Patrice Ayme's Thoughts

Abstract: Specialists know that the foundations of logic are probably not on a sound basis. We propose a new approach: NEW LOGIC For An Old Problem. We point out that, in its simplest version, the Liar Paradox LP can be seen as an example of failed communication, and incomplete statement. 

However, the suggestions made in the 20th century by Russeell and Tarski, inter alia, for solving the Liar Paradox did not recognize this and instead pretend to solve LP by introducing infinite hierarchies (which we reject, with the rest of the notion of infinity, as ultimately non rigorous solutions, because they are non-physical… while we recognize the utility of “types” and “meta” as forms of mental shorthands).. 

We propose instead a new logic, SEQUENTIAL LOGIC, an alternative to Propositional Logic (also called 0th order logic). Sequential Logic mimics the basic functions of the brain, so it is local, topological, ordered and temporal. It’s also endowed dynamically with truth, falsehood, and incomplete values..

The atomic elements of Sequential Logic are Operations, which are more or less identical to Neural Networks, or strings thereof… Indeed, operations are implemented by Neural Networks and can be identified to them… Those Operations/Neural Networks are the equivalent of the propositions which are the atomic statements of Propositional Logic… The latter are used to make strings in PL. Similarly, strings of Operations/Neural Networks make formulas in SL.

In Propositional Logic, if A is a formula NON A is also a formula (it’s an axiom). Not so necessarily in Sequential Logic. However if A is activated in SL it can be true, false, or incomplete: that has no equivalent in PL (proper comparison ought to be with CPL + QT). The other axioms of PL stay pretty much the same in SL.. (If A → B and B → C, then A → C, etc…). 

Real neural networks in the brain use dopamine bursts in mid -brain, to help ascertain truth in part of the brain and a somewhat related, but opposite, dopamine busts, in a different part of the brain, the pre-frontal cortex, to ascertain falsehood. The brain processes truth pleasantly, differently from falsehood which is so alarming it gets dispatched to more advanced parts of the brain.   

Similarly, the formulas (strings of neural networks) of Sequential Logic can be true or can be false (and if neither, incomplete).

In Sequential Logic, logical evaluation is temporally extended and contextually constructed — reference and predication occur in ordered neural events, not in atemporal structures. In particular, a sequence like A → NON A  → A is possible, which is viewed as a contradiction in traditional logic, it’s not a contradiction in SL, it corresponds to cyclical activity within the brain, a cyclical activity similar to any such activity in nature.

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“The first sentence in this essay is a lie.”
This is the Liar Paradox. Something feels off about saying it, and indeed, the unease it provokes has haunted thinkers for twenty-five centuries — from Eubulides to Buridan,  Russell, Tarski, and Gödel. Here we suggest a new approach: one that takes into account not only formal logic, but also the way logic is physically realized in the brain.

1. The LIAR PARADOX UNRAVELLED:

Take the classic version: “This sentence is false.” Traditionally, “this sentence” is read as referring to the entire utterance. The sentence therefore applies the predicate “false” to itself, generating a loop: if true, then false; if false, then true.

But the wise listener, new to the subject, might demur: In what sense is this sentence a lie? What exactly is meant by “this”? What do we mean by “sentence”? Once those questions are carefully raised, the paradox evaporates. The utterance fails to designate anything definite. “This” is a demonstrative with no stable referent. It does not succeed in pointing to an object — even to itself — that can serve as the argument of “false.” The statement is therefore not a well-formed formula, not because of grammar, but because of referential indeterminacy. The paradox arises only when we pretend that this INCOMPLETENESS is well-formed.

In that sense, the Liar Paradox is not a deep mystery of truth, but a miscommunication. The Liar Paradox attempts to refer before it has anything to refer to.

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2. Classical Responses: The Ban on Self-Reference:

Tarski recognized this to some extent and formalized a rule: truth cannot be defined within the same language that expresses the statements being judged. A meta-language is needed to speak about truth in the object-language. This is an application of Russell’s Vicious Circle Principle: “Whatever involves all of a collection must not be one of the collection.” Russell invented it to avoid the contradiction of sets that contain themselves — such as the set of all sets that are not members of themselves — a logical cousin of the Liar [1].

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3. The PROBLEM OF TOTALITY, ENABLING THE CANTOR DIAGONALIZATION TRICK:

This type of “self-reference exclusion” reappears in Gödel’s incompleteness theorems, Turing’s halting problem, and Tarski’s undefinability theorem. Each involves the same move: consider all formulas or programs; then create one that refers to the totality, and contradiction or incompleteness follows. The structure is always the same: totality → self-reference → impossibility. An essential piece of logic is diagonalization of a two dimensional countable infinite array.  One can always diagonalize — as Cantor showed — to create an element outside any purportedly complete countable infinite list made of countable elements… Because the list is countable  but its infinite character always enables us to exhibit an uncountable infinity of logical mechanisms to build elements outside of that list (that’s called Cantor Diagonalization).

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4. Classical Argument: METAPHYSICS OF INFINITY MADE FINITE:

All these arguments rely on a powerful but metaphysical assumption: that one can meaningfully speak of all propositions, all statements, or all computable programs [2]. But such a totality is not something we can construct; it is an idealization, not a reality. 

The existing laws of physics would make the construction impossible; careful inspection of Gödel’s proof shows that the existence of a number is supposedly demonstrated… But NO process is given to construct that number (cynically one could say that infinity is used to pretend that this number exists, but then the number cannot be rolled out since its existence depends upon something that does not exist, namely the sort of “infinity” used in the proof).

An infinity is not a warehouse of completed objects. In practice, logic and cognition do not operate globally; they operate locally, constructively, context by context… Finitely, as a classical computer, or a biological neural network, must.[3]

Our thinking systems — human brains, or any embodied processors — do not evaluate “all possible propositions.” They operate network by network: finite, temporally ordered, and dependent on prior activations. Each logical evaluation is contextual, feeding forward and backward across connections… But always forward in time (as in Quantum Mechanics). The Liar Paradox arises only when one mistakes this dynamic, sequential process for an instantaneous, self-contained totality.

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5. Sequential Logic Arises From The Nature of Reference

In the brain, logic unfolds in time and place, it’s a fundamental process ordered by time and place (similarly, fundamental processes in Quantum Mechanics are ordered in space and time; and it may be more than an analogy [4]; as it has been discovered in 2025 that the brain CONNECTOME’s involves nanotubes between neurons, an indication that the quantum scale has to be considered in the brain; so if the Quantum turns out to be the fundamental mechanics of the brain, it is not surprising that the latter operates like the former.) 

When a sentence begins, its initial segment (“This sentence…”) triggers predictive neural networks seeking a referent. But the referent does not yet exist; we know that a sentence is going to be pointed at. The crux of the Liar Paradox is that there is nothing there. Pointing out how one communicates with a dog may help: 

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6. Learning Communications From Dogs:

When my dog accomplishes this first step, in what he views as a proposition, he searches my eyes and throws me a special look, which indicates that a “sentence” is going to come and I must then acknowledge, by looking back to him in a special way, that I am ready to receive the pointing data. “This sentence”, from my dog, accomplishes succinctly the first part in the pointing sequence.. 

Then my dog communicates what he wants. The communication is in body language. He turns his head and points at the particular object, say, a door which he wants to see open for him. If I indicate agreement, which he checks by looking back at me, he will wag his tail to indicate his satisfaction for the signing of this deal. I will come and open the door.

The part labelled by “This sentence” animates the general neural networks machinery, it tells that machinery that a sentence, which is another and particular neural network, or strings of neural networks, is going to be pointed at. 

But, in the Liar Paradox, that SUBSEQUENT DESIGNATION WHICH “THIS SENTENCE” WAS A WARNING FOR, NEVER HAPPENS… There is no door being pointed at. 

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7. There Is No Argument In The Liar Paradox:

THUS the paradox of the liar then arises when WHAT DOES NOT EXIST… from the point of view of Sequential Logic, gets qualified NEVERTHELESS (by “is false”). The classical scheme is completely different: Classical Propositional Logic, CPL, + Robinson Q Arithmetic, that’s the simplest, Quaintest arithmetic, without infinite induction, + a Truth Predicate brings us Self Referential Contradictions…

If I sit at the dinner table and I say out of the blue:”This is false”. A polite diner will ask: “Excuse me, what is false?”

So, in the traditional Liar Paradox, “This sentence is false”, the neural network animation sequence to designate and then qualify something is short-circuited while still under construction (because neural networks communications are constructed). 

When the utterance ends (“…is false”) the processing machinery, is implicitly asked, by those who believe there is a paradox, to loop backward and reinterpret the earlier part as a designated sentence in light of the whole, and then decide that’s the only sentence around and thus this is what “This” designated — but the brain’s logic is not timeless. It is sequential, it proceeds in micro-sequences, a progressive evolution, not as a single frozen self-observation after a bit of time travel backwards.

Thus “This sentence is false” cannot stabilize as an object of thought which is a neural circuit going one way. “This sentence is false” is not paradoxical; it is dynamically inconsistent, because it is dynamically incomplete. There is no moment when the sentence, as a complete referential object, made of an ordered succession of networks, and the sentence, as a processed linguistic event, coexist. The paradox presupposes an incomplete description is enough of a reference — something cognition, and physics itself, does not allow. 

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8. From Logic to Physics:

One can make an exact analogy between what neurology has to do with what Quantum Measurement Theory does: contrarily to repute, QMT is very explicit and time ordered. Physicist Michel Devoret, who has conducted experiments describing the SEQUENTIAL nature of fundamental processes in Quantum Physics received the Nobel Prize in 2025; see note [5]

Seen in this light, the Liar Paradox is not just a problem of language. It points to the deep structure of how meaning — and perhaps even reality — is generated. Logic, in its living, physical form, is a process of singularization: local stabilization within a sea of potentialities. Each act of interpretation collapses a cloud of possible meanings into one coherent local state made of a set of connected networks.

In formal logic, paradoxes appear when we deny this locality and demand total, self-encompassing closure. But such closure contradicts the physical nature of cognition.

Quantum Measurement Theory, QMT, analyzes the most exquisite, precise fundamental processes.  Bohr insisted that an experiment could not be separated from the apparatus inside which it happened [5]. Bohr was correct and Einstein took a while to wrap his mind around it. Looking at it a la Tarski, the apparatus ML is the Meta Language of the experiment L (Wigner later introduced the “Wigner Friend Paradox” which is akin to looking at MML, not just ML… In this case ML is the Schrodinger Cat…)

In Quantum Mechanics, time acts as a “one parameter group of transformations” (and not as a sort of space!)… So the Quantum Logic unfolds in a timely manner (time and space are not equivalent dimensions in QMT). Truth, like measurement, or causality in physics, is an act that takes place locally in time and context.

Instead, the Liar Paradox, as initially posited, is analogous to an experiment without an apparatus, not anything being measured inside. Russell’s Vicious Circle Principle is akin to not making an experiment while the first one is not completed, and Tarski’s Meta is akin to adding an apparatus, the context, the Meta Language ML (but then one runs into the “Wigner Friend” hierarchy problem).

There is no “view from nowhere” from which all propositions can be surveyed at once as Propositional Logic pretends there is, causing the LP. 

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9. Sequential Logic Handles the No-No Paradox Beautifully And Practically: It’s A Dog Collar!

The NO-NO paradox involves two mutually referential statements rather than a single self-referential one. It was originally discussed in medieval logic by 14th century thinkers like Thomas Bradwardine and John Buridan. 

So consider two propositions, Left and Right, labelled L and R, which contradict each other in an apparent vicious circle. Bertrand Russell and his predecessors from the 13th century would say that this contradicts the Vicious Circle Principle. Left L sends to Right R by asserting that R is true. R says that L is false.

Sequential Logic handles this by observing that reading L activates at the very least a neural network for reading and another expressing veracity.. Suppose veracity shows up as a green color painting L. Then R is read and claims L is false that is red… But L was green before R was read. Now L is supposed to be red , that is false, if R is green and true, as it is. So then L turns red, that is false. 

Then L says that R is false, that is red so then L is green, and so on. So we just have a dog collar whose color changes periodically from green (true) to red (false), according to neurological time of reading one then the other (the objection that physical time can be made as short as possible is not true in neurology and not even in physics as Planck’s Time sets a lower bound). 

[Other resolutions not using Sequential Logic. Tarski would assign L to a language L and then assign R to a metalanguage, ML. But notice that this is what Sequential Logic does, without metalanguage! 

Saul Kripke proposed to define a single language with a single truth predicate in which anything goes but in which sentences not caught up in this construction, neither true (+1) or false (-1) have an intermediate truth value… In the preceding case each box is half true, half false, and thus has truth value zero.]

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10. Conclusion

The Liar Paradox dissolves once we see that “this sentence” does not designate a static self-reference but a temporal instruction within a sequential interpretive system. The paradox is an illusion born of treating time-bound, embodied logic as if it were timeless and disembodied. Logic in the brain — and perhaps in nature itself — is procedural, not total; contextual. Logic is not absolute, standing outside the material quantum world, like Plato’s “forms” [6]. Just as time orders Quantum Processes, time ought to order logic too, time may be local, but it remains the most basic causality…

Truth emerges from the unfolding of communication, not from standing outside it. The Liar Paradox, then, is not a contradiction in truth itself, but a reminder that truth, like understanding, only exists locally within the flow of interpretation [this is how Category Theory from the 1950s defines truth; 7].

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10. Epilogue SEQUENTIAL LOGIC?: 

One may wonder why it took 25 centuries to come up with the obvious solution to the Liar Paradox by making logic sequential. I do not doubt that many peasants, exposed to it, found the same solution as yours truly. But they were not in academia, which installed the LP as a genuine problem (reflecting an academia serving God and the monarch; those who disagreed may have shared Archimedes’, Hypatia’s or Boethius’ fates, top thinkers who were killed precisely for the reason of their superior intelligence).. 

My own approach is to try to understand how logic and truth work, especially in the source of all we know, the human brain (independently of any other consideration). 

Sequential and local are the important notions in nature, physics, in contrast to instantaneous and global: I had already encountered that methodological confrontation in the Foundation of Quantum Physics, and it is prominent in the invention of Relativity by H. Lorentz and Henri Poincaré, followed by Einstein… An engineer, Poincaré encountered the solution to non-simultaneity in the electromagnetic transmissions of the telegraph, generalizing it to LOCAL TIME theory in Maxwell equations. This locality and delay found in local time theory has been experimentally demonstrated to the greatest extent (GPS depends on it). Locality and delay found with local time is generalized in sequential logic [8].  

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The Liar Paradox does not really resolve by evading self-reference through meta-languages, as this creates an infinite hierarchy of “meta”. Instead we proposed to dissolve the Liar Paradox by recognizing that reference itself is temporal, procedural, explicit and embodied. Logic, in the brain and in the world, operates finitely and sequentially — from interaction to interaction, from potential to singularization.

As it is, the Liar Paradox utterance is just incomplete. Neither true, nor false, just literally nonsensical.

The Liar Paradox only looks like a paradox if we freeze logic into a sculpture. Once we let logic move—as it actually does in brains—the puzzle evaporates. “This sentence is false” is not a self-referential jewel of timeless contradiction; it is a failed temporal program. It begins an instruction (“designate something”) and then never supplies the object to complete it. A brain cannot refer backward in time to repair an incomplete reference any more than a quantum experiment can measure an outcome before the apparatus exists.

The Liar Paradox is like a movie whose last frame is shown before the first one finishes rendering. Static logic treats this as a profound contradiction. Sequential logic recognizes it as a simple glitch: the sentence is not false, nor true, but simply not an executable sequence.

When logic is allowed to have a before and an after, the Liar disappears—not solved, but dissolved.

Patrice Ayme .  

Remark: This sequential, embodied logic can be tied into the broader physical theory of photon and particle singularization and TAU collapse (SQPR) — i.e., a possible analogy between logical and physical “resolution events”… (Another time!)

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[1] Tarski was honest enough to recognize he was inspired by Russell’s Theory of Types. Russell “solves” the Liar by noticing that it has a hidden quantifier (“there exists” (a sentence, etc.)) when interpreted as a truth value (or lack thereof). So the sentence itself and its interpretation as truth value are of different “types”. In other words, L and ML, language and metalanguage… From my point of view this is lots of sophistry hiding a much simpler truth…

Amusingly, the VCP was invented to deny the possibility of what made Russell famous: Russell demolished Fregge with the set S whose elements are not elements of S, a version of the Liar Paradox…

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[2] Standard Zermelo Fraenkel Set Theory, especially with AC, the Axiom Of Choice (“ZFC) assumes the Totality Axiom (my label). TA is necessary to express the AC.

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[3] A first sight, a Quantum Computer, by using Quantum Entanglement, in the classical Copenhagen Interpretation of the Quantum (CIQ) is able to do operations simultaneously at a distance. SQPR denies this… But, in practice, the superluminal speeds involved are so great that the practical difference between CIQ and SQPR will be neglectable.

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[4] Quantum Mechanics treats time as a one parameter group of transformation (fundamentally differently from space, and that’s a crucial difficulty in fusioning Naive Relativity Theory and QM) Logic, as it enfolds in the brain, activating a succession of neural network is also a one parameter group of transformations best identified as time.

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[5] Quantum Measurement proceeds with a wavefunction W submitted to a field F in space, the whole thing evolving in time. “This sentence” means W without designating F and then reaching a conclusion, the measurement act, “false”.  In the process of reaching the conclusion, before collapse, we have an implicit time evolution. So we see that the QM is in exact analogy with what neurology does and has to do.

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[6] Quantum Collapse (aka “Decoherence) is part of QM, but violates causality. This is (one of the reasons) why SQPR (Sub Quantum Physical Reality) was proposed: in SQPR, the Quantum Collapse is causal and progressive. Michel Devoret agrees, and gets the Physics Nobel in 2025…

The Liar Paradox also violates causality… As explained above. 

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[7] One can define truth using Category Theory (many call CT “Abstract Nonsense”, but it is a monument of modern math). Truth in CT is very abstract and sounds quite nonsensical… However, the CT truth is LOCAl, and very much compatible with the spirit of what is suggested in the present essay (which, being thoroughly descriptive, can be read again…).

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[8] The proposed Sub Quantum Physical Reality (SQPR) generalizes locality and delay to Quantum Physics… Immediately predicting Dark Matter, etc… It would be most economical logically if it were true…

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[9] The Revision Theory of Truth (developed by Belnap, Gupta, Herzberger).

• Focus: While Kripke seeks a stable, single fixed-point definition of truth, Revision Theory focuses on the sequence of evaluation itself.
• Outcome for L/R: In Revision Theory, the Liar cycle (or my L/R cycle) is defined as pathological because its truth value never stabilizes. It cycles indefinitely between T and F across the revision stages, just as my “dog collar” changes color. This theory affirms the temporal oscillation as the very nature of the paradox, rather than resolving it with a third value…. But it considers it pathological whereas in Sequential Logic, it’s just natural, and nothing more upsetting than waves coming and going on a shore….

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Historical Note: The Liar Paradox was extensively addressed in the Thirteenth and Fourteenth Centuries, and, unbeknownst to most, those studies were partly duplicated in the 20th century.

Jean Buridan was a 14th-century logician and top philosopher and physicist who discovered inertia, momentum, and the first and second laws of mechanics, plus heliocentrism and geometric calculus (while making fun of the Church). Buridan solved the Liar Paradox to his satisfaction with his innovation of the principle of “virtual implication“.

This principle states that every sentence implicitly asserts its own truth in addition to its explicit content (this principle is explicit in Tarski ML). Buridan says that the explicit claim “A is false” contradicts the virtual implication (“A is true”). This contradiction makes the overall content false, according to him.

Buridan was far from my argument (he anticipated Tarski instead)

In the  century preceding Buridan, Scholastic logicians (Quaestionists) were more in the mood of the present approach. The dominant early theories were Cassatio (cancel the argument) and Restrictio (restrict possible formulas). The Cassationists’ solution is the closest to the one exposed in the present essay; for the Cassationists, the LP “says nothing” (nihil dicit). Buridan made fun of them later…

These theories solved the paradox by declaring the Liar Sentence defective, arguing it was not a genuine proposition or that its self-reference was logically forbidden (here we see Russell’s VCP). The present argument is that the LP says nothing, because the designating sequence (a neural network) is launched… but then does not designate anything: the next neural layer is missing (to use Neural Network theory).

Cassationists pointed out semantic defectiveness in the LP. The present essay’s modern neuroscientific argument argues that the failure is even more basic: the sequential neural network for designation is launched but fails to designate an object.It points out at dynamic incompleteness in the context of neurology, or neural networks. So the medieval intuition of some Medieval logicians is grounded in cognitive structure. .***

Full disclosure: 1) I elaborated my own theory before I became cognizant of the Cassationists’ existence. Considering the simplicity of my argument, though, I was not surprised that some Medieval Logicians found it… (In similar historical research recently I found the proof of  F = ma for gravitation in Buridan’s work!, 300 years before Newton).

2) The theory above may seem close to Temporal Logic (Jerzy Łoś, Prior, Kamp, etc.). There are at least two dozen types of Temporal Logics. Prior’s tense logic (1950s–60s) represents reasoning in time, using operators like “it will be the case that” (F), “it was the case that” (P). Later, Hans Kamp (1971) and van Benthem (1970s–80s) extended this to model natural language semantics.

Connection to Patrice Ayme’s theory:

• Temporal logic treats truth values as indexed to moments in time.

   

• PA’s “logic unfolds in the brain, sequentially and causally” seems reminiscent of that: the evaluation of a statement depends on where and when in the neural process one is, which neural network is actually at work — the “now” of cognitive unfolding in the brain (in the brain this is helped by the fact that biological neural networks operate by separated action potentials, giving a real ticking clock, differently from artificial neural networks, which operate continuously).

   

• However, Patrice Ayme goes further: for PA, time is not just a parameter of logic; it is the substrate of reference itself. The Liar Paradox misfires because it assumes a timeless self-reference, while cognition is temporally constructed as it jumps from neural network to neural network. → In Prior or Kamp, truth varies over time; in PA’s model, truth emerges through time.

There are many other types of logic (van Benthem, dynamic epistemic logic) to check if one wants to make sure that the preceding is really innovative. These comparisons are all very technical, and I may put them in a separate paper (if I find the time).  

Patrice Ayme