P= premises; C= conclusion.
Some Americans are women (True). Brad Pitt is an American (True). Therefore, Brad Pitt is a woman (false)-- invalid.
All dogs are ants (False). All ants are mammals(False). So, all dogs are mammals (True). -- valid, How?
some says "True P and False C is invalid" by definition, I want to know why is that definition.
How can we derive the truth out of falsehood?
"Valid" in logic is just a technical term meaning that if the premises are true, the conclusion logically must be true as well. As you pointed out in another question, a special case of this is when the conclusion is a tautological sentence, in which case it doesn't matter what the premises are. For non-tautological conclusions, though, validity is equivalent to the idea that you can derive the conclusion from the premises using accepted logical rules of inference (here are basic rules of inference expressed in symbolic terms, and here are some additional rules of inference for the universal quantifer and existential quantifier in first-order logic). It doesn't relate to whether the premises are actually true in the real world--a valid argument whose premises are true is said to be a "sound" argument. So, all sound arguments are valid, but not all valid arguments are sound.
Since relations between facts in the real world presumably obey the laws of logic, if you start from premises that are true in the real world and derive a conclusion that's false in the real world, presumably you must not have used the correct logical rules of inference in your derivation, so true premises and false conclusions always implies an invalid argument. On the other hand, in your argument with false premises and a true conclusion, it is possible to prove logically that premises of the form "All things with property A have property B" and "All things with property B have property C" imply conclusions of the form "All things with property A have property C" (see p. 162 of First-Order Logic: An Introduction for a formal proof of this using only the basic rules of inference linked earlier), so the argument is valid even though the premises are false. Of course valid arguments with false premises don't always lead to true conclusions, some of them can lead to false conclusions, like "All dogs are ants (False). All ants are birds (False). So, all dogs are birds (False)."
I take you to be asking why the validity of an argument is defined in such a way that the truth of its premises is irrelevant to its validity. That is a great question.
It is natural to think that an argument at least purports to reveal something as true, so that, if it succeeds, that is just what it does. After all, it is natural to mark the conclusion with «therefore», not «in that case». The same point can be made by noting that stating the premises and the conclusion is usually making claims: This is the case; that is the case; therefore, this is the case. A different way of expressing your worry, if I understand you, is to ask: How can the argument be valid if it doesn’t reveal the claim following «therefore» as an expression of knowledge?
A quick and simple answer would be to say that you are failing to distinguish between the validity and the soundness of an argument, or that you are mixing up argument and proof. That might be, but I don’t think the lesson should be that you are misusing the words «argument» and «validity». What is important is that you become aware of the way you are thinking about these notions. It sounds to me like you are using «argument» as an epistemic notion, in the sense that the following captures the essence of what an argument is: An argument lends credence to its conclusion. A proof can then naturally be understood as a kind of especially strong argument: A proof reveals its conclusion as true (as the way things are). There is nothing wrong with using these words in such a way, as far as I can see, but it is not the only way to use them.
Incidentally, I think these notions are fundamentally epistemic in a way that reveals certain formal methods as unfit for studying them. Or, more carefully put: It could be that that the formal sense of validity of an argument (the one that sees the truth of the premises as irrelevant) doesn’t get to the heart of what an argument really is.
(The long and hard answer would explain why validity and soundness are distinct in formal logic, despite what I have said about the epistemic notions. If I tried to explain it, I would only cause confusion. But it is worth trying to understand.)
Some literature concerning the difference between Frege and more modern logicians is relevant. Frege said that inference must proceed from truth. See Danielle MacBeth's book Frege's Logic and Maria van der Schaar's "Frege on Judgement and the Judging Agent". Warren Goldfarb’s «Frege’s Conception of Logic» is also relevant. One exception among modern logicians might be Per Martin-Löf. I'm on thin ice here, but I think it is fair to say that he is studying logic in a sense that does not fundamentally distinguish it from epistemology.
My lecturer years ago explained it the following way (the specifics might be a bit off but I trust the intent comes through).
He picked a student and said, "If I was a millionaire, I would give you a million dollars".
He then asserted that he wasn't a millionaire and talked about the truth value of the conditional.
The natural sense of the outcome (imagine the student suing the lecturer for withholding promised funds) was that since he wasn't a millionaire, he couldn't be branded a liar about his conditional statement. He could only be branded a liar on the basis of his conditional inly if he was a millionaire but didn't give a million dollars to the student.
In other words, material implication accords with the natural sense when the conditional statement is assigned a truth value of True when the premise is False.
I answer your question using notions from 'mathematical logic', with which many (analytic) Philosophy students are familar from introductory courses. If your question is about a historical conception of logic such as Aristotle's one in particular, I think it would help to edit your question accordingly.
How can we derive the truth out of falsehood?
A valid derivation means that truth is preserved. If you correctly derive q from p, that means if p is true, q is also guaranteed true. It does not say anything about the case that p is false. In this case, q could be true - but only by accident. It could also easily have been false. (Some would say: "Garbage in, garbage out.")
Validity means that in an argument, all derivation steps (there can be many) are truth preserving. So if you have true premises, you have a true conclusion. That is guaranteed. If you have false premises, this guaranty is lost. You can end up with anything - either a true or a false conclusion. Validity is about this guaranty.
"All dogs are ants (False). All ants are mammals(False). So, all dogs are mammals (True). -- valid, How?"
The Premise/Conclusion there is not valid. The last statement, "All dogs are mammals", is only true in and of itself, completely removed the first two statements. In other words, the last statement itself is what is valid, not the connection between the Premise and the Conclusion.
Even if you said:
No dogs are ants and no ants are dogs. All ants are insects. Therefore, all dogs are mammals.
That would be an invalid Premise/Conclusion. Based only on the specified details of the premise, dogs could have been birds, fish, mammals etc.
For example, here is another invalid Premise/Conclusion that ends with a correct statement.
Mercury has a density greater than 1 gram per centimeter cubed. (True)
Mercury has a density less than 100 grams per centimeter cubed. (True)
Therefore, Mercury has a density of 11.3 gram per centimeter cubed. (True)
The last statement is true. However, the Premise/Conclusion logic is invalid because the density could have been many other things based only on the two statements in the premise.
There are a few terms one must know when dealing with syllogisms. They are quite different from Mathematical notation. Here knowing the terminology helps out to solve the problem and give a solution.
The example given is as follows: Some Americans are women. Brad Pitt is a woman. [Therefore,] Brad Pitt is a woman.
The concept of distribution is needed to see why the argument fails in reality. The argument above is missing distribution. Distribution is a concept that expresses the membership of an entire class. By entire I mean 100 percent of the cases fit. For instance, the proposition All dogs are animals expresses every member of the subject must be a member of the predicate or else the proposition is false. The quantifier expresses the distribution. So different quantifier means different distribution. The quantifier named "All" are quantifiers that distribute only the subject term by definition. The quantifier Some is a different story. Some s is p does not say anything about an entire class but a particular individual is a member. So that means there is no distribution for positive particular propositions (I type propositions). The quantifier "No" distributes both the subject and the predicate terms. The only quantifier left is When propositions have the form Some s is NOT P (O type propositions). In this case the entire predicate class is being referenced and thus only the predicate is distributed. No distribution for the subject term in O type propositions.
Mood and figure are also concepts useful to know. Mood expresses the types of propositions being used in the argument. We have only four STANDARD forms: A, E, I, and O. Modern language would need to be modified to fit STANDARD FORM. The figure of an argument expresses where the middle term is placed in the argument. The middle term is what actually relates the premise and the conclusion. Without a middle term you would just have random statements that may or may not be formal propositions. So with knowledge of mood and figure one can tell validity of a syllogism, in some cases, near instantaneously because reasoning like this is repetitive. This is what is supposed to be meant by logic is about FORM. The form of an argument is the figure and not solely about what is literally written as a premise. But one should note some terms require knowledge of content of the premise because some usages of language are tricky. For instance propositions about the future are hard to handle by form alone. Propositions that Express a tautology: i.e., all triangles are shapes with three sides is tricky because form alone does not tell you there is a tautology: you would need to understand the language and meaning to symbolize the argument correctly. Without understanding the language you would be lost. So again form alone will not help you. You will need more than just form alone to reason correctly. In pure deductive reasoning all you need are the information contained in the premises alone -- no outside or other knowledge is needed. The fact you need something not given in the premises proves it is not knowledge based on form alone.
The second example, is valid but you don't see why. All dogs are ants All ants are mammals. [Therefore,] all dogs are mammals.
The mood & figure in this example is a valid form that can be recognized on sight. This is a syllogism with the mood AAA in the fourth figure (AAA-4). All arguments of this form will be valid regardless of the topic. Knowing the mood and figure helps one visually see if an argument is valid or invalid because it has a pattern one recognizes. Some arguments can be tricky to visualize though. There are other tests for validity than just figure and mood.
If you use known valid forms along with premises that are true in reality you will get a SOUND argument. There are no such things as invalid sound arguments.
