If you don't mean an axiomatic system (that is, you don't know what to say after reading the above sentence), then this is a theorem.
It is a theorem, because we can prove it, that is, "(p-> q)& lt； = & gt(-q->； -p) "No matter whether P is true, Q is false, P is false, Q is true, P is false, Q is false (then
I don't know what you want to ask. ...
There is nothing incredible. We see that something is not black (it is white), and then we find that it is not a crow, which means that the crow is probably really black. Isn't this verified from another angle?
Using a more popular example, I want to run for president of the student union: I said, I am the most suitable. On the whole, I am more suitable than others. In other words, refuse me. This is the verification conclusion.
I think this premise is not completely correct.
First of all, what is "truth" is undefined. This is a very profound philosophical problem. Fortunately, logic does not study the truth of one thing, but infers other conclusions from what is known. )
Secondly, the conclusion of mathematics and logic (the so-called "truth") is not drawn through experience, but through reasoning or deduction.
In the proof of mathematics (especially advanced mathematics), a person can prove a proposition strictly through his own thinking logic without relying on observation experience, but these theorems developed by mathematicians are usually completely unobservable.
The same is true of logic. Logical proof is only dealing with symbols, which is very strict, and it is impossible to assert that a conclusion is true by observation.
Third, some so-called truths in natural science and social science are really summed up by observation experience. But when these truths are used to demonstrate, there will be paradoxes. For example, the classic syllogism:
Everyone is mortal, Aristotle is human, so Aristotle is mortal.
At first glance, it seems not bad, but when you think about it, "everyone is going to die" is an inductive conclusion, that is, we don't verify this conclusion until we know that everyone is dead-and then in turn prove that this person is going to die. This is called circular argument.
In other words, what I just discussed is that some truths are not obtained through observation and experience; The so-called truth obtained from observation experience-if it makes sense logically-can only be used to be verified as correct, but not to prove that everything involved is correct. The observed "truth" is useless.
Supplement:
The axiom used in mathematical proof-the so-called axiom system, although it seems that some of it is based on observation experience (such as Euclid system that you are familiar with), in fact, the mathematical community recognizes it as an axiom, not because it conforms to observation experience, but because it has no contradiction. That is to say, even if an axiomatic system completely violates normal common sense, as long as it has no mistakes-that is, no contradictions can be deduced through mathematics-then it is a reasonable axiomatic system. For example, the system of non-Euclidean geometry: "There can be countless straight lines parallel to the known straight line outside the straight line" and "There is no straight line parallel to the known straight line outside the straight line", which is absurd in common sense, but completely correct in mathematics.
The axiom system of mathematics is self-contained, even if it has nothing to do with the outside world, it doesn't matter. For example, the axiomatic system of "real number set" can be seen in this entry:
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I wrote this, too. If we only look at these dozens of so-called "axioms", a layman (whether he has the basic knowledge of real numbers or not) will read it like reading a gobbledygook. However, this is indeed an accurate definition of real numbers. In fact, many of these seemingly simple and similar things observed from nature have precise definitions in mathematics. Even the concept of "set", which seems to be defined only by description, is defined by the axiomatic system (Article 7).
As for Euclid's axiomatic system mentioned at the beginning, we have reason to believe that he did get it through observation at first-otherwise, he would be regarded as a madman rather than a mathematician at that time-but these axiomatic systems perfectly meet the requirements of axiomatic systems within the mathematical system (without contradiction and repetition) and are no longer related to observation.
To put it mildly, we can also think that people who engage in mathematics are just a group of boring YY people who feel good about themselves (how perfect the mathematical system is! ), but outsiders regard it as heterogeneous.
I don't need to answer your second question.
Question 3: I don't study physics, so I don't know much.