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IS RADICAL UNIVERSALISM WHAT IT IS NOT? A preliminary inquiry into the logic of pedagogy


This essay was originally written in June 2023 as a response to J.-P. Caron's seminar Hegelian Logic and Analytic Philosophy at the New Centre for Research and Practice.

It is posted here as a companion piece to the post

"The Human as Transformation: Notes Against a Pedagogy for Labor", found here.


A clarificatory note: though it may seem dissonant to place such posts inside a blog titled "Posthuman Art", our research is done under the assumption that "a proof is a program is an artist". At the end of such an endeavour, we hope to have made clear that the activity constitutive of the form of the human is homomorphic to mathematical proof construction. As such, a form of the human seen as the form of transformation will hopefully lead to a pedagogy which answers positively to the unfortunately commonplace, cynical question: "So what? Are you saying that under communism everyone is going to be an artist?"


IS RADICAL UNIVERSALISM WHAT IT IS NOT?

A preliminary inquiry into the logic of pedagogy



Filipe Felizardo - Untitled, Diapositive film, 2016


Abstract

The narrow aim of this essay is to look at paraconsistent logic in the context of pedagogy: its proposals, and its results. The larger aim will be to check if such a logical framework can be accommodated in a philosophy of pedagogy that hinges upon the meta-concept of radical universalism. Such a philosophy is structured upon what in the context of this essay will have to suffice as a presupposition: that pedagogy is a logical endeavour; that learning is an interpersonal and dialogical endeavour contingent upon the outcome of the development of logical forms. As such, it is claimed to be a dialectical endeavour.

We will start with the context of the larger aim and its main presupposition, by defining radical universalism, and defend it as a meta-concept.

Afterwards, we will briefly look into other meta-concepts broadly considered as contenders for such a status in the context of pedagogy: namely, universalism and constructionist-pluralism. Respectively, both will be seen to be mappable onto pedagogical/logical schools of formalism and intuitionism. It should become clear at this stage of the essay that none of these are fully developed as to be able to fit the demands of radical universalism.

Against the background of these two pedagogical/logical pairs (universalism/formalism, and constructionist-pluralism/intuitionism), paraconsistent logic, as proposed by Graham Priest, will be subjected to an immanent critique, and then be shown to share traits with both pairs, as well as feature underdeveloped characteristics of its own. Throughout these sections, we will use modality as a criterion to assess the intelligibility of each logic's claims and the realizability of its purported pedagogical efficacy when paired with their corresponding metaconcepts.

It will be seen that, along with universalism and constructionist-pluralism, paraconsistent logic falters at its epistemological purchase. In doing so, it is not a strong contender for a logic of dialectics, and is thus insufficient for a radical-universalist philosophy of pedagogy.

At the end of this essay we hope to be able to obtain a more determinate image of the meta-concept of radical universality, by having opposed it to other pedagogical/logical frameworks – in the hopes of having enabled the possibility, or opening up the path, to claiming it as able to be developed through another logical framework, that of linear logic.


Part 0

The context of the larger aim:

A definition of radical universality, and how is it a meta concept for pedagogy.


We have already claimed in the abstract that we are trying to activate a philosophy of pedagogy through the meta-concept of radical universality. In order to make good on this proposal, it must be clarified what is at stake in 'meta-concept' and 'universality'.

Meta-concept will not be novel to readers familiar with Robert Brandom. His simpler definition goes: “there are concepts that play the distinctive expressive role of articulating features of the framework that makes description and explanation possible.”1 These are Kant's categories, the pure concepts of the Understanding, as prefigurations of Hegel's “speculative, logical, or philosophical concepts” Brandom calls both “metaconcepts: concepts whose job it is to express key features of the use and content of the ground-level empirical and practical concepts Hegel calls “determinate” concepts.”2 For Brandom, grasp of meta-concepts is a priori.

For our purposes, a meta-concept is no doubt one which “plays the expressive role of articulating features of the framework that makes description and explanation possible”, and is assuredly a speculative, logical, or philosophical concept. On the other hand, in the context of our philosophy of pedagogy, a meta-concept is not a priori, nor a biological given; it is an historical ideal. In the sense that it is not a given, innate, not even a trans-historical category, the meta-concept is itself the structure and the desired product of pedagogy (and philosophy), and it is through this meta-concept that pedagogy operationalizes – maps and transforms - concepts in the ideal and material personhood of its participants. The name of such a meta-concept is Radical Universality.

Adjectivating universality as radical may seem an unnecessary stylistic affectation, but, at least for the moment, it serves as a mark of sincerity in our eagerness to have the universality defended here as distinct from other, historically deemed innefective or even maleficent, sorts of universality. The foregoing section of this essay might help discern some confusion regarding the scope of the term, but for now, we'll stick to elaborating a working definition of the meta-concept without going into what it is not.

Our notion of universality is a direct offshoot, if not a blatant graft, of Ilyenkov's development of the term. Ilyenkov starts by pointing out how Hegel envisioned a bifurcation in the paths of logic: formal thought separated itself from dialectical (or speculative) thought when taking upon itself the tasking of capturing, in abstract form, the “common element in every single representative of one class”3. In Ilyenkov's reading, Hegel is taken to be showing how, in his framework, Aristotle pointed out the non-existence of abstract universals; how to a variety of particulars we can only oppose an “empty universal, […] that which does not itself exit, or is not itself species”4.

It is in this moment that speculative philosophy can advance an alternative to this formal alley. For Ilyenkov, the operative – mediating – notion in the opposition between universality and particularity is that of commonality, or generality. Unity between various particulars is “created by the attribute that one individuum possesses and another does not. And the absence of a certain attribute binds one individuum to another much more strongly than its equal existence in both.”5 This account of commonality through unity leads Ilyenkov to a tidier concept: “The general is anything but continuously repeated similarity in every single object taken separately and represented by a common attribute and fixed by a sign.”6

This is followed by the universal, as “above all the regular connection of two or more particular individuals that converts them into moments of one and the same concrete, real unity. [Such unity is to be] represented as an aggregate of different, separate moments rather than as an indefinite plurality of units indifferent to one another. Here the general functions as the law or principle of the connection of these details into the make-up of some whole, or totality as Marx preferred to call it, following Hegel.”7 It may be fruitful to see the universal as being mediated by the general.

The main thrust of this development relies on showing that universality is neither originary nor substantially fundamental – but more remarkable is the contention that the universal “which manifests itself precisely in the particularities (…), also exists as a particular alongside other isolated individua derived from it.”8

Both this un-originality of the universal and its Janus-facing as just another particular, are extremely well suited to our purposes. On the one hand, the un-originality of the universal will help us understand just how ineffective is an understanding of universality as fundamental, or grounding - both in its defense and its critique; on the other hand, the universal as just another particular is crucial for our concept of radical universality: it starts to design itself as an open-ended universality, both in its direction, and on its ground.

To close the regimentation of Ilyenkov and match his development of the universal with our framework for pedagogy, it is worth looking at how Ilyenkov operationalizes the meta-concept in the concept of that which pedagogy forms – the human. Following Marx in the claim that “the essence of man is no abstraction inherent in each separate individual[;] in its reality it is the ensemble (aggregate) of social relations”9, Ilyenkov shows that only by analysing what is understood by 'whole aggregate' – what it is composed of, can “the separate individual [be seen as] only human in the exact and strict sense of the word, insofar as he [sic] actualises – and just by his individuality – some ensemble or other of historically developed faculties (specifically human forms of life activity)”10. It is our contention that only after modelling this context can “human personality rightly be considered as an individual embodiment of culture, i.e., of the universal in man.”11

Quoting Ilyenkov at such length may appear merely absconding in authority, but it may be the more abbreviated manner of providing the necessary context for the main developments of this essay. We end this section with the following quote: “Universality so understood is by no means a silent, generic 'sameness' of individuals but reality repeatedly and diversely broken up within itself into particular (separate) spheres mutually complementing each other and in essence mutually dependent on each other.”12

Ilyenkov's account of universality appears to be sufficient for our earlier requirements: it appears to fulfil the role of meta-concept as an expressive mediator of description and explanation. How it does will hopefully be seen at the end of this essay. For now, let us ask ourselves: does it fulfil the role or is it a good candidate for expressively mediating description and explanation? It fulfils the role by being the umbrella-concept or the playground for the description and explanation of concepts; and it is not given in any manner whatsoever (it is not deemed original or grounding). On the other hand, it appears to itself as the condition of the possibility of learning, in the sense that radical universality operationalizes the understanding of reality as a chimera made up of separate, mutually complementing particulars (the learners) which/who realize and become intelligible to themselves in this radical-universalizing act of self-understanding which opens both the particular and the universal to mutual – and thus self- – transformation. Moreover, the meta-concept of Radical Universality as such hinges crucially on opening itself up for transformation, as just another particular, in order to become more intelligible to itself, through the realization of its co-particulars. With this is mind, it may appear at first to be a ground after all, but it is always a provisional ground. Very much like in Lorenz Puntel's system, the ground only shows up to be coherent at the outcome of the theoretical structuration – in this case, learning.

Radical Universalism, then, will be a 'school' of thought on pedagogy, or a thought-framework on the formation of thought. It is in contrast to this framework that we'll look at universalism and pluralism in philosophy of education. It is worth remarking that under this lens, universality is neither generality, nor totality. We repeat: Generality is the mediation between the particular and the universal, it is a moment in their dialectic; totality is a closed system, whereas universality is open on both 'ends'.



Part 1 – Contemporary Meta-Concepts

1.1 – Vulgar Universalism

We will see that both universalism and constructionist-pluralism strongly feature a Kantian and Cartesian mistep at their core – the intelligibility gap between mind and world or appearance and reality. There are strong distinctions between the Kantian and Cartesian versions, but for now what matters is the common critique that Hegel directed to them: they hinge upon a “two-stage representational story that sharply distinguishe[s] between two kinds of things, based on their intrinsic intelligibility. Some things, paradigmatically physical, material, extended things, can by their nature be known only by being represented. Other things, the contents of our own minds, are by nature representings, and are known in another way entirely.”13 The former are to understood as reality, the latter as appearance. Appearances are “intrinsically intelligible”, reality is not. The main difference between the Cartsian and the Kantian accounts is that for Descartes appearances are immediate, whereas for Kant they are mediated. Nonetheless, the gap remains, and any such theory “is doomed to yield skeptical results.”14 We will bear this in mind when looking at how universalism and constructionist-pluralism allocate minds in reality, the degrees of skepticism they generate, and the solutions they provide to not lose epistemic purchase on the world.

Onwards to universalism. We follow Jan Derry when she presents universalism in an elegantly skewed manner - through the eyes of its detractors: “A once prevailing view in analytical philosophy presents rationality as abstract and decontextualised: it relies on the idea that reason is separated from the world and can be applied to it with greater or lesser degrees of adequacy. When applied to education such a position can lead to the most extreme forms of formalised teaching.”15

Universalism can fruitfully be understood with the help of criticisms toward it (despite their originating school's failings). At this point it will be useful to start calling it vulgar universalism, in the hopes of distinguishing it from Ilyenkov's corrected, refined meaning of the universal. Vulgar Universalism is decontextualized rationality and instrumental reason: the failure or blindness to account for the role of contextual elements in the formation or presentation of the human. There are just and fine criticisms of arbitrary misconceptions of the human and of freedom in Enlightenment rationalism (racialism, the purported telos of slavery), and of its closure to social, historical, instititutional functions. Nonetheless, it should be pointed out that despite grave misconceptions, Hegel's system still included the germ for their further development, as Marx perspicuously elaborated on. A certain branch of analytical philosophy in the 20th century has done its initial attempts to do justice to both the harshest critiques of vulgar universalism, and also to counter them by further developing the Enlightenment project and avoiding the pitfalls of scientific deism and a relativizing 'greedy skepticism'. This can be seen in the works of Wilfrid Sellars, John McDowell, Robert Brandom, and on the pedagogical framework, David Bakhurst. We should note that Ilyenkov's solitary research in the hostile environment of Stalinist academia was also striving for such developments upon Spinoza and Hegel, in its unique way. It is notable that Bakhurst successfully incorporates Ilyenkov's research in his McDowellian project.

What still trickles down from this to our everyday discussions of “free-floating abstract reason”16 appears to discard any account of mediation, freedom. Although it has an account of agency, it is impoverished under axiomatic or foundational – categorical – imperatives, which if failed to be followed by genetic conditions, are simply deemed as unintelligible towards the one to whom is refused the status of agency – as if cancelled by the overdeterminations of fate. The mention of fate here is important: abstract rationality or vulgar universality refuses history, and the arising of any category as a product of historic relations. It hinges on the reification of Forms which are deemed eternal truths and as such, un-transformable. Further on we'll see that this is the vitiated game of necessity. Under this meta-concept, proto-persons do not so much learn, but are educated into these truths via formal teaching, which regiments a correspondence theory of truth towards said abstract Forms. Such theory is overtly idealist, in the sense that education structures an account of the world which becomes blind to any sort of falsifiability, and lacks any granularity. Again, what does not conform to the Forms, is deemed unintelligible, and thus untransformable; it is precluded to any sort of possibility or contingency, which in turn prevents any realizability under conditions which steer off the main, eternal path. Any subjects or proto-persons educated under this meta-concept are not historical agents, not even agents – they become laborers.

Most importantly for this section, we see that this is an account of universality which is a mere misnomer for Totality. What happens is that the universal is generalized; the proto-persons are not different moments of a whole, but repetitions of the whole or the totality in a miniaturized shape, which can only amplify itself monotonically. There is a cybernetic short-circuit of sorts here: The general ceases being a law of connection, and becomes the universal-as-law. Detractors of vulgar universalism would come to claim that “abstract axioms were in, concrete diversity was out”17. As a critique of vulgar universalism, we agree with this. On the other hand, it is an unjust critique towards Vygotsky and Ilyenkov: their account of the universal is much more nuanced, and is definitely not the same as that of vulgar universalism.

In pedagogical practices, the metaconcept of vulgar universalism can be seen in features of most of contemporary educational institutions: summative assessment, where the un-networked acquisiton of disparate facts held to be definitively true is quantitatively evaluated; the assumption of crystallised developmental stages in biology/psychology, somewhat arbitrary windows of conformity to the universal ideals which foreclose formation if 'missed'; rote learning with its vulgar behaviouristic tinges; the Givenness of 'natural laws' and its concurrent favoring of causes over reasons; the list could go on.


1.2 – Constructionist Pluralism

Against universalism, the 20th century has seen the development of an alternative meta-concept, that of constructivism allied with pluralism. Given the coincidence of pedagogical constructivism with that of mathematical constructivism – which will become important for us in a moment – we opt here for constructionism, which is also used in pedagogical contexts.

According to Jan Derry, “the giving of attention to the process of meaning-making itself, rather than the outcome of such a process, is often referred to as constructivist theory.”18 In tandem with Derry's critique of constructionist-pluralism, we pay close attention to the 'making' in meaning-making, which will be seen to be perhaps too liberal and, despite its avowals, decontextualized and un-networked from any historical inferential web. Secondly, the mention of 'outcome' refers to the failings of vulgar universalism: conformity to the universal is assessed via outcomes, parrotings of pseudo-learned concepts. In its intentions, constructionist-pluralism is apparently a healthy meta-concept – but as mentioned, the meaning-making is perhaps too arbitrary and a-historical. Indeed, it does not produce vulgar-universalism's parrotings, or mere reliable differential responsive dispositions19, but we will see it conjures a mirror image of off-the-cuff, allegedly creative interjections.

Still, “constructi[onism] has succeeded in designating learning as an active process where meaning is acquired through a process of meaning-making rather than through the simple transmisson of knowledge or through a behaviourist conditioning of response.”20 Moreover, its “emphasis on genetic epistemology” seems appropriate as an alternative to vulgar universalism, where 'genetic' denotes the Vygotskyan emphasis on the historicity permeating the learning environment externally and internally. These intentions should be held in mind when considering how the alleged creativity of this framework is said to overcome vulgar-universalism.

Constructionism is seemingly opposed to 'realism' – where this realism is held to be that of the confused term 'objective world'. It apparently rejects Cartesianism, but implicitly assumes the world is devoid of meaning without a mind which constructs meaning upon it. We see here that this implicit assumption hinges on a separation of mind and world, on the intelligibility gap. Such a “world [is] devoid of meaning without the contextually sustained activities of participants, [assumed] as given outside aand separate from human construction.”21 This foundational assumption of constructionism bores a monumental void in its cornerstone claim that everything is socially constructed, by giving the world a noumenal – ultimately unknowable - status.

Despite this failing right at the starting line, constructionism sets itself out as an alternative meta-concept in the sense that it sets for itself the task of pluralizing epistemology. At a first glance, this seems to strive towards radical universality, where every individuum is determined by difference. But the crucial point here is that difference is affirmed without determination. In the sense that in constructionism, meaning is just 'made', spontaneously, and out of any relation. It is not even made ex nihilo in the learner's godlike mind: it is instead given by biologically overdetermined developmental stages as proposed by Jean Piaget, without any interference by interaction or sociality. This effectuates the reverse of the Given: here, instead of the world impressing itself on the mind as a seal upon wax, the elocubrations of a closed mind imprint themselves on a barely existing world. The most perverse corollary of the constructionist thesis is that it, unbenknown to itself, strongly resembles the totalising act of vulgar universalism – the difference being, though, that constructivism dispenses with any criteria for knowledge, while vulgar universalism overdetermines those criteria. As Derry puts in an illuminating and somewhat terrifying manner, “the possibility of meanign arising in a [world-]historical process, whereby nature is transformed through human activity, simply does not arise”22. We inserted world in this quote in otder to reinforce the notion that the world is excluded from this account by virtue of not being historically synthesised. We should also note that 'nature', in these meta-conceptual operations, becomes reified – again, as much as vulgar universalism does, by holding Nature to be, although ommiting it is contradictorily so, a preordained developmental blanket.

The core of constructionism is then 1) World is devoid of meaning until meaning is constructed and 2) meaning is limited to the constructive activity of individuals.


A brief pause in our sustained demands from constructionism in order to justify this insistence: we are giving more attention to constructionism than universalism given that 1) it is our contemporary, and seen as a definitive winner against vulgar universalism and 2) in the context of this essay, it is relevant to intuitionist and paraconsistent logic.

In pedagogical practices, constructionism is operationalized through a focus in 'knowledge construction', in the pluralisation of knowledge, and relativist approaches to knowledge, moving away from 'mere instruction'. Relativist approaches will be important with regard to our look into paraconsistent logic.

Pluralisation of knowledge has seen the development of a notion of 'implicit knowledge', in which formal education is replaced by an apparently beneficial levelling of learners and teachers – the former are elevated, and the latter's 'authority' is diminished. Student elevation is done by crediting their assertions with contextualised knowledge which univocally asserts itself as 'equal' to what schooling is proposing; not so much by crediting the students ability to formulate questions, we would say. Basically, and contrarily to its claims to 'collaboration', constructionism will fall again on one-sidedness and in foreclosure of true bridge-building.

Rejection of 'instruction' is a rejection of vulgar universalism: mass schooling, transmission approaches, didacticism, even un-dialectical empiricism. The main critique of what can now be named constructionist-pluralism is specifically against transmission and hierarchy. Consequently, it is against didacticism and coarse-grained mass schooling. The transmission framework is seen by constructionist-pluralism as: the learner is held as passive, the teacher as only source of knowledge and onlythrough the teacher can student acquire knwoledge. This last tenet implies that the transmission framework refuses to recognize any body of knowledge in the student's context, and that there is a refusal of any spontaneous construction of knowledge from the student. The issues with this constructionist critique of transmission education, despite its overall just critique of vulgar universalism, lie in the notion of spontaneity of knowledge in the student.

To affirm itself as an alternative meta-concept, constructionism relies on proposing epistemological 'inclusion' and plurality, and designing curricula aimed at the 'individual' needs of learners. The problem or insufficiency of constructionism as an attempt at overcoming vulgar universalism lies herein. The liberal notion of what constitutes 'individual' needs is, if taken to its full development, parallel to totalisation, monotonic amplification of glorified monads; the deathblow to the dreams of an alternative pedagogy is dealt by combining this totalising of the individual with epistemological inclusion and plurality. Inclusion and plurality seem, at a first glance, benign. The question here is how they can work themselves out in a framework of totalised, amplified individuals: does every individual knowledge host sensors which can make other individual knowledges intelligible? Apparently not, when each individual knowledge is self-legitimated, and the 'world' is held as unintelligible: other individuals are not recognized, and can only be 'constructed upon' by arbitrary caprice. This, in effect, is a parody of inclusion, and a victory of overdetermination. Here the general ceases, too, to be a principle of connection mediating the indivual towards universality, and we see here that the individual is overdetermined as a universalising principle.

Despite not assuming this defect (inclusion stopping short of itself, given no monad can host a model of other monads), constructionism holds the pluralisation of knowledge as a solution to the shortcomings of inclusion: through pluralisation, the malefices of hierarchy and one-sided authority are set aside, and all knowledges become 'equal' by conjunction. Paradoxes will grow and multiply, by vice of relativisation: 'no one is wrong, both are right in their own special way'. The most destructive effect of this is an anti-learning stance: if, when confronted with paradox, contradiction, or provisional unintelligibility, a student is invited to not analyze what makes it so, they are being invited to not inquire into the theoretical framework which allows for the paradox. Consequently, the student is being asked to passively universalize such a theoretical framework as transhistorical, unquestionable, eternal – both the framework which invites for this, their 'own' individual framework, and their 'opposing but equal' interlocutor's. This will be crucial for our look into paraconsistent logic.


Part 2 – The Logics of Contemporary Meta-concepts

2.1 – Formalism towars Vulgar Universalism


Having illustrated the cornerstones of the contemporary meta-concepts in pedagogy, we have hopefully garnered an adequate picture of their theoretical frameworks enabling us to propose their operative logics.

In the case of universalism, it is our contention that its concurrent logic is that of formalism. Formalism has been known since the revolutionary days of logic and mathematics through the work of Hilbert, after the astonishing developments of Frege, Russell, and of Wittgenstein in his Tractatus. The latter's famous claim that “The world is everything that is the case.” bears some poetic resonance over the comparison we're about to elaborate.

One caveat should be registered, though: it is not this essay's aim to somehow show that universalism, with its genesis in Kant, Hegel, and German educationalism, definitely birthed formalism. Our main aim is to try to identify formalism as the logic which best explains (vulgar) universalism.

A formal logical system is an abstract structure expressed in a formal language, grounded in axioms which allow the derivation of further theorems through the use of a set of pre-determined rules. Formalisation of the initial axioms is obtained by their formulation through a formal language, deemed sufficiently expressive for the constitution of the system as consistent. Crucially, consistency is held as an ideal – that of the the impossibility of deriving contradictions from the system at hand. In this manner, it was hoped that any theorem derived from the system could be proved through the necessity provided by the interplay of the axioms and the formal language used for this aim.

The use of 'necessity' in the previous locution was deliberate – formal logic is not modal logic, but it is historically chainlinked to Aristotelian logic, through its unquestioned pressuposition of the two of the three main cherished principles of logic. These are usually known as 'laws' – those of Non-Contradiction, and of the Excluded Middle (or Third: tertium non datur). Aristotle's (assertoric) syllogistics was strucured independently of his explorations in modal logic, although the latter work hinges on the previous, through the fact that it is structured upon it: modal qualifications are apposed to (previously merely assertoric) premises, thus creating an interplay between necessity and possibility with their respective outcomes. What interests us here, given logic is held as an organon, an instrument of thought, and thought as an organon of the realization of the human in the world (and vice-versa), is how the outcomes of logical procedures are seen as the final word on the constitution of any subject-matter of choice. In other words, logic, in its ordinary usage, can be seen to claim an extraordinary epistemological and ontological purchase on the world, nature, reality. An unexamined assumption that the conclusion of a correctly derived syllogism is, if valid, also necessary, requests for itself an end-all, be-all status. Usually, the strength of this validity and necessity is given sustenance by hinging on the aforementioned 'laws' of Non-Contradiction and of the Excluded Middle. We are also pointing toward the fact that validity of a syllogism is not identical it its modal necessity, in the existential sense of the latter. It is important, though, to have in mind that we are not claiming in this essay that certain cherry-picked logical systems are simply wrong, but instead that, despite that they are sufficient for their own frameworks, if not only in the historical sense, also in the suspended status of their theoretical claims, they will then be seen to lose much of their philosophical leverage if totalized in everyday interactions.

With this in mind, while admitting that necessity was nowhere in the formal logic project 'manifesto', we still insist that there is some sort of necessity codified in, or is at least isomorphic to the ideal of consistency strongly featured in it, when it trickles down to the theoretical framework's openness to its bridging with further historical developments and further realizations of intelligibility. Namely, if such a theoretical framework is seen to underlie forms of life and sociality – theoretical practices - such as our main interest, pedagogy.

The formalist ideal of consistency is an ideal of absence of contradictions. At a first glance, there may be no issue with this: a formal logical system can be structured, abstractly, and account for many avenues of mathematical knowledge. Those unaccounted for are deemed problematic – problems. The fracturing point lies precisely here: are these problems, by virtue of their constitution as such under an axiomatically grounded formal system, as yet unsolved, or are they simply unsolvable? We will see in the next section how the intuitionistic school grabbed the answer to this question as its banner, but for now we will not enter into how they deemed it answerable, beyond saying they chose the 'as yet unsolved' formulation of the question in order to structure an intelligible answer. At this moment, what interests us is to see how the choice of the 'unsolvable' formulation illustrates the vulgar-universalism meta-conceptual practice.

From the outset, we should say that the grounding axioms of a formal logical system are presupposed as consistent in themselves – as necessary, and as non-contradictory between themselves. What this entails is that any theorem derived from here that is seen to be contradictory with the remainder of the structure, is deemed unintelligible to the structure. A contradictory statement has no place in the structure. It is our contention that a consistent formal system is only composed of necessary statements and a certain interplay of possibility and impossibility which closes off the system to further intelligibilities. This is reinforced by the unexamined presence of the principles of non-contradiction and the excluded middle. In the case of the latter, we see an issue arise: that of the transhistoricity of a formal logical system. The principle of the excluded middle, in the shape of “either(true/false)/or(false/true)”, excludes any third statement or term from a theorem – not even until the end of time, but beyond it. In its negative formulation, “neither/nor” will prevail as well, thus excluding not only the so-called third, but even both statements or terms. What this entails is that a formal logical system automatically – almost preemptively - decides what is intelligible or unintelligble under its grid, and refuses itself to pursue – to solve and make intelligible – any contradictions. This refusal implies retrospectively that a consistent formal system has no tools (axioms or rules) which allow it to revise itself in order to unveil and accommodate further intelligibilities.

The point at which we can start mapping an isomorphism between formalism and vulgar-universalism is precisely what starts off the previous paragraph. Vulgar-universalism sees itself as an abstract rationality grounded on its consistent, necessary, and non-contradictory principles. Such principles are also abstract, and formalised under a language deemed as the most expressive for universalising purposes. Our most stringent critique of it, it must be said, is that it is a static, transhistorical ideal. Claiming it is static is relevant to our critique of it as a meta-concept: it can only express those concepts it can articulate formally, but it cannot express concepts which call it into question; moreover, it cannot call itself into question. This allows us to claim that such a meta-concept becomes unintelligible to itself, for lack of conditions to express the possibility of its normativity, and merely affirming it as lawful.

A pedagogy developed under such axioms will necessarily conform to the fixed price of non-contradictoriness: any error is immediate, and therefore, unintelligible. Facing the unintelligibility of error, the only avenue of practice left to such a pedagogy is not quite a practice, but a reification of a theory. In other words, the organons left are basically those of rote-learning and summative assessment. This can be seen in the fact that to a formalised pedagogy, any learning error is defined as contradiction, a closed alley wherefrom the instructor sees no possibility of retracing the steps of error and re-developing them into the knowledge of the subject-matter at hand. Furthermore, it excludes the possibility of practically using the contradiction to enlighten the learner that their error may inform them of another theoretical framework which is determinately negated from the one under the which the error was occasioned.

In sum, such a pedagogy, conversely, forecloses the possibility of mediation. Let us suppose we are closed off in a context where what is understood by “the world” has not yet been called into question, and the statement “the world is everything that is the case” is a picture of the meta-concept of a theoretical practice which aims for the formation of minds-in-the-world. If what such proto-minds state is deemed “not the case”, such minds are foreclosed to being a constitutive part of the world. In this scenario, as we've claimed, the theoretical practice is seen to be impractical. The intelligibility gap affirms itself, even to the point of placing the learner's own mind beyond their reach.

Until now we have refrained to mention that formalism did not historically perpetuate itself without being called into question at the fundamental level of the matter of consistency. Perhaps we should have also noted earlier that Hilbert's criterion of consistency was not completely arbitrary: finitary arithmetic was chosen by reason of being a system proven to be consistent in order to ground the formalisation of all mathematical systems.

Not long after formalism's development by Hilbert, and independently of intuitionism's qualms with it, Gödel showed precisely that consistency breaks down – is unprovable – under axioms capable of expressing classical arithmetic. The issue lies in the formal language chosen for the structuration of the system: it cannot prove its own consistency. In plain terms, as we've seen above, sticking to a formal logical system as such practically entails omitting its own inconsistency while demanding consistency from everything else; the formalisation is hiding its arbitrariness and masking itself as lawfulness. This last point is crucial to a supposed isomorphism between vulgar-universalism and formalism, for this is how a meta-concept is shown to not be able to call itself into question. As a corollary, such a pedagogy cannot provide tools and form individuals which/who can critically perform that same task to themselves and the system which formed them. It is our contention that in this manner, vulgar-universalism behaves as a formal logical system when it generalizes the universal, reaffirming itself as totality in unquestioned reproduction of its axioms and refusal of the intelligibility of contradiction.

2.2 Intuitionism towards Constructionist-pluralism

Pursuing the isomorphic thread, we will now consider a logic operative through the meta-concept of pluralism. The intuitionist school of mathemathical thought founded by L.E.J. Brouwer and its logical investigations pursued by Arendt Heyting, attacked the aforementioned issues of formalism through a revolutionary perspective on the interrelated questions of 1) principles of logic (destituting them as laws), 2) their abstract permanence above a demonstration of a proof, and 3) the issue of dismissing the semantic over preference of the syntactic.

Regarding the latter question 3), which we've barely hinted in the previous subsection, there is something to be said: the formalists reinforced the givenness of axioms through the abstraction of meaning from the symbolic structure of a formal system. AA Cavia describes this succinctly: “for the formalist, the meaninglessness of the symbols is assured by their lack of referent, denoting nothing in themselves, but instead embodying a purely analytical practice”23. This emphasis on analysis would later be called into question, through the revelation of the implicit issue in how meaninglessness itself is dependent on a metalinguistic referential frame: that of the metalanguage in which the logical principles and axioms are formulated. In order to avoid this incipient loop, intuitionism defended a constructive attitude: the semantic façade of a mathematical construction is structured into it anew, through rejection of 2) the transhistoric lawfulness of 1) the principles of logic. In other words, this was done by rejecting the Principle of the Excluded Middle, making the truth or falsity of a statement contingent upon how this semantic criterion is embedded in the construction of a proof of that statement. While still conceding that “either true or false” holds locally for a single assertion, intuitionism refuses the totalizing axiomaticity of the PEM. This has profound practical relevance to the creative aspects of mathematics, and the role of logic in even the most mundane activities – as it is our similar contention when claiming these logics trickle down to everyday reasoning and institutionalised practices.

At the core of intuitionistic logic there is a rejection of formalism's propensity to abstraction. The purely analytical practice hinging on the 'platonistic', independent existence of real mathematical objects whose supposed transhistorical meaning buttresses the meaninglessness-cum-impartiality of a formal system, is bypassed in favor of a constructive attitude where “proof construction [for a statement is] an ontologically ampliative exercise which brings a truth into being.”24 Here we should remark that the intuitionists are closer to a figure who is of importance to radical universalism: Hegel. The becoming of a truth in intuitionist practice is resonant with Hegel's stance on truth being immanent to an inference, and not to an abstracted, free-floating assertion.

This emphasis on construction has serious implications on 2) the abstract permanence, or timelessness of a demonstration of a proof. This hits precisely at our distinction between 'as yet unsolved' and 'unsolveable' problems in the formalist framework. In it, a demonstration of the truth or falsity of a statement holds eternally, so to speak. The same goes for its unintelligibility – we now see how the failed expulsion of semantics, at the outset of the formalist project, allows for the coarse binarity of intelligible/non-intelligible, when the semantics of the laws of logic, in fact, remain in place. On the other hand, the intuitionist framework, which starts out also without a semantics, insists that until a semantic framework is constructed into a formal system, “if no proof currently exists for an arbitrary assertion A, then no guarantee can be made in advance regarding its decidability.”25 It is important to note that our description of the behaviour of formalism facing the meaning of a statement is not quite the same as what is exemplified in our Cavia quote for intuitionism. Theirs is talking about the (proof by) demonstration of a statement. Our claim is talking about an as yet unconstructed proof. The crucial difference lies in the methodology. Formalism is demonstration oriented via de-semantification; intuitionism is construction-oriented before semantification. Formalists opt for demonstration under the aegis of abstraction, under the outlines of the looming shadows of the axiomatic and laws-of-logic building. Such outlines predetermine what falls in or out the demonstrability net, so to speak. With this image in mind, we can say that the meaning of a statement is demonstrated, pre-ordained, even before it is uttered or inscribed. Tragically, a statement has already been demonstrated before it has been demonstrated. We repeat that this happens by vice of the fact that meaning has been embedded, hidden in the formal system, from the outset – hidden even to the formal system itself. And as such, problems under a formalist lens may quickly look as merely, eternally, unsolvable. It is against this picture that we can see that in the intuitionist framework there is an active and even creative interest in tackling the issue from the 'as yet unsolved' outlook.

It is our contention that intuitionism holds a partial acceptance of contradiction which was excluded from the formalist framework. Mechanically, so to speak, this can be seen in how the PEM is excluded from a construction until the intuitionist creates an inferential rule which then makes it hold over the local of the construction where it is applied. This means that a construction is open to contradiction until it is not. On the other hand, when contradiction appears in a construct, it is now held as a door to the necessity of the possibility of further construction. In other words, an undecided statement's proferred unintelligibility is now held as possibly intelligible, when a future construction may decide it.

This generosity, at a first glance, is as productive and creative as it seems. It is also where we find our isomorphism between intuitionist constructivism and pedagogical constructivism and pluralism. It seems clear to us how the pitting of construction against demonstration correspondingly maps in harmony with the opposition between the pedagogical meta-concepts of vulgar-universalism and constructionist-pluralism. A cursory look at the latter pair, intuitionistic constructivism and constructionist-pluralism, shows us that the rejection of the PEM is in tandem with the rejection of a totalized pedagogical framework: both entail the rejection of abstract, universal, purportedly de-semanticized and de-semantifying axioms. On the other hand, both accuse axiomaticity of carrying a disguised transhistorical semantics into their practices. One way to illustrate this is by seeing how a vulgar-universalist pedagogy is isomorphic to proof demonstration: when the former has recourse to no other practice but learning-by-rote and summative assessment, it is instituting the timelessness of the demonstrability net where the meaning of a statement or a subject-matter's fact is always-already demonstrated. A learner's practice will only entail to reproducing the universality in a particular performance. Consequently, there will be no talk of provisional undecidability regarding the statement or subject-matter; the learner is foreclosed any agency regarding their ability to even construct a question about the necessity and the meaning, that is, of the purported consistency, of the framework. Against this, the intuitionistic constructivism and constructionist-pluralism pair appears much more fruitful. Both schools of thought hold that construction is “ontologically ampliative” and that meaning is to be constructed by the individual which/who performs it. From this perspective, the learner gains agency against the universal or totalizing axiomatic framework: they now have the creative responsibility to construct their own axiomatics under which concepts will become meaningful through an inferential process. In a pedagogical context, this implies a new degree of trust from the instructor towards the learner. And upon the learner now falls the responsibility of constructing a proof of their meanings.

It is here, though, that the mapping between the intuitionist constructivism and constructionist pluralism also reveals how both fall short of the demands of radical universalism. Note how in the previous paragraph the choice of words went for “their own axiomatics”. We should also be reminded that constructionist-pluralism holds the owlrd to be de-semanticized, while it still refuses any de-semantifying, abstract universality. Holding that in mind, we should now mention that after the intuitionist logic revolution, the bulk of criticisms toward it went over, almost comically, the fact that under such a framework, the requirement of constructing a proof for every statement would entail too much work. But there was another criticism which was in fact a self-avowed tenet of the intuitionist school's founder, Brouwer: mathematical activity is a purely subjective performance, an apex of creative solipsism. Against this, critics who were even minimally 'platonistic' about the reality of mathematical objects had the necessary rhetorical purchase to dismiss intuitionism with a laugh. This is not our position, principally given we are not discussing ontology in this essay, but also because taking either position – the platonistic-realist or the solipsist one – would cancel any critical leverage we might have with regard to the candidacy of intuitionism to a logic of a pedagogical meta-concept.

That said, we think there is an important interplay between the laziness and anti-solipsist critiques: it hinges, respectively, on the issues of trust and of how universality is organized in a pedagogical situation. Intuitionists contend that behind an assertion or a statement A is implicit another assertion, going somewhat like “i have performed necessary construction in order to assert that A”. Under this assumption, an interlocutor is entitled to trust the assertion A, but can also ask for the reasons – the constructive steps – buttressing the assertion. Opting for not doing so can – but is not necessarily so – a carte blanche for the possibility of asserting something without all the construction work required in order to do so. We are no strangers to this. Everyday practices go on under the assumption our interlocutors are performing under a framework which semanticizes their claims. A certain practicality, or what has been incorrectly called 'pragmatism', invites us to not spend a lot of time asking one another to illustrate our everyday assertions with a whole theoretical framework. In other words, we live by allowing for a partial suspension of disbelief in our interactions and thus, authorizing a fraction of solipsism in one another. Nonetheless, as we all know by experience, semantic glitches arise. It is one thing to unravel them, and it is another to leave them unexamined for the sake of practicality or a liberal pacification of dissension.

It is our contention that the constructionist-pluralist metaconcept, despite its constructive ideal, favors the laziness or suspension of disbelief and a solipsist framework. Pedagogical settings are a critical juncture where this tolerance can be embedded in our practices. We have seen above how the constructionist-pluralist metaconcept favors a spontaneity of knowledge in a learner. It is crucial to have in mind that creativity is not identical with spontaneity of knowledge. The former is an agnostic process on the way to knowledge, whereas the former implies that an individual purportedly carries biologically triggered universals in their mind, or that whatever socially acquired knowledge from their individual history (family, cultural identity) is immediately translatable as identical with the knowledge being presented in a pedagogical setting. Intuitionism subscribes creativity, not spontaneity of knowledge. Still, it is by conflation of these two categories that constructionism applies intuitionist logic in its program, along the unquestioned assumption that a learner has performed all the required construction and semantification of their assertions by the light of their individual axiomatics. In the name of epistemological inclusion and plurality of knowledge, a learner is trusted to have legitimacy to regard an instructor's or a peer's statement as false, and sustain theirs as true. As we have already hinted at, pluralisation of knowledge is effected through the generalisation of solipsism. If every individual knowledge is preemptively, abstractly legitimised as universal in its own constructed personal, unique way, we have indeed gone around the totality of demonstration. But on the other hand, all knowledge(s) have become equal: each personal semantic framework adds up to a totality of arbitrary desemantification of the 'outside' world. The constructionist-pluralist pedagogy's solution to this epistemological and ontological conundrum hinges, after all, on the purportedly benign unsolvability of interpersonal contradictions, and consequently, on the unintelligibility of error as the possibility of realizing new knowledge in each other.

An interesting way to look at this issue is again under the lens of modality. Intuitionism's break with totalizing axiomatics allowed for the novel notion of the necessity of possibility (it is necessary that an undecidable statement is possibly decidable). Bizarrely, what happens under the constructionist-pluralist metaconcept, is the seemingly innocuous notion of the possibility of necessity. Given any learner's purportedly constructed framework is deemed possibly necessary by its own lights, their individual axiomatics are reified as necessary in their own closed-off pseudo-universality, forgetting the fact of the locality of such necessity. What we see here is that contradictions between personal pseudo-universes, although internally legitimated, are externally de-semanticized. As such, they cease to be contradictions and become merely competing self-identical affirmations, oblivious to whatever else that does not fall under their constructed net. Vulgar-universalism comes back, with a vengeance.

As we have seen above, this fails the requirements of radical universality, where the unity between particulars – in this case, learner's constructed frameworks and the intructor's own – is “created by the attribute that one individuum possess and the other does not. And the absence of a certain atttribute binds one individuum to another much more strongly than its equal existence in both.”26. The intuitionist/constructionist metaconceptual framework refuses the bindedness of one particular to another, given each of them is a monotonically amplified universe – in fact, a totality necessitating itself. In order to occlude from itself this proliferation of totalities, and specifically in its tolerance of de-semanticized contradiction, such a metaconceptual framework in fact has opened the door for yet another logic – paraconsistent logic.


Part 3 – Paraconsistent Logic

In this section of the essay we are about to consider the main tenets of paraconsistent logic, as proposed by Graham Priest. Our focus will be on one of Priest's earlier essays on the subject, “Dialectic and Dialetheic”27, from 1989. Although Priest has considerably developed the paraconsistent logic framework since the paper in question was written, given the bulk of said paper is dedicated to the claim that “dialectics requires dialetheism”, we will take such a claim as fundamental for an immanent critique of the paraconsistent logic project as held to be required for dialectics. This stems from the fact that, as hinted throughout this essay, the metaconcept of radical universalism is contingent upon dialectics; in other words, that such a metaconcept requires a logic which can adequately accommodate dialectics as proposed by Hegel and, further on, Ilyenkov. In order to do so, such a logic is expected to be able to include determinate negation among its organon.

Some initial assumptions as proposed by Priest:

I) A dialetheia is a “true contradiction, a true statement of the form 'A and Not A'”.

II) Dialetheism is this view that there are true contradictions.

III) Hegel and Marx's dialectics were based on dialetheism.


Against someone who might think it unnecessary to argue so, Priest alleges there have been historical misapprehensions of dialectics which have distracted us from the main tenet of his paper: that dialectics requires dialetheism, and concurrently, that dialetheism is inscribed in dialectics from the outset of the Hegelian treatment of the subject. Priest regiments an example given in Hegel's Science of Logic: “ . . . common experience . . . says that . . . there is a host of contradictory things, contradictory arrangements, whose contradiction exists not merely in external reflection, but in themselves. [...] External sensuous motion is contradiction's immediate existence. Something moves, not because at one moment it is here and at another there, but because at one and the same moment it is here and not here, because in this "here" it at once is and is not.”28

It is absolutely crucial to note that in the coda of the last sentence, “because at one and the same moment it is here and not here, because in this “here” it at once is and is not”, the quotes for the term here are Hegel's, and the use of and underlined bold font for the demonstrative determiner this is ours.

At a first glance, it may seem that starting our critique right at this point is done in bad faith, but it is important to at least note the minutest glitch at the outset in order to understand how its consequences have unraveled later on. That said, we must pay attention to the iterated quote: this is reminiscent, at once, of the opening of Hegel's Phenomenology of Spirit, on the issue of indexicals. We see that there is already at work a question of scale: the scope of the concept “here” between quotes; not the scope of a flickering truth-value.


Hegel is claiming:

1) It is Here1-because-not-Here2;

Hegel is not claiming:

2) “It is here-and-not-here”


The demonstrative “here” between Hegel's quotes is interchangeably applicable to Here1 or Here2. If Here1 is “this here”, Here2 becomes “not-this here” and thus “not-Here1”; if Here2 is “this here”, Here1 becomes “not-this-here”, and thus “not-Here2”. In simpler words, the stating of a “this here” turns the other moment of the indexical here into “that here”.

The 1) claim opens up to determinate negation and something newly intelligible; the 2) claim preserves the matter undecided and gives no path to get out of deadlock. The former is an implication; the latter is only an assertion.

With the quote from Science of Logic at hand, Priest provides some examples of Hegelian philology, some of which suit his purposes, others that don't. It is quite alright that there are many different historical interpretations and some of them wrong, but this does not immediately imply that Priest's own interpretation is correct. It is interesting to note that this methodological assumption mirrors his own argument. When Priest quotes Sheptulin claiming that: “Aspects in which changes move in opposite directions and which have opposite trends of functioning and development are called opposites, while the interaction of these aspects constitutes a contradiction”29, he is correct in pointing out that it is an exageration to describe a relation of inverse proportionality as constitutive of contradiction. Still, Priest does not say, but we do, that it is from contradiction that we beget the relation.

The historical consensus evoked seems to be that contradiction is logical, occuring only in thought, but not in reality. Priest is against this, proposing, within a dialectic/dialetheism framework, that:

3) Contradictions occur in reality.

But he's also claiming that:

4) Dialetheias (A and NOT A) occur in reality.

Being ourselves informed by a dialectical (but not dialetheic) framework, claim 3) is acceptable, to the extent that such a framework, briefly put, takes reality as a subcategory of material reality which is continuously realized by synthetic activity of minds-in-the-world or minds-in-the-real. In this sense, contradictions do occur in reality – moreover, they are precisely produced by the intellectual and manual transformation of material reality by the minds which constitute it. It is relevant here to recall that intellectual activity need not be considered exclusively immaterial, or purely abstract. It is, after all, embedded in the concrete.

These provisions allow us to have a closer look at claim 4) and note that it becomes stranger than the previous. A weak version of it would be that it is true of reality that in it there is a logic that can grasp A and can grasp Not-A. We can concede that a weak dialetheia occurs in reality by reason of an abstract, contradictory moment in our mind being surely buttressed in reality. But a strong version, which we take to be Priest's, is that the laptop where this essay is being read both is and is not in reality, is and is not true of this world, flickering in and out of existence.

Priest's argument against the “there are no real dialetheias” interpretation seems based on defending the strong version of “dialetheias occur in reality”. At the base level, given that, for example, there are schools of thought – such as Cartesian dualism, or even Kant's idealism - that reject the both the weak claim (contradictions are real) and thus the strong claim (dialetheias are real) as well, Priest is claiming both versions by betting on the strong version.

The rejection of the reality of contradictions or dialetheias is historically grounded – but shaky: Hegel claimed dialectics is not formal logic, and rightly so, given formal logic in his days was only Aristotelian logic: term logic, and the principles, which until recently were laws: the Principle of Identity, the Principle of the Excluded Middle, and the Principle of Non-Contradiction – which are all well for a static reality, but cumbersome for a changing one. Contemporary formal logic is more fine-grained in Frege and Russell, and we'd add, Hilbert. It rules out dialetheias, while during the same historical period other people adapted or hygienized Hegel and Marx's dialectics. We must be reminded that Russel's anti-idealism is a sort of hack materialism, and note that both anti-contradiction (and thus anti-dialetheist) formalists and Priest forget that a main thrust in Hegel and Marx was that no theory is transhistorical. This is to say that although, in the philosophical mainstream, dialectics apparently has not yet been seen to be fully formalized (translated into a formal language), it is neither non-formalizable, nor is it foundationally averse to being so as a purported tenet of its own. It is this essay's tangential contention that dialectics is formalizable – and perhaps has already been so. For the moment, what matters most is how it is formalised, and which assumptions are embedded in this procedure.

For Priest's paraconsistent logic, the embedded assumption is that if one accepts true contradictions, one accepts some things are both True and False; and thus one does not accept Aristotle, Frege, Russel et al, in saying that True and False are mutually exclusive. Again we must mention that there is a slight confusion in matters of scale:


The dialetheic assertion:

5) “X is both True and False”


is not the same as the dialectical inference:

6) “if X is True and X is False, then there is something of X which is not-X”.


It will be fruitful to look at this under the criterion-lens of modality, as previously: in 5) we see that the term X - and only X as itself, not any other possible predication of it - is said to convey simultaneously both the meaning True and the meaning False; it is necessarily both. If the truth of X is possible, its falsity remains necessary, and conversely so. If both values are arbitrarily deemed possible, the claim becomes that it is necessary that X is possibly true and possibly false; this does not entail that X is possibly true and false. What it entails is that the truth or falsity of X is as of yet undecided. We would risk claiming that in these circumstances, 5) is not a dialetheia anymore – and never was – but it is a contradiction.

Of 6) we can see that if, in a dialetheic fashion, X is considered to be necessarily true and necessarily false, then we are immediately faced with a contradiction. We have just seen above that a dialetheia is not sufficient for claiming a contradiction; that only a contradiction can realize itself. It should be said that it seems healthier to identify a contradiction so early in this analysis. Moving on: if X is possibly true and possibly false, it necessarily remains undecided in this respect. If the truth of X is necessary, its falsity is impossible. If X is necessarily false, it is impossible that it is true. At a larger scale: if it is possible that X is necessarily true and necessarily false, this begets again the possibility of a contradiction. Moreover, if it is necessary that X is possibly true and possibly false, it is necessary that the statement's meaning, again, is as of yet undecided, and remains a contradiction until resolved. What matters most here is that the undecidability of 6) – and of 5! – show us the revolutionary aspect of dialectics: facing undecidability, it is legitimate to inquire if there is not some unexamined singular predication of X which is tying the (k)not which leaves it undecided. That is to say, there is an openness to develop a further intelligibility from what is, for now, a paradox. Moreover, this presents itself as springboard for a rational agent to be entitled to inquire what is it about the theoretical framework which originates the contradiction that has to be developed, refined, in order to clarify this ambiguity.

In order to understand how semantic (truth) values vary greatly depending on the scale of their application, let us take an example which mixes the mundane with the scientific: Denis Villeneuve, the director of the film Dune, says that watching it on a cellphone is like placing a speedboat on a bathtub. The argument one learns in film school (or elsewhere!) is that when watching Dune on a cellphone, for example, a large and somewhat blurry spaceship will seem to be on top of a not-blurry gigantic sandworm, and that when watched in a theater's screen, the ship is seen as farther away, in the background, where as the sandworm is in the foreground. This example illustrates that there is something clearly lurking in the question of scale, something which is to be developed from a contradiction, which then becomes known – in this case, cinematographic depth of field.

Against dialectics - or dialetheism if understood as identical to it - Karl Popper's inference “A & not A entail B”, despite perpetrating in bad-faith the same confusion that Priest is nonetheless trying to develop constructively towards a benign outcome, has a grain of intelligibility. The issue, it seems, is in how Priest turns a contradiction into a conjunction. True and False being mutually exclusive, if negated, does not entail they are the same. It does entail that logic – and reality – are richer than that. What Priest is bypassing is the inferential step of saying “No, A is not equal to A”. Instead, Priest opts for re-assertion: “Yes, A is equal to not A.”

This is a mirrored formalism, and again, an issue of locality and scale. The more cogent question would be: “for A to be true here, it has to be false over there. If it is false over there, then what is it that is true over there?”

Priest claims that paraconsistent logic, by their semantics that are alternative to formalist ones, make Popper's inference fail, thus preventing logical explosion. But the question then is: how broad is this prevention? In its supposed overcoming the problem of explosion, paraconsistent logic unfortunately errs precisely on defending itself against Popper's accusation. Pedagogically, we actually 'need' the possibility of logical explosion - in the dialogical transition of a concept from person to person. Getting ahead of ourselves, we can say that this seems to be active in the elimination of 'cuts' in linear logic's account of natural deduction. The un-cut terms locate the concept's meaning in each interlocutor, and necessitate the possibly positive outcomes of misapprehension and creative insight into the semantic dissonance of a contradiction. The prevention of logical explosion in the name of paraconsistency seems instead to reinforce the dead-ends of formalist eternal consistency. With contradiction, we can at least make future knowledge indefinite, instead of making present knowledge – and non-knowledge –infinite. According to Jean-Yves Girard, “logic does not ensure termination, but absence of deadlock.”


Priest's formalization of dialetheic logic will illustrate this further. As we've seen, Frege and Russell logic assigns to each sentence either value True or value False. Dialetheic logic may assign, in addition, both values True and False. In dialetheic logic, a sentence being true does not rule out its being false – taken by itself and seen from a radical universalist lens which sees meaning as historical, this is not polemical. But can it be expanded?

We will not replicate Priest's formalisation here in full, but only comment that superficially, the logic seems richer than classical or formal logics. But it remains formalist at heart: True and False here are not mutually exclusive, but they are mutually inclusive from the outset – until when? Again, intuitionism's critique of Hilbert's axiomatics seem legitimate here: can the simultaneous values be applied into the far future?

Priest informs us that paraconsistent semantics creates an “unorthodox” logical consequence: a false statement B is not a consequence of a true-and-false statement, A and Not-A. But then what has been learned? Only that a true-and-false statement is inferentially cut-off from an only-false statement. This then suggests the question: what else, beyond another true-and-false statement, can be inferentially linked to the previous one? If it is to be a true-only statement, then it seems that a dialetheic statement is uncapable of leading to a contradiction – at best, to a re-assertion. Again, this points towards our suggestion that a dialetheic statement is not a contradiction. Moreover, it seems that dialetheic logic cannot accommodate contradiction, despite its weak and strong claims on the interchangeability of dialetheia and contradiction. This is also another point of contact with formalism: contradiction is prevented, and undecidability is reified in the decided ambivalence of dialetheia.

Priest also claims that in the paraconsistent framework:

7.1) A not being equal to B


Is identical with claiming that

7.2) Not-A is equal to B.


But to us it seems that claiming:

8.1) “A is not-equal to B”

Does not mean that:

8.2) “Not-A equals B.”

To clarify:

8.3) A not being equal to B is just:

8.3.1) [“A is not B” is equal to “B is not A”.]

8.3.2) It does not entail that “B is everything that is not A.”

Apparently, Priest is, in 7.1 and 7.2, dissolving determinate negation into abstract negation.

Further on, Priest claims that sentences in dialetheic logic are to be nominalized: “John is happy” becomes “that John is happy”, or “John is being happy”. In this manner a sentence becomes a noun-phrase, therefore denoting an object. It seems to us that this a way of smuggling Strong dialetheias into reality: That-A and That-Not-A are opposites. An object is not the same as its opposite, so That-A is not equal to That-Not-A. We can see here Frege's Sense and Reference at work, and Priest claims something similar, but that it covers two-valued cases. Formal logic requires consistency, the absence of dialetheias; Priest's logic can apparently dispense with consistency in order to accommodate dialetheia, but we have already seen that in fact it still keeps consistency, although in an ambivalent guise. When claiming that “dialectic/dialetheic” logic “generalizes” formal logic, Priest is, perhaps unbeknown to himself, confessing to the restyling of formalist consistency. We have already seen that generalisation is the step where any logic becomes abstracted from epistemic purchase, no matter if it claims to be doing so in the name of abstraction, or against it. So far, the gains of generalisation only seem to amplify the formal into the concrete, delivering death to the real through asphyxiation by generality.


So far we have been following the thread of Priest's “Dialectic and Dialetheic” paper in order to pursue an immanent critique of its proposals. Its 4th section is an attempt to illustrate the described paraconsistent logic, and will prove very illuminating – especially given we think it is necessary to intersperse the technical sections of this essay with tangential arguments intended as useful for the lay reader. Pedagogy is, after all, the transformative synoptic realm where the layperson's world can enter into dialogue with the scientific.

The story goes:

Your car ran out of gas; you ask me where's the nearest gas station, and I reply:

“there is a gas station around the corner

but it is closed.”

Priest breaks down the reply in two parts: The first, “there is a gas station around the corner”, is said to be True. Priest also claims that the second part, “but it is closed”, makes the first part only False; that if he ommits the second part he is making the first part False – therefore, the whole sentence is both True and False.

But the issue is that the second part – “it is closed” – is not the falsity of there being a garage around the corner – it is a determination, a third piece of knowledge.

That the gas station is closed does not make it any farther. There is no essential connection between the two parts. The interesting thing is that to the question: “where is the nearest garage”, the answer “it is around the corner but it is closed” is dialectics in practice: the answer replies correctly in its first part, but it does more: it shows the asker that the question could have been refined and it also elicits a further question to a further answer. For us, having already proclaimed that radical universalism is contigent upon dialectics, the contradiction in fact lies in the car which does not work without gas – in these circumstances, it is not quite a full-bloomed car. The conversation works out, practically activates, determinate negations that will make the car finally start moving thanks to both interlocutors' inferential – universalising – mapping of previously particularized knowledge.

It should be noted for later exploration, that this illustration's shortcomings and our analysis of them seem profoundly relevant to the question of learning how to bring out fruitful answers from Chat GPT, or even the story of anamnesis in Plato's Meno.


Further on, Priest insists that dialectical contradictions are dialetheias and argues historically, by saying Hegel was influenced by Neoplatonism's tenets that held contradictory things to be true of the One (everything and nothing, everywhere and nowhere; god is being and not being; god is reconciliation of all opposites). But these examples so far only show a sophistical abstract negation. Priest claims the neoplatonist One is Hegel's Absolute, and we cannot but be reminded of how Russell's rejection of Hegel is precisely based on an identification of this sort. This essay is not in agreement with such an identification, deeming it unproductive for both a critique of Hegel or his purported rehabilitation by dialetheism and paraconsistent logic.

Priest picks out Hegel's infamous aphorism, 1) “Being is Nothing”, in order to allow paraconsistent logic to identify it with 2) “Being is Not-Being”. But “Being is Not-Being” is not “Being is Nothing” - if for no other reason, by virtue of the fact that a) “not-being” does not denote the same as “nothing” and b), that when we read or hear or say “not-being” we are not reading or hearing or saying “nothing”.

Claim 2) is an abstract negation, claim 1) is a determinate negation. “Nothing” is neither being nor not-being, it is another, and the contradiction is the flow from the abstract negation to the determinate negation. In this sense, it is a true contradiction.

Hegel did continue Kant and Fichte's thought, but it was by developing it, not restating it. Priest takes Hegel to be validating Fichte in a claim such as “the self is both self and not-self”, but this is a completely static assertion. Curiously, Priest's examples here are reminiscent of Ilyenkov's Dialectical Logic book, which pursues the same history, but in it we see that the contradictions are the movement of thought, whereas Priest is only selecting dialetheias as static moments. Simply put, Priest is claiming Hegel is a dialetheist. Opposing this, we consider that Hegel did precisely go beyond dialetheism.

When looking at Hegel's dialectics, Priest says dialetheia plays a central role in it, and proceeds by showing it to be present in his ambivalent translations of Hegel's statements. In this case, we are using his translation of “Being is nothing” into “Being is Not-Being” and adapting it into paraconsistent logic's formalisation, calling it a Priest-statement:


9) Being is Nothing

9.1) Being is both Being and Not-Being

9.2) B = B & B≠B


Our formulation of the Priest-statement 9.2) follows to the letter paraconsistent logic's formalisation in the paper being discussed. We think it is then legitimate to ask why is it that, in both the sentence which translates Hegel's and the formalism, the determinate negation of 9) is altered into a sentence 9.1) that contains the abstract negation of its subject, moreover through a conjunction with the assertion of the subject's identity with itself. This is seen more clearly in the Priest-statement 9.2) where the conjunction is further evinced. The Priest-statement obtains the aim of paraconsistent logic, of course: its subject features two opposed meanings. But when the statement conjoins the identity of the subject with the abstract negation of it, the sentence ends how it started, and nothing determinately new can be gleaned from the totality composed by the conjunction. It is semantically preordainend into ambivalence, and nothing else. This is not how a contradiction performs.

Below is our attempt to formalise an alternative to a Priest-statement translation of Hegel's claim that “Being is Nothing.”

Bear in mind that the following is not intended as an alternative internal to paraconsistent logic, nor that 10.1 does not “follow” from 10.


10) Being is Nothing.

10.1) B = B &¬B


In this very blunt attempt of ours, we'd like to point out that at least here we can see immediately that the right side of the equation is “larger” than the left side. This is contradictory, and is not dialetheic. What this formalisation suggests is that there is reason to inquire about such unevenness, precisely when the conjuction preserves the left side's object, and by the introduction of the negation symbol – the latter being nowhere present in a Priest-statement. When Priest formalizes Not-Being, it is through the inequality of Being with itself (B≠B), and not through the negation of Being (¬B). These are very distinct procedures, and produce very distinct outcomes. If even abstract negation is nowhere present in a formalisation of dialectics, as paraconsistent logic or dialetheism purport to be, determinate negation seems barely attainable in it, if at all. Stating a previous point in a different manner, the conjunction in a Priest-statement is purely static, and nothing new becomes known from it, which is to say that not even “nothing” was learned.


For the purpose of this essay, it seems untimely to look into Priest's exegesis of Marxist and Sartrean dialectics under the lens of paraconsistency and dialetheism. Nonetheless, a prefatory note should be left here as a reminder for further development in the larger project in which this essay is being developed.

In a succint fashion: it is our contention that there is a fruitful distinction to be made between labour and activity. The former is opressive and totalizing, the latter is emancipatory and universalising. In a pedagogical context, this distinction should also hold, especially under the lens of the distinct metaconcepts of vulgar universalism and constructionist pluralism as opposed to radical universalism. We will not be able to justify here and now the foregoing, but it should be said that such a larger project hinges on the isomorphism between labor and the vulgar universalism/formalism pair (itself a purported isomorphism), and also on an isomorphism between activity and radical universalism.

With this in mind, and having suggested that paraconsistent logic and dialetheism are a mirror of formalism, it is our contention that when Priest takes unemployment to be a tragic condition – contradictory, even – he seems to be conforming to the notion that employment or labor is not a tragic, but an emancipatory condition. This precisely what a radical universalist pedagogy antagonizes: a formalist education is an education in labor and for labor, preventing the flourishing of rational agents through activity, by vice of the fact that labor abstractly negates activity; labor totalizes itself in the individual and the individual becomes identical with the totality. On the other hand, activity allows the individual to determinately negate and realize themself and others, in a truly universalizing procedure. Under the metaconcept of radical universality, labour is the tragic condition.


Although inserted in a framework which is definitely underdeveloped here, the above claim regarding the identity of the individual with the totality is relevant to our closing remarks on paraconsistent logic and dialetheism.

Priest interprets Hegel under Paraconsistency. We should remember for that for him, dialectical contradictions are dialetheias; that dialetheias are “A = B & A≠B”. But we have already seen that dialectical contradictions do not work on the equal sign; they work by negation working out the identity of opposites.

Paraconsistency is said to solve the puzzle of identity in difference. But identity in difference is not identity of opposites. Priest says that “being free is identical to being bound (not free)”. As with the earlier case of “nothing” being immediately identical with “not-being”, we see no reason for the synonym “bound” showing up beside the now parenthesised “not free”. From the identification of differences and opposites, Priest “gains” the conjunction, which leaves us where we started. And “bound” was just smuggled in along the way without having been developed from anywhere. Let us remember the issue of mirrored formalism. In a blunt analogy, we'd say that Hegel showed that the incompleteness of the Absolute could be made intelligible, prefiguring Gödel's incompleteness theorems – thus allowing contradiction to, in its turn, allow the development of consistency out of totalizing axiomatics; out of inconsistency.

Paraconsistent logic seems to conflate opposition with difference, and totalize difference by affirming it axiomatically in a sophistical manner. Priest takes dialectical contradiction to be a “general form” of identity in diference (the conjunction), and this is not the case: a contradiction is an inference, not a trivial conjunction of necessary assertions without any possibility of novelty.

Epistemologically, conflating identity with conjunction only gets us a bundle of disparate facts, untranslatable to each other, un-scaleable, semantically closed, providing no intelligibility of error. Going back to the film Dune: remember that Villeneuve claimed watching it on a cellphone was the same as taking a cruisehip in a bathtub. Priest is saying that watching in the theatre and in the cellphone is the same. Hegel is saying that watching it in a cellphone placed in a miniature speedboat, in a bathtub, is another singular thing.


We have seen evidence that paraconsistent logic shares traits with formalist logic. It is now upon us the task of showing that it also shares traits it intuitionist logic, although it is so in a rarefied fashion. This is our entry point into the next and last section, where we will see if paraconsistent logic is suficient for a radical universalist metaconceptual program.

Paraconsistent logic's commonalities with intuitionism are grounded on the fact the former, too, accepts semantic plurality and preserves solipsism. Regarding the first trait, we have seen that paraconsistent logic proposes that a statement can feature two truth-values simultaneously. In this manner, it seems to reinforce or even exponentiate the arbitrariness of co-existing semantics which was already featured in intuitionism. If a statement is both true and false, it will merely coincide by sheer chance with an interlocutor's interpretation of the concepts deployed in the statement – or never. It becomes a matter of preference to conform the truth-values deployed by oneself to the truth-values deployed by another, and never a question of developing universality through opposition. In such conditions, plurality is a caprice of selecting intelligibilities which may reinforce our echo-chamber.

It is here that the second trait of intuitionism is further amplified in dialetheism: in its omission of negation, paraconsistent logic will never provide the conditions for the world to impinge itself on one's mind or vice-versa, much less provide for such a mind to go philosophically further and reason itself as a different part of and in the world which can transformatively oppose itself to it. In intuitionism, the solipsist mind is at least working under the assumption that there is itself and its meaning-constructive performance, opposed to the meaningless world. In paraconsistent logic, meaning is equalized in the mind up to the saturation point where there barely is any world.


Part 4 – Paraconsistency and Radical Universality? Or not.


We hope that this long and somewhat castigating treatment of Priest's “Dialectic and Dialetheic” was sufficient to hopefully and nonetheless accurately depict paraconsistent logic's conditions of eligibility as a logic for a radical universalist metaconceptual persuasion.

The claim that Hegelian logic or that Hegelian dialectics is a conceptual logic may be eligible for discussion and profound analysis, but we'd like to assume it is so for the remainder of this essay. Radical universalism's metaconceptual strategies depend on Hegelian determinate negation.

This far gone, it will be helpful to remind ourselves of what is at stake in making a pedagogical program hinge on a 'metaconcept'. Recall that for Robert Brandom “there are concepts that play the distinctive expressive role of articulating features of the framework that makes description and explanation possible.” In the Hegelian framework, these are “speculative, logical, or philosophical concepts”. Brandom calls both “metaconcepts: concepts whose job it is to express key features of the use and content of the ground-level empirical and practical concepts Hegel calls “determinate” concepts.” Hegelian determinate concepts are developed through determinate negation. They gain their meaning through allocation in an inferential web which is itself articulated or logically designed for making intelligible the expression of the practical and semantic features of the thus-determined concepts.

Determinate negation is the organon for making these features intelligible, by virtue of its workings as a scales (a balance) of necessity and possibility, and by its capacity to realize the intelligibility of error. For a radical-universalist pedagogical framework, this is to be desired: it is through determinate negation that novel concepts are to be allocated in a learner's inferential web, structuring their meaning through the ever ongoing distillation of error and incompossibility or incompatibility with meanings previously held as true. In this process, these same meanings are seen to be insufficient for the learner's mind to navigate the world, and in opening up to the possibility of such insuffiency, the learner practically necessitates their radical-universalisation. In other words, both instructor and learner welcome contradiction between their inferential webs, for it is only so that a concept's meaning can be determinately negated, revised, and possibly explicated into another new and meaningful, singular concept. In yet another way to say it, we see that it is in this manner that the universal can make itself intelligible to the particular, and again, can become yet another particular in order to unfold itself back into a richer, more variegated and differentiated web of universality.

If paraconsistent logic cannot accommodate contradiction, and, as we've seen in detail, it only deals in abstract negation, it cannot feature determinate negation. In the manner in which it has been illustrated how it reinforces the aspects which destabilize formalism and intuitionism's status as logics for radical universality, it seems even antagonistic to the activity of learning. As such, it does not seem the ideal organon for the metaconcept of radical-universality. Moreover, it is a fundamental constitutive characteristic of this metaconcept that it needs a logic which can make its errors intelligible to itself.

A future essay will try to suggest linear logic as the organon for this task. It will aim to show that linear logic locally conserves useful aspects of both formalist and intuitionist frameworks, but produces a global outcome which is no mere monotonic amplification of their methods, by virtue of being able to also accommodate determinate negation.


Conclusion

Perhaps it is now the moment to insert a clarification: throughout this essay, by using the term 'instructor' we have meant “a person who partakes of a vaster inferential web of knowledge than a learner, and is thus able to dialectically develop concepts with the former”. An instructor is an agent of radical universality – it can be a school teacher, a parent, a guardian, a peer, a neighbour... we have avoided thus far to be exhortative in tone, but under this definition, it seems that an instructor could and should be whomever crosses our path, precisely by way of the fact that an instructor is also a learner. This goes beyond mere hierarchical equalization. It is not so. On the contrary, two radical-universalist interlocutors bootstrap one another into a reciprocal and interchangeable hierarchy structured upon the semantic noise which they unfold as meaning and knowledge in their theoretical-practice. After all, it is a radical-universalist core tenet that we can only be formed as rational self-determining agents who realize their knowledge of the world, if we determinately negate, that is, transform oneself and one another, into true, just, and beautiful universality.



1 Robert Brandom, in Spirit of Trust, 4-5

2 Idem.

3 Evald Ilyenkov, in Dialectical Logic, 349

4 G.W.F. Hegel, Lectures on the History of Philosophy Vol. II, 185-196; quoted in Ilyenkov op cit, idem.

5 Ilyenkov, 350

6 Idem.

7 Idem.

8 Ilyenkov, 355

9 Karl Marx, Theses on Feuerbach, 198; quoted in Ilyenkov op cit, 358

10 Ilyenkov, op cit 359

11 Idem.

12 Idem.

13 Brandom, op cit, 40

14 Brandom, op cit, 41

15 Jan Derry, Vygotsky, Philosophy and Education, 3

16 Derry, op cit, 25

17 Wertsch, 1998, 67; cited in Derry op cit 45

18 Derry, op cit 45

19 Robert Brandom, Articulating Reasons, 162

20 Derry, idem.

21 Derry, op cit 46

22 Derry, op cit 48

23 AA Cavia, Logiciel, 25

24 Cavia, op cit, 27

25 Cavia, idem

26 Ilyenkov, op cit, 350

27 Graham Priest, Dialectic and Dialetheic, Science and Society, Vol 53, 348-415

28 Hegel, Science of Logic, 440

29 A. P. Sheptulin, in Marxist-Leninist Philosophy, cited in Priest, op cit.

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