is commonly held further that true (tautological) analytic propositions are necessarily true, or true of all possible situations, while true synthetic propositions are only contingently or probably true. All formal tautologies, mathematical propositions and linguistic conventions are contained in the former class while all, if not only, empirically true propositions belong in the latter.
Critics of this view, rather than attempting to undermine its foundations, have tended only to blur the analytic/synthetic distinction by contending that (a) propositions may function ambiguously in one mode or the other or change their allegiance with the passage of time; or (b) especially in scientific contexts, the two kinds depend inextricably upon each other for meaningfulness. Thus although they question the stability of propositional roles, they do not, I think, altogether succeed in escaping from the conceptual environment delineated by the analytic/synthetic dichotomy: they quietly suffer the two kinds, while remaining reluctant to point to a statement that epitomises either one of them. The prevailing attitude has sometimes been represented in the following way: provided that the meaning postulates of a language are specified, then we know what's analytic and what's synthetic; however, meaning postulates incline towards arbitrariness and are open to revision. In a sense this is so gross an understatement as to be worthless, since the meaning postulates in my language, anyhow, are apt to change whenever I walk from one room to another.
If we accept this dualism, however, we are compelled also to admit that any two empirical propositions are logically independent. For each proposition is separately capable of verification or falsification, and there could never be any logical necessity that the outcome of an experiment to evaluate either one would affect the outcome of an experiment to evaluate the other. What could be plainer, therefore, than the logical independence of the statements in each of the following pairs?
(a) Joe hit the ball. Share prices collapsed in Tokyo.
(b) Joe hit the ball. Flo missed the catch.
(c) Joe hit the ball. The ball didn't move.
(d) The ball is moving upwards. The ball is moving sideways.
(e) The ball is red. The ball is blue.
(f) The cat is in the cupboard. The cat is outside the cupboard.
(g) The cat is in the cupboard. The cat is not in the cupboard.
(h) The electron took this route from A to B. The electron took another route from A to B.
But while almost everyone would agree that the statements in the couplets (a) and (b) are logically independent, and many would say the same of (c) and (d), I've little doubt that most would find increasing difficulty with (e)-(g), ), even though they might hold the opinion that the statements in the analogous example (h) are, at least, not plainly contradictory. And although the logical independence of empirical statements appears to be an inescapable consequence of the analytic/synthetic dichotomy which is the cornerstone of logical positivism, astonishingly it is just the proponents of this creed who have contended most strongly that statements like those in the couplets (e), (f) and (g) are logically incompatible (which is to say that the proposition which expresses the negation of their conjunction is tautological).
Hahn (1933), for example, asserts: "Like the statements 'every snow rose is a helleborus niger': and 'nothing is both red and not red', the statement 'nothing is both blue and red' says nothing at all about the nature of things; it likewise refers only to our proposed manner of speaking about objects, of applying designations to them". But although it's possible to give academic credence to this aloofness of language from the world of facts, the practical absurdities inherent in this approach are all too obvious: a language with no points of contact with reality can never speak about anything. There appears to be a lamentable confusion between matters of nomenclature, such as the definition of the snow rose as helleborus niger, and possible states of affairs, such as that the cat cannot be both inside the cupboard and outside the cupboard at the same time.
Rationalists of a certain breed are inclined to demand the best of both worlds by insisting that any two apparently substantial but incompatible propositions such as (g) are at once factual and logically (hence "necessarily") contradictory. But their standpoint leads, in my view inevitably, to the unacceptable extravagances of a kind of "global essentialism" (wherein any attribute or state assumed by an object is "necessarily" tied to it): since they would, I think, be obliged to turn our list of examples on its head and thus finally to admit that not only (g) but all the pairs of propositions with the possible exception of (h) are logically connected, while doubtless they would feel compelled to translate (h) into more subtle language. Yet there's a corollary to this outlook, namely that physically unrelated propositions cannot consort in a single cohesive argument, which I shall support in §7.
My view, then, is that "The ball is red" and "the ball is blue" are logically independent empirical propositions; so "The ball is not both red and blue" expresses an empirical relationship and not a tautology or a linguistic convention. For the reality appears to be such whether or not we possess the language to express it: language conforms with the facts and not vice-versa. Still, it can hardly be doubted that this proposition expresses a fact of a strikingly different kind from the mere fact that the ball is red. It is different for just those reasons which the positivists customarily offer: it contains what looks like a logical relation and it's not the sort of fact that one would ordinarily verify by physical examination. And these are just the qualities that imbue propositions expressing facts of this sort with a distinctive air of necessity.
An air of necessity the likes of which can be found nowhere else. In comparison, the definitions of language are mere artefacts that can blow away in the wind, while many scientific laws seem fragile alongside these megaliths. In the ideal gas laws, for example, one can imagine the constants having different values; one can even imagine that gases might contract in volume when heated at constant pressure. But one cannot imagine a surface that's red and blue all over at once, or a cat that exists in two places at once. The laws in this category, one might say, are indispensable and inescapable, and in that sense "more certain" than many scientific laws. However, there are no sharp boundaries between the two kinds, and one can discern a gradation of necessity through both kinds. (Consider, e.g. "No person can survive without oxygen" and "No person can survive unprotected at the centre of the Sun".) And this admission of degrees of certainty must create doubts about the inescapability of any physical laws. Many fundamental concepts must have appeared inescapable to one generation or another - Newton's Laws, for example - but the progress of science has pushed them aside (see §3.1). This is further evidence that the apparent indubitability of the most obdurate of these concepts is not logical in character; so what is its origin?
An answer given by J.S. Mill (1884) was that such laws are inferred from large numbers of observations of supporting instances together with a total absence of conflicting observations. The surface of every object we have ever seen has been of just one colour in the same part at the same time; so if this ball is red all over the chances are extremely high that no part of it is also blue. The weakness of this answer is its failure to distinguish this type of case from the more ordinary: "Every sheep we have ever seen eats grass; so there's a good chance that this sheep eats grass". For while we might not be all that surprised to learn that a particular sheep doesn't eat grass, we should be incredulous of any suggestion that the ball is red and blue all over at the same time. The difference is that we can imagine sheep that don't eat grass but we cannot imagine surfaces that are coloured in more than one way in the same part at the same time. Nor can we imagine a cat that is both in the cupboard and out of the cupboard, or an electron that went through two holes at once, or parallel lines that meet, or a straight line that is not the shortest distance between two points, or.... Regardless of the practical utility of any of these "concepts", the truth is that we cannot conceive of them at all, because the world of our immediate experience accommodates no such phenomena. The weird monsters of modern science and the contortions of pure geometry provide no grounds for censure of the Kantian notion of a priori synthetic propositions. For the illusion of necessity inherent in propositions of this type derives entirely from our inability to form a picture of any state of affairs that doesn't comply with them. Whether this is due to our mental constitution or to the structure of objective reality or to some other cause is not now at issue (and perhaps it isn't a genuine issue anyway). What is clear, however, is that the relationships expressed by these propositions are not purely formal.
Particular statements that express impossibilities in the real world, such as "The ball is red all over and blue all over" and "The cat is in the cupboard and out of the cupboard" and "The ball is not moving up or down, left or right, back or forth, nor is it motionless", all exemplify general propositions representing the way things in the real world are not. These propositions can be re-stated (essentially by negating them) as natural laws setting out the way things are, and can only be, thus: "No discrete surface can be covered all over with more than one visible colour at once"; "No organic individual can be in two places at the same time"; "No macroscopic object can move in more than three dimensions". All such expressions, as well as all specific cases that exemplify them, may be called constitutive propositions - intrinsic if positive (seemingly tautologous) and inconceivable if negative (seemingly inconsistent) (see Endnote 1). Constitutive propositions, it could be said, frame the conditions for observation or, more correctly, for exploration, for dynamic interaction between persons and their surroundings. Or you might say they are propositions which declare the pre-conditions for thought and experience, or which depict instances of such. In §7 we will consider these propositions further, along with the (trite) question whether we should call them "logical" or not. Although it must be admitted the boundaries of this class of propositions are quite vague, there are, I think, many unambiguous examples that can be clearly distinguished from other kinds of proposition which one might also be tempted to call "logical". And despite the inherent weakness of the constitutive category, I believe much philosophical dialogue to this day labours uncompromisingly within its walls, a situation which is to some extent exemplified by recent discussion of the status of scientific unobservables.
3.1 Unobservables
The constitutive concept might appear to some to be open to broader and narrower interpretations. Thinkers in many fields and in every age have recognised in their own ways, implicitly or explicitly, the existence of fundamental principles governing our universe: Euclid with his axioms, in particular the infamous parallel postulate; Locke with his primary qualities; Newton his laws of dynamics; Kant his forms of intuition and the somewhat over-zealous list of categories; and more recently philosophers of science in their discussions of unobservables. But philosophers of science often construe the concept of unobservability so widely as to deprive many quite ordinary entities of their passports to the domain where ordinary rules hold sway. Among scientific unobservables they might well include, for example, microwave radiation and magnetic fields, just because we do not possess the sensory apparatus for directly detecting these phenomena. On this account they would also be compelled to admit that, for a person blind since birth, light (or what it reveals or seems to reveal to a sighted person) is a scientific unobservable. And I do not think they could then escape from the primitive doctrine that nothing is observable apart from "pure sensations"; whales and queens would be banished to the underworld of wormholes and quarks.
Clearly this will not do: the class of observables includes at least whales and queens, and issues arising out of consideration of the roles of specific sensory mechanisms are philosophically rather uninteresting. Thus the inclination of some philosophers to apply the "unobservable" and "theoretical entity" tags so much more readily than do scientists themselves is apt to lead to absurdities. The great majority of today's physicists will continue to pursue their investigations using instrumentation and methodologies that comply with the same conditions of exploration as those that have constrained adventurers throughout time, from Archimedes and the Argonauts to Avogadro and Apollo 11; and they will continue to interpret their results in ways that relate them intelligibly, if not intrusively, to the world of common experience. For scientists can only do what they do and find what they find - their discoveries only have scientific meaning in so far as they are useful in the real world. Which is to say, at least, that no descriptions of experimental methods or of observable effects conflict with any constitutive proposition.
This still appears, to most of us, to leave science with a formidable exotic zoo; and, even though the gates may one-day swing open, one wonders whether there will ever be customers able to pay the entry fee. But if takers there are none, Kant should feel no duty to revise his treatise. The onus falls, rather, on the shoulders of the new generation of philosophers of science to test the locks on these gates. I believe the best opportunity for rapid progress in this direction lies in acquiring a deeper understanding of the interaction between mathematics and empirical reality (§s 8 and 11).
4. DIAGNOSTIC PROPOSITIONS
Their acceptance of the analytic/synthetic dichotomy has led many philosophers (e.g. Ayer, 1973) to sanction a quite extraordinary jumble of concepts under the "analytic" umbrella. I've called attention to one of the most pernicious of these confusions - that between constitutive and verbal propositions. But what is a "verbal" proposition? Traditionally it is one which, though employing words whose meanings more often than not relate to the real world, lacks any reportive intent: its truth depends only on the supposed logical relationships of the concepts expressed, and not on the correspondence of these concepts with observable facts. This "semantical" notion of analyticity has of late come increasingly under fire - and not without cause, for it is a mixed concept and logically too much has been expected of it. I shall argue that "verbal" statements fall into two distinct categories: diagnostic, which delimit the meanings of terms, and generative, which expand or contract linguistic frameworks by providing alternative terminology. This section deals with the first category.
There are two common, but mistaken, ways in which a proposition might be thought to be "analytic in virtue of meaning". One way falsely ascribes an underlying analytic meaning to a sentence, whose proper meaning is thereby suppressed or corrupted. The other way correctly ascribes a certain meaning to (for example) the subject of a statement - that is, it verbally defines or characterises the subject - but falsely charges that ascription with redundancy, tautologousness and necessity. The first of these is a ploy frequently adopted by positivists and their heirs in respect of sentences such as
(h) The internal angles of a triangle sum to two right angles
in which, according to them, the only admissible sense of truth is the logico-mathematical sense: the sentence yields a true statement only by virtue of the axioms of Euclidean geometry (or some other geometry in which (h) is a theorem). Moreover, they say, we do not even need to invoke the concept of what we should ordinarily call a triangle - that is, a three-sided planar figure - in order to establish its truth: we can use pure algebra to guide our thoughts instead. (Notice, here, the preference for algebraic formulae over geometrical figures and the unquestioned assumption that our thoughts - whatever they may be - are better guided by that picture than by the latter.)
But now how can it be maintained that a collection of algebraic expressions (or whatever it is that makes them tick) is in any sense a triangle? In what sense is the general concept of the one equatable to the general concept of the other? One might as well ask how a receipt for ten bails of wool is to be identified with the consignment of wool itself. Clearly those who use this argument to persuade us of the analyticity of (h) are not talking about the same class of objects that we normally have in mind when we speak of triangles. They mean a certain species of abstract mathematical entity; we mean a three-sided two dimensional figure such as we might imagine to exist in real space; much as by a sheep we mean a four-legged woolly herbivorous creature. And although there exist computer models describing the nutrition and wool production of sheep, there appears to be no more and no less merit in preferring the model to the real animal than there is in preferring the algebraic formula to the genuine triangle. The difference is that we should not dream of substituting the model for the sheep (though science fiction writers might). And for the vast majority of our everyday concepts we do not entertain mathematical parallels at all. Nonetheless there are expressions, like (h), which can be construed as propositions of pure mathematics or logic, and which are usually held to be analytic. These receive consideration in §s 7 and 8.
Although perhaps reluctant to describe sheep in terms of some underlying mathematical structure, as they seem all too willing to do for triangles, still most philosophers of the analytic school would say that characterising descriptions such as:
(i) A ewe-lamb is a young female sheep
(j) A sheep is a herbivorous mammal
(k) Sheep are herbivorous
are really tautologous and consequently that they express necessary truths. While the routes taken to justify this extraordinary belief vary among its devotees, they all lead us through essentially the same Carrollian terrain, dotted with the spectral landmarks of redundancy, semantic analyticity, logical reducibility and logical necessity.
By a diagnostic or characterising (see Endnote 1) statement, in its simplest form, I mean one that expounds the meaning of a term, either determinately (i) or indeterminately (j, k); which is to say that it presents, in one way or another, one or several or all of the salient features of whatever objects fall under a given name. Even when determinate, characterisation is not a symmetrical relation: if an expression B characterises a term A, A does not characterise B. Nor is it a property of the verb "to mean" that if B means A, A and B have the same meaning.
Although no diagnostic statement purports to depict any contingent association of observable facts, it would be absurd to conclude that none has any experiential content. For we normally come to understand the connotational limits of a word just because the properties connoted do, in the real world, occur in some sort of more or less regular association. By defining the boundaries of organic processes, or what have come to be known as "natural kinds", most diagnostic propositions reflect the organisation of reality, as we perceive it and in so far as it has any practical importance for us. Their utility consists in assembling and preserving associations of ideas which tend to optimise the ways in which we interact with our environment - and this explains the tendency for definitions to become increasingly scientific. Nevertheless, while these complexes of attributes (relations etc) are in themselves of an evidential nature, the establishment of the boundary of a set of characteristics which is to be included under a given name must always be to some extent a matter of convention. Here I intend to convey no more than what is implied by the following example: I cannot tell, equipped with a certain limited knowledge of language and context, whether the statement "Sheep are herbivorous" delimits the use of the word "sheep" or hypothesises that sheep (defined somehow else) are in fact universally herbivorous. The second alternative, however, strikes me as being decidedly incoherent (§9).
On the other hand, it isn't a diagnostic property of diagnostic propositions that the classes to which they relate are organic processes: they may describe quite arbitrary collections. The crucial feature of diagnostic propositions is just that they set limits on the meanings of words, as employed in particular contexts. And it is clear that language, as we know it, would be of no service if words failed to acquire specific meanings pertinent to given contexts, no matter what the kind. As a basic ingredient of language in all its applications, therefore, characterisation is entirely independent of any particular theory of practical utility or kinds.
4.1 Analyticity
Like other descriptive propositions, diagnostic descriptions may be general or particular, and both forms are subject to a similar degree of woolliness. Thus while today both
(l) Sugar is sweet, and
(m) The Sydney Harbour Bridge is in Sydney
might well purport to be characterising, it's conceivable that tomorrow either could become material but false ("misrepresentative" is the term I employ - see chart). The possibility of a term changing its sense, however, is of no relevance to the question whether the proposition that communicates the sense of a term is analytic. Our language contract terminates with the clause that states that this is what the term means in the present context, any speculative extensions of the contract being liable to render it useless. If meaning were not both context relative and stable within a given context, everyday conversation could not exist and it's hard to imagine how we should succeed in communicating at all. Linguistic symbols, whether words or sentences, adapt themselves to particular tasks just as do the p's and q's of formal logic. It is the concept of analyticity within an agreed context which we now question.
If "analyticity" be considered a logical property, then this label is inappropriately applied to diagnostic propositions, which make no appeal whatever to logic: they are entirely independent of formal rules or axioms and devoid of any calculative means of expanding or contracting them. Nor can I begin to understand why anyone should hold them to be "reducible" to propositions of the latter kind, any more than they would think this of ordinary reportive or adventitious propositions (§6). I can only surmise that their belief has its origins in the same everglade that spawns the phantom of redundancy, which we shall now examine.
A proposition is redundant, I presume, if it is uninformative, superfluous, useless. Since it is immediately clear that this notion is accountable to the states of knowledge of the recipients of propositions, perhaps some of us would feel more comfortable if we said that "statements" or utterances can be redundant. But why should anyone suppose that a characterising statement, uttered with the best of intentions as a characterisation, is more prone to redundancy than any other kind? "Sheep are herbivorous mammals" is redundant if it conveys information that we already possess, but by the same token so is "Cook landed in Australia in 1770" and for someone somewhere doubtless so is "Grandma just swallowed a blowfly". Well, maybe redundancy is seen by some as a pre-requisite for semantic analyticity, if the latter concept is understood as implying that the meaning of the predicate of a proposition is contained in the meaning of its subject. For if a diagnostic proposition is not redundant (that is, if it does usefully characterise) it is simply false that the meaning of the predicate is contained in the subject, since the function of the proposition is precisely to attribute meaning to the subject term. But if it is redundant, it might be thought, this meaning relationship is already understood and the notion of semantic analyticity is seemingly saved.
The problem with this suggestion is that, so far as it goes, it is no less valid for adventitious than for diagnostic propositions. For the argument hinges, not on the distinction between characterisations and factual reports, but on the role of "containment". I do not believe we can ascribe a deeper meaning to this term than can be wrested from the notion of redundancy, namely that we are already familiar with what the proposition says about its subject. If an adventitious description of subject/predicate form is redundant, its subject too "contains" the predicate, in just this limited sense.
It might be protested, however, that the redundancy of "Sheep are herbivorous mammals" has little to do with anyone's state of knowledge, but consists only in the fact that nothing incidental or extrinsic is said about sheep - nothing, that is, beyond what is implied by the meaning of the word "sheep". Although I believe this claim is confused, even if it were true it would not be pertinent, since the statement never purported to convey this sort of information, but only to amplify the meaning of a term; and in this regard it is not redundant. Of course, if anyone mistakenly believed its purportment to be adventitious, he would also hold the statement redundant. I don't know what might persuade him to think this way, unless it be that language contains overtly existential propositions such as "These sheep are herbivorous mammals", whose predicate appears to be characterising despite the expectations of contingency attending the subject term. But if "These sheep are herbivorous mammals" is a fair statement (and assuming it doesn't refer to a subclass) it is indeed adventitious: it asserts that certain members of the class "sheep", which has been characterised by criteria other than that its members are mammalian and herbivorous, are now observed to possess in addition mammalian and herbivorous attributes. That there might exist other characterisations of sheep which do include these criteria is of no relevance.
One might imagine, therefore, that the proponents of the "necessity" of characterising propositions engage in some such train of thought as the following:
1. A characterising proposition is redundant because it makes no contingent report.
2. If a characterising proposition is to be held "analytic" (simplistically, the predicate is contained in the subject), then it must be redundant.
3. If we substitute an adequate definition of the subject for the subject itself, any characterisation of the subject will result in a tautology. So a characterisation is "reducible" to a tautology.
4. A tautology is a logically necessary proposition, so characterising propositions express necessary truths.
But we have seen that characterising propositions do not pretend to report adventitious facts and that, with respect to the kind of information which they do convey, there are no grounds for supposing them to be any more liable to redundancy than other propositions. Moreover it could be said of any redundant proposition of subject/predicate form, whether characterising or adventitious, that the subject contains the predicate, and to this degree one could maintain, somewhat pointlessly, that either kind of proposition is potentially "semantically analytic". And while both kinds are equally amenable to processing by formal logic, there's no reason to suppose them ever to be inherently reducible to tautologies. Not that I find this conclusion noteworthy since, if a tautology is taken to be a logically necessary proposition, I believe there exists no such species (§7).
5. GENERATIVE PROPOSITIONS - TRANSLATIVE SYNONYMY
5.1 Synonymy
It has sometimes been supposed that the concept of synonymy is pivotal to that of semantic analyticity. If it were correct to say that any two expressions are synonymous only if they have the same "meaning", this supposition might deserve attention, even though the prospects for headway in this direction appear dismal. It seems to me, however, that there are just as many basic kinds of "synonymy" as there are basic kinds of proposition: two sentences may be said to be type-x synonymous if they convey the same type-x information. But it seems to me also that none of these kinds of synonymy either expresses a genuine (let alone "necessary") equivalence or identity or plays a crucial part in explaining any other semantical or logical function. If there's a noteworthy feature of synonymy, it is just that different word-forms/sentences may represent a single meaning/ proposition, of whatever type: because success in communicating the same information in different circumstances often depends as much on idiom and perspective as on "saying what you mean". The reason why problems of synonymy, and of translation in general, arise at all has very little to do with the meanings of words and propositions, but everything to do with the who, when and where of their production. If "I'm counting on you" expresses the same proposition as "You were counting on me", then the second sentence is a correct translation of the first. And if "Ich zähle auf Sie" doesn't look like an accurate rendition of "You were counting on me", it may only be because of an initial unwillingness to untangle the confusion that results from the switch of person and tense compounded by the need to be understood by members of a different community. (We too easily assume that the German sentence must always be a correct translation of the English.) Again, if "I'm counting on you" seems not to mean the same as "You were counting on me", this may only be due to an inherent tendency to think of meaning in terms of the most general uses of the words composing each sentence. As in bygone centuries, philosophers of the twentieth have found it hard to kick the dictionary habit.
So far my own use of the word "meaning" also displays the lexicographer's stamp, and suggests an affinity, if not identity, with characterisation. The predicate of a typical determinately characterising proposition supplies the meaning of the subject; often this is achieved by means of a verbally diverse predicate and it is assumed that there is a recipient who knows how to apply the predicate but not the subject. "Meaning", in this sense, could perhaps be construed essentially as a synthesis of words, and we might have managed to grasp the concept of characterisation without ever alluding to the psychological notion of meaning. This approach would doubtless be appealing if it were possible to understand verbal diversity without reference to conceptual diversity, and if it were not the case that (as I think) archetypal characterising propositions do not have verbal predicates at all, but teach meaning ostensively. Thus although the meaning of "meaning" may remain vague, it is not so vague as to prevent us from seeing that characterisation involves more than mere verbal juggling, and depends for its success on some relationship (whatever it may be) between words and objects or ideas.
The concept of translation, on the other hand, pertains to a class of propositions which have no concern with meanings but speak only about the interchangeability of linguistic symbols between occasions on which they may be usefully produced. For the sake of economy and clarity I shall consider only word-for-word synonymy. In "Every snow rose is a helleborus niger" we have already encountered an example of what might well be taken to be, in a suitably bland context, a statement of unblemished synonymy - that is, an identity of verbal function in which neither term expounds the meaning of the other but, rather, both terms are tokens of the same word. To say that the proposition expresses only sameness of word is to suggest that the form "snow rose" = "helleborus niger" is an appropriate representation; while (for example) the inappropriateness of "open" = "not closed" reflects the fact that "If and only if anything is open it is not closed" is not about word usage but about experience. The first proposition differs from the second in that the question whether we can imagine a state of affairs that doesn't comply with the proposition does not arise. It also differs from characterising propositions such as "Ewe-lambs are young female sheep" because the predicate does not elucidate the meaning of the subject. In fact it's possible to invent translative synonymies in which neither subject nor predicate has any assigned meaning: which is to say that the only "meaning" (utility) possessed by the synonymous terms is that which ensues from their interchangeability in some context.
Now if synonymy could be explained entirely in terms of syntax, irrespective of the objective meanings of words, the perception that there's a special problem of synonymy would surely be less prevalent. For to my knowledge most theorists in this field ride quite happily with the general notion of sameness and recognition which underlies not only the possibility of communication and argument but the whole gamut of interactions with our environment. They do not question, for example, that in the formula q  (pq) the second occurrence of q represents the same object as the first occurrence of q, or that various auditory renditions of the formula, such as "Kue entails pee entails kue" (which may well be how it automatically registers with them), function in precisely the same way as a variety of printed and hand written versions, which in turn function in the same way as one another.
But the concept of translative synonymy appears to be exactly of this kind. The analogue of conversational synonymy in formal logic is a so-called "defining" expression (actually a prescriptive translation) such as Q =def q, which decrees that the symbols q and Q are to function equivalently. One might re-phrase this: Q =def q merely instructs us to broaden our field of recognition so as to include capital as well as small q's among the symbols that are to count as q. The formula could as well have been written "Q" = "q". There seemed to be no need, however, to provide the instruction "q" = "q", as it was taken for granted that our early education furnished us with at least the capacity to recognise all occurrences of signs of that sort sufficiently well to appreciate their sameness and to distinguish them from signs of other sorts. And "q" = "q" is clearly absurd if it's meant to inform us that successive uses of q are uses of the same sign, since the defining expression itself contains successive uses of q. Likewise in botanical parlance we have no need of "snow rose" = "snow rose", nor of "SNOW ROSE" = "snow rose", but we might well need "snow rose" = "helleborus niger". For unless Linnaeus was our father we did not learn that when we were four years old. But we could have done. After all we did make the far more remarkable discovery that the graphical symbols C-A-T represent the same word as the sound kat. Until then words were just sounds. Yet for most of us now, any attempt to formally capture that identity would seem hardly less absurd than it was in the case of "q" = "q".
The fact that translative and characterising propositions have traditionally been grouped together in the analytic category partly explains the persistent tendency to confuse them. A more basic reason for the confusion is that most translative propositions express equivalences, a peculiarity which hides the logical properties of translation in general and may lead one to commit the error of thinking that simple translations are really determinate characterisations, or vice-versa. This common mistake can often be avoided by considering the effects of syntactical devices such as quotes and quantifiers. (For example, to say that all sheep are animals is not at all the same as saying that whenever we use the word "sheep" we could use "animal", since, going by dictionary meanings, not all statements of fact about sheep are also true of every animal. Yet there may be circumstances in which "animal" is a perfectly adequate translation of "sheep".) In classical (but, I think, woefully regressive) terms the difference between the two types could be summarised thus: while characterising propositions range over the objects of language (the things we talk about), translative propositions range over linguistic contexts and statement situations.
Though permeating every fibre of language, the concept of translative synonymy rarely finds overt expression in propositions. One reason for this sparsity is the exceedingly casual nature of everyday conversation. We have seen how the repetitive use of "identical" signs, whether in formal or natural languages, is taken for granted, and how the attempt to crystallise this usage in definitions would be unfeasible. Similar considerations apply to signs which are less obviously alike but which are used in parallel ways in different linguistic environments. We just do use them one for the other without fuss as occasion demands, and that is what constitutes their synonymity. To formalise this usage seems impossible because in the act of formalisation we make a deliberate attempt to alter the nature of the beast: we imagine ourselves to be exhorting the signs to behave to order, so to speak, even though they may not, and even though there can be no guarantee that our directive will be heeded in future. Synonymy is a relation we can understand and use but cannot dictate. And where else, in all this, can we detect any evidence of structural perfection, of necessity? The informality and logical weakness of the concept is epitomised in the thought that translation consists in nothing more than this: we use a spade to dig the garden, but on occasion we may use a cultivator because it's more efficient, more appropriate or just more fun.
To use words, phrases or sentences synonymously, then, is to count them as being the same verbal sign, and if there are clouds veiling this notion they are no darker than those surrounding the mystery of how we come to use any expression repetitively. Why, then, do I consider translative synonymy to be of fundamental significance? The reason is that it holds not only the keys to the doors of language synthesis and enrichment but the pestle to crush a popular view in the theory of meaning and truth, namely that propositions necessarily possesses distinct extensional and intensional aspects.
Since translative synonymy implies verbal correspondence without consideration of existing meanings, its effect is not to relate word meanings within the current language frame but to modify the language frame itself. Aside from obligatory applications such as change of tense, there are essentially three directions which translation can take: (1) it can convert an expression in the language we are using to an analogous one in a substantially different ethnic language (which, presumably, we propose to use); (2) it can expand the language we are using by enlarging its vocabulary; or (3) it can restrict the use of an expression to specific contexts, thereby helping to define the boundaries of a sub-language or "language game". In each case we find ourselves using a language that is new and different from the original, even if only in a small way; so this type of activity, if successful, is innovative and generative - not of ideas but of uses of words. Thus translation, like characterisation, is not a strictly reflexive relation, and we cannot properly claim that two words are synonymous, in this sense, just because either one can be freely substituted for the other. We might say, rather, that translative synonymy consists in the fact that a word in a home idiom has its analogue in a target idiom. However, the target idiom may comprise no more than the home idiom plus the analogue.
5.2 Intension and extension
Because synonymy is not only language-relative but language-game-relative, the verbal economy and expressive richness of language both owe much to that concept. Translative synonymy ranges between more or less universal (I should rather say encyclopaedic) and extremely domestic contexts, providing language with an additional, exceedingly fertile dimension. For nothing could be more stifling of progress in linguistic philosophy than the belief that words and propositions stand for completely fixed concepts, or uniquely represent fixed atoms and molecules of reality. (Beliefs of this kind have led to much redundant philosophy even by those who have tried to overthrow them, for example Wittgenstein (1953) on "language-games" and, in more popular vein, Pirsig (1974) on "Quality"). No language is a single, static entity with defined boundaries. Language comprises an indefinitely large, continuously evolving array of overlapping idioms, almost every situation requiring a different one, and of which different ethnic languages are just the extreme cases with least overlap. Through the creative process of translation, we discover new uses for old words and new words to fit old uses. But while the latter aspect may be of greater interest to linguists, it is the former that is philosophically the more important.
The elementary observation that one and the same word can have extremely narrow, as well as broad, applications supplies the principal means to eradicate one of the most persistent myths of philosophy, ancient and modern, namely that we necessarily comprehend the world in terms of two distinct kinds of entity corresponding to two distinct logical parameters: on the one hand universals or attributes (the sense or intension of an expression) represented by logical predicates or relational functions, and on the other particulars or individuals (the reference or extension of an expression) represented by individual terms or variables ranging over the elements of a domain. For it seems to me only that we use words sometimes in a more encyclopaedic and sometimes in a more domestic fashion and that this largely explains the intension/extension dualism. Consequently, though undoubtedly convenient, this division is entirely relative and of extremely limited metaphysical and logical significance.
This same viewpoint can be reached from another angle. What has been said about word-for-word translation applies equally to propositions. While compound propositions that relate other propositions to one another translatively are extremely rare, two simple propositions such as "In Australia the sky's the limit" and "Australien ist das Land der unbegrenzten Möglichkeiten" may stand to one another in the relationship of translative synonymy. A complication appears to arise, however, in the case of propositions which, though essentially translative, include overt or tacit reference to the circumstances of their production: for example, "I have a headache" and (said by you to me) "You have a headache", or "It will rain" and (said later) "It's raining". Although in each example both utterances may in fact be making the same statement, one might think that they have different meanings. From the perspective of someone outside these situations, of course, they do have different meanings, different uses (as a consequence of which they translate differently into German, for example). But philosophers have tended to place too much importance on this point, even to the extent of regarding context-dependent language as logically untidy and replacing it by "eternal sentences". To my mind, both eternal sentences and uniquely referring sentences are mythical entities; the progression from narrower to broader perspectives is a gradual one, any profit realised by distinguishing between sense and reference deriving only from this process of linguistic re-orientation.
6. MATERIAL AND MENTAL PROPOSITIONS
Language comprises, pre-eminently and primordially, a hard core of adventitious, material (or physical) propositions, the bricks and mortar of everyday conversation and of all science. Even though most material propositions are instantly recognisable as such and uncontroversial, the boundaries of this class remain vague, no satisfactory account of the criteria by which their truth is to be evaluated having yet been devised. Nor will any be offered here. My chief purpose is to sustain the "common sense" view of existence as promoted, for example, by G.E. Moore (1953), and doubtless embraced tacitly by the vast majority of ordinary people. According to this view, the world consists primarily of two distinct types of entities - material objects and "ideas" (or "acts of consciousness"). One might therefore expect there to be two types of proposition, material and mental, dealing with entities in each category. Support for this outlook comes both from its practical indispensability and from the fact that historically much of the labour of philosophy has been aimed at bridging the gap between these two realms, so far with little success.
It is probably misleading to suggest that there exists a "common sense" notion of mental propositions, since very few people are disposed to think objectively about matters of the mind. Mental experiences come in a very wide range of colours, encompassing such varieties as bodily sensations, percepts, creative concepts (including uses of language), emotions, memories, hallucinations and so on, and the boundaries between them are exceedingly hazy. I, for one, find it difficult to make head or tail of this lot, so in what follows I shall consider mental propositions relating to only two kinds of experience - bodily sensations and percepts (predominantly sensory information originating outside the body).
On the other hand material propositions, though just as diverse, readily succumb to a "common sense" point of view. This rests on the belief in a world of physical phenomena which are either "directly" detectable or whose existence is inferable from observational evidence, and which exist independently of any person's awareness of them. The strength of this belief is evinced by an unquestioning reliance upon one's personal capacity to relate to and interact with this world and these phenomena. Thus an elementary material proposition, such as "There are cod in the river Murray", is simply one which, if true, is representative of some fact which is presumed to exist in the real world whether witnessed or not. (I won't indulge in speculation concerning the existence of objective facts about which no one ever could, even "in principle", acquire observational evidence.) In addition to elementary propositions, however, there are others, such as "If the rain stays away the Socceroos will win" and "The cat's in the cupboard and share prices are up in Tokyo", which appear to be composed of propositional elements joined by logical connectives. I shall propose in §7 that these are either wholly experiential or else not propositions at all.
Even the most rudimentary sorts of material proposition, however, seem to perform a variety of functions. For example, one might distinguish between the following kinds:
(1) incidental - "The kangaroos are eating the avocados"
(2) attributive - "This sheep is a purebred Merino" (given that "purebred Merino" here behaves descriptively and not as a term whose meaning is being delimited by exemplification)
(3) actualising - "Non-polluting automobiles exist"
However, the differences between these kinds are quite innocuous and they are readily interconvertible (considerations of time and generality often decide which form is the more appropriate). I draw attention to them chiefly because of the risk of confusion with non-material types of proposition (e.g. attributive with characterising) .
While some philosophers have sought to defend material propositions against the inroads of metaphysics, others have contrived to distinguish them from statements about purely mental phenomena and so, perhaps, to repudiate those doctrines that embrace some sort of reduction of physical to mental processes or vice-versa. It seems to me that the "pre-scientific" body of metaphysics which properly educated people recognise as consisting of nonsense and falsehoods has received due punishment, to the detriment of much needed honest endeavours in "post-scientific" speculative metaphysics. In what follows, therefore, I wish chiefly to add my support to the physical/mental distinction, though not by assent to the hypotheses apparently most favoured by academics.
6.1 Investigability (verification)
Although the layman may well feel no difficulty in distinguishing his own thoughts from the facts of the external world, a little reflection suggests it is a fuzzy line that runs between the collection of propositions variously described as objective, physical, material, external, independent of consciousness, public, investigable or fallible; and the collection described as subjective, mental, private, internal, immediate, basic, uninvestigable or incorrigible. Using familiar arguments somewhat sparsely and in piecemeal fashion, I shall attempt to dismiss many of the shades of distinction implied by this colourful assortment of adjectives, beginning with the doctrine of empirical verifiability or, as I prefer to call it, investigability. In using another name, it was my optimistic intention firstly to encompass classical verificationism and all its children (mostly born out of indiscretion, I dare say) and secondly to emphasise that the attempt to verify (or falsify) is suspect, as much as the possibility of success. I take it that underlying the various forms of physical verificationism there's a vague principle which seems to rely on the assumption, roughly, that worldly facts can be observed only by worldly means, and if this principle founders then of course the entire brood goes down with it - notwithstanding that there may be many other reasons for thinking any particular version of the doctrine objectionable.
The usefulness of the investigability principle might well be questioned on account of the great diversity of propositions whose objectivity and truth assessability we are willing to respect. We need only reflect upon the pragmatists' notions that we buy truth "on credit"; that much of our objective knowledge is second hand; that propositions are often the only evidence we can muster for the existence of the facts they depict; that most of them occur as unique utterances whose truth we take for granted; and that many quite ordinary looking propositions appear to be truth functionless and consequently, some would say, unverifiable. In this disarray, however, we shall find no good reason for rejecting the investigability criterion. Nor should we be unduly troubled by the threat to verificationism of the alleged theory-dependence of scientific facts (a grossly overworked tale), or by the charge that positivist verificationism suffers irremediably in the hands of positivist antimetaphysicalism, or by the more explicit tenet that no proposition can be confirmed by particular experiences owing to the universal nature of its descriptive terms (a suggestion that begs the question whether language can ever speak of particulars). Finally I shall ignore the application of investigability to universal hypotheses, as it seems to me that these are not, when taken literally, sensible propositions at all (see §9).
No, the reason why the investigability principle cannot advance our present purpose is just that it lacks any independent explanatory power. Because the world of objective facts is precisely the world in which people live, it's tritely true to say that a fact might have been verified had someone ("the resourceful observer") been in a position to investigate it. But to say that that is what constitutes its being an objective fact is circular, since the proposition that someone is in a position to investigate a fact of this sort expresses a fact of a similar sort, and must be similarly amenable to investigation. The possibility of verification, corroboration or falsification is contained within every presupposition of objectivity, and cannot prop it up from the outside. Verification moreover implies evidential checking of either a pre-existing factual statement or a pre-existing hypothesis or belief. But many propositions are individual reports which either defy the possibility of, or carry no expectation of, subsequent confirmation: the witnessing of the fact precedes the statement, the statement merely reports the fact as experienced and the experience does not verify the statement.
Furthermore the supposition that objective statements must be capable of objective verification would be pointless unless they were in need of verification. An implication of the verification hypothesis, therefore, is that such statements are wanting - that they somehow lack meaning and truth unless verified. Supposing you told me that yesterday you shook hands with the Prime Minister and I said I believed you. What, exactly, would I be believing? If the statement were in need of confirmation, not only would it be unbelievable: it would be incomprehensible. But while it is undoubtedly a requirement of certain "difficult" propositions that they be evidentially checked to bolster their credibility, this is by no means true of the vast majority of factual propositions, which must and do stand on their own merit. Reliance on personal reports is an uneliminable feature of dialogue about the outside world. And although, at the sub-philosophical level, appeal to investigability is often necessary when the practical and social consequences of material statements are significant (I make ample use of the criterion in this article), as often as not the events recalled by such statements have archival or narrative value only.
Still, verificationism has done a great service to philosophy, and its enthusiasts probably discarded it too easily under criticism, not all of which was justified. Its claim to distinguish between meaningful and metaphysical statements was not pushed far enough and I believe it had more substance than is now usually accredited to it. For example (and in absurdly simple terms), electrons, radio waves and gravity are meaningful concepts, if not "real" entities, because they have highly predictable, useful effects in the real world - television, computers, nuclear power plants and space missions would be impossible without them. On the other hand God the holy ghost, demons and souls are metaphysical because they have no predicive power atall. But verificationist theory became too cluttered with ifs and buts. I now sometimes wonder whether all this complexity could be replaced by a simpler idea. If material propositions seem to be endowed with some kind of necessity or certainty, doesn’t this ensue from the conviction that the facts related by historical propositions cannot be undone? Captain Cook has sailed and that's that. Electrons have worked miracles for medicine, who can deny that? Perhaps the verification hypothesis is essentially just a way of saying “Let’s wait and see what time tells”. Isn't that good enough?
6.2 Private and public
Even though the investigability method of truth determination be rejected, one might still think that objective or "external" propositions must be publicly accountable, in a way which apparently distinguishes them from subjective or "internal" propositions (such as "I have a toothache", as well as "You have a toothache"). Every internal proposition, it might seem, can properly be confirmed or denied only by the particular individual to whom it relates. Other people must judge the proposition on the testimony of the subject undergoing the experience, or else by physiological or behavioural investigations which might allude to, but seemingly cannot ascertain, its truth (see 'Mental and physical' below).
That a person can have a toothache and refer to just that experience by the statement "I have a toothache" cannot be seriously questioned. Nor would it materially alter the nature of the case if it turned out that some people have some internal experiences in common, whether through extrasensory perception or, conceivably, because their nervous systems have been connected in such a way that two or more brains register the same impulses. In a certain sense it seems possible to have someone else's toothache. But in addition to propositions referring to these bodily experiences, this class might be thought to include others, each of which refers to an evidently physical situation observed by a particular individual but which could not possibly be observed by any other individual. Thus the proposition "When I went for a walk by myself in a remote forest on this date last year I picked a wild apple from a tree and ate it", even if somehow publicly useful, is arguably not even in principle representative or misrepresentative of any publicly accessible fact; this statement, it might be thought, reports a private fact, although of course (no less than "I have a toothache") it is couched in the terms of a public language. Besides these, however, a much larger number of factual reports might be said to be "contingently private", either because they just happen to represent unique observations (of publicly accessible facts) by isolated individuals or because the presence of other onlookers is inconsequential.
This line of thought advances the view that the integrity of material statements is not compromised by lack of public access to the particular facts that they convey: there are statements which provide objective information; which are private in a similar sense to those which record immediate bodily sensations; and which earn instant recognition as describing facts in the physical category and not in the mental category. The objectivity of the material statement is thus independent of the number of people able to substantiate it - much less the number who happen to hold it true (for nonsense and falsehoods are quite as popular as truths). The number of person-observations required to establish a fact depends on the complexity of the fact, its "historical background" and relation to other facts, the difficulties encountered in assessing it and the kinds of project which it is expected to service. Over and above that minimum number, which in the majority of instances is just one, additional witnesses can serve only to vouch for the integrity of previous observers and not the nature of what is observed. Other people do not play an important role in determining the objectivity of our experiences: quite obviously, most of the time we are very capable of judging for ourselves and of acting accordingly. So, although it must be conceded that much of our objective knowledge is acquired through trust in reports, witnesses of reports and witnesses of witnesses, our basic concept of the material world includes the idea of a public domain and does not rely on it.
The distinction between subjective and objective propositions is possible, one might surmise, because often there's a noticeable time-lag between the publicly accessible cause of an experience and the private effect - the having of the experience itself; thus it seems possible to refer to the psychological effect without implicating the cause, and vice-versa. And one might be inclined to assume that these two aspects always do retain distinct identities, even when there's no discernible interval separating them. On the other hand it's also possible to herd causes and effects (if that is what they are) into one fold, so to speak, though the name of the fold depends on the manner of herding. Suppose, for example, that whenever one saw a black cat one immediately felt a sharp pain in the head. Wouldn't one then call black cats "headachy" cats, thereby bringing the headache into the same objective camp as the cat? Maybe, or maybe not: depending, perhaps, on factors such as the ubiquity of the black cat/headache phenomenon and the degree of spatial associativity attaching to headaches under these hypothetical conditions. But whether or not there's a delay between physical cause and mental effect, it's possible to regard sensations and affections as being public and investigable in the following way:
(o) I enter a room full of smoke; next day I have a headache. Whoever enters this smoke-filled room will before long have a headache. (This is a headachy situation.)
(p) I look at the sun; I'm dazzled. Whoever looks at the sun is dazzled. (The sun is dazzling.)
(q) I look at a poinciana bloom; I have a sensation of orange. Whoever looks at the poinciana bloom will have a sensation of orange. (This poinciana bloom is orange.)
(r) I look over there; I experience an image of a tree-like form combined with sensations of green, grey and especially orange... Whoever looks over there will experience an image of a tree-like form combined with sensations of green, grey and especially orange... (The poinciana is in flower.)
Despite the traumas suffered by language here in performing the duties required of it, the ease with which one can draw up lists of graded analogies of this sort appears to provide considerable support for the view that if we are to make a clear distinction between subjective and objective propositions it will not be on the grounds that only the latter are public. And, as we have seen, nor will it be on the grounds that only the former are private. Both types appear to have both public and private aspects.
6.3 Immediacy and incorrigibility
Still, one might think, mental phenomena surely convey a sense of immediacy and incontrovertibility which is absent from physical events. This notion presumably supplies the incentive for the theory that material propositions can be reduced to, or constructed from, "basic" (or "protocol") statements describing personal experiences: that is, a statement such as "The poinciana is in flower" can somehow be translated into logically equivalent assemblages of statements about private sensations, which (it may also be alleged) are "incorrigible" in the sense that we might suppose "I'm in pain" to be; and therefore, possibly, that a statement which can be doubted is reducible to statements which cannot, at least by someone at some time.
This is a point of view to which the examples (o) to (r) might be thought to lend some credence. However, I presume the first (reductionist/phenomenalist) aspect of this position is no longer widely considered to be defensible, for well-known reasons. But even if reductionism were tenable, it could not achieve its apparent objective of transforming a statement which is allegedly on probation into ones which are allegedly unimpeachable. For if the original material statement is dubitable and its reduction indubitable, then its reduction is surely incomplete: while it may take account of sensory experience and perhaps spatial relations, it doesn't properly accommodate a variety of other relations and external factors. And if it did, we should soon see that the original statement and its translation are incorrigible (or not) to the same degree.
If feeling certain has anything to do with certainty, however, I'd be inclined to dismiss any further discussion of this point, since, while I write this essay, I cannot assent to feeling any less certain that there's a flowering poinciana outside my office window than that I have a headache. Nothing seems to be gained by rebutting with arguments about hallucinating or anything else designed to destroy my confidence about what I am perceiving. In this instance, anyway, no one could convince me that my experience is not of a genuine physical object. They might persuade me that I have incorrectly identified the tree or misnamed the genus, or that the poinciana is a fake - but not that it has no material existence. If I'm dreaming now, then life is nothing but a dream. Life depends on certainty at this level - and death frequently results from the imprudence that spurns it. But while I'm disposed not to question whether there exist certain experiential facts which I believe the statements "There's a flowering poinciana outside my office window" and "I have a headache" adequately describe, it appears to be quite another matter whether these descriptions too are reliable. In fact this plea for certainty has only a very peculiar point to make, namely that its claims should remain dependable even when inexpressible.
It is of course possible that another person, sitting next to me in my office and apparently looking in the same direction, should deny the existence of the poinciana. And although in that case I'd spare him not a little more scorn than if he had denied the existence of my headache, still, I would accord him the right to doubt whether I have a headache, and I ought to accord him a similar right to doubt that there's a poinciana over there, even when I feel certain there are no factors that might lead him to interpret the situation differently from the way I do. It is indeed central to my overall thesis, as I submitted in §3, that two apparently incompatible material statements (such as "There's a poinciana growing a few metres away from this window" and "There's no poinciana growing a few metres away from this window") are not logically contradictory but only experientially opposed: which is to say that although we may not now be able to imagine any circumstances in which both are true, the possibility of such circumstances arising remains open. But the assertion of this possibility carries no weight, it might be thought, unless it can also be claimed that both statements are infallible, as otherwise one could presumably find cause for discounting one of them.
This expectation of a cast-iron guarantee for the truth of the verbal concepts related by constitutive propositions is, however, just as gratuitous as the penchant for logical necessity which it replaces. It is thus not my view that there exist propositions which are incorrigible in some favoured, bullet-proof sense, but only that in the personal confidence stakes some of those we take to be objective are at least as safe a bet as the most intransigent of the subjective kind. Even without the buttress of incorrigibility, the consequences of this position are significant. For if some of the facts we normally call "physical" are just as indubitable as many of those we call "mental", then neither the conventional distinction between these categories nor the classical relations among physical phenomena can be upheld. We shall find ourselves unable to agree upon a method of deciding which of two or more incompatible sets of observations records the true state of affairs: which is only to say that no classical state of affairs is represented by all the observations jointly.
Karl Popper (1959) crisply dismisses the quest for verbal incorrigibility with the proposal that, because the terms employed in all statements are universal, our knowledge of particular facts can never justify the truth of any statement. At face value, this appears to imply more than perhaps it ought - that language by its very nature can never get to grips with particulars, that it can't really talk about particulars at all. Although this seems quite wrong to me, still I suspect the point Popper intends to make here is broadly correct - a point which crops up everywhere in different guises and which I touch upon in §9.5 and §9.6 in discussing attitudinal statements and fiction. In the present context perhaps it could be put like this: it is only in the act of contemplating our experiences that we face the problem of incorrigibility, a problem that seems to me to disappear as soon as it is confronted. For, the moment we start to think about our experiences, we create opportunities for doubting them. Even if we admit the possibility that thinking about experience doesn't entail using linguistic concepts, it surely involves a generalising faculty of some sort - the capacity to recognise particulars as being of certain kinds. And it seems to be of the essence of recognition that it leaves room for doubt. I think this is also the main point made (in a much more roundabout way than Popper) by Austin (1946). For my part, I'm fond of the two epigrams: No perception without conception and No pain without brain, but these imply nothing about certainty.
In so far as they capture some condition of the mind, all propositions might be thought to contain elements of "incorrigibility". I remain puzzled by claims that these elements are utterly indubitable. If we concur with the view that language contains expressions which are bound this tightly to experience, we shall surely be driven to the further conclusion that we cannot even have the experiences which the incorrigible propositions depict unless we do formulate those propositions; which is to say that whoever cannot speak, or, at least, think in broadly linguistic terms, has none of the common feelings and experiences which the rest of us say we have. For my own part (as a chiefly non-verbal dreamer) this conjecture seems preposterous: even if, as I think, perception is impossible without conception, experience is by no means impossible without language.
Many other objections have been raised against incorrigibility, both experiential and verbal. Some philosophers claim we can be mistaken about our own conscious states for reasons that include misunderstandings about how we really feel, inconsistencies between our apparent intentions and our behaviour, and plain self-deception. To my mind most of these arguments either miss the point or they are radically mistaken. In the end, it seems enough to remark, first, that most if not all of our sensory information is "interpreted": we become aware of it only after it has been processed by machinery and supplemented by data which were not acquired along with the (alleged) "bare" sense impressions. And secondly that no propositions can be formulated in the absence of such interpretation. It seems far from certain that even a mental grimace could be taken to be an "incorrigible" expression. Assuming it could, however, the next question would be whether "I have a crashing headache" says anything more than the grimace. If it does, I can doubt it. But it seems clear that every factual proposition, private or not, mental or physical, is in this position. The notion of incorrigibility cannot supply a criterion for differentiating between types of experiential proposition, since none possess that property.
6.4 Mental and physical
Then what grounds are there for the popular distinction between statements about our own experiences and feelings and statements about material objects and events? It might be helpful to return to a simplified version of the table used earlier on:
....................Objective language:.........Subjective language:
Mental.......A. This is a headachy..........B. I have a headache
topic................situation
Physical...C. The poinciana is...............D. I experience an image of
topic................in flower..................................grey, green, orange etc.
Although language can be modified according to whether we wish to emphasise the public consequences of a subjective experience or the personal aspects of an objective phenomenon, in practice it's difficult to imagine circumstances in which there's any more use in replacing B by A than there might be, for example, in going full circle with C and D by substituting "This is a flowering-poinciana-image-causing situation". The ungainliness of the latter expression doubly underscores the fact that excellent linguistic mechanisms have evolved to deal specifically with mental and physical topics, reflecting the natural division between these types - a division which to my mind (but apparently not to every one's) seems quite adequately exemplified by the glib remark that it makes sense to say "I was bashed on the head but was unaware of it" but never to say "I had a crashing headache but was unaware of it".
It is, of course, a stubborn peculiarity of mental phenomena that, in the conventional domain of which we speak, they cannot be properly described out of sight of physical phenomena. Every mental statement apparently refers to a mental event, implicitly or explicitly bound to an individual, a time and a situation. And although it's by no means clear that there are no mental phenomena that are not confined in this way, the language of western culture at the present time barely entertains this possibility. The role played by these physical accessories in simple mental propositions, however, appears to be purely presuppositional: for example, in "Lucy has a headache" the issue is whether or not Lucy suffers a headache, and not whether it's Lucy or Len or some one else that has the headache. On the other hand many conventional experiential propositions assert, rather than merely presuppose, physical facts in parallel with the mental. It can hardly be doubted, for example, that part of the propositional purportment of "Lucy had a headache while she was at the farm today" is the fact that Lucy was at the farm today. So the proposition could only be regarded as wholly true if (1) Lucy was at the farm today (which is apparently physical) and (2) Lucy had a headache (which is apparently mental) and (3) she had the headache while she was there (which alludes to the puzzling link between the two types).
One might suppose that the important question here is: how is it established that Lucy's headache occurred at the alleged time and place? - anticipating that the problem might succumb to either a wholly material or a wholly mental solution. If Lucy herself was aware that she was at the farm today whilst having a headache, and a personal statement from her is all there is to go on, one might get the impression that the latter is wholly mental, since it is only in virtue of her own psychological cohesion that the proposition has sense. This response, however, is at odds with my conviction, alluded to earlier, that the one-off reports of individuals about their experiences of reality are self-sufficiently physical. (Which is only to say: Lucy is just as capable of objective observation as anyone else!). That being the case, if Lucy utters the statement in question it is irrevocably mongrel in character.
If, on the other hand, time and place are determined empirically and independently of input from Lucy, and the presence of the headache is similarly established, for example by means of behavioural or physiological observations, then the statement that Lucy had a headache might seem to be entirely material, bearing no reference to any person's feelings. Here again, though, I very much doubt that the immediate method by which the presence of Lucy's headache is established determines the quality of the proposition(s) reporting that fact. For there appears to be no real difference in the quality of the verdict no matter how it has been reached. That is, there's no palpable difference between the statements "Lucy has a headache" (this having been established by scientific investigation) and "Lucy has a headache" (because Lucy said so). Surely the proposition (that Lucy has a headache) is mental even though it has been inferred entirely from the results of empirical investigation. Nor is the diagnosis any less reliable just because it isn't the direct report of the subject. Given an adequate scientific and communicative protocol, one might well find reason to place more trust in a judgment about the subject's state of mind inferred from the record of an independent scientific investigation than in the subject's own report. Thus the immediate methodology alone doesn't determine the quality of the proposition. But this conclusion is of little significance, as it's obvious that the immediate (empirical) methodology alone never does supply the entitlement to infer anything about anyone's state of mind. This entitlement derives, rather, from the combined historical power of an elaborate scientific wisdom, close communication with other people and personal experience of phenomena such as headaches. There's more to methodology than meets the eye. (Also see §9.5 on attitudinal statements.)
6.5 Space
I might have begun this discussion with the naive proposal that physical objects and events are extended in space while thoughts are not. Today, I presume, almost no one would disagree that this embodies a serious misunderstanding of the relationship of these categories. The amusing regalia of modern technology have conspired to ensure that there are few who cannot interiorise spatial ideas to whatever degree suits their purpose. It is not just that we can speak as if external phenomena are internal, but that we can, at least when in a state of passive observation or aesthetic reverie, mentally align physical objects and events with our most immediate sensations and affections. Surely it's enough to remind ourselves how easily we can be tricked into believing that certain phenomena have concrete existence, in a sense in which they really do not. But while the senses of sight, hearing, touch, balance and so on supply us with particular impressions of space, which seemingly bear no relation to one another except that they are taken to represent aspects "of the same space", and can individually mislead us with regard to the presence of objects or the occurrence of events in that space, we might find it more difficult to imagine how we could be fooled by the coordinated stimuli of all or most of the senses acting concertedly during a period of purposeful interaction with our surroundings (or apparent surroundings!) Though I can't help thinking the boffins of "virtual reality" would rise to this challenge, I doubt that in the long run the "virtual" is eliminable from the worlds of their making. (Will the virtual grenade ever kill?) When we include in this vista the whole milieu of accumulated effects and expectations that contribute to our knowledge of the physical world, between womb and grave, nothing seems capable of shaking our faith in the exclusiveness of that category, even though from time to time we may err about what properly belongs in it.
6.6 Personal interactivity
It is, therefore, our personal appreciation of the nature of the subject matter of which experiential propositions speak which supplies the grounds for distinguishing objective from subjective types. How do we arrive at this understanding? Although the classical idea of public verification has little bearing on this central question, we can surely put into service the notion of immediate or short-term personal verification - a kind of almost instantaneous or ongoing accreditation of objectivity achieved by the cerebral integration of stimuli produced through the interaction of one's body with its environment. (In short, people know what they are doing, and that's enough!) This, however, is the fountain of our objective knowledge and not its ratification. Granted such a diminutive suggestion would be of little worth in celebrating the acquisition of scientific knowledge. But then philosophers of the twentieth century empirical tradition have fostered overly credulous sentiments concerning the relevance of science to the concept of existence and the formation of objective values - sentiments with which scientists themselves rarely consort.
Are explanations of this kind what we are really seeking? If people almost never report any difficulty in distinguishing personal experiences from objective events; if there just are these two camps lying on either side of the boundary between oneself and the world in which one lives, moves and communicates; and if all we need is a nudge now and again to correct our terminology when speaking of them; then perhaps philosophers have been asking all the wrong questions. Until someone with a more inventive mind can make us see the problem from a radically different perspective, I think we must content ourselves with the bald conclusions that there are material and mental phenomena and corresponding statements which refer to them; that the phenomena in the former category are distinguished from those in the latter primarily by our personal, often immediate, comprehension of their relationship to the environment with which we interact; and that the two kinds of proposition cannot be distinguished in terms of verifiability, fallibility, extensionality or public accountability. Ultimately the factors that count for most in communicating either kind of information are personal interrelationships founded on integrity and trust. Somewhere in this forsaken territory lurks the essence of reality, and finding it remains one of philosophy's most challenging quests.
7. SOLUBLE PROPOSITIONS - PURE LOGIC REJECTED
7.1 Interpretations of formal logic
The backbone of modern logic is an uninterpreted algebraic syntax, some of whose formulae are soluble, which is to say, in classical terms, that they are either tautological or inconsistent or that they can be reduced by a specified calculative procedure to one of the values 1 or 0. Logical atomism and positivism, with their notion that the variables - and only the variables - of this syntax can be replaced by unanalysable propositions representing basic facts in, or impressions of, the real world, have promoted the belief in a system of reasoning that retains its logical rigorousness even when interpreted. Thus it's fashionable to think of "ordinary" logic roughly as comprising:
1. unassailable rules of thought or canons of pure reason, faithfully represented by
2. symbolic expressions which expand "analytically" (or "constructively") from more or less classical axioms or in accordance with classical rules of computation, and
3. the logical features of whose formulae remain intact when appropriate meaningful expressions are uniformly substituted for variables of the formulae.
I shall refer to formal systems which operate in all three modes as formal but "topic-neutral" (a term borrowed from Gilbert Ryle), reserving the names pure and systematic for systems whose interpretation (3) is supposed to be confined linguistically to (2) and ontologically to (1). Which is to say that the variables in a system of pure logic may be substituted only by appropriate well-formed expressions of the system itself, while the connectives are interpreted in overtly logical terms, implying, presumably, that they stand for the basic devices of reason and that we reason correctly only if the devices we use conform to the "analytic" conventions of the calculative structure. I shall use the term formalist for any kind of allegedly analytic, symbolic system which is either reckoned to be ontologically vacuous or whose "ontology" is confined to well-formed formulae of the system. In what follows we shall largely be concerned with questioning the worth of each of these three perspectives.
We shall assume, first, that we can think logically, regardless of the status or derivation of this talent. It is widely acknowledged, though - one might be forgiven for thinking - scarcely believed, that the syntax of classical logic is open to non-logical interpretations, or to no interpretation at all. The set of procedures which I often adopt to solve problems in formal logic, for example, treats standard propositional calculus as if it were a binary modular arithmetical algebra (see Endnote 2). One of the several advantages of this method is its psychological friendliness, in that the elementary arithmetical calculations encouraged by its symbology are more familiar than the Boolean-style calculations presented in most textbooks on logic. In doing this binary arithmetic one soon becomes aware that what one is doing is nothing but arithmetic and involves no trace of logic. Psychologically, one follows a calculative procedure which is entirely devoid of any "argumentive" undercurrent, its utility consisting precisely in the ability to completely replace argumentive by arithmetical deliberations. And although one could develop such a system by the conspicuously logical technique of deduction from axioms, this seems a presumptuous and unnatural approach. The system evolves more spontaneously from a fairly arbitrary series of demonstrations, in which one learns to recognise the general features and behaviour of "well-formed formulae" - a strategy that relies on a less painfully acquired cultural repository than the comprehension of the concepts of lemma and deductive proof demanded by the standard approach to logic (which, alas, also depend on pattern recognition).
But if a given structural scheme - even though apparently well suited to logic - is open to interpretations having no logical content whatsoever, we shall want to ask, first, whether there are any general principles for expanding, contracting and maintaining the integrity of the structure (transformation rules) and, if so, whether they are principles of "reason"; and, secondly, what is the relationship between the structure and any interpretation we choose to give it - in particular, the "logical" interpretation. Though for the most part beyond the scope of this article, the first question surely demands special attention; for here we have the chance to confront some principles of thought which are more fundamental than those of logic, and which have relevance both to computational reasoning and to our primary experience of the physical world - notions such as recurrence and similarity, recognition and expectation, differentiation and association, form and symmetry, abstraction and interpolation, substitution and extraction, and above all the difference between organisation and chaos. I take the view that the study and exposition of most, if not all, of these structural principles fall within the province of mathematics, that "logical" structures and "logical" reasoning form only a small part of this field and that mathematicians might do well to break their ties with the traditional logic-based methods of doing maths. As an analytic discipline, however, mathematics in all its aspects must eventually endure the same censure as logic (see §8).
It is the second question that concerns us now. The symbolic expressions employed by logicians comprise only patterns and sequences of marks and evidently have no intrinsic logical import. A formula or finite sequence of formulae which possesses a certain kind of symmetry (as determined by proof, for example) may properly be said to be correct ("orthologous" in my terminology), but only to depict logical validity. It's difficult to see how such representations in themselves might be considered fundamentally logical. This remains true even if we suppose that the symbols dictate their own functional possibilities, the way they are to be used, the rules for manipulating them - which, after all, merely chase the patterns that can be constructed from them. The same rules apply, the same possibilities for pattern-building occur, whether or not we interpret these constructions "logically". Even so, it might be argued that this symbology and these rules do accurately reflect whatever it is we are doing when we do logic; or that they faithfully represent a non-mental metaphysical realm of logic that is somehow distinct from these underlying patterns. For a moment let us admit this possibility, viz there is an activity or realm of formal, topic-neutral logic whose properties can be faithfully represented by symbolic structures developed by logicians.
7.2 Logic, meaning and relevance
In §3 I posed the question: if "The cat is in the cupboard" and "The cat is not in the cupboard" are both observationally testable propositions, how can it also be maintained that these very same two propositions are logically contradictory? There are a number of ways of visualising, and perhaps attempting to resolve, this quandary. One approach is to deny that there is any connection between the propositions as such and the abstract treatment they receive in systematic arguments. We simply replace one proposition by p and the other by ¬p, where "¬" is a strictly logical operator. But then of course we fail to mark any relation whatever between the original material propositions, much less to offer any warranty that the facts will comply with the formal design invoked to speak about them. If this position is extended consistently to all argumentation, the queer consequence appears to be that rational discourse has nothing to say about reality. For how can logic both maintain its regal aloofness and apply to practical situations? - as if there were an infallible structure of pure reason, on the one hand, and a less than perfect real structure to which it corresponds, on the other. But supposing, regardless, that someone had reason to subscribe to this view: one of the principles they would seem to be endorsing is that there is a pure logic comprising formal relations, though the related terms be meaningless.
Alternatively we could interpret the p's and q's of logical formulae empirically while giving the connectives (including the one-place connective, "¬") systematic interpretations. This stance, in which the bones of logic retain their respective bits of flesh, so to speak, seems to be especially prevalent despite the philosophers' recognition of its shortcomings. In particular, one faces the practical difficulty of how to work with the idea that the basic operator "¬" remains purely logical in interpreted propositions of the type ¬p while the contained proposition p is experiential. This leads to hopeless tangles. "The door is open" would have to be translated as "It's not the case that the door is closed" (or vice versa, but not both). We should have to deny that we could investigate both whether the door is open and whether it's closed. We could not reasonably substitute unnegated terms for negated ones in arguments. Clearly the notion that a proposition of the form p or ¬p is analytic, though p be experiential, is nonsensical.
A step forward is to exclude "basic" uses of negation from logic. On this account truth and falsehood in relation to simple variables are regarded as empirical notions, ¬p is interpreted as "p is false" and so negation in this situation has an empirical meaning. All other connectives, however, are regarded as logical, and expressions such as p & ¬p are regarded as representing analytic inconsistencies. But this only moves the difficulties a little further up the line, in addition creating incurable problems with the interpretation of negation in complex formulae and at the same time accentuating the underlying conceptual dilemma with all these approaches, which is as follows.
Given that if "p is true" and "p is false" both express experiential propositions they cannot also be logically contradictory, it would be capricious to ignore this by positing a formal relation of contradiction between them. Furthermore this care must be extended to all pairs of apparently contradictory basic propositions whose truth is judged by meaning criteria other than those of pure logic. Only after eliminating all meaning and substituting a systematic notion of truth might we become entitled to engage in systematic argument. But now supposing we do eliminate all meaning, as suggested in the first approach. It is one thing to posit, as a formality, that two propositions be contradictory, quite another to appreciate that they really are so: only if we understand the propositions can we then understand how they conflict with one another. Throw out the meaning and the possibility of contradiction goes with it.
The concept of contradiction - one of the most fundamental in logic, along with identity - thus relies on the interdependency of meaning between the two opposing terms. It is surely just as obvious that no relationship among experiential propositions carries anything like the force of logic unless those propositions are not only meaningful, but meaningful in a way that makes them relevant to one another. Without mutual relevance there can be no "logic". In particular, no argument is valid unless its terms are relevant both to one another and to the purportment of the argument. I'll try to explain this principle only in relation to material propositions (similar considerations apply to propositions of other types).
There are of course numerous ways in which material propositions may be mutually relevant. Most philosophers appear to acknowledge the indispensability of the notion of relevance in "law-like" conditionals, including not only causal, subjunctive and counterfactual conditionals but all conditionals which contain any trace of a law-like connection, and which, therefore, one can at least find excuses for declining to call "truth-functional". And while most philosophers would also admit that there are also molecular propositions of non-conditional form which are incapable of a systematic interpretation, I doubt whether many would find this weakness in all material propositions. It seems to me, however, that "relevance" is a general prerequisite for valid argument, being primarily a condition among propositions that attends the rejection of logical atomism and the notion of pure logical relations that goes along with it. Formal relations are displaced by experiential relations. In contrast, classical logic treats "material" relations (in the sense of "material" implication) as if all propositions are experientially independent, failing to recognise that the semantic continuity that underpins the fundamental concepts of identity and contradiction extends to all other relations of experiential propositions.
No one can doubt that when a proposition such as "Had you doused the fire nobody would have been harmed" is represented as a conditional (p
q) it is non-truth-functional, the truth of the whole relying upon a certain kind of dependency of meaning between antecedent and consequent. But what of "John Howard is Prime Minister of Australia and the moon is made of marshmallow"? This might well be taken to be a conjunction of two physically independent propositions which happens to be false as a whole just because one of the conjuncts is false. For, although there may be both philosophers and politicians who might think otherwise, the composition of the moon appears to have no bearing on the fact that John Howard is Prime Minister of Australia. But if these propositions are really unrelated we might well ask: why would we ever want to, and how could we, include them in the same argument, and how could we allot truth values to the propositions, on a par with one another, as it were? In what circumstances would we unflinchingly call the one proposition true and the other false, without changing pace or perspective?
In any ordinary conversation about the status of John Howard we would indeed have to alter our perspective if the subject of the moon's composition was abruptly introduced. But academic conversation tends to be unordinary, taking into its fold everything that exists or might exist anywhere and anywhen, and freely dishing out truth values all from the one pot. Not surprisingly, when it invents sentences the interconnectedness of whose components is obscure, it finds instead only paradox. The present example is surely not that "far out": both its conjuncts relate to our planetary system in 1997, and both represent facts which are investigable by the resourceful observer. Here the academic's concern just marks the perimeter of the domain in which the clauses are materially relevant to one another. As practical conversationalists, on the other hand, we are interested in whether both clauses are relevant to any useful argument (and, usually, to some specific argument). For I take it that no one except the odd philosopher or astrologer would ever want to argue empirically about propositions whose material connections are so remote. Should it happen that we do decide to talk about the composition of the moon, the Prime Minister's eyebrows and the price of chips in Moscow all in one breath, then hopefully we shall be aware of the limited nature of the association between these topics and not find paradox where really there is none. At the same time we shall appreciate that we cannot talk about such matters at all unless we recognise some connectedness of meaning - an interdependency that is sufficient unto the argument, so to speak.
This, then, is the dilemma: propositions are either meaningful and not formally related or else meaningless in which case it might be presumed that they could be formally related. But if they are meaningless it becomes impossible to see what is involved in such a relationship: meaningfulness is required for relationships such as contradiction to be comprehensible. Therefore either we must be "purists-gone-bust" and deny that there are logical relations, or, as ascendant pragmatists, admit the use of the word "logical" on the understanding that such relations draw their life from the meanings of the related terms and not from a conspiracy among mathematicians.
Because relevancy comes in a multitude of colours, there's no place for a unique mode of reasoning that is independent of the subject matter with which it deals. Take away relevancy and the relevant logic goes with it - there's no such being as a pure, topic-trading but topic-neutral logic. On the contrary, far from being topic-neutral, logic appears to be remarkably topic-dedicated. This does not of course imply that we cannot assort logics, in a rough and ready manner, into general classes appropriate for different types of purpose. It is one thing to abstract from uses of reason, quite another to install a formal procedure that regulates reason. We can recognise chairs and tables and use them effectively without appointing a warden to control their use (as if there were not enough to go around).
7.3 Systematic logic
Although it may be conceded that when logic has to deal with the inexactitudes of the real world it must follow a muddy, twisting track, still one might imagine that there is a pure, analytic logic which consists in the application of typically logical reasoning to the constructions of mathematical logic. This seems plausible because it's possible to substitute the variables in a logical system by well-formed expressions of the system itself, thereby endowing them (so one might think) with sufficient meaning to preserve the quintessentially logical features of logic, such as contradiction and deducibility, which apparently ensue from the use of appropriate syntax. But as intimated at the beginning of this section, this ploy alone does not genuinely fulfil the semantic requirement for systematic formulae to be interpreted as "logical". For it is impossible to eliminate the p's and q's (which are unanalysable and therefore meaningless) from molecular propositions. Replacing them with more complex formulae doesn't help, because when the proposition is reduced to normal form meaningless terms remain as basic components, so the logic as such disappears. And even if it could somehow be supposed that every variable tacitly represents a structural expression, this would bring us no closer to justifying our belief in a logical interpretation.
Consequently a semantical idea of logical relations is a prerequisite for beginning logic. I don't know what the exact nature of this idea would be, but we might imagine that it is an abstraction from the material relations of which we have been speaking (rather as the concept "jump" might be regarded as an abstraction from a large number of instances of jumping). The only hope of reforming this street-wise generalisation lies in persuading it to follow unswervingly the path dictated by an axiomatic structure. Provided our logic conforms rigidly to a mathematical pattern, will it not then be analytic? Wishful thinking indeed! The question is grounded in pure empiricism and presents no genuine challenge to any of our previous positions.
7.4 Formalist logic
I therefore conclude that the three aspects of logic with which we began cannot be combined and that logic as an argumentive process is thoroughly empirical. However, there are formalist systems employed by logicians in the study of their subject, and by others as aids to thinking. The well-formed formulae of these systems belong in the general category of structural or mathematical propositions, whose calculative properties appear to provide some of them with the credentials for analyticity. Soluble propositions are one such kind (see §9). I now wish to argue, by example rather than by principle, that not even these formalist structures (or those of them which are supposed to qualify) are analytic. The symbolic constructs of mathematicians are themselves "impure", in as much as they are restricted by the same experiential conditions that shape every other aspect of our lives. Far from dealing with the indubitable, mathematicians inhabit a fairyland whose images are frequently as nebulous as any in the most recondite of metaphysical systems. (Not that mathematics is any worse off for this, as we shall see.)
In attempting to promote this evaluation, instead of logic I've chosen to consider arithmetic, in particular some aspects of the concept of number. Like symbolic logic, arithmetic is essentially formalist and calculative, but unlike symbolic logic it doesn't scream out for interpretation beyond its own symbology: it deals primarily with the element of recurrence within a calculative system. It therefore lends itself to a more incisive impression of syntactic structures and their empirical constraints. At the same time, I hope that, by applying a little pressure to the borders of normality, we shall uncover something of the nature of arithmetic itself.
8. ARITHMETIC AND STRUCTURE
Arithmetic, like geometry, is a curiously mixed pursuit. Few would dispute that a proposition such as "There are about 20,000,000 people living in Australia" is entirely empirical, because (provided one does not think too hard about it!) it contains no trace of the calculative procedures that characterise the science of mathematics. Number, as used in this proposition, is apparently a purely empirical concept. As soon as one begins to calculate, however, the character of number seems to change. That the population of Australia is close to 20,000,000 is a fact, but that 400 x 50,000 is exactly equal to 20,000,000 is a fact of a very different kind. Propositions of the second kind have long been considered analytic, in the strongest sense of that word. In fact it has sometimes been held that arithmetic tows the analytic line, as it were - it is the archetypal system of analytic propositions. While this may be going too far, it is apparent, at least, that the philosophers of mathematics and logic have done their best to divest number of its empirical connections and transform arithmetic - indeed the whole of mathematics - into an all-embracing analytic discipline. In spite of some devastating snags identified by Russel, Godel and others, this is pretty much how maths is envisaged by most people today. Yet many of the foundational concepts of maths are blatantly empirical, while many of its fundamental axioms (such as that every number has a successor) seem to beg the question of empirical status. My aim in what follows is to give credence to the opinion that, even in its most formal aspects, arithmetic cannot throw off its empirical chains.
A problem for many philosophers, but few mathematicians, with modern arithmetic is the riddle of the meaning of infinite numbers. In fact Georg Cantor, one of the founders of number theory, divined that there's an infinite hierarchy of infinite sets, but, thank goodness, we shall barely need to touch one of them. While of course I don't question the indispensability of the systematic concept of infinities to mathematics, I do subscribe to a certain intuitionist/formalist precept which grants priority to finitary concepts and syntax, crediting them with a type of meaning or "contentfulness" (David Hilbert's term) which is absent from notions of the infinite and from incompletable expressions. The thrust of my concern, however, is with the concepts of the finite and the denumerable, not with the infinite. It seems to me that two grey areas have been neglected, lying between the infinite and the observably finite or numerically accountable - one of them at the immensely extensive or multitudinous end of the scale, the other at the minutely divisible or infinitesimal end. It is here that I believe mathematics fails, that its failure is in principle demonstrable and that its weakness in this area impinges upon the whole of mathematics and rational thinking.
Although it is now well established that mathematics is "weak" in the sense that it contains uneliminable elements of randomness, uncomputability, paradox and unprovability, I don't know whether these failings are related to those which we are about to consider. I suspect that, despite their acknowledgment of the former kinds of weakness, many mathematicians would reject the notion of empirical conformity, and would doubtless think my conjectures are intuitionism-gone-crazy (although, so far as it goes, the "theory" is essentially realist). Should they remain of that opinion, however, they will one day be jolted into submitting alternative theories when the inevitable happens: mathematics will become so far stretched that it will cease to produce consistent results for reasons which seem non-systematic. While doubtless much of my story is old hat, I'll try to arouse a glimmer of interest by adopting a speculative but graphic approach. In offering some rather secular and simplistic illustrations, I hope to supply clues as to how it might be possible to test the central idea; for it must be assumed that this is either a cosmological theory with practical consequences or else metaphysical nonsense.
8.1 Space as structure
First, a little speculation about the nature of space is in order. When I say "space", I mean local or domestic space, the space that we live in and about which you and I, as well as the Euclids and the Newtons, tend to form intuitive, pragmatic judgments. It is widely appreciated, I think, that the impressions of space received through the various sense organs are entirely different in kind from one another: there's no resemblance between visual space, auditory space and tactile space, and no inherent reason why one should suppose any object in a visual field to be identical with one perceived via any other sensory route. What, then, is the binding force that unites and coordinates these different kinds of sense data, along with their peculiar space-like perspectives, causing us to embrace them as representations of a single space? A very plausible answer is that they are related by a mathematical or logical structure. But once having brought ourselves around to that outlook, it begs little further insight to reach the judgment that space consists of nothing more than a mathematical structure. For nothing is required besides mathematics to unify these different sensory perspectives; the assumption that there is a "noumenal" physical space apart from pure structure is metaphysical and needless. Space is nothing but mathematics.
Since the human brain is a structure in space, and has evolved primarily as a mechanism for sustaining itself in its spatial environment, one might suppose that the conditions of pragmatic thought in general, and of logical and mathematical thought in particular, are themselves influenced by the nature of space. And unless there are aspects of thought which are independent of the physical existence of the brain, conceived as a spatial object, it would be very surprising indeed if the topology of human thought turned out to be unrelated to that of space. This is of course a chicken-and-egg situation, but one that is of no immediate consequence: the significant idea is just that local space and the way we think have a predominantly common structure. (Admittedly the equation must be extended to include at least space, time and force in a broadly Newtonian structure, but the resultant complications would not further our cause.) The crucial point to grasp is that there's a bigger chicken and a bigger egg: the logical and mathematical scope of the human mind is subject to the very same constraints that the mind attributes to the local universe. Conversely, the local structure of the universe as comprehended by the human mind is limited by the same intrinsic topological factors that determine the kind of logic and mathematics of which human thought is capable. Mathematics, the mind and the local universe have a common foundation.
The reality of the situation, however, is that the physical universe does not (according to present-day reckoning) possess the intuitive Euclidean/Newtonian structure we once believed in. It's different, and therefore one should expect the foundations of mathematics to differ in a related way, or to collapse in certain circumstances. The situation is evidently complicated by the fact that the mathematics we have available to undertake cosmological enquiries is just the mathematics whose foundations are in question: it seems we need to understand the nature of mathematics before we can describe the universe, but there's a vicious circle involved. Nevertheless, I believe the hypothesis that the foundations of mathematics are awry is in principle testable, and, if true, there are many far-reaching consequences, some of which will be noted in the Postscript.
8.2 Formalist arithmetic
There are, of course, countless "theories" about the nature of arithmetic, ranging from the purest formalism through various psychologistic interpretations to the most implacable realism. While my own view leans toward formalism, this is not so much a philosophy of arithmetic as a judgment that all calculative symbolic systems are intrinsically arithmetical. Arithmetic is the science of recurrence. But while the notion of recurrence is crucial to understanding symbolic systems it applies well beyond that sphere, being an integral component of almost every aspect of human experience. Consequently this predilection for formalism is of little significance.
Furthermore the question of the ontology of arithmetic is of no immediate importance. For there could surely be an agenda - call it "arithmetic" or not - which is formalist, as well as one in which the symbols are objectively or psychologistically interpreted (so that the meaning of the symbols is not just more symbols in the same system). In respect of the latter discipline, however, one might expect to find a variety of interpretations of "arithmetic-like" syntax and hopefully a reasonable explanation as to why any of them be regarded as distinctively mathematical; in particular one would feel entitled to an explanation of how the analytic character of such a discipline is conserved in its ontology rather than merely in its syntax. Regardless, I shall initially assume that arithmetic is formalist (i.e. it comprises nothing but symbols - I don't think this coincides with formalism in the modern sense, which puts more emphasis on meaninglessness and rigid proofs), so that our discussion can easily be related to the question of "pure logical syntax" which is our primary interest.
Within a broadly formalist framework one can still approach a number of issues from different angles: specifically, it looks as if one can adopt attitudes which might appear, to the sophisticated, to be ontologically different. For example, one might take the view that arithmetical syntax comprises nothing but transformations of utterly meaningless symbols; or one might say that the system contains names or tokens which denote other systematic objects (which in turn can be used as names or tokens), thereby giving the impression that the syntax is after all meaningful. Although this particular distinction (which may be verbal only) doesn't affect my case, I prefer the second approach because it buoys my view that arithmetic is best envisaged as possessing a "fluid hierarchical" structure (see below).
8.3 Number and tokens
Arithmetic begins at the small end, the human end of the mathematical spectrum. It begins with counting. This truism (the cradle in which, it might be thought, the avowed intuitionist chooses to spend his life), seems to have been respected by the inventors of number theory, notably Cantor and Giuseppe Peano. And how could things have been otherwise? Who would have taken any notice if the theory had had no footing in the nuts-and-bolts concept of natural number with which we are all familiar? Are we not entitled to assume that this is what the theory is about? Unfortunately when arithmetic is stretched into the realms of the uncountable, we can no longer take this assumption for granted.
I know I'm not alone in harbouring the feeling of being duped by Cantor's theory of sets and numbers, the feeling that it's somehow circular and fails to get to grips with number as such. (This is not a psychological problem, but rather a problem about the validity of real-world proofs for mathematics and, conversely, of the existential status of the mathematical objects so defined.) Although Cantor himself was obviously no formalist, his explanations are uncompromisingly syntactical in as much as they depend on establishing one-to-one correspondence relationships between series of numeral-tokens, that is, between symbols occupying a small zone of Euclidean space. Its assumption that these series of demonstrations are indefinitely extendable and/or indefinitely interpolatable appears to rely on the assumed topological properties of infinite extension and infinite divisibility of the space in which they are represented. And although, under a less formalist interpretation, it might be held that these representations do not depend on real space for their actualisation, it would still appear that whatever it is that's supposed to be going on requires a logical space with similar properties. For without this assumed space, one could never predict that every supposed one-to-one correspondence would in fact be unique, or even possible, when the series is "represented at length". Thus Cantor's definitions of numerical infinities rely on an undefined notion of spatial or logical infinity. The acceptance of his technique as a valid mathematical method depends on the unfounded and improbable belief that what can be physically demonstrated on a piece of paper can be extrapolated ad infinitum to increasingly unwieldy gesticulations that cannot actually be symbolised anywhere or anyhow.
As Cantor draws upon ever more picturesque techniques, the limitations of the page become increasingly bothersome and the proofs less convincing. For example, in the procedure that's supposed to show that the number of proper fractions (rationals) is the same as the number of cardinals, Cantor introduces two complications. First, in order to deal with the the fact that every fraction can be represented in an infinite number of ways, he has to delete an infinitude of irrelevant fractions (namely, all those whose numerator and denominator have a common factor). Secondly, he has to coax us along a zigzagging path through his two-dimensional array of fractions, skipping the irrelevant ones on the way. This is an extremely "spatial", undependable looking procedure (see Endnote 4).
There is of course no difficulty with the notion of correspondence of relatively small, finite sets which can in fact be matched and whose number might feasibly be ascertained. But the extrapolation of the notion to larger sets involves the use of synoptic tokens which do not themselves possess the properties of the sets referred to. This, however, is a feature of arithmetical syntax in general (irrespective of ontological presuppositions).
Yes, arithmetic does begin at the human end of the mathematical spectrum, with counting. Formalist accounts may accommodate this maxim by recognising that some expressions in arithmetic are relatively "primitive", while others are really names or tokens for (often exceedingly extensive or infinite) collections of primitive symbols. In other words, there are synoptic tokens whose meanings are complexes of more primitive signs, and which in principle can be expanded analytically, by correct calculation, into the complexes that they represent. Naturally there are degrees of primitiveness, degrees of complexity and little inclination to single out any particular set of signs as being "the meaning" of any other set. But one can surely sympathise with the idea that a token such as 123 can be expanded to one of the form 1+1+1+....+1 which better captures the literal meaning of the original token in so far as it contains just as many 1's as the number signified by 123: it exemplifies the number and does not merely signify it. But now what of the token 219937-1? Evidently this expression could not be expanded to one of exemplificatory form (1+1+1+....+1) even though one had begun to compute it at the beginning of time using every available minuscule in the universe. Yet more than twenty-five years ago this number was proven to be a prime, and very much larger primes have since been discovered. (And of course still larger numbers can be expressed in token form. For example, there's a number called a moser which is unimaginably huge, but which is easily defined using only the number 2, a few geometrical symbols and the concept of exponent.) So, on this view, a proposition such as 101000 = 10500 x 10500 is analytic but meaningless because the most primitive forms of expression betokened by the terms of the equation cannot in fact be completed. But regardless of whether some signs are more basic than others, it remains true that arithmetic alludes to some translations of signs which can never be depicted or used because they are too extensive or, rather, their components are too numerous! Since no such translations can possibly exist, there is strong justification for the claim that any reference to them is "uncontentful".
If this is right, then the sign 219937-1 cannot be used to refer to an expansion of the form 1+1+1+....+1 and remain a bona fide component of an analytic system. Owing to its central concern with the rudimentary concept of number, however, conventional arithmetic does contain references to uncompletable signs. Consequently arithmetic as a whole lacks the credentials for analyticity. And (for those whose concept of number is not tied to mere symbols) it's clear also that no token in an arithmetical system can denote an extrasystematic occurrence of a precise number (such as a counting of objects) if such an occurrence does not, nor ever could, exist.
Thus arithmetic contains expressions denoting either incomplete (and uncompletable) tokens or uncountable sets of objects, or both. Although, so far as I know, this verdict doesn't at present detract from the utility of the analytic craft of sign juggling, it might be prudent to keep in mind both that sign juggling is not necessarily the same as number crunching and that the analyticity of mathematics is vulnerable just to the degree that it projects its language beyond the reaches of the conceivable. Much as Newtonian physics is vulnerable to the degree that its domestic language of space and time loses meaning when we want to converse with the electrons and the stars. (Is it simply the physics or are there already signs of the maths going wrong?) As with other sciences, mathematics holds no built-in guarantees of performance.
I have used a spatial picture to show that we cannot predict how numerical signs behave when we try to imagine an extrapolation of a series into regions beyond the immediate environment which establishes the conditions of sign writing. It cannot be assumed that the framework and assumptions applicable to manageable numbers has legitimacy for gigantic numbers. Had we been more adventurous and chosen an illustration befitting our times, such as the way that computers handle arithmetical information (as has been done, for example, by Rolf Landauer, 1986), we should have arrived at just the same conclusion - that arithmetic is empirically constrained. A more anthropocentric illustration, however, is provided by the concept of counting.
8.4 Counting
Numbers begin with counting! According to the axioms of natural number ascribed to Peano (1908), every number, n, has a unique successor, n+. An intuitionist might take this to mean: take any particular number, there is just one number that is one greater than it. But how would you in practice take any particular number? Suppose it was a very large number whose value could not feasibly be checked by counting. How would you then know that you had the number you intended to take? The notion of identifying an uncountable number as being a particular number is incomprehensible. On this account, Peano's axiom seems meaningless because it doesn't satisfy the criterion of real countability.
If counting is to be explained in terms of putting signs and objects into one-to-one correspondence with one another, then nothing more need be said: we have seen, first, that this is an empirical procedure and, secondly, that it cannot actually be done with very large numbers. But is it not also possible to count by rote, as school children often do, by learning the sequence of signs without attaching them to physical objects? Well, how can we distinguish counting by rote from counting things? The first kind of counting seems to consist only in reproducing the conventional tokens for successive numbers in the series of ordinals, while the second kind involves both reproducing those tokens and placing them in one-to-one correspondence with the members of a set of objects. It seems to me, however, that the distinction lacks substance: when counting by rote, we do in fact put different signs into correspondence with instances of something, even if only intervals of time. Of course, if we are counting events, or just counting off definite intervals of time, such as seconds, it's easy to contend that we are employing the correspondence procedure. But it might seem that counting by rote lacks objectivity, that it doesn't involve events and that the time intervals are somehow too arbitrary and inconsequential. I doubt this: the activity of counting itself supplies the events and thereby demarcates and orders the intervals.
One possible objection is that we cannot in principle do a recount: objects can be recounted, events can be recorded and recounted, but how does one recapture a counting per se? Well, couldn't we replay a recording of our counting, assuming we counted aloud, and count our counting again, so to speak? Suppose we just recount the noises as such, without taking notice of their form. Then surely we could be said to be recounting our original count - which was, so to speak, a labelling of noises contrived by giving a particular shape to each noise. I think the following consideration completely justifies this view. If we count by rote, say, from 1 to 20, it would be in order for anyone to ask if we have counted right. If we did not count right, then at some stage in our counting the number of noises delivered up to that stage would not correspond to the meaning of the noise uttered at that stage. (I say "at some stage", not necessarily upon reaching 20, for we could have made two or more mistakes which cancelled each other out, and so have made 20 noises yet not have counted right.) Now suppose there was in principle no way of checking the count. In what sense could we then be said to be counting at all? How could we ever be sure that we were not simply uttering noises at random? Counting by rote entails counting objects, namely the signs that constitute the counting; the signs are labelled by giving each of them an unique, conventional form; if the counting is right, the form of every relevant token corresponds to the number of tokens delivered up to that stage. Counting cannot take place in a void. Even when "nothing is being counted", counting is an obstinately experiential process, temporal and psychological. And since every sign that represents a natural number must represent a countable number, or else be meaningless, arithmetic in general is empirical. Its applicability to the real world is irrelevant to this argument. Arithmetic is inherently real.
If counting is experiential, the proposition that every number is countable in principle is meaningless: something, even if only the numeral tokens themselves, must be countable in practice. A numeral token such as 101000 in isolation cannot stand on its own feet - we cannot tell whether it stands for "the number" it's supposed to, nor conceive of its basic meaning, nor ascertain whether any such number exists, since it is not and never will be literally countable even on the fastest computer that could in principle be designed. On the other hand, the number 1000 is meaningful because it's countable in practice and there are sets of objects or events that can be placed in one-to-one correspondence with the series of natural numbers up to and including 1000.
How else might we attempt to count things? Although it isn't necessary to literally count the members of sets to compare their number, some method of matching them is required. It's easy to show, however, that any procedure for matching sets or patterns at some stage involves at least as many discrete operations as there are objects common to both sets (or elements common to patterns), and therefore a similar number of operations as would be involved in doing a literal count. The "contentfulness" of number depends on this pragmatic potentiality and cannot be captured by shortcut techniques. It's of little account whether we imagine these operations to be essentially spatial, temporal, psychological or belonging in some more abstruse logical space; it matters little in what framework we conceive of the existence of numbers. Given a coherent view of space, time and "psychological space", we shall find that the various formalist and psychologistic concepts of number are practically identical. Somewhere along the line we turned the manuscript upside down: number itself calls the tune, erratic though it be, and both space and time dance to its strange music.
8.5 Mathematics and meaning
Is there a unique category of mathematical, as opposed to structural, propositions? The breadth and complexity of the subject-matter of mathematics make this extremely unlikely. Without getting involved in questions about the nature of mind and artificial intelligence, one might presume that an effective test of whether mathematics comprises only pure structure or demands a psychological interpretation is whether computers can do it. But the test is marred by the need to resolve, in turn, what it is that computers do, and (having decided that) by the purely terminological question of whether it's proper to call what they do "mathematics". In regard to classical logic, for example, computers can perform the appropriate systematic operations, but I doubt whether they can do logic. I don't hold quite the same doubts about their capacity to do mathematics, or some mathematics, anyway.
Much depends on what one makes of the business of interchanging symbols that have complicated meanings. For while the principal domain of mathematics is structure of all kinds, it seems to me that as a formal system it comprises nothing but a sign language, employing signs and about signs. Its alleged analyticity consists in its calculative procedures for translating signs into other signs. Mathematics is everywhere hierarchical, it employs an indefinite series of types. A clear-cut distinction between mathematical and metamathematical levels is nowhere to be found, but the difference invades mathematical systems in subtle ways at every point. Signs (tokens) are objects and vice-versa: the objects that are the meanings of signs are themselves signs: the distinction is relative. Essentially there are symbols that are more synoptic and symbols that are more expansive (such as strings of operations or sets characterised denotatively). But the analytic nature of mathematics becomes suspect whenever a sign cannot be completed, or when the objects in a set that a sign stands for cannot be listed, or when the operations required to expand an expression cannot be enumerated. One can only say that an expression (such as that 219937-1 is a prime) is meaningful if one can envisage a practical, "uncompressed" procedure employing a finite number of steps to prove it. And then it is meaningful just "to the extent of the procedure" and not in any sense that exceeds it. Thus the number of operations required to disclose the primitive or expansive meaning of an expression is of paramount importance. Number is of the essence.
Mathematics in the large therefore differs little from physics: it has inherent experiential limits and it is subject to similar conditions of uncertainty and relativity. Expressions with colossal or infinitesimal denotation, however, might be amenable to statistical treatment, so that mathematics could in some measure solve its own problems by adopting the same kinds of procedures for itself as it does for problems of physics. But I believe also that mathematics will turn out to be "assessor-relative" or subject-dependent. No procedures exist that will equally well serve every one in every time, place and condition. There is no almighty algebraist, no mystical realm embracing every possible mathematical construction. On the contrary, all mathematical structures, no matter how complex or inevitable - or how paradoxical or uncomputable! - they may seem, come into existence as the creations of calculating people, and all mathematical solutions must be understood by them. At the same time, this seemingly constructivist activity possesses an objectivity that is scarcely distinguishable from physical reality; and I can see little value in the belief that mathematics is endowed with a special sort of reality of its own.
I hope that now no difficulty will be encountered in transferring the example of arithmetic to logical syntax and similar constructions. All formalist systems operate within similar experiential constraints, and although undoubtedly the numerical aspect of symbolic structure is not the only one calling for consideration, like arithmetic, other formalist systems embody recurrence and a hierarchical symbolic structure comprising a gradation of signs from more primitive to more synoptic. Of any such system it can be said that meaninglessness and uncertainty grow as we attempt to extrapolate to increasingly remote cases a prescription that executes successfully within the local environment in which it was designed. Repeatability, "correct calculability" cannot be guaranteed.
8.6 Logic as inductive thinking
The position with systematic logic, then, is as follows. The uninterpreted syntax of classical (or any other) formal "logic" is but one kind of mathematical structure or calculative system. As soon as we attempt to enliven the syntax with the breath of logic, with the potential for genuine argument, we face a dilemma. Any instillation of meaning into argument blocks the possibility of pure logic, while the absence of meaning renders logic totally redundant: any calculative interpretation - or none - will then suffice. So for any argument one can say: either one is thinking logically but one can never know whether the logic will work, or one is performing a calculation but it doesn't matter what one is thinking; the computations can be executed mechanically. What one cannot do is either think or perform "analytically" - for neither reason nor symbology can engage the required infallibility.
What can it mean to be thinking logically when the paradigm - pure syntax - is devoid of logic? Well, clearly the source of logic is inductive thought, not deductive. It springs from expectation, our continual confrontation with the recurrent associations inherent in organic processes. And I do not believe any thought process can have greater strength than inductive strength. There can be no seventh category dealing in pure logic because there are no propositions of that sort. Of course, the reduction of symbolic patterns to forms that permit their status to be assessed requires some kind of intellectual effort. And although, as I pointed out, this need not be a logical chore at all, one would very likely opt for a procedure which one believed to be logical. What is it that one does in using this option? One simply draws upon the inductive mode of reasoning* one uses in solving many a real-world problem (consciously or unconsciously) and applies it to this particular practical problem. Doing symbolic logic per se, or any other "abstract" calculation, is just another empirical problem to work out, and like every other problem it is subject to experiential restrictions. (* I refer to induction as a mode of reasoning reluctantly. Induction does not involve any kind of straight-jacket reasoning at all, but combines an appreciation of continuity and regularity in nature with a feel for the relevance of events and situations drawn from all quarters of one's experience. So far, attempts to explain and formalise induction have all been futile, most of them pitiable. But I am claiming that one kind of reasoning is to be explained in terms of another which apparently has few of the characteristics normally associated with reason. Either we must understand "reason" more broadly or give it another name - Wisdom? Intuition? Common sense? Induction, understood as reasoning from the specific to the universal, won't do because (a) it begs the question and (b) the universal is mythical.)
The invention of mathematical logic has set all literate people thinking in reverse gear, and, because the authority of a respectable education has etched this discipline deep in their minds, it's going to be difficult to get them to drive forwards again. They have it in their heads that there is a pure logic which is somehow applied to objective facts, whereas the truth is that reason merely reflects the possibilities inherent in real-world situations, whether natural or engineered: we think correctly if we perceive these possibilities clearly. There is no formal logic - the logic of any situation emanates from the situation itself and the way it can develop in its environment, as seen through the eyes of survivors. So if we are to retain the name "logic" for the kind of rational thought that embodies inference and the sorts of linguistic devices associated with the classical discipline, we must say, not that it's a foolproof analytic technique, but that it reflects the relationships among actualities and possibilities in the real world.
Their trust in the necessity of mathematical logic has made it especially difficult for some philosophers to come to terms with relativity and quantum theory and some of their more exotic implications. As well, they have failed to understand and promote the thoroughly existential nature of the logics underlying both the old and the new sciences. This is a pity, because the scientific developments of the twentieth century provide a clear omen that the children of the twenty-first will succeed in escaping the shell of conventional existence which has been the prison, the fortress and the home of their ancestors for many generations past.
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