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.....SIX KINDS OF PROPOSITION - Sections 9-13

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9.1 The propositional scheme

By way of summary, the accompanying chart (Classification of Conventional Propositions) displays a useful arrangement of the propositional types. This scheme comprises a series of "hexagons of opposition", each of which can be explained by analogy with the single hexagon depicted below (a modified square of opposition) that can be constructed from the familiar classical varieties.
An analogy for any of the hexagons in the Chart of propositional
types (please click on a text link)
example hexagon
Each arrow is read as "included in"; for example only the tautological and synthetic varieties are included among those that are consistent; propositions that are not synthetic are analytic and these are either tautological or inconsistent (in Ayer's (1973) terms - true or false "solely in virtue of the meanings of the signs which express them"). In a restricted or model argument a negated tautological proposition is inconsistent, while in a liberal argument (one that makes room for the possibility of failure other than falsity within type) it is either inconsistent or synthetic (an argument that is not valid is "illogical" but not necessarily inconsistent). Any of the six varietal names could be used as propositional modifiers in formulae, e.g. taut p = inc(¬p), con p = ¬(inc p).

Each hexagon of the proposed scheme contains just one of the basic propositional types, at the peak of the hexagon. The class named in the centre of the hexagon includes all six of the surrounding classes, as well as all the classes in the hexagons beneath it, while the category name outside each hexagon refers only to the set of propositional kinds at that level, considered as an independent system (see Endnote 4). (I leave it an open question whether certain classes in the scheme are really identical, e.g. the existential and reportive classes).

The hexagons are arranged so that there's a gradation from the most structural class at the head of the chart to the most experience-related at the foot. The thought behind this arrangement is that a basic proposition in any given category is presuppositionally dependent on concepts expressible within its own category and those represented by hexagons above it, but not on concepts expressible in any of those below it. That is, the competence (the capability of being true or false) of a basic proposition relies tacitly upon the truth of basic propositions of the same and higher types. So, for example, a material proposition might fail because its constitutive limits are disrespected, its terms meaningless or its structure incoherent, but it should not fail on account of any purely mental consideration. Again, according to this account, a test of the intelligibility of every mental proposition is that one can find true material propositions upon which it relies, while on the other hand mathematical propositions make no appeal to any higher category (unless the scheme is really a closed loop, with maths depending on mind; but a different approach is suggested below).

According to the concept of propositions advocated in §2, every proposition purports to be true, in some unique sense of "truth": a generative proposition, for example, purports to be translative and a material proposition purports to be representative (of some physical state of affairs). So if truth is a quality only of propositions, there are, according to the present scheme, just six kinds of truth; which is to say, the word "true" has just six functional meanings. Although ordinary language contains other types of occurrences of "true", these are either descriptive of functional uses of "true" in general or else functionally redundant. Is there a "higher truth" outside this scheme, or do all propositions somehow ultimately answer to one of the kinds of truth within the scheme? Clearly this question demands an answer, for the fact that we use the word "true" to cover all types of truth suggests there's an ecumenical concept of truth.

9.2 Truth hierarchies

One approach to this problem is to consider the question: how should we interpret "higher level" truth valuations such as the ascription of truth in the following example? Let p be the proposition "Koalas are marsupials"; let q be the proposition "p is characterising"; then q is true. Here q treats the proposition p objectively and correctly allots it to a type. This is clearly a material proposition of the attributive kind (see §6). Consequently it might be argued that there's a sense in which the ultimate criterion of truth is empirical, a point which is more clearly displayed in extended examples such as: Let p be the proposition "There are no gum trees in Queensland"; let q be the proposition "p is orthologous"; let r be the proposition "q is misrepresentative"; then r is representative and so are the propositions, s: "r is representative" and t: "s is representative".... Every genuine proposition, it seems, does or could end up by being "metalinguistically" representative in a hierarchy employing only the classes in the scheme.

A difficulty with this is that as one ascends the ladder the criteria involved in deciding whether a proposition is indeed representative (or anything else) become increasingly obscure. To check the representativeness of t, for example, one would surely have to work backwards until one reached the bottom of the ladder where the critical propositions reside (... and where our positivist instincts might have us probing deeper still, for those mythical incorrigible treasures lurking underground). Thus redundancy looms so swiftly that one wonders whether any purpose is served by treading even the first rung, a hunch that will be defended when we discuss purportment and closure (§9.) . Still, if (as I should presume, given that the ladder exists) the "metalinguistic" character of the propositional scheme is material, those who are bothered by language hierarchies, paradoxical sentences and related matters will find that representation has an elevated status among types of truth.

Personally I am less than enamoured by mesmeric manoeuvres of this sort. Theories of truth such as Tarski's seem to me to be (in this respect) perversely circular and reliant upon a formal hierarchy of discrete languages which in truth does not exist - not in science, not in mathematics and least of all in common speech. Language in the raw is so thoroughly steeped in "metalanguage" from the outset (though never regimented by it) that it's impossible to conceive how a formal appeal to linguistic hierarchies could wield any explanatory power. A comparison of the propositional types reveals an increasingly metalinguistic complexion as one moves upwards through the categories. But the suggestion of a language hierarchy begins even before vacating any given category, with the simple observation that there's an imbalance between truth and falsehood. Propositions purport to be true and cannot purport to be false. But if p is the proposition "The mouse ran up the clock", what is being asserted is p, not "p is representative"; because the function of the proposition is to state some fact and not to classify p! The purportment of type is tacit only. On the other hand one might be hard pressed to find any practical difference between ¬p and "p is misrepresentative". For some it is doubtless a question of philosophical temperament; for me it depends, rather, on argumentive context. One could, of course, assert the purportedly representative negative proposition "The mouse did not run up the clock", in which (I've argued elsewhere) "not" is being used empirically; at any rate, it is not being used to reclassify "The mouse ran up the clock". However, if the mouse did not run up the clock then "The mouse did not run up the clock" is representative and "The mouse ran up the clock" is misrepresentative. But while small children the world over know very well that the mouse did run up the clock, it's extremely doubtful whether the argumentive context of "representative" will ever escape the confines of these pages. And better it should not, if the alternative is to banish quoted propositions and formal hierarchies from theories of truth henceforth.

9.3.1 Theories of truth

One might be excused for thinking that any theory which claimed to be a theory of truth would be of little value if it did not indicate how one tells whether a proposition is true, or how propositions relate to the various functions they perform. Many recent theories, however, altogether neglect this problem. Despite the deceptively contrary appearances of some, they are really not accounts of truth at all, but accounts of argumentive consistency. Indeed some of these theories appear to have been devised expressly for the purpose of resolving or circumventing paradoxes of argument such as those of the "liar" breed. The most economical of the "argumentive" theories is the original redundancy hypothesis of F.P. Ramsey, and, although its invention seems not to have been motivated by the need to avoid paradox, to my mind (given the conventional notion of propositions) it provides for the clearest disparagement of liar-type paradoxes. According to the redundancy theory, it is true that p means the same as p, and it is false that p means the same as ¬p. Moreover the need for quotes in concoctions such as "'p is true' is false" disappears along with true and false. It seems reasonable to me to suppose, in that case, that "This proposition is false" can be represented as p = ¬p, and (for example) "The next proposition is false; the previous proposition is true" can be represented as
p = ¬q; q = p. So the "paradoxes" translate into straightforwardly inconsistent arguments.

But what exactly has been eliminated by the redundancy explanation? Not, by any means, the idea of what it takes for a proposition to be adjudged true or false, but only the use of truth states as argumentive devices. It is just that a proposition is thought of as being false relative to some other, and therefore nothing ensues from this concept of truth and falsity save the relation of contradiction. And in my view all coherence and "semantic" theories of truth suffer from a similar kind of relativity, even if less flagrant. But if one were committed to a theory that confined itself to a relativistic account of truth one would be wrong to think of a proposition as a sentence hanging in limbo, so to speak, as if waiting to be proven true or false. For, on the redundancy theory, to assert the truth of a proposition (with or without "proof"!) is just to assert the proposition, and to proclaim it false is to assert a contradictory proposition. This, however, is a position that I believe to be entirely mistaken, as I intimated in §2. Surely if is true is redundant in p is true, then p is true, and any further argument must proceed on that basis*. To assert a proposition is not (necessarily) to assert a true proposition, but only one that (necessarily) purports to be true. What determines its truth value, at least in the case of material propositions, is its "correspondence with the facts" - for want of a better phrase. (* For example "if p then q" would have to be read either subjunctively as "if p were (or had been) true then q would be true" or assertorically as "p therefore q".)

I do not wish to suggest that each of the six propositional types calls for a specific, dedicated theory of truth or meaning. (Indeed for some of the types it might be difficult to put forward a theory that does not suffer from the kind of triviality I'm complaining about - see below.) But if one could conceive of a single theory, as has generally been the philosopher's ambition, it should obviously be one that endeavours to define truth in all its aspects. In particular it should square up to the traditional problem of the truth of material propositions. On this question it seems to me that coherence and so-called "semantic" theories have very little of significance to say. Tarski's contribution to this topic, for example, provides, at best, only a grammar of "true", given a correspondence theory of truth (or rather, given a way of speaking as if ...) Since his account offers no clues as to how one can actually tell whether a simple proposition is true or not, it cannot be regarded as a definition of truth at all, but only as a kind of guide to the mechanics of applying the word "true" in a specified language. For myself, I don't understand how it can succeed in this. Indeed I think it would be difficult for anyone except academic philosophers of a certain peculiar species to understand the point of Tarski's idea of characterising the truth of a statement in a given language in terms of a metalanguage. To me, Tarski's theory is just a tautology with excuses - feeble excuses at that.

In contrast to this approach, consider how a scientist might go about defining "green". It would be very surprising, anyway, if he thought fit to include in his set of criteria such matters as: if x is green and y is green then both x and y are green; or if x is green then there's a sentence in German that says the same thing.

Recently (a decade or so after the first draft of this article) variants of the "deflationary" and "disquotational" theories of truth have become popular. Deflationary theories cover a wide field and live up to their name - they knock the stuffing out of truth, or rather, out of the predicate "is true". Disquotation (a term invented by Quine, and now existing as a number of variants) appears to be a modification of Tarski's basic scheme so that it looks more like the redundancy theory, while retaining a use for "is true". Disquotational theories have in common the schema "S" is true iff S, exemplified by "Snow is white" is true if and only if snow is white, the idea being that if you remove both is true and the quotes around S from the antecedent clause you're left with the consequent clause, a result that can be grasped only by the brightest of intellects! But disquotationists add that attaching is true to the quotation of a sentence has the same effect as would be obtained by erasing the quotation marks. How this is interpreted is, and has been, anybody's guess, but usually it appears to mean simply that truth drops out of the picture altogether. Colin McGinn begins a book chapter entitled "The truth about truth" with the proclamation "Disquotation is the essence of truth". Fiddlesticks! As it stands the formula seems utterly trivial, even in the context of the commonly accepted view of propositions.

So what smidgen of charity can theories which indulge in diversions of this kind extend towards a proposition such as: "If you lie unprotected on a concrete pad and a road roller runs over you, you will be squashed to death"? Those who maintain that the sense of truth associated with this statement can be captured by their favourite coherence or minimalist theory appear to be guilty of intellectual tomfoolery, philosophical wimpishness and a rude indifference to the human predicament. For their attitude puts wordplay ahead of the business of survival, ahead of life's thrust, its solid batterings and its genuine accomplishments. In withdrawing from the front where the real action goes on, the promoters of linguistic theories of truth take the meaning out of the world and condemn it to a closed circuit of symbolism - as if existence were nothing but a serial told in books. There are books indeed, and scraps of paper to boot, but they too are in the world, which would not be all that much poorer without them. It's time to snap out of this academic slumber! No theory of truth can afford to ignore recent developments in the neurological and behavioural sciences, on the basis of which a very comprehensive picture of the associations between words, ideas, objects and actions is beginning to take shape. The outstanding task might then appear to be to upgrade this physicalist description to an epistemological version that avoids the absurdities of reductionism. It is my belief, however, that a conventional description of these relationships will never be completed, nor prove adequate, and that during the developmental process a quantum leap will occur, taking our conception of brain and mind beyond the edges of normality. And once again science will leave philosophy staggering in the doldrums. (Why! - philosophers of the present era cannot even agree on what it is that scientists do, let alone match their vision of society and the universe.)

9.3.2 Purportment and the deflation of deflationism

Having so fervently belittled semantic theories of truth, it must seem remarkably disingenuous of me to join the fray by arguing for a point of view that looks for all the world like another of the same genre. Nor is it a satisfactory excuse to insist that my proposal is not a theory of truth but (as I see it) an irrefutable truth about propositions. A better excuse is that I believe the proposal gets rid of deflationism and "propositional grounding hierarchies" and engages more naturally with objective theories of truth. And it is certainly not an “account of argumentive consistency”, as I dubbed some of these minimalist dogmas (§9.3.1).

It's possible to think of all deflationary so-called theories of truth - including and perhaps especially Tarski's - as expressing essentially the same thing: that a proposition or utterance says what it says, and this is very obviously not the same as saying that it is true: they are of little help in explaining that connection. And this goes for the purportment approach too. The main difference between deflationary theories (notably the redundancy theory) and the notion of purportment is that, while in neither account does the predicate is true in the expression p is true add anything to the meaning of p, in the purportment account the predicate assigns a truth value while in the deflationary accounts it does not.

The principal idea signified by "purportment" is that a proposition is necessarily advanced as, and understood as (but not necessarily accepted as), a truth and not as a falsehood or a mere sentence with no clout. And this is to say no more than that a proposition means what it says, not its opposite or some other meaning. The value "true" is not automatically allotted to a proposition just because it has been asserted. It may well be false. Its truth value depends on its connection with reality, and the nature of this connection depends partly on the type of proposition.

Imagine a language in which propositions do not purport to be true. Suppose all propositions meant the opposite of what they mean now. So if you said “Mary did not go to the party” you really meant “Mary went to the party” and if you said “Some cows eat aspirins” you really meant “No cows eat aspirins”. All propositions would thus purport to be false. But this seems so ridiculous that one would think that what now look like negatives are really, in the hypothetical language, affirmatives. (An analogy is the experimental finding that subjects who wear spectacles that turn everything upside down soon adapt so everything looks the right way up again.) Again, suppose every sentence is in the form of a question, which must be accompanied by a qualifier to indicate whether it is intended as a true or a false assertion: "Do bats have wings? Yes" or "Do bats have wings? No". Or, in similar vein, you might think of a proposition as being a kind of hypothesis or probability statement: "Assuming bats have wings...". (This is akin to the "in limbo" supposition dismissed in §9.3.1.) Formats of this sort might seem otiose due to the fact that the content of the question is just that bats have wings, or, to put it in a less assertorical mode, the having of wings by bats, and that dressing the sentence up as a question or a hypothesis doesn't alter the fact that "what is meant is what is meant". A more appropriate format for questions and conjectures might be: “Bats have wings, don’t they?”

If a proposition only purports to be true then its truth value is undecided. It's another thing to say that it actually is true, that it has been granted the value "true". If "truth" is to retain objectivity and authority, then to proclaim a proposition true is to place its truth beyond doubt - to certify, conclude, verify, resolve, discover, sanction, prove, confirm ... that it's true, implying that there's nothing further to be gained by attempting to deny its truth: you'd need a kind of appeal, so to speak, a review or reconstruction of the whole situation, a mini-paradigm shift. And if truth does not possess that kind of robustness one might as well succumb to the uncivilised whims of relativism and deconstructionism. Under this interpretation, therefore, the deflationary formula  "S" is true iff S  is incorrect: to assert p is not the same as asserting "p is true". The first is a proposal (a proposition in fact!), the second a resolution, acceptance... or, as I shall call it, a closure, justified, if necessary, by appraising whatever criteria are appropriate for the kind and context of the proposition. In classical symbolic terms, this is close to saying that a variable, p, becomes a constant, P. And indeed this is how I propose to handle the matter. When a proposition is resolved it becomes an entity with a fixed truth value; it is no longer a proposition but a closure. This approach construes propositions primarily as instruments of communication and brings truth down to earth, but of course it has its worries (some of which are discussed later on) and sheds no light on the central problem of truth and meaning - the question how propositions relate to real-world phenomena. As we shall discover, this vewpoint may be inconvenient but I am confident it is essentially correct and accordant with all natural languages. (Note: Apart from signifying closure, “is true” has various auxiliary uses which may not involve closure - such as indicating the truth of indirect statements and applying emphasis.)

The most obvious kind of closure is the result, 1 or 0, of a correctly calculated valid formal argument. But if the argument does not reduce to 1 or 0 it is not closed with respect to the test for solubility, that is, it remains an insoluble structural proposition (see chart). However, it may be closed with respect to a lower category: for example, the proposition "If it rains the cucumbers will grow faster" represented by p → q  might well be closed with respect to the objective category but is obviously insoluble. (Please note that the scheme represented in the chart was not designed to take account of purportment, which is a later addition to this article. Under the purportment interpretation the three kinds of proposition at the top of each hexagon would be classified as closures and doubtless the entire scheme would change shape.)

A difficulty that apparently arises in the purportment account (as in other related kinds of proposal) concerns the status of the proposition p is true, for, as things stand, it seems this proposition only purports to be true, and likewise the proposition "p is true" is true, triggering a regress and forcing the use of quotes. However, given the "hard" conditions for truth indicated above, it's clear that "p is true" purports to be true makes no sense*, and that assigning a truth value hardly warrants launching a regress. On the contrary, the assignment of a truth value brings closure to that proposition. Of course, with various kinds of exception, when a proposition is posited it is in fact either true or false. But (remembering that propositions are linguistic/communicative devices) if no-one knows whether a given proposition is true or not - if it hasn't been established one way or the other - the comment that it is either true or false is a mere tautology. (At the end of §9.5 I put a different complexion on this by claiming that all propositions are purportmental and attitudinal. I'm not sure whether closures are exempt from this verdict.)

* Since purportment is an intrinsic property of propositions, explicit statements of puportment such as this are redundant: In a logical context, having defined p, q ... as propositional variables, "It is not the case that p purports to be true" is senseless, and "p purports to be true" is a tautology of the same kind as "p is a proposition".

But this by no means disposes of the problem of the status of closures. For surely one needs to be able to utter or inscribe closures just as one does “ordinary” propositions. In that case it seems we are committed to a two-tier arrangement of statements: those that are propositions as I’ve defined them and those that express closures. The trouble is, in ordinary conversation it would usually be difficult to tell them apart. I have no immediate answer for this, but see §9.6 below for more comments. (Also note: 1. It’s possible some propositions, such as trivial tautologies and perhaps many of the utterances of smalltalk, may be considered to be spontaneously true or false. 2. There are presumably countless "facts" concerning which no propositions have been, or ever will be, formulated - though exactly how one is supposed to read this I haven't the slightest idea. It’s pretty much an act of faith that the facts we don’t know don’t impinge on the facts we presume to know and for which there are statements expressing closures.)

There's no particular reason why ordinary logical arguments should be affected by this revised view of propositions. The stepwise proof and syllogistic styles of argument, following preordained rules of inference, make the assumption that the premises are true, while in the formal inferential model p → q  the truth of the consequent depends on the truth of the antecedent. In both cases the traditional true/false dichotomy applies, but in neither case does the argument itself depend on the existence of any true proposition. Doubtless, though, it would help many debaters to make their case if the key elements of their argument, and indeed the structure of the argument itself, were supported by established facts and principles - closures? And could closures serve as points of contact with reality?

One could draw a comparison between the evaluation of propositions and the marking of students' answers in a test paper. Their answers are put forward as being correct, but they are actually neither correct nor incorrect until marked with a tick or a cross by the examiner. Likewise with propositions: they purport to be true but the stamp of authority or concensus is needed to settle their truth or falsity. This might appear to warrant a 3-value system of formal logic, but I can't see anything to be gained by going down that track. Again, consider the notion that propositions, as defined here, cannot be nested: i.e. while a formula such as p ⊃ q  might well be a proposition (which of course purports to be true), the internal p's and q's cannot have the same status. This would be a mistaken presumption, no doubt elicited by a failure to internalise purportment. "If p then q" does not mean "If p purports to be true then q [or then q purports to be true]". The notion of purportment is firmly embedded within every proposition and cannot be externalised in this way. The logic goes around it, so to speak; hence a more accurate representation of this example is along the lines of: "If p (which purports to be true) is true then q (which purports to be true) is true". But even this bracketed externalisation is acceptable only for the sake of explanation. (Also see §8.6 Logic as inductive thinking.)

A variation of the purporpment view is the “innocent until proven guilty” account. In this version the fundamental notion of purportment remains intact but in addition propositions are granted the benefit of the doubt: that is, they are assumed to be true until proven false. With this measure, it might be thought, the semantic concept of truth would stand a chance of keeping its usual deflated complexion. But if propositions still await proof of falsity they surely await proof of truth too, in other words they await closure. On the other hand, if false propositions are afforded the same degree of innocence as true ones then the “benefit of the doubt” statement would read “propositions are assumed to be true until assumed false”, closure drops out of the picture but, in effect, so does truth. We again become imprisoned in a virtual world of never ending procrastination.

9.4 Preservation of type - universals and quantification

As I've already suggested, the theory of propositions advocated here raises questions about argumentation as well as about truth, owing both to its rejection of the analytic concept of logic and to the conceptual incompatibility of its propositional types. It is my contention that all model arguments comprise basic propositions that preserve their type under any valuation of the argument. But I should certainly not wish to say that every argument that fails to meet this condition is unintelligible. What it then might do, rather, is expose one or more of its constituent propositions as "ungrounded": the proposition is liable to fail, not because it is straightforwardly false, but because it isn't of an agreed type or perhaps for other similar reasons (such as the non-currency of the grammatical subject).

The ostensible reason for allotting type-preserving arguments a favoured status, therefore, is just that they are clearly oriented and much less likely to be misleading. But if, as I think, changing type implies switching between "logically" unrelated categories, then the customary style of logical argument is in yet more trouble. Standard formal logic, for example, could no longer provide a template for reasoning about extrasystematic matters, most conspicuously when material expressions are substituted for its variables. Under the conventional interpretation of predicate calculus, the negation of an existentially quantified expression is equivalent to an unnegated universal expression. The status of most universal propositions, however, has been a bone of contention. It is widely held that there are wholly meaningful universal material propositions, which is to say, at least, that specific empirical observations may serve to discredit or reinforce them but not to verify them. In my view this is outright nonsense, reflecting a wholly mistaken view of reality: there are no universal propositions conforming to this criterion, call them hypotheses or anything else. General expressive propositions (see chart) may belong to one of four classes, listed below with examples:

(1) Material categorical: Every sheep now inhabiting the Redland shire is a Merino
(2) Material probability: All matches ignite when struck (meaning only that the probability of finding one that does not ignite is small)
(3) Constitutive: All soccer balls are able to move only up or down, frontwards or backwards, left or right
(4) Diagnostic: All (intact adult) birds have a beak and feathers

Only 1 is disprovable by searching for contrary instances, and this is not a universal proposition because every object to which the statement refers can be identified; in principle the same observational procedures may be used to test its falsity or its truth. But a logic with quantifiers reflecting these properties has different rules from those of standard predicate calculus. And (in spite of examples quoted by some writers) it would be absurd to suppose that universal expressions might be interpreted diagnostically (i.e. as definitions) while existential expressions remain contingent. For then the negation of "All Union Jacks are red, white and blue" would be "Some Union Jacks are not red, white and blue", which is nonsensical unless it too is diagnostic (i.e. it relates to a subclass or kind). Arguments with quantifiers, like those without, must remain true to type - or else lead to bedlam. Unfortunately standard predicate logic will not succumb to any unified empirical argument, and this fact has been the chief source of an extraordinary legacy of puzzlement and paradox, reconstructionism and ingenious epistemological conjectures, from post-Meinongianisms to Popperisms, and in particular some extraordinary analyses of inductive inference attempting to justify it or undermine it. (I have joined the fray in A Ravenous Tale - notes on the induction and falsifiability illusions.)

Some may argue that there are "scientific" or "law-like" propositions, i.e. universal empirical propositions not falling into any of the four categories above - the kind of fundamental propositions which Popper says can be falsified but not verified, the kind which Khun says get replaced when there is a "paradigm shift", and perhaps some other well established scientific principles. A supposed distinguishing feature of law-like propositions (as opposed to "accidentally true" propositions) is that they uphold counterfactual propositions that exemplify instances of the law. For example, if "All mercury is liquid at normal room temperatures" is a scientific law, then it is upheld by the counterfactual "If the locks on my bedroom door were made of mercury they would be liquid". Whether or not law-like propsitions are a distinct species, this criterion is circular in the extreme - worse, considering the obscure nature of counterfactuals. Examples frequently offered as non-law-like general propositions include statements like the one I've given above to illustrate class 1, and statements one could only describe as flippant, such as "When the sun is shining the socceroos always win" with the corresponding pseudo-counterfactual "If the sun had been shining the socceroos would have won".

But induction and its soul-mate, prediction, don't even require a large number of past observations: a single observation may suffice. If you once stick your hand in a fire and get burnt, you are unlikely to try it again. You enter a hotel room, wonder which switch turns the main light on, you find it by turning it on; next time you enter the room you know straight away which switch to press. Induction? Maybe, but a host of underlying factors come into play: implicit background knowledge, circumstances/context, perceived causal connections, analogous usages/instances ..... and always guided by one's intentions. And this is true of inductive inference in general, law-like or not. Regardless, how can we put our trust in any kind of induction or prediction, given that the future for every one of us is so uncertain?

The difference between constitutive and allegedly scientific facts is that the former cannot be conceived to be otherwise while the latter can (for example the values of the fine structure constant or the speed of light could conceivably be quite different to their observed values). Still, the idea of a law that applies everywhere and for all time (whatever that might mean) is very strange, and I don't think most scientists ever deliberate in those terms. The fact that laws remain open to the possibility of revision explains our affection for their technical utility rather than for their aspirations toward absolute truth. There's no good reason for supposing that such laws differ logically from category 1 or 2. And whether or not we think the world is coming to an end, there's no good reason for rejecting their unquestionable utility. For more remarks on this topic and Goodman's "grue" see A Ravenous Tale (including footnote 5).

9.5 Assessment of type

Although of course many quite ordinary arguments do contain a mixture of types of proposition, the respectability of mixed arguments depends on the existence of a uniform operational context in which the truth or falsity of the mixture can be evaluated. The purportment of the argument as a whole must therefore be of an acceptable type. Since in my view there are (not only at least six but) probably no more than six propositional types, then these mixed arguments can be evaluated as wholes only if the whole can reasonably be allocated to one of these six categories. The following simple example lends itself to this treatment:

(r) There are koalas in the forest (material); the koala is a marsupial (diagnostic); therefore there are marsupials in the forest (material).
The trick is to see clearly the purportment of the argument as a whole - often only a matter of picking out the "operative" term (or the implicit rationale if the operative term is missing). To assist with this a number of techniques are available. One method is to consider the inverse argument, taking care to avoid changing the type of the constituent propositions:

(s) There are no marsupials in the forest (material); therefore either the koala is not a marsupial (diagnostic) or there are no koalas in the forest (material).
The purportment of the inverse argument is determined by that disjunct in the consequent which one would expect to be denied in countering the argument (presumably, in this example, the diagnostic expression). Another method is to reduce the argument to "implicit" form by removing (what one believes to be) the operative term. Compare:

(t) If there are koalas in the forest there are marsupials in the forest
(u) If there are koalas in the forest there are gum trees in the forest
The rationale of the first argument is diagnostic while that of the second is material. If only every case were that simple!

In addition to overtly mixed arguments, language is replete with quite innocuous looking mongrel expressions which resist cleavage into components of different types. We came across some of these in considering the status of mental propositions. Words like "maturity", "work" and "love", for example, generally encompass both physical and psychological aspects. This plurality of type is partly to blame for the inability of such words and phrases to form definitively true or false predicates, a point that seems to be tacitly understood even by those of a non-analytical temperament. Propositions with structural content are less easily explained. Even if

(v) The total weight of the cabbages is 1874 kilograms minus the weight of the container (425 kilograms)
is not about cabbages or containers, it is about physical quantities (weights) and has a more experiential than mathematical colour, while

(w) The standard deviation of the weights of the cabbages is 0.2968
seems distinctly more mathematical than experiential, even though nothing explicitly calculative is contained in it.

Mixed arguments like these, as well as statements employing mongrel words, obviously hang together in some way - perhaps, as I believe, different ways for different combinations of types. But the fabric of this coordination is often by no means easy to unravel, and one might be inclined to think only that, in the last resort, it is pragmatic, purpose-oriented and to that extent empirical; or even just "logical". And despite my call to maintain respect for type, it remains a fact of life that evaluations in one category (e.g. structural or subjective) may have relevance in another (e.g. objective).

Doubtless much of this ambiguity ensues from the fact that conventional ways of categorising our ideas are to some extent arbitrary. While it is in our interest to enumerate and define these categories as effectively as possible, it is more important to appreciate not only that the walls between them are vulnerable, but that this vulnerability begs exploitation. One example of this potential fusion is the empirico-structural theory pursued in §s 8 and 11. Another pertains to the interaction between verbal and material propositions, as suggested by the remark (§4) that meaning is or can be learnt ostensively, and the claim (§5) that the apparent difference between denotative and connotative aspects of propositions is explainable in terms of language usage. In addition, there's a point of view, with which I have some sympathy, that language in use is best thought of objectively, as a material association between word-tokens and the objects that they depict.

For, although it's arguable that many words and other semantic units have no referents, most are in fact signs of the things they mean - rather as effects are signs of causes, a swarm of bees a sign of honey or an ocean buoy a sign of a sandbank. By stepping outside language so as to place its utterances in the same arena as the objects with which it deals, we can develop a more pragmatic awareness of how all these elements interact with one another, and avoid that intensely parochial attitude that denies meaning in objects and objectivity in speech.

The world spoke to us long before we could speak back, and it taught us to reply in its own vernacular. The workface of language is the stratum of communication between people and objects, wherein a more or less spontaneous release of speech signs occurs in reaction to changes in a person's surroundings, effecting, in turn, immediate behavioural responses which lead to further modifications of the environment...and so on. From this angle, the purely linguistic concepts of connotation and denotation lose their significance and there emerges instead the single idea of signification, an activity which embraces both language and objects in an unbroken field.

If philosophy would ever get to grips with existence, it must learn its meaning, its predicative value. The analytic offensive has managed only to drive existence deeper and deeper into the sands of empty symbolism, culminating in the most barren and dispiriting expression in the recent history of philosophy: To be is to be the value of a (bound) variable (Quine, 1948). Is there a Heidegger in the house?

I haven't touched on a whole range of other sorts of propositions which might often appear to be of mixed type and which, moreover, raise various other issues that are relevant to one's understanding of truth and meaning. Donald Davidson (1967) includes the following in a list of "difficulties and conundrums": counterfactuals, subjunctives, probability statements, causal statements, adverbs, attributive adjectives, mass terms, verbs of belief, perception, intention, action. Most important among them, I think, are those which fall under the broad heading of attitudinal (or "intensional") propositions (Larry believes that..., Anita wishes that, many economists expect that... etc), and, seemingly less problematic, the indirect speech form (Chloe said that...). Many attitudinal statements cause a kind of paradox of identity, which can be dismissed quite simply. Consider: Lois thinks Superman is strong; Lois thinks Clerk Gable is weak. But Superman and Clerk Gable are the same person... The "paradox" disappears as soon as one realises that Lois does not think Superman and Clerk Gable are the same person. Thus "In the mind of Lois, Superman is strong; Clerk Gable is weak; Superman and Clerk Gable are not the same person" is unproblematic (apart from the problem of identity or co-extension which of course is not specific to attitudinal statements). The classical problem of "Oedipus hopes to marry Iocasta/his mother" and many related examples can be similarly handled. I doubt that most attitudinal statements are mongrels - at least, not on account of their containing a secondary element (the noun-clause or "sub-statement" that is believed, wished for... or said by the subject of the main statement), which one can regard as being merely the object-term in the main statement. Sometimes they are material propositions, sometimes mental. As for the potpourri of other difficulties associated with these speech forms, this is not the place to discuss them, nor am I equipped to do so.

I can, however, do two small things. First, I will register the passive observation that many philosophers make very heavy weather of some of this stuff, dragging in all manner of irrelevant and exotic ideas - not least, that ogre of modern logic, the possible worlds conspiracy. It seems not to have occurred to some of these metaphysicians that in any statement of the form "x thinks that y", anything whatsoever can be substituted for y, regardless of what goes on or makes sense in the real world. The kinds of links that many of them seek between what x thinks and the real-world status of y are illusory. As for statements of the "x said that y" kind, a common problem is that the main verb refers ambiguously to the form (and/or sense) of words used by the subject and to the content expressed by those words. So if the "subproposition" implicit in y is expressed using a different form of words, the extent to which the truth of the main proposition is affected is often unclear. Why the simplest instances of these indirect statements should cause so much commotion, however, is perplexing: it would appear to be only a question of deciding the intended meaning of the predicate of the main proposition (or of realising that two propositions are really intended concurrently). For sure, in the pages of philosophy journals, if not in everyday talk, things are rarely that simple. But this does not excuse the terrible confusion that prevails between problems that typically arise in "attitudinal" and indirect discourse and problems that occur just as frequently in direct discourse.

It seems to me that, in particular, the main problems raised in relation to attitudinal and indirect statements cannot be usefully discussed until one has established a clear position on one of the most fundamental issues in the philosophy of logic and language, namely the nature of the distinction between referring and decribing and the fact that some propositions seem to actively refer and describe at the same time. And, as far as I can tell, few, if any, philosophers do have a clear position on this issue - not, anyway, one that stands up to scrutiny. The whole dispute takes on ludicrous proportions when it comes to singular terms (of the Morning star and author of Waverley variety). My own approach here, as I've hinted elsewhere, is to deny that propositions contain denoting terms, in any sense of denoting that differs significantly from connoting. This is in sharp opposition to Kripke, Putnam and others who speak of directly referring terms and (worse still) rigid designators, not only for individuals but for substances and "natural" classes.

But now we have apparently broken away from indirect and attitudinal statements, and this brings us, rather impetuously, to my second "small thing": all propositions are indirect and attitudinal! For every proposition is both a creation of intelligent beings and understood only by intelligent beings. Some propositions are understood widely, others (the majority) more narrowly, in restricted contexts or by a narrow circle of people. And some propositions gain the backing of many, others of only a few. Does this imply that any widely accepted proposition could be re-hashed as "In general, people believe that p?" Well, of course nobody would argue that "p is true" is equivalent to "Everyone believes that p". But, on the other hand, no proposition can guarantee the truth of what it says, nor stand above the judgment of people. Every proposition is in some sense an opinion, a hypothesis - a purportment, if I may stretch the meaning of that word even further - whether in the mouths of many or in the mouth of one. And I believe that if we approach the more stubborn problems of indirect and attitudinal statements from this perspective, we shall find solutions that fall easily into a coherent view of meaning and truth.

9.6 Small talk, fiction and the testimony of witnesses

Everyday conversation generally consists of a giddy jumble of statements, questions, recommendations, exclamations, indirect reports and various other forms of speech, ranging in import from casual remarks and humour to painstaking instruction. How can we bring some semblance of orderliness to this hotchpotch? We cannot. But let's consider the declarative utterances of casual conversation from our present standpoint: they are often accepted as true with very little support, creating ambiguity between purportment, supposition and closure. Well, except in more formal circumstances, most of the time nobody is either offering or seeking concrete evidence. So most small talk consists of propositions with no expectation of closure: the whole of any conversation occurs at the "purportmental" level, with a liberal admixture of utterances that are not propositions. Those propositions that occur as premises in arguments are "weak" and can be regarded as assumptions or as antecedents of conditional statements. Should closures turn up, they either put an end to the line of conversation that initiated them or serve as "hard" premises (immutable "facts") for use in further discussion about the topic in hand or some new topic.

Although the logical fuzziness of small talk is only to be expected and from the present standpoint may be mostly of limited interest, in the fantasy world of the creative writer we find an important branch of small talk that deserves further scrutiny. It’s possible the six-kinds scheme is mistaken in confining the number of categories to six, though much depends on what one is prepared to call a proposition. "What about the propositions that fill the pages of novels?", some one might ask, perhaps believing them to be in some sense analytic. The answer to this is that, if the declarative sentences constituting novels are in fact genuine propositions (discounting those which do not qualify for reasons other than the mere fact that they occur in novels), they belong to one or another of the classes I have described. But according to the criteria I have adopted, most of them may not be genuine propositions, since the latter purport to be true while the fictional kind, it might be thought, only pretend to be true - they are bogus propositions. The suggestion of pretence might seem all the more attractive when conventional fiction is bracketed with other kinds of fantasy, such as the make-believe of children and religious worship. This supposition, however, seems to have the unfortunate consequence that purportment comes in different flavours. Well, when you’re wrapped up in a gripping novel would you want to keep reminding yourself that the bulk of it is a sham? Of course not, it would no longer be gripping! For all intents and purposes the propositions in novels are just like those in real-life stories. Still, the idea that they only “pretend” to be true cannot be dismissed until the concepts of pretence and bogus proposition have been made clearer.

The chief characteristic of bogus propositions is the ungroundedness of their subjects, or of the context in which they occur. Most seemingly relevant questions about the subject or context have no answers, so most potential statements about the entities of which bogus propositions speak are neither true nor false, either in fact or in fiction. A novel is, so to speak, an extended joke at the expense of the reader - or to the delight of the reader, if it's a good joke. Fictional statements cover the same broad spectrum of language games as those enacted in real life situations, they are neither confined to any particular one of the six classes of genuine propositions (such as diagnostic) nor do they comprise a special class of their own on a par with the other classes.

One might be disposed to think that, in the absence of sufficient knowledge, one could not tell whether a book was a work of fact or fiction, and might conclude that language doesn't contain the means to make this distinction; that language has no points of contact with reality and that meaning and truth are to be explained in terms of a coherence theory. This seems to me (at times!) to be a mistake. The difference between factual and fictional statements is (1) that the former are grounded - somewhere back along the line - in ostensive "definitions" or explanations (this is my house, this baby shall be called Anabelle, let's call these things quarks etc), or more gradually by a kind of social "catching on" process, or in a multitude of connections with the historical universe; and especially (2) that the former are mostly "domestic" expressions that take part in everyday activities (where's the vacuum cleaner, the soup needs more salt in it, put the car into reverse gear etc), uttered in association with specific real-life situations, used for practical purposes and thus firmly grounded in the real world. In short, given that all propositions purport to be true, the salient characteristic of fictional propositions is that it’s impossible for them ever to attain closure. But all fantasy is in the same situation while all “mental” statements (such as “I have a headache”) appear to be ambiguous depending on who has the reported experience and who does the reporting. (It may be preferable to assume all such statements remain purportmental, though one might think that termination of the experience itself would bring some kind of closure).

I'm therefore inclined to play down the notions of pretence and bogus propositions and to reaffirm the thesis that propositions purport to be true and may convert to closures; and while utterances may purport to be true if, and in so far as, they are utterances of propositions, the idea that bare sentences as such are “truth bearers” is one for the scrap-heap. But, although sentences often don't articulate propositions, we should not forget the very significant fact that they can nonetheless be strung together to create a fictional dialogue which, in the hands of an accomplished writer, carries just as much conviction as a factual report. Nor has the author only himself to thank. For the power of a good novel comes not so much from the greatness of the story-telling as from the willingness of the readers to believe in the reality of the parallel universe which it seems to depict, on the one hand, and the inherent weakness of real-life reporting, on the other. The unfeigned report seems no more able to capture or construct reality than the novel - it too is only a story. And story-telling in the large accounts for a huge chunk of the average person’s verbalization and activity. This partly explains why fact and fantasy are so easily confused - language alone, it seems, cannot draw the distinction.

In the present era television, cinema and computer games have blurred the dividing line between fact and fiction to an unprecedented degree, not merely by expanding and entangling the uses of language but by embedding the verbal mode of communication in their own worlds of compelling imagery, worlds that have evolved to leave less and less to the imagination and to resemble the real world more and more. The worlds of the movie theatre typically occupy two levels: on the first level we find audio-visual representations of the settings and actors, all of whom are real (computer generated animation is another kettle of fish), while at the second level we find the characters and events, whether factual or fictional, that comprise the story line of the movie. The first level includes the soundtrack of the movie, so all the utterances populating the movie, inasmuch as they are verbal tokens, occupy this level, but the communicative significance of these utterances clearly engages the second level, the storyline. So far as those utterances that are material propositions are concerned, their truth or falsehood is determined by criteria explored above and elsewhere, so these criteria may be invoked to ascertain the verbal elements of a movie that are factual. But in extending the discussion of fiction to the movie scene my intention was not to delve into the intricacies of the depiction of fact and fiction therein, but to work from back to front by suggesting that we might learn something of the relationship of propositions to the real world by heeding their status in the screen world. If we take a step back to see what our brains are processing when we receive a spoken message in some real-world situation we shall find at least that they are handling (1) representations of the word-tokens and relevant parts of the environment in which the message is produced and (2) the meaning of the message. It seems to me that the significance of the first aspect - the "statement tokens" - for theories of truth has generally been greatly underestimated by philosophers interested in these affairs. As these tokens are very much a part of our objective environment, we may discover in them a road to "hard" truth. I have flagged one possible road in a protracted footnote to My Philosophical Outlook. Perhaps we should also take on board how chidren learn language, then reverse the process. To give a bit more idea of the general direction I'm heading (some will say downhill!) I've also added a page of extracts from rough notes in my computer.

I suggested above that language alone cannot draw the distinction between fact and fantasy. But this is not to say we cannot make the distinction. Of course we usually manage to do so very easily, but this ensues from our inexorable belief that we are indeed real people embedded in a real world - a world which empowered even the novelist with the wealth of experience and the words to create his narrative. Whether this foundation is sufficient to differentiate the fiction and non-fiction sections of libraries throughout the world is debatable. Despite its terrestrial origins, I have a feeling that if ever language should manage to get an unshakable grip on reality, it will be through channels yet to be discovered. At present I can only conclude that if we want to know the meaning and application of an utterance we should look very closely at the circumstances of its production; and I'm strongly inclined to extend this prescription to mathematics too.

Doubtless human beings could pursue their lives well enough without propositions, without any structured language at all. But, as I hinted in §6.2 Private and public, paragraph 3, one's personal experience of the world and that of other people are what constitute the primary subject matter of propositions. Deeper than this, propositions may not burrow. There may indeed be untellable truths, but the truth of material (if not all) propositions appears to close with the testimony of witnesses. Their interpretations of reality will differ as much as the philosophers', and the search for the right answer to this question is of far greater importance than delvings into the structure of languages. My own views on the nature of "reality" are summarised in My philosophical position... Aha!, paragraphs 2-4., and to some extent in the Postscript below. Some other aspects of the relationship of propositions to reality are noted in Domestic and dictionary parlance and What Existence is not, and what it might be


If the critical feature of propositions is their subject matter, what is to prevent us from multiplying the number of types indefinitely? - there could be propositions about witches and propositions about washing machines, for example. The case for distinguishing at least six types - and hopefully not too many more - rests on the endorsement of a hierarchy of six distinctively different aspects of ordinary verbal communication. A fully developed language system requires these six functions and the means of expressing them. It doesn't need witches or statements about witches, but it does need, for example, meanings and statements about meanings. Each of these needs is attended by an ambience of necessity. So far as ordinary language is concerned, it had been my hope to defend the need and denounce the necessity; for, in spite of its commitment to the telling of home truths, language must be allowed the freedom to advance in line with the expanding horizons of our understanding. In so weakening the boundaries of the propositional types, however, I accept both that the way is made easier for the introduction of propositions about witches and that any of the already proposed types is susceptible to the charge of witchcraft.

My aim has been to paint a picture of a highly informal language structure, yet one which, by attempting to do justice to reality, experience and reason, provides a foundation for honesty in communicating our ideas. In this I concur with the main goals of logical empiricism: to determine the boundaries between sense and nonsense, truth and falsehood.

The touchstone I offer is this: to speak the simple truth is to utter a basic proposition that can be shown to be true, and to speak with integrity is to speak in terms of pragmatically coordinated, experientially effective basic propositions which one believes to be true. There's little more bewildering than the spectacle of otherwise educated people battling to defend philosophies of life based on untruths, muddlement and gobbledegook; and little more threatening than the perpetuation of these clouds of darkness throughout the world. Among them may be counted not only the more vulgar pretensions of the religious and cultural heritages of the various nations, but the frightening trends in modern sociology which draw sustenance from the turbid shallows of behaviourism and consequentialism, shunning the principles of human rationality and crushing individual responsibility at every turn. In dissension from the positivist tradition, however, I'm convinced that morality, human dignity and perhaps even religion are not for dumping in the never-never land that lies beyond truth and common sense but must be firmly grounded in these concepts. While we might well be perplexed by the uncertainties in these foundations, let's not be disheartened by them. They are the liberators of reason, enterprise, artistry, magnanimity and compassion.

The preservation of society in something like its present form will depend, first, on fostering these values through well considered, genuine and compelling applications of conventional language, aimed primarily at moderating the lamentable behavioural tendencies of our species (to multiply, to abuse and to destroy). And, secondly, it will depend on the development of concepts and benevolent technologies that transcend the boundaries of common reality, providing the means to confront the hidden gremlins of human existence in their own back yard. Modified language structures will play a crucial role in this enterprise; in particular, like the Greeks of another age, we shall want to learn whether our language can act as well as speak. But for the present we shall do well to contain our communication within the shores of our homeland - if we are wise, speaking hesitantly but comprehensibly according to our purpose, and resting content with the realisation that the truth we tell can never be stronger than the telling of it.


It is a curious fact that the most abstruse physics, plying the kind of stuff that philosophers are most disposed to call "unobservable", presents the greatest potential to turn our lives upside down, if it has not done so already. Out of the no-man's land of subatomic particles and mc2 looms the capacity to obliterate all physics, all philosophy, all life. What at first we cannot touch turns out to impose the most tyrannical presence, holding both the outside world and our inner senses in an awesome grip. How much stronger, then, must be the command of mathematics alone, flying free of all commitments to physical description! Clearly, in the physical sciences, effects have the punch but mathematics has the power. And there's absolutely nothing in between.

If, as I've suggested, space is mathematical, then at least some mathematics is spatial, and if space is also in some sense physical, so is at least some mathematics. If we can extend this alliance to time and mass (and perhaps even if we cannot), the following possibilities arise: some mathematical laws might be verifiable by observation; some mathematical laws might be modifiable by experiment; and some physical situations might be responsive to mathematical activity alone.

But surely facts can only agree or disagree with mathematical models! I believe this is an old-fashioned sentiment, one which defies all reason and which can no longer be of great service to mankind. In the first place, certain twists, paradoxes and puzzles of nature can be regarded as, or explained in terms of, peculiarities of the fundamental structure of mathematics; secondly, under certain conditions some theoretically computable problems should yield unpredictable or unexpected results, regardless of the materials used in producing and chronicling these results; and thirdly some mathematical models that do have sufficient integrity might interfere directly with reality.

These proposals imply that some of the future strategies of mathematicians will not succeed, simply because the maths will not find the room to work. In due course this hypothesis could be tested either by carrying out sufficiently complex calculations requiring sufficiently precise answers or by experimentally enforcing conditions which compromise the reliability of the calculations. On the other hand the most ingenious mathematical schemes will succeed not only in managing their physical constraints, but in altering the environment in which they are produced. Is it possible that there's already evidence of the ability of mathematical models to influence reality?

Up to this point in time, few scientific experiments have produced results that might lead one to make such an improbable (yet obvious) suggestion, so we are compelled to turn to the shadowy pages of popular "metaphysics" for some indicators. This genre, of course, thrives on all things supernatural, from ESP and necromancy to UFOs and teletransportation. It eschews the boundary between the mental and physical categories, delving into a world of pseudo-existence that combines aspects of both but failing abysmally either to convince intellectuals of its legitimacy or to find a unified explanation for the endless variety of apparitions that dwell there. Academics just might gain something from a more earnest inquiry into this realm. But, if we are to take any of it seriously, should we not also be seeking interpretations lying outside the immediate psycho-physical domain in which these phenomena seem to occur? The following examples (if any of them have any credibility) suggest a mathematical explanation:

The street map used by the psychic to locate the scene of a crime is a model of the real situation.

The effigy used by the voodooist to gain revenge is a model of the victim.

The mental image (or its cerebral counterpart) induced by a remote physical event or idea is a structural copy of the remote phenomenon.

The physical state of affairs induced by mental concentration is a structural copy of the mental (or cerebral) state.

The UFO "observed" in the sky is nothing but a mathematical model, inducing effects which could be taken to be "out there" or in the mind. The model could have originated anywhere.

Besides these (admittedly fanciful) paranormal examples, the universe is rife with mathematical structures, which appear again and again, reproducing themselves in countless organisations of stars, atoms and genes and their associated effects. Some examples of convergent evolution, for instance, could be attributed to mathematical effects alone, as could many of the examples cited by Sheldrake (1981) in support of his theory of morphic resonance (according to which all self-regulating organisations, ranging from molecular to social systems, respond to "morphic fields" that provide templates for the development of each type of organism. Lyall Watson's "contingent system" (1979) is a similar but broader, evolution-based theory.) In many areas of science it's becoming easier to accept the mathematical model and harder to understand why there are so many and varied instances of it.

Added to this is the spectacle of the "scientific zoo", whose inhabitants seem half mathematical, half empirical in character. But isn't it possible that these creatures have become too much of a distraction? Instead of hunting down ephemeral particles and pondering over their inherent nature as opposed to their observable effects, we might do better to inquire into existent mathematics and ascertain its empirical effects as opposed to its abstract properties.

In the world of computers and electronic engineering, hardware is increasingly being displaced by software, the actual by the virtual. Computing tasks that used to depend on moving tapes, discs or other physical devices and took months to complete are now performed in a flash by the motion of electrons in microchips. Cumbersome, grooved records that reproduced three minutes of song when revolved on turntables have given way to tiny MP3 players that can store and deliver hours of entertainment, with no moving parts and no evidence of any physical “thing” corresponding to the sound patterns that are produced. Computer programs can mimic many of the functions of hardware such as sound cards. The trend is obvious and we can only wait for the ultimate artificial mind – a computer that has no existence, except as information inscribed on a ball of virtual reality.

Mathematicians explore structural possibilities, and structural possibilities delineate reality, the cosmos. The aims of mathematics and cosmology are essentially similar. Moreover the productions of mathematics and those of technology are convergent. Always at the forefront of technological progress, the military will surely be the first to recognise and capitalise upon the possibilities of mathematical models, aiming for the capacity to build weapons of remote destruction. If my thesis is right, and its principles are open to controlled exploitation, military developments will present the main threat to an otherwise philanthropic, invigorating era of "mathtech". While nuclear weapons are very much a result of mathematical thinking, future weapons will actually comprise mathematical models themselves - the bomb will be replaced by the computer. Whereas in nuclear weapons the mathematics is confined within the explosive device which must be located at the target, in the next generation of weapons the mathematical trigger will be external to the device. In the third generation there will be no device - only a model existing in some computer remote from the target. And in the final generation even the computer may not exist as such.

Of course the target is likely to be some sort of recurring well-defined structure, either man-made or naturally occurring, and the method of destruction a model that alters some or all instances of the structure. Presumably the artefacts of high technology, including other computer systems, which are themselves largely the products of sophisticated modelling techniques, would be especially susceptible to direct mathematical manipulation. In the biological world the obvious targets will be specific kinds of DNA molecules - mutations at the press of a button anywhere in the universe! (Could it be happening already?)  Similar considerations apply to the evolution of benevolent uses of mathtech.

When physicists speak of laws of nature and believe in them, they can only mean one thing - that nature is mathematical. Which implies something like this: the universe is a theory, a mathematical model, a tautology. Perhaps it's a kind of self-perpetuating model, a model that must expand its axioms indefinitely, a kind of computer that must keep on working. Paul Davies (1992), in a section of his book that explores the limits of computation, points out that such a conjecture was mooted as early as the mid nineteenth century by Charles Babbage, the father of modern computers, and I believe it has since been echoed and elaborated by many a physicist. So one ought to be able to "seed" a universe just by formulating the "axioms". The inventor would not need to construct the whole universe if the axioms contained the means of self expansion. It would create itself, rather like the chain reaction of a nuclear explosion. Therefore, if our own universe is like this, to say that it started "with a big bang out of nothing" is to say that it started with a formula. Obviously this says nothing whatever about certainty, predestination or free-will: the model could allow for totally random syntheses. All it need contain are the essential conditions for existence. But whoever discovers the complete formula owns the genetic material of the universe - not just an image in the head or on paper, but the real McCoy.

What is the main stumbling block to attaining these designs? Just that the mathematician is confined within the walls of the very universe which he seeks to explain. He needs a kind of "twister" theory of mathematics to get out of the bind. Such a concept, I think, would provide a more useful avenue to a "grand unifying theory" than would a twister theory of physical forces! (However Stephen Hawking thinks that neither the universe nor human endeavour has walls. If language can take this on board, escaping the blind rant of politicians and ecclesiasticals and the bias and lack of courage of the media, in fifty years’ time people may have difficulty understanding what the present age has to say.)

So, from deriding rationalism, my story turns full circle, attributing nothing but structure to the whole of existence. Where it differs sharply from the more extreme forms of rationalism, especially the higher grade essentialist doctrines, is in its denial of the necessity of any proposition or connection of facts. Structure is not pure. When all is said and done, it remains only useful, a conjunction of many approaches and attitudes. The genius who would profess to hold a Theory of Everything must understand perfectly the fusion of mathematical, physical, psychological, even biological and ethical perspectives. It's unlikely that any such mastermind will ever exist but, well, the universe exists, doesn't it?


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Wittgenstein, L. (1953). Philosophical Investigations. Trans. G.E.M. Anscombe. 2nd edition 1958/63. Blackwell, Oxford.


1. Although in this essay the word "meaning" probably occurs more often than any other noun, nowhere have I attempted to justify my various uses of the word or to explain the meaning of "meaning", largely because it is problematic. There are, according to most sources, two (or more) unrelated areas of debate about meaning, semantic and foundational, neither of which has been visited by any satisfactory theory. I suspect they are paralleled by equally distinct problems of truth, and that answering the first kind of problem depends on answers to the second. But in so far as I concur with this division, my own approach to the semantical problem is intimated in §5.2, and to the foundational problem in an extended footnote to this article and elsewhere.

2. Historically the term "analytic" has sometimes been used to mean only "tautological" (analytically true) and sometimes "tautological or inconsistent" (analytically true or false). I have tried to avoid this ambiguity in the propositional classification depicted in the diagram by giving, for each type of proposition, a different name for each of the two cases. For example, a constitutive proposition is intrinsic if true (and inconceivable if constitutive but false). For most purposes, however, just as in the case of "analytic", it doesn't matter whether one implies the broader or the narrower meaning, so it doesn't matter which of the two terms (constitutive or intrinsic) is used. This is to be expected, given that every proposition purports to be true, whether in fact it is true or false. (Please use back-arrow to return to where you were).

3. In this system the values true and false are respectively represented by 0 and 1 (even and odd), "+" and "=" both represent unbracketed equivalence and "x" (the sign usually omitted) represents disjunction. A proposition can be represented arithmetically in "summative normal form" (SNF) as a sum of products of variables and reduces to 0 (tautology), 1 (inconsistency) or a contingency in simpler form; e.g. the SNF of "A implies B" is AB+B. The scheme can be expanded into a first order predicate calculus by using exponent variables. For the development of this system out of "nothing" and some of its advantages see this article.

4. The position of soluble propositions in this scheme perhaps needs clarification. They are the formulae of classical symbolic logic that are reducible by calculation to 1 or 0 (T or F), and as such they represent just one kind of mathematical structure. However, while there are a great many mathematical expressions that are conventionally called "propositions", many are distinctly non-propositional, and for this reason I do not include mathematics as a whole in the propositional scheme. The choice of classical symbolic logic as the syntactical model is quite arbitrary. Along with set theory, it appears to play a fundamental role in much current mathematics, and it appears still to be regarded as the system that most closely models the basic patterns of reason (Kant thought the same of Aristotelian logic). But I have no better reason for excluding non-classical logics from this position. Indeed I should much prefer to replace the conventional system with one that combines a "relevance" calculus of propositions with a non-classical (existence-free) predicate/relational annex (see this article).

5. This paragraph replaces a clearly incorrect paragraph in an earlier version of this paper.

6. In adapting this scheme to standard formal logic, one could use symbols representing the categories in the top half of each hexagon thus: (O)p means orthologous or true in S (soluble category), (T)p means translative or true in G (generative category) etc, or in general (Θ)p means true in Ψ. So (Θ)¬p means false in Ψ. Thus (O)¬p means (U)p or p is untenable, (T)¬p means (M)p or p is mistranslative, and so on. Note: it appears to make no sense to say that an orthologous proposition may be true or false: it simply is a true soluble proposition, with no possibility of falsification; while an untenable proposition is a false soluble proposition. Likewise for the corresponding elements in the other hexagons. If that's so, one seems to be restricted to using (Θ)p as a premise in a stepped (syllogism-style) argument and one would need to be careful about representing such arguments in the form: p & (p→q)→q, where the first occurrence of p is, in a sense, "supra-argumental". (Also see A logic of concepts that reduces paradox - section on RELEVANCE).

It's also possible to give a "sweeping" or "nesting" translation, in which j)p entails n)p for all n lower in the scheme than j and j)~p entails n)~p for all n higher in the scheme than j. This ploy would of course alter the entire structure, destroying the independence of the six types and causing unnecessary confusion.

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