Formal proof of Ontological Holism?

Metaphysics & Epistemology
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The Purpose of Providing the Proof

The only legitimate purpose of providing a formal proof for the three fundamental laws of logic as absolutely interdependent as a holism is systemic mapping.

Within the strict, isolated sandbox of Classical Propositional Logic, the proof serves to demonstrate structural symmetry. It shows that if you map these laws onto a linear syntax using truth-functional connectives, they share a relationship of semantic equivalence (A \\equiv B \\equiv C). It is an exercise in tracing the interior plumbing of an abstract system to show that its foundational pillars are perfectly balanced and interconnected.

Why the Proof is Ultimately Superfluous

Outside of that narrow exercise in mapping, providing a formal proof for these laws is completely superfluous and philosophically incoherent for two definitive reasons:

1. It Commits a Category Error (The Map vs. The Territory)

A formal proof is a downstream product of linear, transitive, and syntactic rules. The three fundamental laws (Identity, Non-Contradiction, Excluded Middle) are the upstream, objective, ontological conditions that allow a formal system to exist in the first place.

• To demand a formal proof for the laws of logic is like demanding a architectural blueprint of the earth before you are allowed to build a house.

• The earth provides the physical ground that makes the house possible; it does not need the house’s blueprints to justify its existence.

2. It Triggers an Infinite Regress or Fatal Circularity
If a critic demands a formal proof to take these laws seriously, they must immediately answer a fatal question: What laws govern the validity of that proof?

• If the proof uses the three laws to validate its own steps, it is viciously circular.

• If the proof uses a different, “deeper” set of laws, then those new laws must be proven by a third set of laws, and so on, escaping into an infinite regress (the Achilles and the Tortoise paradox).

The Verdict
The three fundamental laws do not require a proof because they are transcendental presuppositions. They are self-evident, not because they are simple, but because any attempt to deny, prove, or even articulate them silently relies upon them.

The formal proof is not a “birth certificate” creating the laws; it is merely a mirror reflecting their simultaneous, holistic existence. The proof is superfluous because the laws are already vindicated by the very existence of reality and thought.

The three fundamental laws are not circular because they do not derive their validity from one another in a linear chain of inference. Instead, as a holism they exist simultaneously and interdependently, each already containing the other two within its own structure. This triadic interdependence provides the minimum stable perspective required for triangulation, allowing any objective observation or coherent thought to take place at all.

The Formal Interdependence
Here is how each law inherently incorporates and requires the other two.

  1. Identity Formally Requires Non-Contradiction and Excluded Middle [2]
    The Law of Identity is stated as:
    P → P (If P, then P) [3]
    By the rule of Material Implication (α → β ≡ ¬α ∨ β), we rewrite it:
    ¬P ∨ P
    By the rule of Commutation (α ∨ β ≡ β ∨ α), this becomes:
    P ∨ ¬P (The Law of Excluded Middle)
    By the rule of De Morgan’s Laws and Double Negation (α ∨ β ≡ ¬(¬α ∧ ¬β)), we rewrite Excluded Middle as: [4]
    ¬(P ∧ ¬P) (The Law of Non-Contradiction)
    Conclusion: You cannot state that a thing is itself (P → P) without simultaneously asserting that it cannot be both itself and not itself ¬(P ∧ ¬P), and that it must be one or the other (P ∨ ¬P).

  2. Non-Contradiction Formally Requires Excluded Middle and Identity
    The Law of Non-Contradiction is stated as:
    ¬(P ∧ ¬P) (It is not the case that P and not-P)
    Apply De Morgan’s Law to distribute the negation:
    ¬P ∨ ¬(¬P)
    Apply Double Negation (¬¬P ≡ P):
    ¬P ∨ P
    Apply Commutation:
    P ∨ ¬P (The Law of Excluded Middle)
    Apply Material Implication in reverse to P ∨ ¬P (rewriting it as ¬(¬P) ∨ ¬P, which becomes ¬P → ¬P):
    P → P (The Law of Identity)
    Conclusion: Guarding against contradiction ¬(P ∧ ¬P) is mathematically identical to establishing boundaries (P ∨ ¬P) and ensuring self-consistency (P → P).

  3. Excluded Middle Formally Requires Identity and Non-Contradiction [5]
    The Law of Excluded Middle is stated as:
    P ∨ ¬P (Either P or not-P)
    Apply Double Negation to the first term (¬¬P ∨ ¬P) and use Material Implication in reverse:
    ¬P → ¬P
    By the rule of Contraposition (α → β ≡ ¬β → ¬α), this flips to:
    P → P (The Law of Identity)
    Apply De Morgan’s Law directly to the original statement (P ∨ ¬P):
    ¬(¬P ∧ ¬(¬P))
    Apply Double Negation:
    ¬(¬P ∧ P)
    Apply Commutation inside the parenthesis:
    ¬(P ∧ ¬P) (The Law of Non-Contradiction)

1 Like
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Im out of my depth but ill give it a go. Ill focus on this part. (\(P \to P\) becoming \(P \lor \neg P\)) fails to prove holism but is logically sound.

Reductionism says that the whole is just the sum of its parts whereas Holism says the whole has properties that the individual parts do not have on their own.

Example: In quantum entanglement, using bells theorem, you cannot describe the state of particle A, without paritcle B, which meets the standard of physical holism because our system contains information that does not exist as individual parts.

If X is true, then Y must happen is exactly the same as saying Either X isn’t true, or y happened. This does not prove that the particle and the rest of the universe are physically or ontologically bound together as a single, indivisible whole. A reductionist can still accept your proof while still maintaining that a particle is just a separate, independent pile of quarks, among other things.

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You’re right that the step from P → P to P ∨ ¬P is logically sound inside the system, but that’s exactly the point—those rules only work because the three laws are already doing their work underneath.

The derivation doesn’t create the interdependence; it reveals it.

I only posted this because a critic elsewhere demanded a formal proof or he’d refuse to take the claim seriously. The irony is that every formal system already rests on the three laws as its unspoken ground, so the demand itself was circular. I supplied the proof anyway to meet him on his own terms, but the ontological priority of the laws doesn’t depend on playing that game.

A reductionist can accept the equivalence while still treating the laws as separate parts, but that misses the deeper claim: you can’t even state one without the other two already being present, because the laws aren’t assembled from independent pieces that could exist on their own.

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I believe I misunderstood you…

Thanks for giving it a go but I think your reductionist argument actually misses the specific target of the post.

You are completely right that a reductionist can view a particle as an independent pile of quarks, and that logical equivalence doesn’t prove the physical universe is an indivisible, mystical whole.

I wasn’t making a cosmological argument about physical matter, nor was I denying that contingent, created categories have distinct constituent parts.

My argument is strictly about the infrastructure of the 3 fundamental laws, not the physical layout of the cosmos.

When I say logic is a holism, I mean that the three fundamental laws cannot be treated as separate modular pieces that you can pick and choose from. You cannot have a universe with ‘just the Law of Identity’ but without the Law of Non-Contradiction.

As the syntax shows, each law structurally and semantically requires the other two to even be stated.

So while a reductionist can absolutely be a reductionist about physical particles, they cannot be a reductionist about the laws of logic themselves.

Logic operates as an indivisible, holistic triad, providing the absolute minimum framework required for us to even have a debate about quarks in the first place.

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To put your quantum mechanics angle into perspective, it actually serves as a perfect illustration of how this logical holism operates.

In a Trinitarian ontological reality, the three fundamental laws of logic describe the necessary, objective, transcendental boundaries that everything else must align with in order to be actualized as ‘real.’

1. Superposition as Freedom Within Boundaries

A quantum superposition represents a form of subjective freedom or indeterminacy, but it can only exist within holistic boundaries. The superposition is not lawless chaos; it is a real, highly structured set of possibilities defined by the wave function equations. Those equations are themselves made ‘real’ and consistent by the 3 laws.

2. Collapse as Ontological Certification

The moment a measurement is made, the state becomes classical. It collapses from a cloud of potential into a definite, localized fact. At that exact moment of measurement, it must immediately align with the three laws to be recognized as objective reality: it becomes itself (Identity), it cannot be both its measured state and not its measured state (Non-Contradiction), and the ambiguity is resolved (Excluded Middle).

The three laws act as the absolute certifier of objective fact.

3. The Intuitionistic Parallel

We see this exact same dynamic in intuitionistic logicalsystems.

In intuitionism, you can refuse a binary choice of outcomes—effectively stepping back into a cloud of uncommitted potential. But the moment you articulate that choice, you merely create a new binary boundary.

For instance, you can refuse to choose between chocolate or vanilla ice cream and choose to have both. However, by doing so, you have instantly and absolutely excluded the choice of having only one or the other. You haven’t escaped binary exclusion; you’ve just shifted where the boundary line is drawn.

Even the freedom to hold a superposition requires a structural, holistic baseline to define what is being held in tension.

So the fundamental ontological structure of reality within the laws does not determine everything. It actually makes freedom possible and stable by making our actual choices real.

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Let me review a different law then. In classical logic, a part and the whole must have separate identities, but in a hologram, the part is the whole while simultaneously remaining a distinct part such as a single pixel that is simultaneously itself and the entire image. Therefore, you can no longer say P = P. Under a holographic framework, it must be P = W.

! = NOT

  1. Under the law of identity P is P

  2. In a holographic model, you have to review how every part contains the whole. P = W.

  3. However it is a fragment and does not incorporate the full magnitude of the whole. P = !W

  4. By subsituting our example into your proof we are left with the following. W OR !W

  5. Under the law of non contradiction, we substitute our truth into the formula. !(W OR W)

  6. Apply the impodent law of W OR W = W and we are left with !W.

Instead of balancing the formula the system has erased the existence of the whole. This is because if you take a apple your formula balances. When a holographic apple is used though, and lets say you cut it, unlike in the base apple.example you do not get two halfs. Instead you get a half that contains the whole. Even if you break it down into a million pixels, every part holds the information of the whole apple through spatial redudency of the distrubuted information otherwise known as the holographic principle using fractal geometry.

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I understand what you’re saying and it is a misconception that underpins the paradigm in understanding itself.

Look at an object in your room. I’ll use a coffee table as an example.

As you look at it, you are seeing it from one perspective. Yet, you are looking at the whole table.

If you don’t mind, it was Jesus who articulated and gave us the information to rightly understand hollism. The textbook definition is misleading because it speaks of ‘the sum of its parts’.

Jesus said to Phillip, “Anyone who has seen me has seen the Father.” Yet, Jesus also distinguished himself from ‘the Father’ and prayed to him.

Very weird from the position of a reductionist paradigm.

The 3 laws are not the ‘whole’ of logic on their own. And neither is Jesus the whole of God on his own.

There is no ‘on their own’. The laws are wholly and completely INTERdependent. They are not INdependent.

In other words, to see one law is to see them all. To see Jesus is to see the whole Trinity. They are not divisible. They are not ‘parts’ of a whole. They consitute the whole.

The confusion comes because we are finite in spacetime and perspective. We can physically only see one law at a time. Its very much like the coffee table analogy. You can ‘perceive the whole’ table even from only one perspective. But not with empirical eyesight.

The mathematics in the other thread turn out to describe a tetrahedron. That was a surprise. But what’s more interesting is that the math describes the holism as isotropic.

It confirms that no matter which law (vertice of the tetrahedron) you observe from a 3 dimensional vantage point, you always perceive the whole through node 0.

Is that helpful? Christians make this mistake often.

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This became a spiritual talk huh?

Im against this model personally because it takes away rights from deities, demons, angels, or whatever the spirit is being labeled as from my point of view. In this case, you have made a rule that your Jesus has to follow now from my point of view. Through free will, assuming one has it, you placed it inside a box, well call that box a lamp, and unless the condition that you laid out for it is honored, you have taken away its right to be summoned. It has to appear as the trinity, whereas to me it will decide those answers for itself unless it decides otherwise. A example. If Jesus decides to show up without being the trinity, perhaps it adds or subtracts, im perfectly fine.

I do question peoples intensions in these scenarios. If you were a priest in this hypothetical example in service to a community, this possible perversion, its not my right to determine what is or isnt a perversion, would spread into the community unknowingly. What if they to lay conditions for it. God has to be a he. God had to be a human and so on. The accumulation would shackle the very thing that you would pretend to worship all so that you could steal its power, or what you call have more control over it. A recipe of how gods fall and become Djinn in my opinion.

Ill add your rule to my journal and study of demononolgy. A new rule to ground the spirit.

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You’re misunderstanding the argument Jonathan. The laws are not a prison. They are the essential boundaries that make a real and stable reality possible.

Imagine a video game with no rules where everyone hacks the system at will.

Chaos. System crash.

It is the laws of identity, distinction, and exclusion that make a game or world possible.

You for instance, cannot truly exist unless you are identified (Jonathan). And identity has no meaning apart from being distinguished from ‘another’ (Rob). Only then can you exclude what your identity is ‘not’. The 3 laws operate simultaneously.

The rules are what make subjective freedom and individuality possible. Within the laws of physics for instance, you are free to do all kinds of things. No physical laws = no physical freedom.

Jesus isn’t BOUND by these rules, he is the SOURCE of them. That was his claim. He claimed to BE the Trinitarian truth. And he articulated this Trinitarian structure of reality.

Hope that helps.

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All you have shown is that these three are among the tautologies of propositional logic. Along with denumerably many other tautologies.

Even if we accepted these as axioms, (and they are usually taken as theorems, not axioms) they are insufficient, as they stand, to the task you set; they cannot demonstrate the consistency and completeness of propositional logic without additional axioms.

For example, from

  • P→P
  • P∨¬P
  • ¬(P∧¬P)

plus Modus Ponens, you cannot derive P→(Q→P), or (P→(Q→R))→((P→Q)→(P→R)), or (P∧Q)→P. Your need more than three axioms. There goes your trinity.

And further, listing theorems does not imply anything about ontology. Nothing about how things are in the world follows from P→P.

And again, as has been pointed out to you previously, there are logical systems that do not use these theorems. Intuitionist logic rejects P∨¬P. Minimal logic uses only identity. Paraconsistent logics work with variations on ¬(P∧¬P). That’s onyl the beginning of the list.

Your argument is a series of misunderstandings tied together with shoelaces in order to bolster a Trinity that you hold true on faith.

Dreadful stuff.

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I try to honor all faiths and beliefs when I can. In your universe, the structure of reality must honor this truth. Got it. Ill pass on being born as a character like newton or einstein though for its history. No particular reason.

And yet im drawn to discover what type of life forms would manifest out of it due to what physics would end up rewriting itself to. If their is a opening for a sentient ameoba ill consider it when you’re taking applications from souls wandering the ethos, assumming you get to that stage.

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You move FAR too quickly.

I already agree with you that any formal proof of Ontological Holism is circular and ‘superfluous’. Its actually Gödel’s Incompleteness theorem extended to any formal logical system we create.

Gödel showed that any sufficiently powerful formal mathematical system can’t prove its own consistency from within. This point extends that: any attempt to ground the three laws inside a system that already relies on them is the same kind of self-referential loop. The laws are prior to and the ground the formal system cannot account for.

It applies to logic itself, not just arithmetic. The circularity isn’t a flaw in the proof. It’s the inevitable limit of what any formal system can do for its own foundations.

The foundation for those systems is the 3 fundamental laws operating as 1 unified objective ontological reality. They are the objective ground of everything else.

Your appeal to other logics does not land the way it claims to. It is a very common misunderstanding that fails to distinguish between objective and subjective categories.

Intuitionistic logic for instance is a subjective logical system. I can walk up to an Ice Cream cart downtown and refuse the binary option of either Chocolate or Vanilla. Perhaps I pay a premium to have both. But as soon as I make that choice, I have now excluded the alternative of only having one flavor.

All of these subjective systems produce the same result. As soon as the function is complete, the measurement made, or choice decided, the output of the system is made ‘real’ by the identification, distinction, and exclusion.

There cannot be an actual reality apart from all the laws.

Even choosing to remain undecided is made real by the laws. A quantum superposition for instance can be identified, distinguished from, and excludes an actual classical state. The potential is ‘real’ as defined and bounded by the Schrödinger equations.

An unmeasured superposition is not fundamental. Nor is it chaos, because the equations are formal mathematical descriptions built from the objective laws that make mathematics possible.

Superposition represents the subjective freedom within the boundaries of the objective reality.

All of your arguments are ultimately dependent upon your faith in the 3 fundamental laws same as myself.

The only difference is that I recognize the implications of their Trinitarian relationship and I am not afraid of those implications. You seem in a rush to dismiss them.

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It is a little unerving. I have struggled with that same reluctance.

But then I realized that in doing so, I was Identifying, Distinguishing, and Excluding the universe I did not desire and for my own self interest.

I realized I was ALREADY inside the universe I did not prefer. I was employing the laws myself in order to escape them.

Took me awhile to get my bearings. I had a lot of misunderstandings about God to overcome. I didn’t then realize that God didn’t condemn me, but actually took RESPONSIBILITY for me.

He just wanted to teach me. But I was afraid of Him.

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Well, that had me smiling.

I notice that you didn’t directly address any of the criticisms I offered. I suspect that’s because you haven’t quite understood them.

You begin with your faith in the Trinity and work towards a justification from logic; what follows is that there is little point in a critique of your reasoning, when your faith is indubitable. Reasoning is, for you, a pretence; you’re doing theology, not philosophy. You already have the answer and are spending your time looking for questions, which is, of course, arse about, and disingenuous.

I’ll leave you to it.

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Haha, we’re back to the same impasse from the other thread.

You’re committed to an impersonal ultimate reality and treat logic as if it floats free of any personal ground in some Platonic dimension. I’m committed to a personal ultimate reality and treat logic as if it is a manifestation of mind.

Neither of us is omniscient, so both positions are starting assumptions, faith commitments about the nature of logic and reality that cannot be proven.

The difference is that I recognize we’re both persons doing philosophy. Only minds do philosophy. Rocks don’t.

Just because you don’t name your starting assumptions does not mean you are operating from neutral ground. In fact, it is disingenuous because by pretending your position is neutral, you’re hedging.

Given that, your accusation that I’m “doing theology, not philosophy” loses its force. We’re both doing philosophy from different starting assumptions. I’m simply operating in total transparency about my assumptions. You’re hoisting fig leaves.

I answered all of your points with devastating clarity and fairness. Just because I don’t play your game of pretense, doesn’t mean I didn’t answer you. It just means I didn’t give you the answers you expected.

You stacked the deck. I un-stacked it because that’s what good philosophers do! :wink:

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Was re-reading your comment and I was reminded of C.S. Lewis’s illustration of the Obstinate Tin Soldiers. The soldiers were given the opportunity to become real men. It had kind of a Pinocchio vibe.

The soldiers were resistant because all they could imagine from their point of view was that the tin would be ruined. It is difficult to trust what we can only partially perceive.

We wonder and even long for and fantasize about a different world, but simultaneously fear anything truly alien.

If you’ve never read John’s gospel you should. You’re smarter than you think. Its not the alien visitation anyone expects. That’s what makes it really alien…

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I see this is how the laws of thought look like in sentential logic. I don’t think ontological holism is a proper term for the intricate relationship demonstrated between LI, LNC and LEM.

To me, these laws are truly at the very foundation of thought.

There are different types of logic. Schrodinger logic disregards the LI. Polyvalent logic the LEM and paraconsistent logic the LNC.

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Thanks for that reply. It really helps us understand why this is important. All of us have been so busy reasoning, that we failed to really stop to examine what enables reasoning itself. Epistemology and ontology are crucial.

Yes. You’re correct. Sentential logic is propositional logic. Most people familiar with the 3 laws have learned about them in the context of propositional logic.

Mathematical statements are also propositions, 1+2=3. But the point is deeper than that.

The term ‘ontological holism’ fits because I am looking at the actual relationship of the laws themselves, not just how they are applied propositionally at the practical level of application.

Your appeal to other logics is a very common misunderstanding that fails to distinguish between objective and subjective categories. I’ve made this mistake myself.

Intuitionistic logic, for instance, is a subjective logical system. I can walk up to an Ice Cream cart downtown and refuse the binary option of either Chocolate or Vanilla. Perhaps I pay a premium to have both. But as soon as I make that choice, I have now excluded the alternative of only having one flavor.

All of these subjective systems produce the same result. As soon as the function is complete, the measurement made, or choice decided, the output of the system is made ‘real’ by the identification, distinction, and exclusion.

There cannot be an actual reality apart from all 3 laws.

Even choosing to remain undecided is made real by the laws. A quantum superposition, for instance, can be identified, distinguished from, and excludes an actual classical state. The potential is ‘real’ as defined and bound by the Schrödinger equations.

An unmeasured superposition is subjective, not fundamental. Nor is it chaos, because the equations are formal mathematical descriptions built from the objective laws that make mathematics possible.

Superposition represents the subjective freedom within the boundaries of the objective reality.