Valid reasoning is successful reasoning, reasoning that preserves truth from its premises to its conclusion.
Example of a Valid Argument
Premise 1: Socrates is a man.The relationship between the premises and the conclusion is such that, if the premises are true, the conclusion must also be true. The truth of the premises guarantees the truth of the conclusion. This argument (i.e., this act of reasoning) preserves truth from its premises to its conclusion.
Premise 2: All men are mortal.
Conclusion: Therefore, Socrates is mortal.
(In logical terms -- terms such as "necessary conditions" and "sufficient conditions" -- we can say that the truth of the premises is sufficient to guarantee the truth of the conclusion. Or, we can say that it the truth of the conclusion is a necessary condition of the truth of the premises.)
As it happens, the premises in the above argument are true, so the conclusion is also true.
Example of an Invalid Argument
Now consider a different argument:
Premise 1: Socrates is a man.This argument (in other words, this act of reasoning) does not preserve truth from its premises to its conclusion. The premises are true, and so is the conclusion, but it is not the case that the premises -– "Socrates is a man" and "All men are mortal" -– count in favor of the conclusion -– "Socrates is Greek". The truth of the premises does not guarantee that the conclusion must also be true. This argument is an example of invalid reasoning.
Premise 2: All men are mortal.
Conclusion: Therefore, Socrates is Greek.
Sound reasoning -- or, to put it another way, a sound argument -- is the combination of valid reasoning with true premises. And a valid argument with true premises (i.e., a sound argument) will have a true conclusion.
Deductive Vs Inductive Reasoning
There is a further clarification that should be mentioned regarding the above rules: the difference between, on the one hand, deductive reasoning and validity, and, on the other hand, inductive reasoning and validity.
To put it briefly -- and at the risk of oversimplifying things -- the difference between deductive validity and inductive validity comes down to the strength of the guarantee, the degree to which you can be sure that truth is preserved from the premises to the conclusion. In a deductively valid argument, if the premises are true, the conclusion must be true, it could not possibly be false. In an inductively valid argument, if the premises are true, the conclusion is likely to be true. It may be false, but probably isn't.
Premise 1: The sun rose today.This argument is not deductively valid. The conclusion could be false even if the premises were all true. But it does seem to have some merit, doesn't it? That is, the argument does seem to give us some reason -- even if not an iron-clad, flawless reason -- to believe that the conclusion is true, right? It is an example of an inductive argument, and a valid one (at least to some degree, though what constitutes inductive validity is vague at best).
Premise 2: The sun rose yesterday.
Premise 3: The sun rose the day before yesterday.
Premise 4: The sun rose the day before that.
Premise 5: The sun rose still another time the day before that.
Conclusion: Therefore, the sun will rise tomorrow.
When we argue and debate, we're trying to figure out the truth.
Logic is the means by which we do this, by crafting valid arguments that preserve truth from their premises to their conclusion.
EXAMPLES AND ANALYSIS
The gay rights movement cannot abide a middle ground and a free exercise of religion for a simple reason — homosexuality is not normal in nature, in historic relationships, or in the sacred texts of almost all religions. The gay rights movement must therefore censor and subjugate dissent. Any who point out the lack of historic or religious acceptance or the lack of its ready existence in nature or, for that matter, the lack of scientific evidence showing homosexuality is a birth trait as opposed to a choice or external factors, must be shut up.-- Pundit Erick Erickson, March 31, 2015. His remarks concerned the Religious Freedom Restoration Act (RFRA, also known as Indiana Senate Bill 101).
Comment: The claim that being gay is not normal suffers from ambiguity: does "normal" in this context mean typical? Or does it mean acceptable? Or does it mean something else? At any rate, it's questionable to reason from the presence or absence of homosexuality in nature and religious scripture to whether homosexuality should be legally or morally acceptable. More, how is it that the entire gay rights movement is opposed to the free exercise of religion related to the RFRA? Nobody supports both gay rights and religious freedom at the same time in this instance? That seems like an exaggeration or a distortion.
HOWARD DEAN: This [his declining to answer question on evolution] is a particular problem for Scott Walker which has not been an issue yet but it will. Scott Walker, were he to become president, would be the first president in many generations that did not have a college degree. He's never finished. The issue here is not just an issue of dancing around the question of evolution for political reasons, the issue is how well educated is this guy? And that's a problem.-- Former Gov. Howard Dean (D-VT), February 12, 2015, appearing on Morning Joe with host Joe Scarborough and pundit Donny Deutsch.
JOE SCARBOROUGH: Are you serious?
DEAN: I am absolutely serious.
DONNY DEUTSCH: I like that. I like that line: "are you serious?" He didn't finish his senior year so he didn't get all that stuff.
SCARBOROUGH: Are you serious? You're saying he might not be qualified because he didn't finish college?
DEAN: I think there are going to be a lot of people who worry about that.
SCARBOROUGH: Do you worry about people that don't finish college?
DEAN: I worry about people being President of the United States not knowing much about the world and not knowing much about science. I worry about that.
SCARBOROUGH: Oh, God. Let's name the people that didn't finish college that have changed this world.
DEAN: Harry Truman, who was a great president, there's no question about it.
SCARBOROUGH: Did Bill Gates finish college?
DEAN: I think Bill Gates is a little on a different --nobody is accusing Scott Walker of having the intellect of Bill Gates.
SCARBOROUGH: Well, nobody is accusing Scott Walker of being dumb because he didn't graduate from college except you.
DEAN: I didn't say dumb, I said unknowledgeable.
Comment: This is very poor reasoning. First, how does the fact that someone lacks a college degree make it more likely that they don't know much "about the world and about science"? Second, as Deutsch points out, Gov. Scott Walker (R-WI) did go to college, but didn't graduate. Does that mean Walker gained no knowledge while at college?
(The list above is not intended to be a comprehensive record of all relevant examples.)