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For beginning, it is arguable that for example logic knows the POSSIBILITY of the existence of matter and the result of a 'deterministic' 'causal' 'mechanical' system.

PART 2: FEATURES OF LIFE OR SOUL

But I think among the fundamental things that logic cannot know are the possibilities of 'ABILITY TO MOVE', 'SENSING' (and different sensations), 'INTELLIGENCE', and maybe 'free will' and 'memory'.

Consider 'ability to move' from intelligence or free will. We can create a physically-indeterministic physical 'motion or change' from nothing physical. We eat food, but by our logic there cannot be any PHYSICAL relation between the food we eat and our motions, because when we move from intelligence or free will, our motions are 'physically' 'indeterministic' (since our intelligence or free will is physically 'indeterministic'). Similar to what our logic says about 'conservation of amount of matter', MAYBE it also says about 'conservation of motion' (see also 'conservation of energy'). Therefore our ability to move MAY be even a proof that 'violation of logic' or magic is possible.

Note I: Regardless of we think that our motions come from the motion of the food, or even the matter of the food (see 'theory of mass-energy conversion'), by our logic we find out that it is impossible that a pre-existing thing is responsible for an indeterministic motion; that for a physically indeterministic motion to happen at the right time and at the right direction, we or something must be able of creating or causing a completely new physically-indeterministic physical change (from no pre-existing physical thing).

Then consider mind, especially an indeterministic mind. We may think that mind is a 'property' of matter; but an open question may be that 'is a non-physical property any less strange than a non-physical substance?'; and then another question: Is mind, especially an INDETERMINISTIC INTELLIGENCE, a proof of violation of logic?

Another thing that may be strange is that we do not have one sensation, but we have different-quality sensations (or qualia) such as vision and hearing.

Also logic may not say anything about for example our 'speed of thinking', since for example there is no 'physical' relation between the speed of our intelligence and our amount of matter. There cannot be any physical law, and we don't know if there is any metaphysical law about it. (Talking of speed, though it is not a mental speed, I wondered by 'cpu clock speed' that is for example 1/1,000,000,000 of a second.)

Note II: 'why is there anything at all?' may be a question that logic cannot answer.

Note III: Also I think by the definition of 'fundamental forces' in physics, they are created from nothing physical.

Note IV: If our intelligence is physically indeterministic, then it may mean that 'artificial intelligence' is impossible.


PART 3: A PROBLEM WITH 'TIME'

By the existence of time and universe, we have a number of alternatives. Two of them are:

Time existed form an infinite past, and universe existed from an infinite past (TU).

Time existed form an infinite past, and universe existed from a finite past (Tu).

All alternatives are by our logic impossible.

For more information, see my question: Is a finite or infinite past about time or universe a sign of 'violation of logic' or magic?.

Note V: We may first, mistake; for example in that 'it is possible that an infinite summation does not get over a particular finite number'; but if we know that something is wrong and search for it, in that example, we can find our mistake and the correct answer. Also we may not be able to solve 'mathematically complicated' problems. But the point is that, despite that we know something is wrong, and despite that we only assume 'uncomplicated' things, we cannot find what is wrong in this time-universe problem; and therefore this item 'may' show that there is something 'fundamental' we don't know.


PART 4: OTHER THINGS

Considering that there an infinite number of numbers between two numbers, a progressive motion of a certain length must pass an infinite number of lengths, which is by our logic impossible. See 'zeno paradox of motion'.


Another example that I cannot understand is that, considering that there an infinite numbers between two numbers, why an infinite summation of equal terms cannot produce a finite result? why getting smaller is the only way?


I am not aware of any logical reason that space should be three-dimensional. (Why for example it is not one-dimensional or five-dimensional?)


Although Wikipedia said that the 'law of the lever' has a logical proof, I think I have not found any correct proof, and it may be another thing that our logic fundamentally doesn't explain.


Another example is that 'fundamental interactions' are not a part of logic. (Is a magnet that attracts and repels specific poles alive or metaphysical?)


Another example is that I think the force and momentum definitions and formulas are not complete: Imagine someone is pressing a cube with his thumb and index finger, and there is no motion in the system. We cannot justify that pressure while nothing moves, because the force and momentum definitions and formulas are based on motion.


BACK TO QUESTION

I am looking for strange things, things that logic can't know, metaphysics, another dimension, the illogical, things that violate the law of logic or mathematics, or magic, especially those that are FUNDAMENTAL.

Note VI: Maybe we should define 'original logic' and 'new logic' and distinguish between them.

What things can't logic or doesn't our logic know, see or explain?

REFERENCES

For a defense of 'mind' in the 'mind-body problem', or that 'the sensation of color cannot be fully explained by any mechanism', see 'HARD PROBLEM OF CONSCIOUSNESS', or what the philosopher J.S. Mill wrote (under the title 'Attempts to resolve radically different phenomena into the same' under the title 'Fallacies of generalization'):

Now I am far from pretending that it may not be capable of proof, or that it is not an important addition to our knowledge if proved, that certain motions in the particles of bodies are the conditions of the production of heat or light; that certain assignable physical modifications of the nerves may be the conditions not only of our sensations or emotions, but even of our thoughts; that certain mechanical and chemical conditions may, in the order of nature, be sufficient to determine to action the physiological laws of life. All I insist upon, in common with every thinker who entertains any clear idea of the logic of science, is, that it shall not be supposed that by proving these things one step would be made towards a real explanation of heat, light, or sensation; or that the generic peculiarity of those phenomena can be in the least degree evaded by any such discoveries, however well established. Let it be shown, for instance, that the most complex series of physical causes and effects succeed one another in the eye and in the brain to produce a sensation of colour; rays falling on the eye, refracted, converging, crossing one another, making an inverted image on the retina, and after this a motion—let it be a vibration, or a rush of nervous fluid, or whatever else you are pleased to suppose, along the optic nerve—a propagation of this motion to the brain itself, and as many more different motions as you choose; still, at the end of these motions, there is something which is not motion, there is a feeling or sensation of colour. Whatever number of motions we may be able to interpolate, and whether they be real or imaginary, we shall still find, at the end of the series, a motion antecedent and a colour consequent. The mode in which any one of the motions produces the next, may possibly be susceptible of explanation by some general law of motion: but the mode in which the last motion produces the sensation of colour, cannot be explained by any law of motion; it is the law of colour: which is, and must always remain, a peculiar thing. Where our consciousness recognises between two phenomena an inherent distinction; where we are sensible of a difference which is not merely of degree, and feel that no adding one of the phenomena to itself would produce the other; any theory which attempts to bring either under the laws of the other must be false; though a theory which merely treats the one as a cause or condition of the other, may possibly be true.

— A System Of Logic (1843), Book V, Chapter V, section 3

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    There is nothing in logic that says the universe has existed from an infinite past. Also, you will never find anything that directly violates the laws of logic or mathematics because those laws are necessary--they apply to everything. However, what you may be able to find is exceptions based on factors that were not considered. When such things are found, the laws of logic or mathematics are modified to bring them into line again, so there aren't any currently known. If there were, the laws would be modified to remove them. Commented Jun 6, 2021 at 20:48
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    Logic cannot see or predict anything at all, that is not its function. Its function is to validly derive conclusions from premises, seeing and predicting comes from premises. Whether the universe existed for infinite time before now or not is a question of physics, not of logic. And the meaning of 'infinite' does not make it impossible. Logic has no problem whatsover with the infinite, which is why it is used ubiquitously in mathematics.
    – Conifold
    Commented Jun 6, 2021 at 20:49
  • What does it mean that "logic see" something? In what sense are you using the verb "seeing"? Commented Jun 9, 2021 at 6:58
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    Mill may be using 'logic' somewhat differently here from its modern sense. Based on the passage quoted, he is saying that no amount of scientific understanding of the physics or chemistry of light and nerves, etc., can help to explain our subjective experiences. This is what we now call the problem of qualia. It is still a live philosophical issue, but we don't associate it with logic, which now has a much narrower sense.
    – Bumble
    Commented Jul 26, 2021 at 16:34
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    @CriglCragl, I've tried as hard as I can to read your comment charitably, but as far as I can tell, you just don't know what "the laws of logic" means or understand what it would mean to violate the laws of logic. People may not be logical, but that doesn't mean that they violate the laws of logic. Axioms are not part of the laws of logic. The existence of multiple systems of logic does not mean that some must be wrong any more than the different branches of mathematics means that some branches must be wrong. Commented Jul 28, 2021 at 4:46

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The most basic answer to your question is that Logic is unable to predict or know about the Metaphysical/Supernatural. The very words, "Metaphysical" and "Supernatural", are, definitionally, "beyond" the physical or natural world. Logic, both historically and contemporarily, was and is, an intellectual system, that can affirm the validity of a particular statement through the system of proof and disproof.

Geometric Theorems, Archeology, as well as a variety and diversity of Scientific experiments, are empirically and epistemologically provable intellectual exercises, because they often provide us with a tangible and perceptible result. How can one possibly begin to prove the existence or nonexistence of something that is definitionally and inherently, NOT subordinate to the laws of nature? Even the most brilliant and most erudite of humans-(then and now), were and are still unable to transcend....the transcendental; it is simply "out of our league".

In other words, let Logic continue to function as the most sophisticated intellectual tool for providing us with a more coherent understanding of complex matters. However, Logic's primary aim, should remain within the parameters of reality....and NOT beyond.

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    Up to the Scientific Revolution, the supernatural was probably the primary focus of logic. Logic is not limited to the natural. Commented Jun 7, 2021 at 8:05
  • Thakns for the comment.
    – Alex
    Commented Jun 7, 2021 at 20:34
  • I am a former History Instructor and can certainly appreciate what you are saying; however, I am not sure that your statement is totally correct. For example, if you take Aristotle-(who is often associated as the Founder of "Logic"), while he tried-(rather erroneously, in my opinion), to use Logic to prove the existence of an "Unmoved Mover"-(i.e. The Metaphysical), he and his likely team of Student Researchers, also produced a vast literature that examined a variety of non-Supernatural oriented disciplines from a purely Logical perspective.
    – Alex
    Commented Jun 7, 2021 at 20:47
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    In other words, Aristotle, his Student Researchers, as well as generations of Aristotelians-(with the notable exceptions of Aquinas, Maimonides and Averroes), have utilized the system of Logic to better understand terrestrially oriented matters.
    – Alex
    Commented Jun 7, 2021 at 20:55
  • good points. I was improperly joining metaphysics and the supernatural under the single term "supernatural". Sometimes brevity is the enemy of clarity. Commented Jun 8, 2021 at 9:16
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just personal thought* logic in my opinion are system to make sense of something based on information you already know beforehand. you need to know the cause/premise of something that's happening to cause it be logical ,for example: -someone who have a dog will know why their dog bark compared to just random stranger -a magician know how their trick are performed , while the watcher most often logically don't know how to do the trick ,so anything that logic cannot predict are something that doesn't have accurate information of why it's happening

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  • You are describing experience or knowledge, not logic. Commented Jun 7, 2021 at 8:05
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The answer depends heavily on the meaning of "logic," a word that is frustratingly difficult to define with logical phrasings. Arguably any sentence defining a truth can be added as an axiom to a system of logic to make it provable. .

However, if I pick and choose word choices from the quote you provide, I notice an early predilection towards the word "proof." Personally, I recognize this as something I consider a fundamental aspect of logic, which is the ability to prove knowledge using other knowledge.

I have to take a bit of a circuitous approach. Best to cut off the known false paths first, and then start looking at the tricky fundamentals that may make one bolt down said paths if they aren't secured first

In mathematics, we have Model Theory. In model theory, we talk of a model, ℳ, which "entails" that some sentence is true. We might write ℳ⊨"The sky is blue" to state that our model asserts that "the sky is blue" is a true sentence. This is a particularly convenient theory for this question because model theory assumes the meaning of the logical symbols has their accepted pre-defined meaning. It also connects the concepts of something being true to the idea of a sentence being a "true sentence." This is necessary as logic operates on symbols. If we cannot put something in symbols, it cannot be acted upon by logic. Thus, as long as your definition of "logic" is in line with matheamtics' definition of "logic," this theory will be a good topic for exploration.

We also have the idea of implication, ⊢. This is the concept of a "logical proof." {"a = 2", "2 is even", "adding 2 to an even number results in an even number"} ⊢ "a+2 is even". You can prove it symbolically, without having to consider the meaning.

Note that there are two different symbols here. We say a logical system L is sound with respect to a model ℳ if, given any sentence s, (LS⊢s)→(ℳ⊨s). In other words, a logical system is sound if the logical system proves something to be true, then the model entails that it really is true. We then say a logical system is complete if (ℳ⊨s)→(LS⊢s). In other words, a logical system is complete if it can prove every true statement in the model.

I call these out because mathematicians have spent a great deal of effort on these topics. It shows that the answer you seek is almost impossible to answer in general. You have to pick the model and the logical system. Speaking to the model "reality" is extremely difficult, but we can speak to much simpler models, such as "all true statements in arithmetic." And we can speak to popular logic systems, such as First Order Logic. And when we do, we come up against undesirability. Any First Order Logic that strives to describe the laws of arithmetic must be incomplete. There must be something that cannot be proven. And there's similarily frustrating phrasings about other popular systems of logic.

Even arithmetic itself defies logic in its own peculiar way. So its easy to suggest that reality may have something which is itself defying logic. This is hardly a deductive logical proof, but I'd argue its an interesting line to think about logically, and see where it leads.

Now these theories focus on proving there is something that cannot be known/seen by logic. You ask for something fundamental. That is another tricky word to define. Rice's Theorem might suggest we can't even phrase that concept of "fundamental" in a computable way. But of interest may be the Aggripan Trilemma, which suggests there are only three ways to complete a proof:

  • The circular argument, in which the proof of some proposition is supported only by that proposition
  • The regressive argument, in which each proof requires a further proof, ad infinitum
  • The dogmatic argument, which rests on accepted precepts which are merely asserted rather than defended

Of these, logic typically skirts around circular arguments and regressive arguments, as they are very difficult to work into consistent theories. Dogmatic arguments are most common, known as "axioms" in the world of logic.

So your question would be is there any axiom which cannot be admitted into a logical system, or a statement which requires "too many" axioms for your logical system (recursively enumerable is a popular limit).

And that question really is personal. What languages do you admit into your logic? If you only admit things that can be defined this way, then you will never find a fundamental thing which cannot be penned in logic (you will still face the wrath of incompleteness as you look beyond fundamental statements). On the other hand, if you look at other people's language, the "dao" is famously ineffable. Those who believe in that concept believe it cannot possibly be put into words/symbols. And if you cannot put it in symbols, then you cannot apply logic to it.

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The only thing that one can know for certain is what one is experiencing, and only at the moment that it is experienced.

This excludes any certainty about the interpretation of the experience (because we might just be a brain in a jar of some science experiment in the future). It excludes knowing what we experienced in the past (since our memory may be faulty). It excludes knowing the result of any logical deduction (since we might have made a mistake).

All other "knowledge" is based on varying degrees of certainty less than absolute certainty.

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Oh yes, it is true that logic doesn't know many things, but this is a trivial claim as logic it not the sort of thing that could possible know anything.

Presumably, you use the word "logic" to mean something like the human rational mind. Yet, it is also trivial to claim that we cannot use rationality to solve every problem and answer all questions. We cannot explain consciousness (in the sense of subjective experience) and we cannot explain reality taken as a whole. So what?

This is nothing magical, metaphysical or somehow mistic about explaining things. Logic and perception are natural abilities. They demonstrably help us survive, prosper and reproduce in our natural environment, not in any imaginary world. They are selective advantages, which explains that the human species still exists and even exists at all. It is irresponsible to dismiss logic on the ground that we are unable to use it to explain why reality exists. Modern civilisation was only possible because of the development of rational disciplines such as Ancient Greek philosophy, mathematics, the sciences and engineering, and rational here means logic plus empirical facts. Without rationality, there would simply be no civilisation as we know it today.

Yes, rationality is limited. It is even extremely limited. It is more limited than most people realise. Yet, it appears that we don't have anything else we could use instead to do the things we want to do. We do plenty of things without resorting to rationality. Indeed, we do most of the things we do in the course of our lives without using rationality. There is nothing remotely surprising in that, man is an animal and also the only rational animal, which means that all other animals are not rational, and yet they survive and prosper and reproduce, which proves survival, prosperity and reproduction do not require rationality. However, it is also true that no other animal can do what we do: science, engineering, philosophy, mathematics and more. So rationality works, it works remarkably well, but it would be wishful to expect that it should be able to square the circle or dismiss it because it doesn't.

We can only expect to be able to use logic to solve problems of the real world. Accordingly, the proof is in the pudding. Demonstrate that the law of the lever is wrong by designing a machine that works on the basis of a different law of the lever. Short of that, it is simply wishful thinking to quip that it may be an example of the shortcoming of logic.

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I understand better your question looking at it from a psychologic point of view. Husserl spoke about the intentionality of conscience: conscience is always conscience of something. Carl Jung's phenomenology of conscience goes a step further. He says that when we are conscious of something we are conscious through a certain type of conscience: through thinking, feeling, sensation or intuition. If I am using thinking I am looking at logic and causality. I am not evaluating if something is good or bad and what that thing means to me. That means I am using feeling. I cannot do both things at the same time. When I am using sensation I am looking at details. Intuition is related to the big picture. Nietzsche said that philosophy is autobiography, in the sense that it expresses in a deep unconscious way the personality of the author. Some of us are better at thinking and others at feeling. Many strata of our society overvalue thinking. Logic cannot deal with sensations and values. Many ethical systems are attempts to deal with values using only the thinking function. That doesn't work well. The way I read your statements you are not vary happy with that overvaluation. That makes sense to me.

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