Paraconsistency

Negation and Contradiction in the Works of Graham Priest and AI (Jam Sessions on Negation, Part 3)

Prelude

Over the last two weeks, we’ve been using the Jam Sessions on negation to make one thing concrete: a “negative view” in Khora that doesn’t just block or filter, but actively surfaces opposition—and does so in ways that help users think otherwise. In Part 1 (Sebastian) treated negation as a generative gap that can disclose meaning. In Part 2 (Edward) argued for an elenchic design that resists premature closure and keeps live questions alive inside the graph.

Today, in our final installment, Stephen Rego turns to paraconsistency, to ask what it would mean for Khora to tolerate contradiction without collapsing. -Eli Kramer

By Stephen Rego

Negation has many forms, modes, and applications throughout the table in the history of philosophy; it is not merely a logical tool - it’s a gateway to:

• Freedom (Sartre)

• Transcendence (Pseudo-Dionysius)

• Transformation (Hegel)

• Emptiness (Nāgārjuna)

• Mystery (Heidegger)

In short, negation allows philosophy to probe the limits of thought, being, and language.

However, today, I want to very briefly talk about the logic of negation, focusing on one type of non-classical (or non-standard) logic called paraconsistent logic (PL), and how it might be leveraged within Khora’s functionality.

PL allows a theory to contain contradictions without entailing every proposition and collapsing into triviality - that is, the situation where a formal system becomes so permissive that every proposition is provable, meaning that the system can no longer distinguish between true and false statements and reasoning collapses - this “collapse” is also known as the Principle of Explosion (PE).

PE states that that once a contradiction is true in a deductive system, any proposition can be derived from that contradiction - indeed, it is sometimes known by its Latin formulation ex contradictione quodlibet (“from a contradiction, anything follows”), or ex falso quodlibet (“from falsehood, anything follows”) - thereby “exploding” the system’s integrity.

Negation in paraconsistent systems can behave so that a sentence and its negation may both be held without entailing arbitrary conclusions, and so PL also allows reasoning in the presence of inconsistencies while maintaining meaningful inferential capabilities.

There are examples of PL through the history of philosophy, both West, being traced back as far as the proto-dialectical approach of Heraclitus, and Non-Western, such as is found in Nāgārjuna, and the Japanese Zen Buddhist teacher Dōgen; in fact, Non-Western philosophies more-readily place contradiction at heart of their thinking.

Graham Priest (https://www.gc.cuny.edu/people/graham-priest) is an English philosopher, logician, and expert on paraconsistency and other non-classical logics, including an interest in the Mahāyāna Buddhist logic of Nāgārjuna and has recently written an article that is recommended as very useful background reading:

~ G. Priest, Beyond True and False (2014

Priest’s work during the 1980s is regarded as a key development in PL within the Western philosophical/logical tradition. He supports a position called dialetheism, which is the philosophical view that some contradictions are true - that is, there exist statements that are both true and false simultaneously. Thus, dialetheists accept that certain paradoxical statements - such as the Liar Paradox (“This sentence is false”) - are both true and false. This challenges one of the foundational principles of classical logic: the Law of Non-Contradiction (as well as the concomitant Law of the Excluded Middle - both of which were first formulated by Aristotle (Three Fundamental Laws of Thought: https://www.britannica.com/topic/laws-of-thought), which states that a proposition cannot be both true and false at the same time.

Paraconsistent logic has become increasingly valuable in AI systems, especially as they grapple with inconsistent, ambiguous, or contradictory information. Real-world data is messy: conflicting inputs, and uncertain sources, and overlapping expert opinions are common. Paraconsistent logic offers a way to reason through contradictions without trivializing the system. In AI it serves as a foundational tool for reasoning systems, particularly in environments characterized by incomplete, uncertain, or inconsistent data. Its primary roles include:

• Knowledge representation and inconsistency-tolerant reasoning: allowing the integration of conflicting expert reports or datasets while preserving useful inferences; PL enables automated reasoning without discarding potentially valuable inconsistent data
  

• Multi-agent systems: PL allows agents in a distributed environment to maintain local inconsistent beliefs while permitting effective global coordination and decision-making

The following is an idealized outline showing how PL can work in an AI environment:

(Note 1: the symbol “¬“ represents the logical operation of negation and means “not”.)

• Input A = X

• Input B = Y = ¬X → contradicts Input A

• Input C = Z → supports Input A again.

In classical logic, the contradiction between Input A and Input B could trigger explosion (PE).

• From X and ¬X the system could infer anything, rendering the system trivial and effectively useless.

Using a paraconsistent logic engine (e.g., Graham Priest’s Logic of Paradox), the system:

• Accepts both X and ¬X as coexisting truths

• Flags the contradiction but does not infer arbitrary (i.e., non-defined) cases

• Continues reasoning with other inputs.

(Note 2: In Priest’s Logic of Paradox, the Law of Non-Contradiction is rejected to accommodate paradoxes (e.g., the Liar paradox), while the Law of the Excluded Middle may still be retained in a modified form.)

In the first instance, this approach to contradiction made me think of the links (or, to use the technical term, ‘edges’) between nodes on Khōra’s knowledge graphs, that are weighted with a positive or negative value used to indicate the affinity (or “gravity”) between nodes and, as a subjectively-determined value that varies from user to user, paraconsistency is one way of understanding the logic of contradiction inherent in this type of configuration. On the other hand and thinking ahead, I feel that it is something that could perhaps be leveraged even further to add to the end-user experience (such as, for example, the Elenchus feature), and this is something I hope we can explore some more in future Jam Sessions.