Can science be used to prove that assertion is true?
Honestly, people who turn science into scientism are so funny that it reverts around and becomes sad. People who base their lives on a system of naturalistic methods made to study nature and who do not accept anything else as truth never can quite see that their own system cannot even be proved to be accurate by using that system!
Can you scientifically prove logic exists, or that something is logical? No. It's common sense that logic exists, but if you apply its existence to the scientific method, you get a giant case of circular reasoning: Using a product of logic (the scientific method) to prove logic. Which is in itself illogical. Any scientific study of logic will come up empty-handed.
I guess logic is an illusion.
Science is a method of applying methodological naturalism to a series of tests and scrutiny in order to affirm a theory or hypothesis about a natural occurrence. By definition science cannot say "yes" or "no" on metaphysical issues (no matter how much the neo atheists or fundamentalist theists want to say otherwise). For those issues, we turn to another method, one that's been around longer than humans: logic. Logic is behind everything we have ever done, from mathematics to philosophy, but now there are people who want to replace logic with science, or scientism, as they turn it into by claiming that science is the only way to know anything.
Of course, when asked to prove this claim using the scientific method, all one gets is a drool and a rant about how you're stupid for opposing the almighty science. The almighty science cannot go against the almighty logic, and since proving science with science is circular reasoning (a logical fallacy), it is impossible to prove science is the only good method of observation with the scientific method.
But logic has its flaws too. Theoretically, one can come to any conclusion, no matter how ridiculous, perfectly logically. This is why we have science and philosophy; to separate the good conclusions from the bad, either with empirical evidence or logical proofs. It is these logical proofs that make up almost all of mathematics and philosophy. A proof can go like this:
If a, then b.
If b, then c.
Therefore, x.
This type of proof is used most often in philosophical discussions and theories. The first two steps are called premises. The premises are the steps you take to get to the last step, called the conclusion. If the premises are correct, and the conclusion follows from the premises, then the proof is valid and can be used as evidence for a position in a philosophical matter.
When a proof is invalid, a fallacy has been committed. One of example of a fallacy is this:
My ice cream is melty.
Therefore God does not exist.
This fallacy is called a non sequitur, which means "does not follow." It is applied to arguments that have conclusions that do not follow from the premises. What does melty ice cream have to do with the existence of God?
For a list of logical fallacies and examples of how they are applied, see here.
My point with all of this is to show that science is not the be-all end-all of knowledge; logic is. To treat science the way some people do is turning it into a religion, that awful plague upon humanity that science has come to cure us of. But then, what shall be done with scientism?
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