Graham Priest, Paraconsistent Logic, and Philosophy,
Or, Logic and Reality

by Ralph Dumain

Review of: Priest, Graham. Beyond the Limits of Thought. 2nd ed. Oxford: Clarendon Press; New York: Oxford University Press, 2002. Table of contents.

Where then, lies the duty of surrationalism? It is to take over those formulas, well purged and economically ordered by the logicians, and recharge them psychologically, put them back into motion and into life . . . In teaching a revolution of reason, one would multiply the reasons for spiritual revolutions."

Gaston Bachelard


This birthday gift proved to be almost as stimulating as a related gift, the Funkadelic anthology Music for Your Mother. The second edition includes three supplementary chapters: one on Nagarjuna (also published as a journal article), a chapter on Heidegger, and a response to critics. My review of the Nagarjuna essay follows this section.  Also of note is a more recent article by Priest:

"Where is Philosophy at the Start of the Twenty-First Century?", Proceedings of the Aristotelian Society. 103 (2003): 85-99.

Abstract: This paper sketches an analysis of the development of 20th-century philosophy. Starting with the foundational work of Frege and Husserl, the paper traces two parallel strands of philosophy developing from their work. It diagnoses three phases of development: the optimistic phase, the pessimistic phase, and finally, the phase of fragmentation. The paper ends with some speculations as to where philosophy will go this century.

I presumed from his publications in the journal Science & Society that Priest was some kind of Marxist.  Yet this is curiously muted in both the aforementioned publications.  This doesn't matter to me at all from a political standpoint.  I am interested in the question from a purely philosophical standpoint, i.e. on the question of the relation between logic and reality, ontological commitment, and orientation towards philosophy's past and future.  I am both inspired and perplexed by Priest's book: there's a shoe somewhere I'm waiting for to drop.  I am much less satisfied by Priest's article, which both discusses and effectively dismisses Marxism from his historical purview, though Priest claims a much wider intellectual territory than the usual pigeonholes Anglophone philosophers customarily drop into.  Priest's breadth combined with his intellectual niche as a logician both does him good and yet makes me wonder what he has to say on the translogical side of affairs, and by this I include a perspective on how logic fits into the totality of human behavior and the history of alienated social being.

Also to be factored in are the history of logic in various philosophical traditions (India, China, Greece, medieval Europe, modernity) and recent local discussions of the Tao Te Ching (Dao De Jing), which very much relate to the subject of Priest's book.  This all feeds into our local group discussion around the question: "What is the relation between logic & reality?" 

Interestingly, Priest disclaims the need for a technical background in logic.  When he writes in prose, he's an exceptionally clear and crisp writer.  However, he does delve into logical notation, and those of us not well grounded in logic will have a problem with these passages. 

(26 July 2005)


Garfield, Jay L.; Priest, Graham. “Nagarjuna and the Limits of Thought,” Philosophy East and West, January 2003; 53(1): 1-21.

Abstract: (Jay L.) Nagarjuna seems willing to embrace contradictions while at the same time making use of classic reductio arguments. He asserts that he rejects all philosophical views including his own—that he asserts nothing—and appears to mean it. It is argued here that he, like many philosophers discovers and explores true contradictions arising at the limits of thought. For those who share a dialetheist's comfort with the possibility of true contradictions commanding rational assent, for Nagarjuna to endorse such contradictions would not undermine but instead confirm the impression that he is indeed a highly rational thinker.

This was published after the second edition of this book.  I did not check to see whether this version is identical to the book chapter.

Nagarjuna is one of India's most influential philosophers and has also attracted attention in the West.  Priest (shorthand for Priest and Garfield), an advocate of dialetheism, or the admission of contradictions in logic (his field is logic, particularly paraconsistent or transconsistent logic), suggests that Nagarjuna is not an irrationalist as commonly thought, but an implicit dialetheist.  Nagarjuna's radical self-contradictory statements are analyzed.  Priest puts him in the company of Kant, Hegel, Wittgenstein, Heidegger, and Derrida.

One way in which Nagarjuna does differ from the philosophers we have so far mentioned, though, is that he does not try to avoid the contradiction at the limits of thought.  He both sees it clearly and endorses it.  In the Western tradition, few philosophers other than Hegel and some of his successors have done this.)  Moreover, Nagarjuna seems to have hip upon a limit contradiction unknown in the West, and to suggest connections between ontological and semantic contradictions worthy of attention. (p. 4)

Nagarjuna both distinguishes conventional from ultimate truth and then obliterates the distinction.  The ultimate truth is that there is no ultimate truth.  "All things have one nature, that is, no nature."  (Note: In classical Indian logic, there are four, not two, truth values.)   The paradox is that Nagarjuna's fundamental ontology shows that fundamental ontology is impossible.  Nagarjuna does not just attribute paradox to the nature of language.

But Nagarjuna's system provides and ontological explanation and a very different attitude toward these paradoxes, and, hence, to language.  Reality has no nature.  Ultimately, it is not in any way at all.  So nothing can be said about it.  Essencelessness thus includes non-characterizability.  But, on the other side of the street, emptiness is an ultimate character of things.  And this fact can ground the (ultimate) truth of what we have just said.  The paradoxical linguistic utterances are therefore grounded in the contradictory nature of reality.(p. 16)

Oy, I have a headache!

Priest then gives a formal logical treatment of "inclosure", which models this paradoxical scenario.

While Nagarjuna's paradoxical doctrine overlaps that of various Western philosophers, there is a unique, ontological "Nagarjuna's Paradox."

If Nagarjuna is correct in his critique of essence, and if it thus turns out that all things lack fundamental natures, it turns out that they all have the same nature, that is, emptiness; and hence both have and lack that very nature.  This is a direct consequence of the purely negative character of the property of emptiness, a property Nagarjuna first fully characterizes, and the centrality of which to philosophy he first demonstrates.  Most dramatically, Nagarjuna demonstrates that the emptiness of emptiness permits the 'collapse' of the distinction between the two truths, revealing the empty to be simply the everyday, and so saves his ontology from a simple-minded dualism. (pp. 18-19)

Such is Priest's conclusion.  But this seems rather inconclusive to me.  Do we take logical metaphysical reasoning at face value, dismiss it as meaningless, go in for therapy for this bewitching linguistic (and metaphysical, according to Nagarjuna) disease?  All of these solutions strike me as unsatisfactory.  I believe that limit situations (totalities and infinities) do generate contradictions.  Here we are talking about fundamental categories and their logical relationships.  But I would ask another question, as I did in our discussions of the Tao Te Ching: how to relate an enclosed categorial schema (which in that discussion had to do with the relation of being and non-being, etc.) to actual empirical reality?  There is a mismatch.  Metaphysical constructs are neither meaningless nor to be taken as is.  The resolution of metaphysical paradox in the "everyday" seems to be to be as closed, vacuous, and 'metaphysical' as the situation crying for such resolution.  Something is missing here.  But I'm also wondering: where does Priest stand?

(4 July 2005)


This is truly a remarkable book in many ways, though at times it makes me uneasy.  It is amazing to see a logician pursue the illogical purely via logical means, driving certain notions to their logical and paralogical limits.  Secondly, Priest is remarkably non-provincial, ignoring the barriers that have been set up between analytical and 'continental' philosophy, and between Western and Eastern philosophies.  Thirdly,  Priest gives us some new tasks for the 21st century, while also giving us a grand view by which to reevaluate all the philosophies of the past.

Though he only makes brief appearances, Hegel seems to be the hero of this book, with his notion of the true infinite and his forthright engagement with contradiction.  Priest, though, pretty much ignores the structure of Hegel's logic and thought as a whole, and his take on mathematics including his view of mathematics as a poor basis for philosophy.

(When I review Zeleny's critique of Priest, perhaps I'll have more to say.  It will be interesting to see if Priest has anything of a dialectical nature to offer now that he has established to his satisfaction the ineliminibility of contradiction from the world-picture.  In any case, from within the history of logic and analytical philosophy he has attained a level of sophistication about these matters lacking in the history of Marxist thought.  His attraction to oriental philosophy can be seen in his interest in the martial arts, Taoism, etc., but hopefully he can transcend that as well.)

The historical overview has essentially four components:

(1) Pre-Kantian (ancient Greece, some medieval philosophers, Berkeley);
(2) Kant and Hegel;
(3) Paradoxes of self-reference in modern logic (Cantor, Russell, Ramsey, Zermelo, Von Neumann, Curry);
(4) Language & its limits (Frege, Wittgenstein, Quine, Davidson, Derrida).

The first part is not difficult to understand.  Kant is the historical watershed in formulating the limits of knowledge and their relationship to generation of paradoxes.  From this point on, reading Graham is more demanding on the reader.  Priest is a very clear, unpretentious writer, but as we read through part three, logical notation and the technical arguments become much more difficult to grapple with for the non-specialist.  I cannot say that I have fully understood this section.

Part four is curious in its own right.  Priest drives all of the arguments past their logical conclusions on a strictly logical basis.  Yet I think there's much unsaid—perhaps material for another book.  Why?  Because I think that, paradox aside, there's something fundamentally misguided about all these thinkers (though I don't react negatively to Frege in the way that I do to the others.)  I am still waiting for the other shoe to drop, as my ultimate question is the schema of relationships connecting concepts, language, logic, and material reality, which I think all the philosophers of language cited have flubbed badly.

Priest's grand project is to unify all of the paradoxes, rejecting even Ramsey's distinctions, and to show how they all fit into what Priest calls the inclosure schema.  (See diagram on p. 156.)  The limits of thought struggled over throughout the history of philosophy involves limits of expression, iteration, cognition, and conception.  Attempts to suppress or resolve one limit also involves the resurrection of the paradox at another limit.  Priest's grand scheme is to show how the fundamental paradoxes involving totalities, infinities, and limits fall into a single pattern and cannot be eliminated.

(2 August 2005)

Graham Priest's Inclosure Schema


I will return to parts 3 and 4 of the book, but now I will discuss the first of the chapters appended to the 2nd edition, chapter 15: "Heidegger and the Grammar of Being."  Priest begins by arguing, as I do, that the very distinction between analytical and continental philosophy is a bogus conceptual artifact of Anglo-American philosophy that cannot hold up under sustained analysis.  (Priest, an Australian, blames only the British here.)  If one wants to pursue this conventionally postulated distinction, one finds that Husserl, Heidegger, Sartre, and Derrida are different from those classified as analytical philosophers, and the obvious difference is that logical grammar does not figure obviously in their work. (p. 237)  However, Heidegger is distinct in that his preoccupation with being incorporates a grammar, and this is the focus of Priest's analysis.

Heidegger's problem is that being turns out to be ineffable, and no predication can be made about being in general without lapsing into nonsense.  Priest find's Heidegger's dilemma analogous to Frege's, in which predications can be made about objects, but concepts are not objects and hence cannot be characterized within Frege's logical apparatus (p. 240). The problem of being turns out to be the same as the problem of nothing.  Priest defends the very assertion by Heidegger that Carnap famously ridiculed: "Nothing nihilates." (p. 240-1)  Paradoxically, we can talk about nothing as if it were something.

Heidegger developed the device later adopted by Derrida: writing 'under erasure'. And here is where Heidegger's system comes under the Inclosure Schema, in a manner structurally the same as Frege (p. 246)

Heidegger eventually turned to poetry as the revealer of being, analogous to Wittgenstein's concern with showing what cannot be said.  But why cannot Heidegger continue with prosaic discourse admitting the presence of contradiction within it?  Precisely because Heidegger accepts the traditional logic with its principle of non-contradiction! (p. 248) Here Heidegger, in the tradition of the entire history of western philosophy, has got logic wrong.  Priest concludes: "Perhaps if Heidegger had been writing later, with a full knowledge of developments in modern logic, he would have said that an adequate thinking of being requires, not simply altheia, but dialetheia."

But can Priest be naive enough to leave it at this?  Perhaps the shoe will drop in his next book, or with luck, in his response to critics.  Do Priest's silences indicate a naivete on his part, or is there more to his agenda?

(2 August 2005)


I will return to parts 3 and 4 of the book, and engage some more general criticisms of Priest's approach, but now I will summarize the final supplementary chapter, Priest's responses to critics—chapter 17: Further Reflections.

Dialetheism is the view that there are true contradictions.  Priest can now count a handful of people who embrace this view.  In this chapter he responds to critics.  Tennant has contributed to paraconsistent logic but is not a dialetheist, but rather holds a rival interpretation.  Priest nonetheless thinks this comes down to an extra-systematic interpretation. (p. 273)  In response to Weir, who objects that dialetheism undercuts the process of theory revision (sounds like Popper), Priest denies this, claiming that has happened before, e.g. in medieval logic, the principle of non-contradiction may be sacrificed in favor of a more satisfactory view on more compelling grounds (p. 275).  In response to another criticism, Priest has no problem admitting of paradoxes (e.g. the Barber paradox) that don't conform to the inclosure schema (277-8).  I don't think I see the point here, though.

Do contradictions exist only at the limits of thought, as some suggest? (279)  Priest argues that this standpoint devalues an understanding of how logical systems really work.  Priest also discusses contradiction in relation to a naive notion of sets, which he accepts (279).

There is also a dispute over the Domain Principle (280ff).  Priest makes something of the sense/reference distinction.  Sense demands the notion of totality.

Zalta has a novel alternative solution to the paradoxes, based on two notions of predication, exemplification and encoding (282ff).  But this also reinstates paradoxes, according to Priest.

Another objection involves a denial of the limit totality (omega) (287ff).  But paradoxes that can be shown to be of one kind mandate a uniform solution.  (See also p. 290: level of abstraction and uniform kind.)

There were also reactions to part 4, on language and meaning (291 ff).  I don't see any objections that parallel mine.

The final section, 17.8, is on "The ontological turn" (294-5).  I will quote it in full, omitting the footnotes:

Finally, let us return, once more, to the subject of dialetheism. The title of the book makes reference to 'the limits of thought'. One reviewer (Evnine (1997)) pointed out that the contradictions with which the book concerns itself are not contradictions in thought, but contradictions in reality, and wondered if the name was appropriate. I think it is. As page 1 points out, the concern with thought here is not with subjective mental states, but with objective content; and one cannot divorce this content of thought from the reality of which the thought is about.

I claim that reality is, in a certain sense, contradictory. I do not, of course, mean that the objects that constitute reality, like chairs and stars, are contradictory. That would simply be a category mistake. What I mean is that there are certain contradictory statements (propositions, sentences—take your pick) about limits, that are true. I am enough of a realist to hold that there must be something about reality that makes them so (though I take no stand on the matter of whether reality comprises entities of the category fact). When I say that reality is contradictory, I mean that it is such as to render those contradictory statements true. If we are to think about that reality in an adequate fashion, it follows that those contradictions must be part of the content of our thought.

The limits of thought that the book deals with are of various kinds—though all satisfying the Inclosure Schema. Some of them may concern human products, like language; the limits discussed in Part 4 of the book are principally of this kind. But some are not. If there were prime matter, in Aristotle's sense, Then this would be contradictory (1.5-1.7), whether or not there had ever been people around to think or talk about it.

This fact is underlined by the two previous chapters. In the taxonomy of limit contradictions given in page 11, the paradoxes in these chapters concern the limits of expression. But they concern limits to expression not because of the nature of what is doing the expressing, language, but because of what is being expressed about—to put it clumsily. For both Heidegger and Nagarjuna, what it is to be, in some most fundamental sense, is contradictory.

Though this is not a novel development in the book, these two new chapters, which put what is at centre stage, certainly mark an ontological turn in focus. The philosophy of language took pride of place in twentieth-century philosophy. Certainly there is no going back to how things were before this. But maybe this century will see a return to the mainstreaming of a more traditional philosophical issue, the nature of reality—and if I am right, a nature that is contradictory.

I have some problems with this position, but first, I'll note that it is not stupid about the problem of contradictions in reality. Such debates are hardly new: more than a century of debate about dialectical materialism and dialectical logic has been devoted to the question as to whether there are contradictions in reality. Like Richard Norman, Priest states the problem initially in an intelligent fashion:

I do not, of course, mean that the objects that constitute reality, like chairs and stars, are contradictory. That would simply be a category mistake. What I mean is that there are certain contradictory statements (propositions, sentences—take your pick) about limits, that are true. I am enough of a realist to hold that there must be something about reality that makes them so . . . . When I say that reality is contradictory, I mean that it is such as to render those contradictory statements true.

Not bad.  But unfortunately, his immediate identification of logic with reality even in this refined sense carries a concomitant failure to make other distinctions that need to be made.  His Inclosure schema unifies a number of paradoxes—logical, semantic, linguistic, even metaphysical, but few of them have any substantive empirical content.  Hence the question is: which are merely formal or artifacts of logical and conceptual systems, and which apply to the universe beyond?  Those discussed by Priest are mostly formal and/or metaphysical in character.  Nearly a century ago, Lenin predicted that the nascent revolution in physics, which was already ideologically resulting in mysticism as an outcome of the prevalent phenomenalism of the late 19th century, would show that reality is fundamentally contradictory, i.e. dialectical.  Yet here metaphysical and logical paradoxes were not even under consideration.  (Earlier, though, Engels had treated motion as a contradiction.)  But what has happened since?  Either the implications of this scientific revolution have been pragmatically swept under the rug, or converted into mysticism, or occasionally inspired attempts at a new logic (in the case of quantum theory).  The most prominent philosophical engagement with contradiction was Bohr's principle of complementarity, inspired not by Hegel or Engels but by Kierkegaard.  This mess has never been straightened out.  Isn't it odd that Priest would say nothing about any of this?  Instead, like a traditional formalist and metaphysician, he wants to claim in the realm of pure philosophical categories that reality is contradictory.  Is this a reversion to (eastern) mysticism?  Citing Nagarjuna and Heidegger in this context does not inspire confidence.  This may give a clue as to what is wrong with the rest of the book.

(2 August 2005)


Part 3 is about the paradoxes of self-reference.  The story begins with Cantor, the great pioneer of the properties of transfinite numbers and sets.  I have a vague knowledge of these notions, i.e. of the orders of infinity: denumerably infinite sets (e.g. the natural numbers) vs. nondenumerable infinite sets (the real numbers or the continuum).  I don't know much more than that.  I never heard of absolute infinity before, but apparently it is a distinct notion from transfinite numbers.  Cantor shows this using ordinals, but I can't say I understand it.  Absolute infinity is a true infinite, a maximum beyond which there can be no greater quantity (p. 116).  Cantor's notion is virtually theological.

Priest then introduces the notion of diagonalization, used inter alia to show the distinction between different cardinalities (orders of transfinites).  While I've heard of this, I didn't get Priest's explication of it.  Anyway, there is no largest set.  Then Priest launches into paradoxes of absolute infinity, but I can't follow.  The Domain Principle is introduced.

The next chapter is on vicious circles, beginning with Russell's Paradox.  The limit of iteration and the limit of conception are united in the Inclosure Schema.

Chapter 10 is about parameterization and begins with Ramsey.  While Russell united the paradoxes of self-reference, Ramsey divided the paradoxes into two families—group A consists of formal logical paradoxes, Group B of linguistic, conceptual, or empirically related paradoxes.  Priest rejects this distinction, and goes on to analyze a subgroup, Bii.  He aims to prove that all paradoxes fit into the Inclosure Schema.  Ramsey's strategy is a case of a more general strategy for disposing of contradictions—parameterization, i.e. to find ambiguities in the offending contradictory expression, or to distinguish one respect from another. (151 ff)  Priest finds this unsatisfactory.  First, he claims, there is no evidence to support the contention that English is a hierarchy of metalanguages (153).  Among other reasons given, parameterisation is said to merely relocate the paradox, not solve it (154).

Chapter 11 is on sets and classes, and begins with a diagram of the Inclosure Schema (156).  This schema relates existence, transcendence, and closure.  Applying transcendence to closure takes us both in and out of closure (a contradiction).  Various strategies have been taken to avoid this problem, e.g. Zermelo, intuitionist quantification, Von Neumann's proper classes . . . Priest introduces the principle of uniform solution (166).  This is all way beyond me.  There is also a technical appendix to this chapter.

(3 August 2005)


In the previous section I summarized Priest's treatment of the set-theoretic paradoxes.  I'll say a few more words about the relation of paradoxes to reality later.  Part 4, beginning with chapter 12, is about language and its limits.

Here, it seems to me, we leap beyond the strict concerns of logic in more ways than one, as at this point we need to relate at least three terms: language <—> logic <—> reality. Ultimately, we need to complexify the set of relations thusly:

    ┌ → ── ──── ────
    ↑ ┌ ←→ ────
    ↑ ↑    
←→ reality ←→
           ↑         ↑
           ↑   thought ←→ language
           ↑        ↕
mathematics ←→ logic ←→ concepts
←→ ────

I will bypass the relation of mathematics to logic, which is easily covered elsewhere and presents the least of my problems.  There is more to be said about the relation of mathematics to reality, as, there (a) that there is a real relationship is more than obvious, (b) yet the situation is now more complex than the ancient Greeks could have known.  I'll venture only a few comments now on these other relationships.  That language is a prerequisite of the manipulation of concepts is indisputable, but the idea that concepts and the structures formed by them is equivalent to the inherited semantic mess of natural language is another ideological notion that must be disputed.  (Note the raw scan of Meaning and Conceptual Systems by R. Pavilionis.)  (Perhaps 'thought' is synonymous with 'concepts', but you never know.) But the main issue here is the language-logic-reality triangle.  I think it was an entirely misbegotten conceit that any of these three terms could be taken as the equivalent of any of the others.  But I'm still getting ahead of myself, so back to Priest.

Priest resumes his story with Frege.  Instead of the traditional subject-predicate structure, Frege gives us 'name' and 'concept-expression'.  You may have already heard of Frege's famous sense-reference distinction.  How do the sense/referents of the parts of a proposition express the meaning of the whole?  A concept is a function that maps an object to a truth value.  Functions, however, leave gaps, and objects (the arguments of the function) fill those gaps.  Functions alone are incomplete, hence 'unsaturated'.  (180-1)  I haven't the slightest idea of what any of this means.

Concepts are not objects, hence cannot be named.  A notorious example: 'the concept horse' cannot refer to a concept.  Other statements become unsayable in this framework.  In order to express his own theory, Frege needs to be able to talk about concepts in order to express his own theory, but his theory forbids him from doing so.  This puts the statement of Frege's theory beyond the limits of the expressible (one of Priest's categories of the paradoxes) (p. 182).  Priest goes on with attempts to repair this embarrassing situation.  But in the end he finds the situation, which I don't quite understand, irreparable.  This is the dilemma inherited by Wittgenstein.

Wittgenstein openly admits the fundamental paradox that in order to think the limits of thought, one would have to think both sides of the limit.  But on language and the world, Wittgenstein claims that the world is the totality of atomic facts.

How anyone could assert such rubbish seriously is beyond me.  I would say the world is the totality of matter, energy, the laws of nature, and emergent phenomena such as mind and society. The very notion of making the world 'facts' and supposing that all ontological questions are reducible to logical ones is a piece of brilliant stupidity that ruined a whole century of philosophy, but I'm getting ahead of myself.

The world is the totality of facts.  Facts are assemblages of objects.  Language is composed of propositions.  "These are all truth-functional compounds of atomic propositions, and hence their truth values are determined by the truth values of the atomic propositions they contain." [quote from Priest] True propositions picture facts. (185)  "Propositions show the logical form of reality." [quote from Wittgenstein's Tractatus.]  Propositions say things about objects.

Priest has a digression on simples and complexes, and all statements about complexes can be analyzed down to simples.  There is chain of dependency of truth and sense.  There is a question here about the determinacy of sense, but I'm having trouble deciphering the argument.

What Priest calls 'structural facts' are not facts in the Tractatus, and attempts to construct propositions about them results in meaningless statements.  This situation parallels claims of other philosophers: form cannot be a constituent like the constituents it holds together.  Wittgenstein says:

"Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent."

Also, names name objects, not propositions.

Hence, logical grammar imposes constraints that cannot be sensibly contravened.

The consequence, however, is that all of the Tractatus attempts to express the inexpressible, thus is, strictly speaking, nonsensical.  Hence Wittgenstein's statement that his book is merely a ladder, and that of whatever we cannot speak, therefore we must remain silent.

In the end, Priest finds that this scenario fits the Inclosure Schema, i.e. the form of ineluctable paradox we come to whenever we come to the limits of thought.

Interesting as this is, Priest's single-minded objective—the formulation and unification of formal paradoxes— disinclines him to pursue other possibilities of what is wrong with all of this.  In the next chapter he will discuss Quine and Davidson,  Then in the next chapter he will go on to Derrida, and then the concluding chapter of the first edition.  In the additional chapters for the second edition (which I've already reviewed, having skipped ahead), he treats of Heidegger, Nagarjuna, and reflections on his critics.  But one thing Priest never ever does is question anything about the philosophies of any of these people as whole—only those elements of them that lead up to the paradoxes at the limits of thought—the Inclosure Schema.  I find this both helpful and harmful.  Helpful in that in ranging over the history of Western philosophy (with a late admixture from India), he brings to light a number of common problems and attempts to deal with them in a unified manner.  Harmful in that in his obsession with logic and paradox he refrains from asking the tough questions about the validity of these philosophies and their ontological commitments.

We could question for example, the tacit assumption that language = logic.  Logic is an attempt to formalize the conceptual content of language, but why should we for one minute restrict the thinkable to the formalizable, yielding the dichotomy logic <—> nonsense?  And while science attempts to mirror reality, why should we assume that the properties of propositional logic—which is by its very nature entirely indifferent to content—are an exact mirror of the statements of act or scientific theory?  The immediate identification—pending the real conceptual unification of a complex interrelationship—of language, logic, and reality—is a piece of sophisticated childishness it's impossible to believe anyone could entertain. And yet the naive positivism of such silliness ends up as the royal road to mysticism and irrationality a la Wittgenstein and eventually Rorty.  There's something very very wrong here.  Why wouldn't Priest, who once wrote an article on Marx and dialetheism, at least show some concern about this issue?

Turning this into a Saturday Nite Live routine a là Mike Myers' "Coffee Talk", I would summarize my frustration thusly: "Talk amongst yourselves. Here, I'll give you a topic: Philosophical Logic is neither philosophical nor logical.  Discuss."

(24 August 2005)


Chapter 13 is titled "Translation, Reference, and Truth."  Here is where Priest engages Quine, Davidson, and others.

The postulation of semantic correlates is deep-sixed by Quine's behaviorism.  Quine trashes any notion of mentalism, any notion of meaning not implicit in overt behavior.  (Cf. Quine's "Ontological Relativity'.)  Down with the 'museum myth' of meaning goes the determinacy of sense.  Another consequence is Quine's thesis of the indeterminacy of translation.  In translating an unknown language, we find that there may be different analytical hypotheses for the language which conform to all observed speech dispositions.  The reason there is no unique solution is the same we know there is none for any observed empirical regularities, after Hume.  And what applies to unknown languages also applies to known languages, such as the English we are using now.

This, however, brings us to a familiar self-referential paradox.  How are we to apply a determinate meaning to Quine's own statements?  Are we all translating Quine's writings into our own idiolects, creating an indeterminate range of personal meaning-schemas?  If this were so, how could we ever understand, or agree or disagree with Quine's views or with the arguments of one another about them?  Yet it seems that Quine's utterances do have determinate sense in spite of his assertions.

Priest cites Searle as making a such an argument against Quine, Searle basing his objections on Quine's behaviorism.  Priest, however, claims that behaviorism is not essential to the indeterminacy argument, as the argument would work just as well with intensional notions. (199)  I made note of this given my absolute lack of respect for behaviorists.

From inscrutability of sense we derive inscrutability of reference, and then not just to the utterance of others but to one's own.

But we are once again led into the paradox of expression: the indeterminacy of reference cannot be expressed, yet here it is expressed perfectly. (201)

Quine is aware of the problem and is forced to conclude that the notion of reference is meaningless.  Quine finds a number of subterfuges to circumvent the undesirable implications of this situation, such as reference to a background language, which Priest finds unsatisfactory.  How then does one relate this background language to reality, and deal with claims about reality?  Here at least Priest shows himself to be aware not just of paradoxes, but of the shortcomings of empiricism (204).

Next comes Davidson, who is neither a behaviorist nor an extensionalist.  Priest outlines Davidson's specs for a theory of language.  He uses Tarskian theory to construct a finitely axiomizable theory of truth for a language.

This would seem to follow in the footsteps of Frege, who suggested that the "meaning of a declarative sentence is its truth conditions."  I fail to see why this is so.  It seems to me a prima facie senseless supposition.

OK, so suppose we can construct a theory of a natural language in the language itself.  Davidson admits of a problem here.  Priest asserts that the attempt to resolve the contradiction at the limit of cognition results in a contradiction at the limit of expression (206-7).

Davidson, in "On the Very Idea of a Conceptual Scheme" finds himself in another quandary, argues that the claim that the conceptual systems of two speakers may be different, or the same, is senseless.

Such is the legacy of the progression Frege—>Wittgenstein—>Quine—>Davidson.

Clearly there is something wrong with all this, but what?  Presumably, Priest is interested in the unavoidability of contradiction at the limits of thought, and thus the universality of his Inclosure Schema.  He never asks the salient question: why equate in the first place language (the linguistic expression of thought) with formal logic?  What is the relationship of both to each other and to reality?  In fact, these questions are evaded from first to last.  Perhaps one might pursue this hypothesis: the logical conclusion of positivism (and hence 'postpositivism') is mysticism.

ADDENDUM: It would be instructive to compare the relation of logic to language as philosophers see it as to the way that real linguists see it.  To be sure, logic becomes central to linguistic theory in the wake of the Chomskyan revolution, but would any linguist interested in the logical structure of natural language venture the claim that the meaning of a sentence is its truth conditions?  Linguistic theory indeed mandates the formulation of natural language sentences in a formal apparatus (hence the interest in Montague grammar, for instance), yet the perspective seems to be different: to stretch logic to fit the facts of natural language, not to assume that language is just logic in longhand.  There's something very very wrong about what analytical philosophers have done in their single-minded reduction of ontological and epistemological questions to linguistic ones, the latter themselves reduced to formal-logical terms.

(25 August 2005)


Chapter 14 covers the later Wittgenstein before turning to Derrida.  As we know, Wittgenstein later repudiated the Tractatus, reverting from mystical logicism to a much more objectionable irrationalist collectivist solipsism.  On this point see Ernest Gellner's Words and Things and Language and Solitude.

Anyway, here Priest follows Kripke in interpreting Wittgenstein.  The question here is the nature of following a rule, language being a rule-governed activity.  How do we follow a rule, though?  Wittgenstein rules out mental states as the determiners of the application of a rule.  Hence it is wrong to call meaning a mental activity.  We are left with the indeterminacy of rule-following and hence meaning.  Rule-following is just playing according to the rules of socially established language games.

Wittgenstein concludes that philosophy is disallowed from interfering with language use; in the end philosophy can only be descriptive and leave everything as is.  Rather than debating what a foul load Wittgenstein has dropped on us, I'll move along with Priest's argument, which is that Wittgenstein's own argument becomes impossible by his own lights.  (213)  Kripke claims that Wittgenstein's views consonant with his style. (213-4)  But for Priest, as Wittgenstein has expressed the inexpressible, we once again come to a paradox at the limit of expression.

Again Priest is interested only in revealing the paradox, and ignores what else might be wrong with this way of thinking.

Derrida's hobbyhorse is the denial of presence, or metaphysics.  Derrida denies that anything can act as a presence to ground meaning.  Derrida departs from the structuralism of Saussure.  Writing, unlike speaking, enables the appropriation of linguistic expressions in different contexts.  All words refer to other words indefinitely, and there is no escape from circularity to find ground in the transcendental signified.  Here we are led to the play of differance.  Deconstruction involves the self-undermining of texts.  A text that endorses some presence x is dependent on the opposite of x, i.e. on the x/non-x distinction.  The distinction is shown to be false, and we are led to undecidability. The emergent undecidable concept undercuts the original distinction, . . . I can't go on.  Read this for yourself (214-8).

The punchline is obvious: how then is Derrida's own argument interpretable?  We are at something like Cratylus' problem, except here it cannot be evaded, given the attack on presence.  Differance is claimed to be unnamable. All language is structured by binary opposition, hence metaphysics is inescapable and hence no way of expressing its negation or an alternative.  One technique for dealing with this situation is to adopt Heidegger's device of writing under erasure.  However, the consequence of Derrida's views are that his writing is meaningless, yet if we understand him, it does have a meaning.

In all these cases we are left with the paradoxes at the limits of the expressible—with ineffability.  The next chapter is the conclusion to the book's first edition.

But let me first venture some other observations.  All of these ventures in the philosophy of language are predicated on much more than the set-theoretic and semantic paradoxes treated earlier.  Sure there is a formal element involved that can be given logical expression.  But yet there are a set of assumptions at work which are not merely intrinsic to the structure of logical assertions.  There are additional assumptions about the nature not just of the logical assertions made with language but about language as a whole, how it's learned, what it expresses, its relation to both reality (ontology) and knowledge (epistemology).  And all of these arguments trade on all the old skeptical conceits that came out of the age of empiricism: i.e. if we don't have absolute proof, we can't know anything.  And yet, the realm of natural language or knowledge claims never has anything to do with absolute proof, of which there is none outside of axiomatic systems.  That is, there never is absolute justification for any knowledge claim.  We've known this for well over two centuries, but instead of concluding that aprioristic philosophy is useless, these philosophers have just turned apriorism on its head and reverted to a skepticism they then struggle to weasel out of, all on the bogus assumption that language, concept formation, epistemology, and ontology can all be reduced to formal logic.  In its positivist phase, bourgeois philosophy pretends it can do everything worth doing via logic or banish all other questions to the realm of meaninglessness.  In its decadent phase, bourgeois philosophy settles in its old folks home, till it finally becomes senile with mysticism and irrationality.  But yet it refuses to die, and the bills are mounting up.  Euthanasia, anyone?

(25 August 2005)


The conclusion to the first edition is subtitled: the persistence of inclosure. Priest summarizes the course he has covered from ancient Greece to the present.  In any situation, when contradictions are suppressed at one point, they come up at another of the four limits of thought—iteration, cognition, conception (definition), expression.

Limitative theorems attempt to suppress the paradoxes of self-reference.  Theories must be prevented from expressing their own truth-predicates.  Hence the theorems of Tarski and Goedel.  As for semantic paradoxes, one might attempt to show that object theory plus metatheory is consistent, but the result would be a regress and failure.  We are back to the failures of parameterization.

If we were to find a solution for the semantic paradoxes of English, then the metalanguage must be one for English and the relevant notions should not be expressible in English, but they are.

I still have questions about the presumption that a natural language is formally identical to a logical language, but I believe Priest makes an argument somewhere that the paradoxes end up being the same.

Priest then discusses a book by P. Grim, The Incomplete Universe: Totality, Knowledge, and Truth. Grim is mainly concerned with the contradictions at the limit of cognition.   There is no totality of truths, as we can diagonalize out of any presumed totality, hence quantification over totalities is impossible.  Priest argues that Grim's thesis is subject to the Inclosure Schema and thus Grim's thesis becomes inexpressible.  (He also contradicts Cantor's Domain Principle.)

Priest concludes this (formerly) concluding chapter with a short section on Hegel.  The historical figure who had the clearest insight into Inclosure is Hegel.  "For him, the moments of totalisation and transcendence are the two moments of the true infinite."

As I mentioned in reference to the concluding chapter of the 2nd edition, Priest sees 21st century philosophy as a prospect for a return to reality, asserting that reality itself will be found to be ultimately contradictory.

However this may be, I would begin tackling this question with a differentiation that Priest himself does not make, i.e., sorting out the cases in which contradictions refer to aspects of physical reality (the nature of time, space, matter, energy), others which are artifacts of the properties of logical reasoning, still others that come from epistemological positions.  The serious 'contradictions in reality' were discovered almost a century ago and canonized in Bohr's principle of complementarity, inspired by the dialectics of Kierkegaard.

In effect, Priest suppresses any calibration of the relationship of metaphysical categorial statements to empirical reality.  In local meetings on the Tao Te Ching (Dao De Jing) I noted the curious disjunct between the metaphysically posited relationship of being to nonbeing in Taoism and its mapping onto empirical knowledge.  My guess is that all categorial schemes—including I would guess Hegel's Absolute Knowing—involve a logical self-reflection (self-consciousness) that cannot be self-contained, that must flow outward in an indeterminate relationship toward empirical knowledge, and for that matter, to the theories founded upon it.

If logic is to be merely a self-contained formal apparatus, as an idealization of the properties of valid inference, and resign all other pretensions about reality, that is its impregnable stronghold.  The minute that logic becomes philosophical, and embroils itself in ontology and epistemology, the rules change.  First, I suggest, that the logic itself becomes more complicated, for logic + reality (real world examples) generates more complex structures than contained explicitly within the logic itself.

I first had this thought when seeing Raymond Smullyan in Buffalo in 1980, giving a talk, I think, on modal logic.  He struck me as a bit of a mountebank—he may have even mentioned once doing magic tricks or working in carnivals—and his examples had a curious New Age flavor: mineral baths with mysterious curative powers, in connection with the logic of belief, etc.  It seemed to me that the situations he described had a more complex structure than the specific logical conundrums explicitly identified.  Perhaps I have notes buried deep somewhere, but this is the best I can do for now.

Now I think of the paradoxes involved in double-bind situations.  You may be old enough to remember an influential novel based on the double-bind: Joseph Heller's Catch-22, a grim piece of black humor taking place in World War II.  I read it as a teenager, and now I think I should read it again.  If you've not read it, you should.  But I have a much stronger, in fact, indelible memory, that also goes back over more than three decades, reading Richard Wright's autobiography (first published in 1945 under the title Black Boy, now restored with previously censored portions under the intended title American Hunger).  Wright's very life was endangered when presented with a Catch-22 situation at his workplace, being threatened by Southern rednecks.  I still recall the gut-wrenching feeling I had reading Wright's account of his attempt to talk his way out of a particular situation.  Here we have logic in real life, but the 'logical' structure of the situation is more than logical, for it also encompasses the 'logic' of the speech act—its intentions, motivations, and the behavioral dispositions of the social actor.

In my summary of the book, I basically glossed over the details of Parts One (Heraclitus, Cratylus, Plato, Aristotle, medieval philosophers and Two (Kant and Hume).  Actually, my concern with the relation between logic (and mathematics) and reality surfaced when reading Priest's account of Aristotle's treatments of Zeno's Paradox and the contradictions in the notion of prima materia (prime matter).  The relations between logic and reality are even more complicated than Aristotle could have known, given revolutionary developments in recent centuries in physics, mathematics, and later, foundations of mathematics and logic.

The next step is to sit back and digest all this before attempting to itemize all the issues as I see them.

(25 August 2005)


Dumain, Ralph. Graham Priest vs Erwin Marquit on Contradiction. 2005.

Garfield, Jay L.; Priest, Graham. “Nagarjuna and the Limits of Thought,” Philosophy East and West, January 2003; 53(1): 1-21.

Grim, Patrick. The Incomplete Universe: Totality, Knowledge, and Truth. Cambridge, MA: MIT Press, 1991.

Norman, Richard; Sayers, Sean. Hegel, Marx, and Dialectic: A Debate. Brighton, Sussex: Harvester Press; Atlantic Highlands, N.J.: Humanities Press, 1980. My review: Dialectics Bout: Richard Norman vs. Sean Sayers.

Pavilionis, Rolandas; translated by H. Campbell Creighton. Meaning and Conceptual Systems. Moscow: Progress, 1990.

Priest, Graham. Beyond the Limits of Thought. 2nd ed. Oxford: Clarendon Press; New York: Oxford University Press, 2002. Table of contents.

___________. "Dialectic and Dialetheic," Science and Society 1990, 53, 388-415.

___________. "Was Marx a Dialetheist?", Science and Society, 1991, 54, 468-75.

___________. "Where is Philosophy at the Start of the Twenty-First Century?", Proceedings of the Aristotelian Society. 103 (2003): 85-99.

Zelený, Jindrich. Paraconsistency and Dialectical Consistency [corrected from original, which appeared in From the Logical Point of View (Prague), Vol. 1, 1994, pp. 35 – 51].

Edited & uploaded 8 February 2007
©2005, 2007 Ralph Dumain

Graham Priest's Inclosure Schema

Philosophy of Paraconsistency & Associated Logics (Web Guide)

Graham Priest vs Erwin Marquit on Contradiction

What is the Relationship Between Logic and Reality?” by R. Dumain

Salvaging Soviet Philosophy (1)

Marx and Marxism Web Guide

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