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?
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