I think among the fundamental things that logic cannot know are 'SENSING' (and different sensations), 'INTELLIGENCE', 'ABILITY TO MOVE', and maybe 'FREE WILL' (see also 'property').

But for example logic knows the POSSIBILITIES of the existence of matter and motion (or a deterministic causal mechanical system).

For reference, that 'the sensation of blue cannot be fully explained by any mechanism', maybe 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

Another example that logic cannot know: Either the universe has EXISTED FROM AN INFINITE PAST, or the universe CAME INTO EXISTENCE FROM NOTHING. Both cases are, at least in human understanding, ILLOGICAL.

For better explanation, see my question: Is an infinite passed time about universe a proof of 'violation of logic' or magic?

Another example MAY be 'elementary particles' and 'fundamental forces'.

I am looking for strange things, things that logic doesn'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 know or see?

  • 2
    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. Jun 6 at 20:48
  • 4
    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
    Jun 6 at 20:49
  • What does it mean that "logic see" something? In what sense are you using the verb "seeing"? Jun 9 at 6:58
  • 1
    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
    Jul 26 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. Jul 28 at 4:46

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.

  • 3
    Up to the Scientific Revolution, the supernatural was probably the primary focus of logic. Logic is not limited to the natural. Jun 7 at 8:05
  • Thakns for the comment.
    – Alex
    Jun 7 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
    Jun 7 at 20:47
  • 1
    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
    Jun 7 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. Jun 8 at 9:16

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

  • You are describing experience or knowledge, not logic. Jun 7 at 8:05

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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