Paraconsistent Logic | Attic Philosophy 

Perfect. I've gone from barely understanding paraconsistent logic to "Oh. Cool. That's simple." and I know darn well that while I might have been the student that was ready for that shift, you were, for me, a spectacularly perfectly paced and pitched teacher. Thank you!

I love your video series on logic. Should have waaaaay more views!

As a masters student who still has no fking clue what to write about for his masters, but massively interested in logics and my mentor suggested Paraconsistent logic and Tarski scheme, i was so lost swarming trough many literatures and opinions and knowing i have no experience with anything besides traditional logic and deduction and doing actual tasks and calculations with deduction, tree, etc, i'm feeling so lost rn and any help would be appreciated guys .__. am i being stupid for wanting to try masters in logics

1.3. Paraconsistency and contradiction Another key feature of the both/and logic is its embrace of paraconsistency, allowing for the toleration and even acceptance of contradictions without trivialism or logical explosion. This stands in contrast to the principle of non-contradiction in classical logic, which holds that no proposition can be both true and false at the same time, and that any contradiction or inconsistency in a logical system leads to its complete collapse or absurdity. The concept of paraconsistency has a relatively recent and controversial history in the tradition of non-classical logics, dating back to the works of Vasiliev, Jaśkowski, and others in the mid-20th century. The basic idea is to develop logical systems that can handle contradictions or inconsistencies in a controlled and localized way, without allowing them to infect or trivialize the entire system. In other words, paraconsistent logics aim to preserve the possibility of meaningful and informative reasoning even in the presence of contradictions, by restricting or modifying the classical inference rules that lead from a contradiction to arbitrary conclusions. One of the most well-known and influential systems of paraconsistent logic is relevance logic, developed by Anderson and Belnap in the 1960s. In relevance logic, the classical inference rule of ex falso quodlibet (from a contradiction, anything follows) is rejected, and replaced by a weaker and more restrictive rule that allows for the derivation of a conclusion from a contradiction only if the conclusion is relevant or related to the premises in a substantive way. This ensures that the presence of a contradiction in one part of the system does not automatically lead to the derivation of arbitrary or unrelated conclusions in other parts of the system. Another important system of paraconsistent logic is dialetheism, developed by Priest and others in the 1970s and 1980s. In dialetheism, the possibility of true contradictions or dialetheia is not only tolerated but actively embraced as a fundamental feature of reality and thought. Dialetheists argue that there are genuine and unavoidable contradictions in various domains, such as self-reference, change, or the foundations of mathematics, and that a consistent and complete logical system is impossible or undesirable. Instead, they propose to develop a logic that can handle contradictions in a meaningful and productive way, by allowing for their local and controlled occurrence within a broader paraconsistent framework. The both/and logic incorporates and extends the principles of paraconsistency from relevance logic, dialetheism, and other non-classical systems, while also grounding them in the metaphysical and ontological framework of the monadological system. In the both/and logic, contradictions are not just formal or linguistic phenomena, but are rooted in the fundamental nature of reality as a complex and dynamic network of interrelated monads. Specifically, the both/and logic allows for the coexistence and mutual implication of seemingly opposing or contradictory properties or relations within the same monadological configuration, without leading to trivialism or absurdity. This is because the monads themselves are not simple or indivisible entities, but are infinitely complex and multifaceted, containing within themselves a diversity of perspectives and potentialities. Each monad reflects the entire universe from its own unique perspective, and thus contains within itself a multiplicity of seemingly incompatible or contradictory aspects, which are nevertheless unified and harmonized within the overall monadological system. Moreover, the contradictions in the both/and logic are not arbitrary or chaotic, but are themselves structured and organized according to the principles of coherence, compatibility, and mutual implication that govern the relations between monads. A contradiction between two propositions A and ¬A is not a simple logical opposition, but a complex and dynamic tension between two aspects or perspectives of the same underlying reality, which can be resolved or transcended through a higher-order synthesis or integration. The formal semantics of the both/and logic can be defined in terms of a paraconsistent truth-value space, such as the four-valued logic of Belnap or the many-valued logics of Priest, which allow for the simultaneous truth and falsity of propositions, as well as their independence or indeterminacy. The inference rules of the both/and logic are designed to preserve or respect the paraconsistent nature of the propositions, by restricting or modifying the classical rules that lead from a contradiction to arbitrary conclusions. For example, the both/and logic rejects the classical principle of explosion, which states that from a contradiction, anything follows (A ∧ ¬A ⊢ B for any B). Instead, it adopts a weaker and more restrictive principle, such as the principle of controlled explosion, which states that from a contradiction, only relevant or related conclusions follow (A ∧ ¬A ⊢ B only if A and B are related in a substantive way). This ensures that the presence of a contradiction in one part of the system does not automatically infect or trivialize the entire system, but can be contained and managed within a local and coherent context. In addition to these basic paraconsistent inference rules, the both/and logic also introduces some novel and distinctive principles and relations that are specific to the monadological framework, such as the principle of holistic contradiction, which states that every monad contains within itself a diversity of seemingly contradictory aspects, which are nevertheless unified and harmonized within the overall system (∀M ∃A, B: M ⊢ A ∧ ¬A and M ⊢ B ∧ ¬B and M ⊢ (A ∧ ¬A) ∧ (B ∧ ¬B)). This allows for a more nuanced and context-sensitive representation of the logical structure of reality, and for the expression of complex and paradoxical patterns of thought and experience. The paraconsistent nature of the both/and logic has important implications for the philosophical and methodological foundations of the monadological framework. It challenges the traditional assumptions and dogmas of Western logic and metaphysics, such as the law of non-contradiction, the principle of sufficient reason, or the dichotomy between appearance and reality. By allowing for the coexistence and mutual implication of opposites, the both/and logic provides a more adequate and flexible framework for capturing the complex and dynamic nature of reality, and for navigating the paradoxical and often ineffable nature of human understanding. Moreover, the paraconsistent approach of the both/and logic has important applications and consequences for various fields and domains, from the natural sciences to the social sciences and humanities. In quantum mechanics, for example, the both/and logic can help to make sense of the seemingly contradictory behavior of subatomic particles, such as the simultaneous wave and particle nature of light, or the entanglement and non-locality of quantum systems. In psychology and cognitive science, the both/and logic can provide a more nuanced and realistic framework for representing the complex and often conflicting nature of human cognition and emotion, such as the coexistence of conscious and unconscious processes, or the interplay of reason and intuition. And in philosophy and theology, the both/and logic can help to articulate and explore the paradoxical and ineffable nature of ultimate reality, such as the unity and diversity of the divine, or the transcendence and immanence of the absolute. In summary, the concept of paraconsistency and contradiction is another central and distinctive feature of the both/and logic, which sets it apart from classical logic and opens up new possibilities for representing and reasoning about the complex and paradoxical nature of reality. By grounding paraconsistency in the metaphysical and ontological principles of the monadological system, and by introducing novel and specific logical principles and relations, the both/and logic provides a powerful and flexible framework for capturing the dynamic and often contradictory nature of thought and being, and for navigating the ineffable and mysterious depths of human understanding. In the following subsections, we will explore in more detail the formal semantics and proof theory of the both/and logic, and demonstrate its coherence, expressiveness, and philosophical significance.

Extremely underrated

Love your videos keep them coming. There aren't many online resources, that make formal logic accessible. I am wondering, if I could use LP, because I have the intuition that explosion isn't a valid principle of inference, without taking the metaphysical position of dialetheism. How would I interpret the third truth value then ? And could I still define logical inference as some sort of truth preservation ?

i just came here to say p = (p = !p). also i don't think the liars paradox is inherently logically inconsistent. the assumption of p's starting with a truth forces the falsity of the end result, and vice versa, which is a logically consistent system.

Yep the principle of explosion is one of the most non-related and confusing name the basic principles of logic.The only reason i'm understand is when i search paraconsitent logic in wikipedia where it state"Since A and not A is inconsistent is it TRIVIAL to show that A and not A implies B" i'm prefer to called it the trivial principle 😂

What about the opposite , when its false preservation , and truth avoidance 1:50

Hi. Are these questions you could help me with? Explain in your own (and simple) words what it means for a (paraconsistent) logic to tolerate, but not to exploit contradictions. Every member of the group should present his or her own version. Also, Within the context of paraconsistent logics, provide examples (at least one for each case) of an impure tautology, an impure contradiction, and an impure contingency

"Lukasiewicz" is to be pronounced "Wookasheeaywiz".