No Form Action Theory Isolation logic (the unification of formal logic and dialectical logic)

Isolation logic (the unification of formal logic and dialectical logic)

Author: Hongbo Sun 2025/02/23

Using isolation dialectical logic to construct isolation logic is to unify dialectical logic and formal logic.

Isolation logic has four values: true, false, T | F, and indeterminate. I believe operations can be performed as follows: true and false are unified in opposition as "indeterminate," and by negating "indeterminate," we can obtain the deterministic true or false (T | F). Expressed symbolically:

Where T=true, F=false, T | F=true or false, Ind=Indeterminate.

(1) (T,F) represents the opposition of true and false

(2) unify(T,F) = Ind

Since true and false are in opposition, through negating this opposition, true and false are unified into indeterminacy, indicating that opposing sides exist in an indeterminate state under certain conditions. This operation is logically similar to a decision-making process or quantum superposition state.

(3) not(Ind) = T | F

By negating "indeterminacy," we obtain deterministic true or false. This operation is equivalent to "selecting" a definite value from an indeterminate state, returning from an indeterminate state to classical binary logic. This process is similar to transforming from a fuzzy state to a clear state in logic, or similar to quantum collapse caused by measurement. Here, "not" negates "indeterminacy."

(4) not unify(Ind) = (T,F)

not unify is oppose; this operation transforms the unity of indeterminacy into the opposition of true and false. Here, "not" negates "unify."

Below, we use Lattice Theory to construct a three-valued lattice. In this lattice operation, it is stipulated that Ind will automatically execute the operation not(Ind) = T | F.

(1) Let our set of logical values be L={T, F, T | F}, including three values:

T represents true;

F represents false;

T | F and F | T are not distinguished in lattice operations.

Define a partial order relation ≤, such that: F≤T | F≤T.

(2) Lattice operations:

Meet (intersection): Defined as the AND operation in logic, but extended here to three-valued operations:

T∧T=T

T∧F=F

T∧Ind=T∧not(Ind)=T∧(T|F)=T|F: Ind as indeterminate will either become T or F; these two cases intersected with T give T and F respectively, therefore, T∧Ind=T|F.

F∧F=F

F∧Ind=F∧(T|F)=F: Ind as indeterminate will either become T or F; these two cases intersected with F give F and F, which is determinate, therefore, F∧Ind=F.

Ind∧Ind=(T|F)∧(T|F)=T|F

This operation is an extension of the "AND" operation in formal logic, but considering the complexity of isolation logic, we have defined the relationship between indeterminacy and other values to ensure consistency in operations.

Join (union): Defined as the OR operation in logic:

T∨T=T

T∨F=T

T∨Ind=T∨(T|F)=T: Ind as indeterminate will either become T or F; these two cases united with T give T and T, which is determinate, therefore, T∨Ind=T.

F∨F=F

F∨Ind=F∨(T|F)=T|F: Ind as indeterminate will either become T or F; these two cases united with F give T and F, therefore, F∨Ind=T|F.

Ind∨Ind=(T|F)∨(T|F)=T|F

This operation is an extension of the "OR" operation in formal logic, handling the interaction between true, false, and indeterminate values.

(3) Negation operations:

Negation in formal logic:

not(T) = F

not(F) = T

not(T | F) = T | F

Negation in dialectical logic:

not(Ind) = T | F

not unify(Ind) = (T,F)

(4) Unify operation:

unify(T, F) = Ind

unify(F, T) = Ind

unify(X, X) = X for X in {T, F, Ind}: unify(X, X) should be equivalent to unify(X), which can be understood as "the unity of a single entity." This indicates that an entity's unity is self-consistent when there is no external opposition. In other words, X itself is unified, and unifying X with itself results in X. This is actually operating on identity, because X itself possesses identity. Therefore, this unify(X) is equivalent to an identity operation, expressing the meaning "X is X," which is distinct from a unification operation with two opposing parameters. For the operation unify(X), the result is X, emphasizing X's internal unity and identity. This operation doesn't involve unifying opposites, but rather confirms X's own state. Thus, unify(X, X) can be abbreviated as unify(X).

unify(T, Ind) = F: Intuitively, the result of unifying T and Ind should be an indeterminate state. However, according to isolation dialectical logic, this unify operation requires T and Ind to be completely opposed, meaning that negating T should yield Ind, and vice versa. But this is impossible. That is, T and Ind do not produce opposition, so the value of unify(T, Ind) is false (F).

unify(F, Ind) = F: The same reasoning applies as for unify(T, Ind) = F.

unify(T, T | F) = F

unify(F, T | F) = F

Ind is a unified state, expressing indeterminacy with a definite value. Whereas T | F is obtaining a definite value, breaking the unified state, but manifesting as uncertainty about which value is obtained—using indeterminacy to acquire a definite value. Therefore, T | F's acquisition of certainty doesn't eliminate the underlying indeterminacy and opposition. Thus, unify(T | F) = Ind can be used to return this indeterminacy and opposition back to indeterminate opposition.

In pure lattice operations, Ind automatically reduces to T | F (that is, it automatically performs a not(T | F) = T | F), because all lattice operations are formal logic operations (although this lattice has three values), and only when performing dialectical logic operations is it necessary to convert to Ind (unify(T | F) = Ind). This separates the operations between the two logics, avoiding confusion. In formal logic operations, the value Ind cannot be directly operated upon because it is a value in dialectical logic; it can only be operated upon after being transformed into T | F in formal logic. This is similar to how in the macroscopic world, we cannot directly operate on quantum particles in the quantum world.

In other words, in pure lattice operations, it's still a three-valued operation, still a three-valued lattice. This way, operations between formal logic and dialectical logic neither interfere with each other nor lose their interconnection.

(4) Properties of Algebraic Structure

After defining these basic operations, we need to ensure they conform to certain algebraic properties, making the entire system consistent. To maintain mathematical rigor, we need to verify that our operations satisfy the following properties:

Idempotency: a∧a=a, a∨a=a

Associativity: (a∧b)∧c=a∧(b∧c), (a∨b)∨c=a∨(b∨c)

Commutativity: a∧b=b∧a, a∨b=b∨a

Absorption: a∧(a∨b)=a, a∨(a∧b)=a

Distributivity: a∧(b∨c)=(a∧b)∨(a∧c), a∨(b∧c)=(a∨b)∧(a∨c)

Duality: not(a∧b)=not(a)∨not(b), not(a∨b)=not(a)∧not(b)

Through verification, the three-valued operations I've defined satisfy the above properties. Therefore, it forms a lattice.

(5) De Morgan Lattice

For all possible combinations of values p,q∈{T,F,Ind}, it satisfies De Morgan's laws:

¬(p∧q)=¬p∨¬q

¬(p∨q)=¬p∧¬q

Therefore, we can conclude: the three-valued lattice I've defined is a De Morgan lattice, meaning that this three-valued lattice maintains logical consistency and symmetry when handling negation, meet, and join operations.

A De Morgan lattice is a generalization of Boolean algebra. Since the three-valued lattice I've defined satisfies duality, it preserves the key structural properties of classical Boolean logic while extending it to handle the additional value "indeterminate." This is an ideal feature, as it means this three-valued lattice is not a completely arbitrary system, but is built upon and generalizes classical logical principles.

Thus, we have defined an isolation logic system that includes a three-valued lattice and dialectical logic. This isolation logic actually combines and unifies dialectical logic and formal logic, forming a unified logical system. This unification is "perfect" because it not only preserves the advantages of both but also compensates for their respective limitations.

This also allows us to see that in formal logic, the intermediate value Ind between true and false does not exist; only in dialectical logic does an intermediate value Ind appear between true and false. Although this value can be called an intermediate value, it is a value that unifies true and false, so this intermediate value is not simply a linear value that appears between true and false. Ind can be viewed as a state that can collapse into either T or F, and this collapse makes us consider Ind as a value between T and F in formal logic.

This approach can also be seen as extending traditional formal logic to isolation logic, providing new tools for handling uncertainty and fuzziness. It is a natural extension of binary logic (true, false), an extension that doesn't violate the law of the excluded middle (that every proposition can only be true or false), but rather expands the range of truth values in logic by introducing indeterminacy (Indeterminate), enabling the logical system to express more complex states. It not only extends classical binary logic but also maintains the core principles of formal logic. This approach provides a logical foundation for understanding uncertainty in the real world. For example, in scientific research, measurement uncertainty, probabilistic events, or unknown variables can be more naturally expressed and processed within this logical framework.

This three-valued lattice satisfies the basic axioms of a lattice (associativity, commutativity, absorption, distributivity, duality, and De Morgan's laws), indicating that it is a mathematically consistent and complete structure. This rigor provides a solid foundation for this isolation logic, making it not merely philosophical speculation but a system that can be formally verified.

This isolation logic is not just a logical system; it provides a mathematical expression for dialectical logic. Through the unification (unify) and negation (not) operations in isolation logic, it can mathematically simulate how truth and falsehood transform through opposition and unification, thus supporting the thinking patterns in dialectical logic. This enables dialectical logic to be realized within a formalized logical framework, rather than remaining solely a philosophical concept.

This isolation logic will change the way people think about problems. Here are a few examples:

(1) Isolation Logic Analysis of the Trolley Problem

The Trolley Problem is a classic ethical thought experiment proposed by Philippa Foot in 1967. Description:

A madman has tied five innocent people to a trolley track. A runaway trolley is heading toward them and will crush them momentarily. Fortunately, you can pull a lever to direct the trolley to another track. However, the problem is that the madman has also tied one person to the other track, which would sacrifice this person on the track.

The question you face:

In this situation, should you pull the lever?

Traditional Binary Logic Analysis:

Within the framework of traditional binary logic, we must make an either-or choice between pulling the lever and not pulling it.

If choosing to pull the lever (T - True):

Sacrificing 1 person to save the lives of 5 people aligns with utilitarian principles: maximizing happiness or minimizing suffering.

Binary logic conclusion: The statement "pull the lever" is true (T).

If choosing not to pull the lever (F - False):

Argument: Pulling the lever is active intervention, directly causing the death of 1 person, while standing by is merely passively accepting fate's arrangement, with less moral responsibility.

Binary logic conclusion: The statement "pull the lever" is false (F), not pulling the lever is true (T).

The Dilemma of Traditional Binary Logic:

Traditional binary logic forces us into a black-and-white choice between "pull" or "don't pull," leading to a binary opposition that cannot express the uncertainty existing in ethical dilemmas.

The Solution of Isolation Logic:

Using isolation logic, we can transcend the limitations of traditional binary logic, appropriately express the trolley problem, and analyze it more profoundly.

Analysis Using Isolation Logic:

unify(T, F) = Ind: Acknowledging the ethical uncertainty of the trolley problem. "Pulling the lever is true" (from a utilitarian perspective) and "pulling the lever is false" (from a moral responsibility perspective) are both "opposing views" that have certain "rationality" in ethics, but also have limitations. By "unifying" (unify) these two opposing views, we arrive at an "ethically indeterminate (Ind)" conclusion. This appropriately expresses our genuine feelings when facing ethical dilemmas: being caught in a dilemma, finding it difficult to make a choice.

Although the trolley problem is "indeterminate" in ethics, this doesn't mean we can avoid making a decision. Negating "indeterminacy" (not(Ind)) means we must make a choice between true (T) and false (F) (T | F). This choice can be made based on different ethical principles, value judgments, or specific situations, for example:

Based on utilitarian principles: Choose to pull the lever T: To maximize the happiness of the majority, sacrifice 1 person to save 5 people.

Based on moral responsibility principles: Choose not to pull the lever F: To uphold the moral responsibility of not actively harming others, not pulling the lever is the more moral choice.

Based on other principles or situations, different "choices" might also be made.

Advantages of Isolation Logic:

Accommodating the "uncertainty" of ethical dilemmas: Isolation logic can directly express and handle the inherent ethical uncertainty in the trolley problem, acknowledging that ethical judgments are not always black and white, but contain gray areas and ambiguity.

Demonstrating the necessity of "choice": Even when facing ethical uncertainty, isolation logic still emphasizes the "necessity of choice." Through the operation not(Ind), it highlights our "responsibility and commitment" to make decisions in ethical dilemmas.

Compared to the simplification and absolutism of traditional binary logic, isolation logic's analysis is more appropriate and humane, closer to our real ethical experiences and emotions. It acknowledges the uncertainty of ethical judgments while respecting the "rationality" of different ethical principles and value judgments.

(2) Judicial Decisions in Legal Cases

A judge is presiding over a complex case where defendant Mr. S is charged with insider trading. However, the evidence is not clear-cut, containing factors for both conviction and acquittal.

Evidence for conviction (possibility of True - T):

- Mr. S traded before a major company announcement that significantly impacted stock prices.

- There were unusual communication patterns between Mr. S and company insiders.

Evidence for acquittal (possibility of False - F):

- Mr. S claims his trading was based on independent market analysis, not insider information.

Formal Logic (Applying Legal Rules and Evidence):

Propositions (advantages of formal logic): The judge first separates key propositions and evaluates them according to formal legal rules and established legal precedents. Definitions:

P1: "Mr. S traded before a major company announcement." (Evaluated as true - T based on factual records)

P2: "There were unusual communications between Mr. S and company insiders." Evaluated as true (T) based on communication records

P3: "Mr. S's trading was entirely based on insider information." Evaluated as indeterminate (Ind), which is the key point of uncertainty. Although there is evidence suggesting insider trading (P1 and P2), there is no conclusive evidence to determine P3 as true beyond reasonable doubt, which is the legal standard.

P4: "Mr. S's trading was completely innocent, based on independent analysis." Evaluated as indeterminate (Ind), despite Mr. S offering this explanation, the judge cannot clearly rule out the possibility of insider trading based on available evidence.

Using Meet (∧) to Evaluate Combined Evidence:

"P1 ∧ P2 ∧ P3" (trading before announcement AND unusual communication AND trading entirely based on insider information). Using Meet for evaluation: T ∧ T ∧ Ind = T | F.

Interpretation of Meet (∧) in Isolation Logic: Although P1 and P2 are true, the entire prosecution case, represented by their conjunction with P3 (indeterminate), becomes indeterminate. This reflects that although there is some evidence suggesting guilt, P3's uncertainty weakens the entire case, placing it in a state of uncertainty. Formal logic here highlights the weakest link in the chain of evidence.

Dialectical Logic:

Identifying Opposing Arguments (the realm of dialectical logic):

Thesis (prosecution argument - guilty): Based on existing evidence, Mr. S may be guilty of insider trading.

Antithesis (defense argument - innocent): Based on the lack of conclusive evidence and alternative explanations, Mr. S may not be guilty of insider trading.

Applying Unify(T, F) = Ind to acknowledge inherent uncertainty: The judge recognizes that based on the available evidence, neither side presents a clear conclusion. They are opposing views of an inherently uncertain situation. Therefore, the judge applies the unify operation to acknowledge this dialectical tension and inherent uncertainty:

unify(thesis: guilty (T), antithesis: innocent (F)) = Ind

Interpretation of the unification operation: The judge concludes that based on the current state of evidence and opposing arguments, the final ruling is "indeterminate." This is not an evasive or failed decision, but a logically reasonable and philosophically nuanced acknowledgment of epistemological limitations and the inherent ambiguity of the situation. In this specific legal context, the truth remains undetermined based on available information.

Applying Not(Ind) = T | F to Reach a Verdict (Making a Choice Under Uncertainty): However, the judge cannot simply remain in a state of "indeterminacy." The legal system requires a clear ruling. Therefore, the judge must now apply the not(Ind) operation to make a "choice" between true (guilty) and false (innocent) based on available information and prevailing legal principles:

not(Ind: ruling is indeterminate) = guilty (T) | innocent (F)

Interpretation of the not(Ind) operation: After negating "indeterminacy," the judge must make a judgment. This choice is not arbitrary but guided by legal principles and the weight of evidence (even if inconclusive):

Possible Choice 1: Innocent (F): If the judge prioritizes the principle of "presumption of innocence" and the higher standard of proof in criminal cases, they might choose innocent (F). This "choice" tends to protect individual rights in the face of uncertainty.

Possible Choice 2: Guilty (T): In different legal systems, or with slightly different weighting of evidence, the judge might choose guilty (T), perhaps emphasizing the seriousness of insider trading and the necessity of deterring such behavior, even without absolute certainty.

This example demonstrates how isolation logic, by combining formal and dialectical elements, provides a powerful and nuanced tool for analyzing complex situations involving uncertainty, opposition, and the need for choice, far beyond the limitations of traditional binary logic.

(3) Paradoxes

The Liar Paradox: This sentence is false (P). If this sentence is true, then it is false; if it is false, then it is true. Traditional binary logic cannot handle this type of self-reference because it leads to logical conflicts.

This type of sentence can be viewed as Ind. Because it simultaneously expresses two opposing meanings, we can say that the truth or falsity of this sentence cannot be determined, thus avoiding logical self-contradiction. By viewing it as indeterminate, we can see that this is an ambiguous, undecidable proposition, rather than a simple opposition.

The Barber Paradox: In a town, the barber announces that he shaves only those who do not shave themselves. Should this barber shave himself? If he shaves himself, then it is contradictory; if he doesn't shave himself, then he can shave himself. These are two opposing outcomes, so this paradox can be seen as indeterminate.

Russell's Paradox: Proposed by Bertrand Russell. Consider a set R defined as "the set of all sets that do not contain themselves as members." That is: R={x|x∉x} (set x does not contain itself).

Proposition: Set R contains itself (R∈R).

Analysis:

If R∈R=T, then R∉R (by definition), contradiction.

If R∈R=F, then R∈R (satisfies the definition), contradiction.

Similarly, R∈R can be set to Ind, indicating that "whether a set can contain itself" is undeterminable.

This not only reflects logical contradiction but also implies the mathematical uncertainty or undefinability of the concept "a set containing itself." From a formal structural perspective, the Liar Paradox and Russell's Paradox are the same, but the barber can shave himself, while whether "a set can contain itself" is indeterminate. This is the essential difference between these two paradoxes. In fact, in the section "Formal Logic," it has already been argued that "a non-empty set cannot contain itself." Isolation logic correctly expresses this indeterminacy.

Unlike classical logic, isolation logic does not collapse or lead to contradictions when facing paradoxes. It provides a logically consistent method to include and analyze paradoxes within the system itself.