(Editorial Board, JU, Director of Humanities M. Phil., Memorial University, on 22 February 2023 - 13 March 2023, via e-mail)

Introduction

This brief e-mail conversation arose from noting a mention of “Janus” in Ludwig Wittgenstein’s Remarks on the Foundations of Mathematics (1967). In the passage cited below, Wittgenstein seems to be considering what is now commonly called the Russell-Zermelo paradox, which arises in naïve set theory from defining a set through all sets that are not members of themselves. Ludwig Wittgenstein speculates that in the absence of variables, a propositional function may be self-contradictory without inconsistency. That is, some propositions may be inconsistent without being incoherent. Such a proposition may tower over all logic like a Janus head, though presumably not looking over both past and future (as Janus is usually interpreted). Instead, Janus is a proposition that is both true and false, overlooking all claims to truth and falsity. The conversation with the ChatGPT 3.5 LLM highlights the heterodoxy of Wittgenstein’s proposal in much mainstream thinking. As shown, ChatGPT acknowledges “the acceptance of ambiguity and paradox in certain contexts” and that “while some degree of inconsistency or uncertainty may be present in the scientific process, the goal is always to minimize these factors.”

22/02: Jay Foster: Hi Peter, … Since it’s Reading Week [semester break], I’ve been reading Wittgenstein’s RPM (because I always need that as an antidote to Bertrand Russell’s PLA).1 This passage may be of interest:

Why should Russell’s contradiction not be conceived as something supra-propositional, something that towers above the propositions and looks in both directions like a Janus head? N.B. the proposition F(F)—in which F(@)=~@ (@)—contains no variables and so might hold as something supra-logical, as something unassailable, whose negation itself in turn only asserts it. Might one not even begin logic with this contradiction? And as it were descend from it to propositions. The proposition that contradicts itself would stand like a monument (with a Janus head) over the propositions of logic.

The pernicious thing is not, to produce a contradiction in the region in which neither the consistent nor the contradictory proposition has any kind of work to accomplish; no, what is pernicious is not to know how one reached the place where contradiction no longer does any harm. (Wittgenstein 1967, III.59-60)

P.P.S. Dipesh Chakrabarty has the most cited paper in academic humanities, “Climates of History” from Critical Inquiry. How about a Janus Unbound edition on Chakrabarty? “From Provincializing Europe to the Anthropocene?”

23/02: Peter Trnka: Hi Jay! Thank you for the brilliant e-mail. The Wittgenstein is wonderful and I will use it—Remarks on the Foundations of Mathematics? I enjoy the brute affirmation of “why not a superproposition?” So you’ve probably noticed we came out with two CFPs, the second of which is with special guest editor, Danine Farquharson. I’d be thrilled to assist with a special guest edited volume by you on something like “Provincializing Europe by way of the Anthropocene.”

8/03: JF: Hey Peter, I came back from analytic [a philosophy undergraduate course at MUN] and found a cryptic note written in blood and stabbed to the door [of my office] with a stiletto! Not quite—but I prefer that story. Yes, sorry, it is RFM not RPM as I cited it. It was late and I might have been cruising by the hour. But, the passage is at III.59. The passage you cite beginning “Each proof ...” is at II.59! So, I am merely a semi-unreliable narrator!

9/03: PT: Thank you—hmmm I’ll check again [regarding the text] at my office—yes blood soaked with the struggle of labour.

[Précis of intervening conversation and investigation: The English-only RFM has various corruptions and the Janus passage is entirely missing.2 Regarding RFM, I had the English-only edition, while Foster was using the German-English-facing-pages edition, which is not, as far as I have been able to make out, corrupt on this issue. There is also an intervening discussion between us concerning ChatGPT and other AI systems and the recent directive from MUN administration regarding risks and benefits. The conversation becomes three-way.]

13/03: JF (asks the AI, ChatGPT 3.5): “Is it possible to consistently maintain a contradiction?”

AI: From a logical perspective, it is not possible to consistently maintain a contradiction, as a contradiction is a statement that cannot be true under any circumstances. If someone were to try to maintain a contradiction, they would eventually run into logical inconsistencies and contradictions that would undermine their argument or belief system. However, there are philosophical and cultural perspectives that recognize the inherent complexity and ambiguity of the world, and accept that seemingly contradictory beliefs or ideas can co-exist in certain contexts. This is sometimes referred to as dialectical thinking, and can be found in a variety of traditions and fields, including Hegelian philosophy, Taoism, and modern psychotherapy. So while it may not be possible to consistently maintain a contradiction in a purely logical sense, there are certain frameworks and perspectives that allow for the acceptance of ambiguity and paradox in certain contexts.

JF: Is not logical inconsistency incompatible with science, or does science require inconsistency to be fruitful?

AI: Logical inconsistency is generally considered incompatible with the scientific method, which relies on logical reasoning and empirical evidence to generate knowledge and make predictions about the natural world. Inconsistencies in scientific theories or models can indicate flaws in the underlying assumptions or methods, and may require revision or replacement to better align with observations and data. That being said, scientific progress often involves testing and refining hypotheses through trial and error, which can involve some degree of inconsistency or uncertainty. Scientific models and theories are often approximations or simplifications of complex phenomena, and may require certain assumptions or idealizations that are not perfectly consistent with all aspects of reality. However, the goal of science is to continually refine and improve our understanding of the world, and to minimize logical inconsistencies or errors in our models and theories as much as possible. So while some degree of inconsistency or uncertainty may be present in the scientific process, the goal is always to minimize these factors and arrive at the most accurate and consistent representation of reality possible.

13/03: PT: Thank you for my first letter to the editor from an AI—or I will choose to take it as something like that. We’ve been comparing Wittgenstein on rule following to Jean-François Lyotard on discourse in the Cornelius Castoriadis class discussion of signification. Thank you for this, and the previous, and the next, however it follows.