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The Logic of Admissible Contradictions

drafted 1970-74

typeset with revisions 1997-8

© 1998 Henry A. Flynt, Jr.




I. Empirical Foundations

A. Logical falsehood in natural language

B. Stationary motion

C. Replying to evasion

D. The tactile illusion of a two-valued integer

E. Dreams of impossible world-states

F. Inconsistent recollected time-assignment

Note on three-valued logic

II. Inferential Relationships

III. A Provisional Axiomatic Treatment

Appendix. Cardinality and Intensions

IV. Absorbing Contradictions



In the discussion of negative afterimages of motion on page 107 of Eye and Brain (McGraw-Hill, 1966), R.L. Gregory states the following about the rotating-spiral version of the waterfall illusion (Plateau’s Spiral).

The illusory movement may be paradoxical: it may expand or shrink, and yet be seen not to get bigger or smaller, but to remain the same size and yet to grow. This sounds impossible, and it is impossible for real objects, but we must always remember that what holds for real objects may not hold for perception once we suffer illusions. We can experience things which are logically impossible when we suffer illusions. [boldface added]

In the present work, I will observe that visual illusions, tactile illusions, dreamed experiences, and temporal memories have logical structures which cannot be modelled with conventional logic. Speculatively, the reason may be that in illusions, in dreams, in and temporal memories, we are disconnected from logical norms which have the character of habits, habits affecting not only thought but perceptual interpretation. When traditional logicians said that something was logically impossible, they meant to imply that it was impossible to imagine or visualize. But this implication was empirically false. The realm of the logically possible is not the entire realm of connotative thought; it is just the realm of normal perceptual routines. When the mind is temporarily freed from normal perceptual routines – especially in perceptual illusions, but also in dreams and even in the use of certain "illogical" natural language phrases – it can imagine and visualize the logically impossible.

But a qualification is in order. The neurological asides in this work are nothing more than a favor to the reader. Received neurology is highly derivative and invested in physical science; and its treatment of anomalous perceptions focuses on their failure to be veridical. It is important to be clear that the phenomenological availability of inconsistency in the sense Gregory delineates (for example) allows an investigation of perceptions and their appellations which is self-contained.

In what follows, I will describe and analyze some of the anomalous logical structures occurring in visual illusions, tactile illusions, dreams, and memories. In the course of modeling these structures, I will propose new logical machinery. So the first major goal of this monograph is to describe four non-vacuous (admissible) contradictions and to outline them in a rough formalism.

In the Fifties, when I received my mathematical education, academic preoccupation with an alternative logic took the form of a preoccupation with many-valued logic. I have no intention here of providing a reprise of that copious literature. At the same time, minimal logic was paraconsistent. Tarksi alluded to paraconsistency in "The Semantic Conception of Truth," Philosophy and Phenomenological Research, 1944 (even though he did not call it that). Infinitary logics allowed theories which were syntactically consistent but had no models. (The era of paraconsistency’s popularity had not yet begun.) What these approaches do – once we understand the lesson codified in the Tarski paper and in Helen Rasiowa and Roman Sikorski, The Mathematics of Metamathematics (1963) – is to give reinterpretations to noncontroversial and commonplace theorems in algebra.

My approach is utterly different; I do not seek to give a different role to theorems in the accredited algebra. In particular, this means that the concepts to be postulated here must have experienceable "pictures" which serve as their meanings. It is not enough for logic to map appellations to other appellations according to a blind formalism; the "other appellations" must have experienceable pictures as a semantic base.

Let me look forward to the end of Chapter I, for the benefit of the open-minded reader. We will offer four contradictions which are non-vacuous (admissible) empirically or phenomenologically. They are from quite disparate experiential realms. Knowing one does not guarantee that you will discover the others. The structure of each case – that is, the structure of the identitarian relationships, the admissible contradiction superimposed on them, and the "primary contradictions" which are not yet admitted – is quite different. I am sure that we will discover more non-vacuous contradictions; and there is no reason why they have to resemble these four. [1997. Written at the beginning of the Seventies, and vindicated since.]

Every step in our argument will run roughshod over territory for which there are intricately nuanced professional theories. And – will elicit doctrinaire complaints when our phraseology is informal or colloquial. Actually, as I proceed, I will offer some guidelines on pragmatics and other sorts of background. But wherein I do not give explicit guidelines, I have no patience with doctrinaire sniping. It is an attempt by bankrupt ideology to hinder us in assimilating evidence nobody has pondered before. I shall strike a balance between casualness and rigor that serves my investigation.

The defenders of the status quo can choose certain definitions and interpretations and thereby evade the threat which admissible contradictions pose to mainstream logic. One can evade our results by making ad hoc prohibitions, such as a prohibition of all natural language predicates. (I will give a more formal reply to attempts to forestall my interpretations in Chapter I, §C.) But I am not even seeking the real truth about the real world here. I wish to expand our instrumental powers. What matters to me is that admissible contradictions are a sufficient justification for the attitude that new logical machinery is desirable. Further, admissible contradictions constitute a rather extensive guide in formulating the new machinery. Why would we want to evade the threat, given the vast expansion of our instrumental capabilities which it affords? Why let pedantic punctiliousness bar the way to expanded instrumental powers?

I express myself as if the overwhelming likelihood is that readers will be hostile. Indeed, every text on perceptual psychology mentions the perceptions in question. And yet, logicians have not embraced their potential. How much more needs to be said? [The first hint of the topic in a mainstream journal came eighteen years after these thoughts were drafted.]

I have not said what logic is. I might say, it is everything which enables the calculus of the consequence-relationship. In Chapter I, we will learn the following, referring to the phenomena Gregory was writing about. When I see the waterfall illusion, I think of the positive qualities, motion and stasis. I definitely do not think of their negatives, non-motion and non-stasis. If there is going to be a visualization of non-stationary non-motion, it is going to have to come from somewhere other than the waterfall illusion. Thus, we cannot yet concede that non-stationary non-motion is an admissible contradiction. Given a universe for which a contradiction is admissible, there can be a contradiction conventionally considered to be identical to it: which – on the basis of the experiences our theory is designed to model – is not equivalent to it and cannot be conceded to be admissible.

There you have a hint of where our machinery needs take us. Given an unexpected phenomenon which vitiates mainstream logic, how far does it spread? – and is the answer to that question empirical, or is it a "deduction"? What are the inferential relations in this unexpected situation? That is what is wanted from this logic. The logically impossible is not a blank; it is a whole layer of meaning and concepts which can be superimposed on conventional logic, but not reduced or assimilated to it. The logician of the future may use a drug or some other method to free themself from normal perceptual routines for a sustained period of time, so they can freely think the logically impossible. They will then perform rigorous deductions and computations in the logic of amcons.

• • •

Polemical note on typesetting and revising the holograph (1997)

My collection Blueprint for a Higher Civilization (1975) contains the essay "The Logic of Admissible Contradictions, Chapter III." By including that piece, I meant to document the existence of a much larger investigation, consisting of four chapters and multiple appendices. The opportunity to publish the entire work was not offered to me. I began work on the manuscript in 1970, and finished it in 1974.

Until now, the unpublished balance of the book has remained handwritten. Herewith I typeset large portions of the unpublished material.

Since the early Seventies, I have had the opportunity to circulate later codifications of my project in typescript, e.g. "Introduction to the Logic of Contradictions." In 1994, more experienced in contending with the academic syllabus, I wrote "Regulating Inference from Authentically Descriptive Inconsistency," whose final section concerned the Chapter III published in Blueprint. I sought to extract a presentable gist of the method of ø-intensions for close study. But the early Seventies manuscript is still needed to ground my project: because of its more painstaking explanation of the empirical motivation of the project. Namely, the non-vacuous inconsistent concepts afforded in vernacular thought – of which the most palatable academically are the negative afterimages of motion. Ultimately, the Seventies manuscript and the 1994 account need to be consolidated.

The Seventies manuscript envisioned a discipline which, if realized, would begin in vernacular thought, underneath the academic paradigms – and consolidate so as to shatter them. There was no question of accepting the shape given to logic by Boole and Frege, and then giving it a little twist to allow certain curiosities as an interpretation. I wish to extirpate the Boole-Frege ideology of scientific truth. I wish to direct "the calculus of the consequence-relationship" into an altogether different channel.

I hate defensive expositions. But it simply was not possible to hide, from myself or from the reader, that I am flying in the face of fundamental professional orientations and intricately nuanced professional codifications. The reader will simply have to decide whether to entertain what I say, or be a loyalist and dismiss it unread.

What is specific to me is: (a) The sort of evidence I insist on. (b) My larger philosophical perspective (which rejects the ideology of scientific truth). (c) The demand for a treatment which rises from vernacular thought.

My perspective and method were shaped by academic logic, analytic philosophy, and perceptual psychology; but I am not a follower of those disciplines. There is no question of diving back into "the mathematics of metamathematics" – the Boole-Frege program as codified by Rasiowa-Sikorski, Lawvere, etc. All the while, I had no compunctions in the Seventies about borrowing a distinction here or a device there from the academic syllabus and giving it a role unique to my enterprise. I used notions from modern "exact science" naively, detached from magisterial doctrine, to depict a phenomenology which the professionals will hate. I was casual in employing these devices – as if I could lift notations from the ideology which had invented them and use them as simple abbreviations or markers.

When the original made errors of method, and I am confident what they were, this typescript ruthlessly retouches the original. I take considerable liberty in revising the vintage manuscript where I know what it needs. But when it is not obvious how to deal with my errors, I have let them stand. I want a chance at extended familiarization with the material before I try to fix it. I do not want to second-guess myself in order to shield myself from the hostility of analytic philosophers (to ingratiate).

I did not recant "Concept Art" or "Primary Study" when I found them to be unpublishable and unmentionable academically; and I did not halt because professors trained in sophistry could pick quarrels with them. A point comes where one who seeks a new instrumental capability will entertain an investigation without requiring that it be perfect, or proof against every wish for a quarrel. I am not going to become an appeaser for the first time in my life by bowdlerizing this work.

An investigation which cuts a road where there has been no road does not have to be perfect, and it should not apologize for not being perfect. The demand "Be perfect or be suppressed" is made just by those who have an ideological animus toward acknowledgement of the new evidence at all.

I offer this document not to those who want to block the threat of the new (as Peter Ungar made it clear to me in 1967 that he wanted to do across the board), but to those who are hungry for whatever clues they can glean for a logic to destroy the ideology of science. Of course an exponent of the ideology of scientific truth will will want to shut this discourse out.

In the Nineties, my exposition faces an additional and ghastly irony. A fad called deconstruction has gained a reputation as a nihilistic philosophy which furnishes a logic of contradictions and proves that truth does not exist. In fact, deconstruction does not confront science frontally, and does not know what such a confrontation would look like. (In 1996, the deconstructionists accredited a rant against physics by printing it in Social Text; the next day it was announced to be a gigantic hoax.) Deconstruction does nothing which an exposition such as this one is required to respond to; and I have no intention of citing deconstruction as if it were a cogent alternative to the truisms of science.

As deconstruction swept the world, academics en masse celebrated the death of truth and logic, although nothing cogent had been said against either. The situation is much worse than any of us were prepared to cope with. After all, I am no friend of scientific truth. But to be embraced by the public as a critic of scientific truth, you have to show that you are a buffoon. The public runs like a beheaded chicken from one babble to the other. The public arena is not worthy; we are reduced to writing for a few friends.