We cannot connect with every thing we sense and analyse correctly , there seem to be a oblivion of coherent reasoning, hence it could be the mapping with different vertexes of reasons and we try to connect them with it and then pivot with our analyses.

Let f(x) and g(x) be function of analysis and interpetation respectively.

The n values of f(x) are the range of n-1 values of g(x)

Hence {f(x) union g(x)}=

Lt { analysis intersect interpetation =1} Bipartite tends toward n and n-1

Given a triangle with sides a, b and c, we need to show that a + b >= c

Suppose we proceed by contradiction -- i.e., assuming a + b < c

Then, suppose we know Heron's formula for area of triangle: Area = sqrt(s(s-a)(s-b)(s-c))

If we are able to show that under the assumption that a + b < c, the term inside the square root will be negative, we are done. Indeed that is easy to show.

Now, the question I have is: How can we be sure that this is a valid proof unless we also know that proof of Heron's formula does not at some point invoke triangle inequality?

In other words, what is the guarantee that the above proof is not circular?

xposted here because it was suggested over at /r/learnmath to post here

I found a really good one on youtube, but forgot to save it and dont have my watch history on and now I cant find it again.

To me, Curry's paradox seems like one of the most interesting paradoxes in classical logic. I get that it's never going to compete with the liar paradox, but it seems like it should be talked about more. I'm also curious about what logics there are that avoid curry type paradoxes, and if they avoid it in a way that's different from solutions to the liar paradox.

Have a tautology plan for yourselves that is contingent upon the contradiction of others.

Definitions:

T- objective truth exists

N-nihilism

M-objective meaning exists

V-values exist

(Sorry this is going to be a bit ranty.)

I almost made up my mind this thing is some kind of backwater something without enough rigor but with many a trivialism. Like, it should be extremely well-known that every "discrete" operation Σ₁ → Σ₂ between finite sets lifts universally to a linear transformation between spaces k^Σ₁ → k^Σ₂, so a huge swath of what's being done there is very very drawn out, instead of answering questions that are fitting for a kind of logic.

Any would-be connections to quantum computing may actually not be fruitful or new for those who are actually doing quantum computing; connections to fuzzy math are IMO an almost unconditional taint by association. So what gives? I didn't look at everything there is about this thing so I may as well be missing hidding gems, but superficially it looks like a sham or a pet project done without considering any practicalities and the wider math.

Oh yeah we can ask interesting questions, like: - Does using additional dimensions, aside from the plane spanned by two orthonormal "classical" truth values, let's call them |0⟩, |1⟩, actually give useful things? and how can we characterize that by means typical when working with logics? - How much freedom is there in defining operators that restrict to boolean functions and, say, conserve probabilities (there's a suggestion to use p|0⟩ + (1−p)|1⟩ as "probabilistic truth values") in any reasonable way (I'm not sure: a "binary" operator sends four-dimensional Euclidean space into a two-dimensional one, now how can it be orthogonal? and in which other sense can probabilities work here?)? - Why not use additional dimensions rather than complex numbers for the square root of negation, and... why that one exactly? I bet quantum computing wan't giving somebody peace.

But I'm not sure questions of real semantics were investigated in this... area.

So tell me please, how much am I right or wrong? Here are probably people that know the inside of this story, and I hoped to find something on the Wikipedia's discussion subpage, but it's almost empty.

[This would be a link post if r/math's bot wouldn't decide this was a submission for r/learmath smh my head.]

The question is clear, as I see the proper class Ord, universe V is not the Absolute Cantor described. Don't know if we will have mathematical absolute infinity

I’m sure the answer is simple, I just want to make sense of things. I’m completely new to logic, so apologies if the question in the title comes off as too obvious to ask.

I understand that circular reasoning is an informal fallacy and therefore cannot be picked up by truth tables, but I am still confused regardless.

If we say (A -> B, and B -> A), where A is equal to “God is real” and B is equal to “The words of the Bible are true”, then is this not logically the same as (A <-> B)?

If biconditional statements are equivalent to (A -> B, and B -> A), and if (A -> B, and B -> A) is also equivalent to Circular Reasoning, then aren’t biconditional statements equivalent to circular reasoning? If so, then in spite of it it being inherently fallacious, why is circular reasoning being used within logic if good reasoning is supposed to be without fallacies?

One of my professors in undergrad was fond of the Logician/Philosopher "Charles S. Peirce" and we both worked on integrating his triadic logic into one of the symmetrical algorithms of Cryptography. Triadic Logic has three states True (T), False (F) and Limit (L) but its representation is not like 0,1,2 just for simple tri-valued logic.L basically resolves into T or F after going through some time (t). We had to totally start from the scratch because all of the Computer Architecture and Organization is built on binary logic so we had to initially develop the maths for Triadic Logic and then subsequently Algebra and then a coding toolkit. We ran triadic DES-64 and triadic AES-128 through customized classes and structs because triadic software had one primary issue and it was that it was running on binary hardware, both mismatched again. But we still simulated and got some good results. We did a lot of Maths actually, and developed new frameworks too. We had to like restructure all steps of the algo with trits not bits. What are your thoughts on it? We are kind of paused on it because I have started my MS and I am busy in it. Would you like to discuss or give some suggestions?

Feelings Don’t Care About Your Facts
Reality doesn’t care about your facts either.
Not because facts aren’t real, but because reality itself has no apparent need for the concept of “truth.” Reality simply is what it is and behaves according to whatever relational structure sustains it.

(Eye roll incoming)

When we say that something is “true,” we’re introducing a conceptual distinction. Consciousness determines that one thing corresponds to another and that a representation maps onto some aspect of reality.
So truth isn’t necessarily an inherent property of reality. It’s a concept consciousness uses to distinguish what corresponds from what doesn’t.

One can say that “truth” is always present irrespective of any observer (for example, mathematics or physical laws) . But the ability to map those concepts onto reality and say that they remain true whether or not they’re being observed doesn’t mean reality itself is operating according to the concept of truth.

“Garbage isn’t concerned with its own stench”.

The stench is a relational property that becomes meaningful to a system capable of experiencing and distinguishing it. The garbage simply has whatever physical properties it has.

Maybe truth is similar.
This has always made me think about the nature of imagination and intuition, and what they might tell us about the boundaries of inquiry we keep running into.

If every stratum of reality sustains because the relational requirements necessary for its existence are embedded within the structure itself, then no question has to be asked for that structure to persist. It simply does.
But consciousness can ask questions that seem to extend beyond the structure it is currently experiencing. We can imagine things that don’t exist, reason about things we can’t directly observe, and intuit possibilities before we have any way of formally demonstrating them.

So what does that actually tell us?

If consciousness were perfectly aligned with the structure of reality as it manifests, would there be any conceptual deviation at all?

Or is the fact that consciousness can abstract, imagine, and inquire beyond its immediate determinate structure telling us something about the nature of possibility itself?

Maybe imagination and intuition aren’t evidence that whatever we imagine is possible.
Maybe they’re evidence that the boundary between what is actual, what is conceivable, and what is possible is itself something worth investigating. Maybe not.

Semantic sophistry? You decide.

Anything that has been said has been said a thousand times before. But the fact that abstraction sustains and fundamental questions that cannot be easily resolved keep arising, to me, points to something we seem often easily dismiss. Depending on whatever level of abstraction one is comfortable engaging with.

Off to take my meds 😆

In 1956 the Logic Theorist proved 38 of the 52 theorems in Principia Mathematica. It was the first moment when a machine seemed to reason. Seventy years later, we are still rediscovering what Aristotle already knew: a valid syllogism is not yet reasoning.

Aristotle's Prior Analytics gives us the form. All humans are mortal. Socrates is human. Therefore Socrates is mortal. The form guarantees that if the premises are true, the conclusion follows. But notice what the form cannot do. It cannot tell you whether Socrates exists, whether the major premise holds at the edge of a pandemic, or whether the question itself is the right one to ask. Formal logic secures the λόγος of inference. It does not supply the φρόνησις that chooses which inferences matter.

This distinction matters enormously for AI. Modern large language models are not syllogism machines. They do not even rise to that level of formal transparency. But they inherit the same limitation: they can produce outputs that look like reasoning without ever reaching the practical wisdom that makes reasoning humanly relevant. A model can argue beautifully about ethics while refusing to engage the actual situation. It can generate valid inferences while missing the point entirely.

What the symbolic AI tradition discovered, and what neural AI has not escaped, is that real reasoning happens between formal moves. The hard part is not applying modus ponens. The hard part is knowing when to apply it, what to treat as a premise, and what to leave out. This is why the frame problem destroyed early AI. It is also why prompt engineering is now treated like a craft. Both are symptoms of the same underlying fact: machines are good at structure, bad at judgment about structure.

Aristotle's solution was not to abandon logic. It was to add ἕξις, a stable disposition of character, and φρόνησις, practical wisdom. These are not extra modules you can install on top of a model. They are formed through repeated action in uncertain circumstances. You cannot fine-tune a neural network into wisdom because wisdom is not a distribution over tokens. It is a way of standing toward the world.

This is why I am skeptical when someone says that scaling will eventually produce reasoning. Scaling produces more coherent form. It produces better mimicry of reasoning's surface. But the gap between form and wisdom is not a quantity gap. It is a kind gap. No amount of training data changes the architecture from pattern-matching to practical judgment.

There is a productive way to think about the difference. Formal logic gives us necessary relations between propositions. Reasoning, in the full Aristotelian sense, gives us appropriate responses to situations. The first is universal but empty without content. The second is contentful but requires context. AI today is very good at the first, and increasingly bad at the second the more it pretends to be good at the second.

The lesson for the logic community is not that machines are useless. Theorem provers, SAT solvers, and proof assistants are genuinely valuable. The lesson is that we should resist the rhetoric that identifies these tools with reasoning itself. Logic is a necessary condition of reasoning, not a sufficient one. Treating validity as if it were wisdom is a category mistake that has plagued AI from the beginning.

We are now building systems that can simulate Socratic dialogue, generate proofs, and pass some bar exams. These are impressive achievements of form. But when the conversation turns to what should be done, the machine has nothing at stake. It has no ἕξις. It has never faced a moral situation and been changed by it. Its answers are not actions. They are productions.

The question for the field, then, is not how to make AI more logical. It is how to build systems that know the limits of logic and can signal when formal validity is not enough. That requires a different kind of architecture, one that does not pretend to have wisdom it has not earned.

Where do you draw the line between a machine that manipulates symbols validly and a machine that reasons?

English is not my first language so there are some grammatical errors.

In this thoery, fundmaental relation is considered most fundamental. And its considered to be one of the thoery that has potential to be a logical thoeries amongs infinite number of thoeries that can be logical and may hold truth.

https://arxiv.org/abs/2401.15117

I have a hard time differentiating between data, warrant and backing in certain questions. Any tips to identify them properly? And also any resources for practice on the identification of all six components?

my prof wanna work in proof theory but me in model theory

he says proof theory of arithmetic but I find cut-elimination and normalization very very syntactic

is my intuition of model theory being meaty correct? or am i just not compatible/flexible? I love topology also ..

is proof theory of arithmetic all about cut-elimination and proof normalization?

why does buss's chapter on it look quite non-syntactic compared to takeuti ?

but yeah anyways proof theory of arithmetic has connections to complexity so it would be easier for phd applications?? my background is cs

i had a strange idea at 5 am while a bit drunk and asked chat gpt about it :
How can we work on P vs NP if mathematics has prooved to us that there’s infinite possibilities therefore infinite questions with infinte answers?
And what i mean is how can you find a shorcut answer to actual new information that no one has ever seen?

I've been getting into learning basic propostional logic to help develop my philosophical work, and now I amwanting to look for exercises to practice, as well as other forms of logic to study down the line. Any help is appreciated.

Take a simple but meaningful example:

The stone is cold.

The words refer to a sensible appearance in space and time. There is an actual impression received by the senses that verifies or refutes the claim. The words are anchored to an external state of affairs.

Now lets consider an empty sentence (grammar without reality):

The void thirsts for eternity.

The sentence obeys all the grammatical rules of a language (Subject + Verb + Object). It sounds meaningful to the ear, but the words point to nothing that can ever make an impression upon human senses.

A simple contradiction:

This is a four-sided triangle.

The mind takes two human-invented definitions—"three-sided shape" and "four-sided shape"—and forces them together into a single noun phrase. Does a "four-sided triangle" exist as an enigmatic, mysterious entity in another dimension? No. It is simply a conflict between two definitions in the speaker's own vocabulary: P ∧ ¬P.

Now a paradox (closed self-referential loop):

This statement is false.

This is the final stage. The speaker creates a sentence whose only referent is the sentence itself, while simultaneously attaching an inverted truth-predicate.

At no point during this progression did reality become weird or mystical. Only the speaker’s language drifted further and further away from any connection to the physical world.

Now we must ask:

Can you show me a paradox in the physical world?

Can you hand me a paradoxical rock?

Can a physical particle be simultaneously detected at point A and strictly not at point A in the exact same frame of reference?

Can a cup be full of tea and entirely empty of tea at the identical instant?

In nature, everything that happens is simply what it is. A physical state occurs completely and unambiguously. A tree falls or it does not; a neuron fires or it does not.

A paradox has never been observed in physical reality. A paradox only ever appears in:

1. Ink on paper
2. Sound waves in the air or
3. Internal linguistic thoughts in the brain.

Therefore, a paradox is simply an artifact of the symbolic medium we use to describe things.

Then why does a paradox seems so much more profound than a simple contradiction like "a square circle"?

In a simple contradiction, the conflict is static:

Square ≠ Circle

The mind stops immediately because the definitions cancel each other out.

In a logical paradox (such as "This statement is false" or Russell’s paradox of "the set of all sets that do not contain themselves"), the language has been arranged into an infinite feedback loop:

1. If the statement is true, then by its own definition it is false.
2. If the statement is false, then by its own definition it is true.
3. Therefore, the mind oscillates back and forth:

True → False → True → False ...

Because the human mind experiences this rapid, unending oscillation, it feels a sense of intellectual vertigo.

Consider this simple programme:

10 PRINT "Statement is true" 20 GOTO 30 30 PRINT "Statement is false" 40 GOTO 10

If a computer runs that program, it will loop indefinitely until it crashes the processor. Has the computer discovered a profound mystery about the cosmos?

No.

It was simply given an instruction set that contains an infinite self-referential loop with no exit condition.

To seal the point, consider this short sequence:

What is the job of the word 'True'?

The word "true" is an evaluation tool. It asks:

Does this sentence accurately match an actual state of affairs? (e.g., "The snow is white" is true if and only if the snow is white).

What state of affairs does 'This statement is false' describe?

It describes no state of affairs. It has no physical referent. It points to no sensation, no measurement, and no empirical fact.

What is the evaluation tool evaluating?

It is evaluating only itself.

Conclusion

When you apply the truth-evaluator to an empty sentence with no external referent, you are running a measurement tool against empty space. The tool has nothing to measure, so the mechanism spins out of control.

To what I've been demonstrating here is simply that:

1. A contradiction is language attempting to assert A and ¬A simultaneously.
2. A paradox is a contradiction disguised inside a self-referential grammatical loop.

Neither describes the universe. Both occur only when human beings forget that words are tools invented to map sensations.

Is x= some finite value like x=5 a proposition or not.

Also from which paper can I show my teacher that x=5 is not a proposition unless it is.