Good day! I would like to present an idea (hopefully, a fresh one) in the field of logic called "Context-Gradual Referential Logic." The theory is described in the author's paper by Maxim Petrov-Verin. I have attached the paper.

Context-Graded Referential Logic — Max. Petrov-Verov. — Self-published, 2025, 4 p.pdf

The author does not claim scientific novelty, academic recognition, or educational value — these are his personal reflections (see disclaimer).

Possible value of the work:

It presents a very simple and intuitive system (accessible even at the high school level) based on well-known principles from modal logic, fuzzy logic, and referential approaches. Combined, these elements form a compiled system that allows for:

1. Paradox resolution: Self-referential statements are marked as false, preserving the system's consistency.

2. Gradual truth value: The parameter k explicitly represents the degree of distortion — from complete truth (k = 0) to absolute falsehood (k = 1).

3. Model flexibility: Does not require prohibitions or complex hierarchies, relying instead on intuitive notions of reference and context.

4. Lie tracking: The marker (⊥₀) explicitly identifies false statements, allowing for analysis of their sources and frequency.

This may be useful to someone... or spark a desire to discuss. This topic has been created solely for informational purposes.

Edited May 19, 2025May 19 by alionaqwe3f8

Not sure why this is in Applied Mathematics, nor why you need to post a link, which is against the rules of this discussion site.

However the subject appears interesting and does highlight a good point that there is much more to 'logic' than simple first order binary logic.

In my experience I don't know of any paradoxes that can be resolved by the simple observation that the paradox is contained in a compound or complex statement that tries to combine two (or more) incompatible simple statements.

Thank you for your interest in the topic. I couldn't find a more suitable section on the forum (though I’m new here). Regarding the link — it’s a link to an attached file, and I wasn't aware that sharing links is against the rules.

13 minutes ago, studiot said:

In my experience I don't know of any paradoxes that can be resolved by the simple observation that the paradox is contained in a compound or complex statement that tries to combine two (or more) incompatible simple statements.

Thank you for your comment! Just to clarify, my model does not propose that paradoxes arise from the combination of incompatible simple statements into a more complex one. Instead, it asserts that self-referential statements cannot be valid in both W' and W_0 imultaneously, and as such, they are marked as false. But perhaps I misunderstood your point.

Just now, alionaqwe3f8 said:

Thank you for your interest in the topic. I couldn't find a more suitable section on the forum (though I’m new here). Regarding the link — it’s a link to an attached file, and I wasn't aware that sharing links is against the rules.

Thank you for your comment! Just to clarify, my model does not propose that paradoxes arise from the combination of incompatible simple statements into a more complex one. Instead, it asserts that self-referential statements cannot be valid in both W' and W_0 imultaneously, and as such, they are marked as false. But perhaps I misunderstood your point.

I haven't read your paper yet, but I feel that if you are simple defining out of existence awkward statements that is bit of a cop out.

Surely if we can't address the difficult stuff then there is no point in logic or philosophy ?

Where would we be if we still insisted that the square roots of negative numbers should be defined out of existence ?

Thank you for your comment!

I believe there may have been a slight misunderstanding of the core idea — perhaps because the paper hasn't been reviewed yet. Of course, I’m not “defining away” complex or paradoxical statements.

The key point is that I consider statements within two distinct contexts: the actual world (W₀) and a possible alternative world (W′). When a statement becomes meaningless or undefined in a possible world — for instance, due to self-reference — it is explicitly marked using a special symbol ⊥₀. This is not deletion or avoidance, but rather a way to formally acknowledge falsehood or paradox as a specific, traceable case.

Additionally, a parameter k ∈ [0,1] is used to represent the degree of distortion — from full truth to full falsehood — allowing for a gradual, fuzzy evaluation of statements. In this system, falsehood isn't removed but integrated and tracked within the formal structure.

It may sound a bit technical at first glance, but the paper is very short (just 2 pages) and explains everything in a concise and intuitive way. I’d be happy to continue the discussion if you have thoughts after giving it a read!

Short Summary of the Theory (for quick understanding):

The core idea behind Context-Graded Referential Logic is to extend classical logic by introducing three main concepts:

1. Contextual Worlds: Every statement is evaluated with respect to a current world (W₀) and a possible world (W′). This helps distinguish between what's actually true, what's potentially true, and what's fundamentally inconsistent (like self-referential paradoxes).

2. Gradual Truth Value: Instead of binary true/false, the system uses a parameter k ∈ [0, 1] to represent the degree of distortion — from full truth (k = 0) to complete falsehood (k = 1), with fuzzy or probabilistic shades in between.

3. Falsehood Marker (⊥₀): When a statement refers to an impossible or invalid context (e.g., due to self-reference), it is not discarded — instead, it's marked as ⊥₀, a special logical value that allows tracking of lies or inconsistencies without breaking the system.

This approach avoids contradictions while still acknowledging problematic or paradoxical statements. It offers a more flexible, intuitive, and analyzable framework — especially useful when modeling how information gets distorted across contexts (like in lies, errors, or misinformation).

The full logic is formalized in the paper, but this should give a general sense of the motivation and structure. If you're curious, the complete system is just a few pages and quite accessible.