For beginning, 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' (see also 'substance theory').
But for example logic knows the possibilities of the existence of matter and motion (or a 'deterministic' 'causal' mechanical system).
By the existence of universe and time, we have two alternatives:
Alternative 1) Universe exists from a finite past = It came into existence from nothing.
Alternative 2) Universe always existed from an infinite past.
Both 1 and 2 alternatives, by our assumptions or logic, are impossible.
Note: It is arguable that, while for some things that may seem strange, it is only a matter of 'reducing' a 'complexity' to see they are not strange, this item is not a problem of reduction, and thus it proves that either one of our 'fundamental' assumptions is wrong, or the 'violation of logic' or magic is possible.
For more information, see my question: Is a finite or infinite past about universe a sign of 'violation of logic' or magic?.
Although Wikipedia said that the 'law of the lever' has a logical proof, I think I have not found any correct proof (in the internet), and it may be another thing that our logic fundamentally doesn't explain.
I am looking for things that logic can't know, things that violate the law of logic or mathematics, the illogical, or magic, especially those that are FUNDAMENTAL.
What things can't logic or doesn't our logic know, see or explain?
For reference 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:
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