Can something be true if it is not logical? Can something be true if it does not follow logic? if I can't deduce it logically? or if it contradicts logic?
In other words: if someone says statement X is true, can it be that X contains some ilogical constructs, and still remains true?
Or again in other words if I find a logical flaw in a sentence which someone says, can it be still true?
Firstly, in a strictly formal setting, we note that Gödel's first incompleteness theorem tells us that truth is not reducible to proof, so there are many truths which are not derivable.
In a formal setting, we chose our axioms because we believe them to be self-evident truths - i.e., true for no (logical) reason. But why do we assume that only self-evident truths are not derivable.
More generally, beyond the formality of mathematics, if one accepts that nature includes random processes, then such processes may provide examples of brute facts which are true simply because they are true and for no other (logical) reason. For example, if one accepts that the human evolutionary process is driven by random mutations of our genetic material, then we are who and what we are for no logical reason. It is true that humans exist on planet Earth, but it is not a logical necessity and it could have been otherwise.
Regarding true statements that contradict logic, one might argue that quantum superpositioning may provide examples. Superpositioning is a phenomenon that is supported by experimental evidence, but it certainly appears to be illogical to assert that a particle can be simultaneously in two different states or two different locations. Having said that, it may follow logically from the formalism of quantum theory that superpositioning is a logical necessity. I'm not a physicist, so I'm not entirely sure.
Perhaps you mean: "Can there be undemonstratively true truths?"
Aristotle, when resolving the infinite regress problem, shows, against those who believed no or all truths are demonstrable, that there are some truths which cannot be demonstrated to be true. Thus, Aristotle could be considered a precursor to Gödel.
See
Aristotle's Posterior Analytics bk. 1 ch. 3-4 (72b5-24)
St. Thomas Aquinas's Expositio Posteriorum, lib. 1 l. 7 et 8
Truth is a property of propositions. A proposition is true if it refers to a fact. Hence truth is a relation between a proposition, a sentence from language, and a fact, a component of the real world.
This kind of truth bears no relation to logic. Whether a single proposition is true or not, cannot be decided by logic. As an example take the proposition "On 1.1.2016 it rained in Manhattan."
On the other hand - and now without looking at the physical world -, one can investigate logical relations between different propositions. E.g., if a proposition is true for all objects of a certain kind, then the proposition remains true when restricted to one particular object. Textbook example: "All humans are mortal." "Socrates is a human." Hence "Socrates is mortal."
According to classical logic it is not possible that two propositions are true if they contradict each other (law of non-contradiction).
Can something be true ... if I can't deduce it logically
Logic in one sense, is truth-preserving; hence it isn't the art or techne by which finds out new truths, or how truths we now hold were first discovered.
This, for example, in Badious philosophy is the role of Love, Art, Science and Politics.
Can something be true ... if it's illogical ... or contradictory
This is the position of dialethism, which holds that there are such things as true contradictions; a position which is more common in Eastern philosophy than Western - for example, see Nagarjunas use of the cakaskoti however Graham Priest is a modern exponent of this, which should be carefully distinguished from his advocacy of non-classical logics, including notions of paraconsistency.
It can be true even if it does not follow logic. Because logic does not cover all aspect of life. The best explanation for logic is, discovering links between two things.
Let us say there exist some thing which you never experienced in real life.
How will you treat the event? Do you really do all logical mind kung fu or treat it as broadening of your awareness?
Logic won't cover every thing unless you know everything which is kind of paradox in itself.
It is the main reason why Indian philosophies look different from western point of view.
It is perfectly possible to come to truthful conclusions while the logical arguments upon which those conclusions are flawed.
Herebelow are three examples of basic propositional logic where the conclusion is correct but something is wrong in the logical argumentation for it.
Sven is a man.
All men have mustaches.
The logical conclusion of propositions (1) and (2) would be that Sven has a mustache.
Now, Sven indeed does happen to have a mustache. However, not all men have mustaches. In this example, proposition (2) is wrong, yet the conclusion based on both propositions happens to be correct.
Consider the following propositions :
Heidi is a man.
All men have breasts.
The logical conclusion of propositions (1) and (2) would be that Heidi has breasts.
Now, Heidi indeed does happen to have breasts. However, she isn't a man. Neither do all men have breasts. In this example, both proposition (1) and (2) are wrong, yet the conclusion based on those propositions happens to be correct.
Sven is a man.
Some men have mustaches
You cannot conclude based on propositions (1) and (2) that Sven has a mustache. Drawing that conclusion based only on those two propositions would be a logical error. However, it just happens to be the case that Sven has a mustache.
In the popular example of "Schrödinger's cat" (see citation below), both polemic states of being alive and dead are accepted as true. Schrödinger presents a situation that, despite the seemingly inherent fact that if one is alive they are not dead (and vice versa), in the mathematical interpretation of a physical experiment you can have a "justified true belief" for both possibilities simultaneously. It is only until one opens the box that one becomes true and the other false.
This epistemological issue exists in the relationship to "Truth" and "Knowledge". While A and "not-A" cannot both be observably true, they can theoretically (mathematically) exist simultaneously given a closed system without an observer. This same issue is brought to light with similar thought experiments.
A man stands in a field full of barn facades. In the field there is one real (non-facade) barn. The man, under the impression that all the barns are real, points to the real barn and says, "That is a barn".
Though the man has a "true belief", the justification of that belief complicates the matter, because he has a false sense of the accuracy of his choice. However it does not in the end diminish the accuracy of his statement. Therefore, even with a fallacious premise ("the man believing all the barns were real") the truth of his statement remains intact. Just as the truth of the cat being simultaneous dead and alive remains intact, until an observer opens the box. These are examples of Logical Truths despite false premises.
...ridiculous cases [like, a] cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.
~ Erwin Schrödinger, Die gegenwärtige Situation in der Quantenmechanik (The present situation in quantum mechanics), Naturwissenschaften (translated by John D. Trimmer in Proceedings of the American Philosophical Society)
What kind of logic? There are many logical systems, devised by human beings to think about and understand different kids of phenomena or experience. For example, the two valued Aristotelian logic. That logic is inadequate to understand the sub-atomic phenomena of quantum physics. Even though sub-atomic phenomena are real, they seem illogical to Aristotelian mind.
here is a religious perspective:
"...Jews demand signs and Greeks search for wisdom, but we preach Christ crucified, a stumbling block to Jews and foolishness to Gentiles, ..."
there appears to be something goofy about seeking the salvation of humankind in that of sacrifice. just doesn't seem logical.
