Sound, Valid, True

"Sound," "Valid" and "True"

In logic, "sound", "valid", and "true" are not synonymous.

A set of premises and a conclusion is called a syllogism:
Premise: If X, then Y
Premise: If Y, then Z
Conclusion: Therefore if X then Z.

What makes syllogisms useful is that they reduce an argument to its simplest form, making it easy to examine for flaws. The premises and conclusion can be "true" or "false"; the chain of reasoning itself can be "valid" or "invalid"; the argument as a whole is either "sound" or "unsound".

Truth refers to the factual accuracy of each premise and conclusion of the argument. It has nothing to do with the validity of the argument.

All dogs are animals. (True premise/All A are X.)
All cats are animals. (True premise/All B are X.)
Therefore, all turtles are animals. (True conclusion: All turtles ARE animals, HOWEVER - Invalid logic: If all A are X, and all B are X, then all C are not necessarily X.)
(The argument as a whole is unsound, even though the premises and conclusion are true, because the logic is invalid. This is not a valid argument, even though it's factually correct.)

Validity refers to the chain of reasoning, the logical part of the argument. An argument is valid if it is impossible for its conclusion to be wrong and the premises true. It has nothing to do with the truth of the conclusion and premises.

All animals are dogs. (False premise/All A are B.)
All dogs are terriers. (False premise/All B are C.)
Therefore, all animals are terriers. (False conclusion: All animals are not terriers, HOWEVER - Valid logic: If all A are B, and all B are C, then all A are C.)
(The argument as a whole is unsound because the premises and conclusion are false, even though the logic is valid. This is a valid argument, even though it's factually wrong)

Soundness refers to the argument as a whole. The premises must be "true" and the logic must be "valid". (Using a fallacy or false premises results in an unsound argument) If these conditions are met then conclusion must be true as well, by the above definition of "valid".

All terriers are dogs.
All dogs are animals.
Therefore, all terriers are animals.
(The argument as a whole is sound because the premises and conclusion are true and the logic is valid. This is a valid and factually correct argument.)

A perfect (deductive) argument. It is true and valid, and therefore sound. In other words, the argument must be based on accurate information and not contain any errors in logic.

Strength and Cogency: Inductive logic

The above only refers to deductive logic. When it comes to induction, things get a bit more dicey. (See what we did there?)

The first thing to note is that all inductive arguments are, by their nature, invalid: induction, by its nature, relies on probability as a central element. Since the definition of validity is that, given true premises, you always end up with a true conclusion, and the definition of a probabilistic premise is that you can feed in true data and still come up with a false answer, inductive arguments are always invalid according to the strict standards of logic. This doesn't make them any less useful. For instance:

Brahim is Moroccan.
98.7% of Moroccans are Muslims.
Therefore, Brahim is Muslim.

This is what is called a strong inductive argument: more likely to be true than false. It's invalid: even if both premises are true, the only thing we know for certain is that Brahim is Moroccan.note  Assuming that he is Moroccan, Brahim could still be one of the 1.1% of Moroccans who are Christian, or one of the 0.2% of Moroccans who are Jewish. Nevertheless, it is highly likely, given no data about Brahim other than that he is Moroccan, that Brahim is in fact Muslim.

If it is true that Brahim is Moroccan and that 98.7% of Moroccans are Muslim, this argument is also cogent. A cogent argument is a strong argument with true premises.

Theoretically, the dividing line between strong and weak inductive arguments is at 50%: at anything above 50%, the argument is strong. This can be a bit counterintuitive:

Pat is human.
50.25% of humans are male.
Therefore, Pat is male.

This is theoretically "strong", and if the premises are correct then it is cogent, but one wouldn't want to rely on that.

Conversely, simply because an inductive argument is weak does not mean that it isn't true. This should be fairly intuitive:

Barack Obama is the President of the United States.
2.32% of Presidents of the United States are mixed-race.
Therefore, Barack Obama is mixed-race.

This is a weak argument, and therefore not cogent, even though all its premises are true and has a true conclusion. On the other hand, this argument would be strong, and it is also cogent even though its conclusion is false:

Barack Obama is the President of the United States.
97.68% of Presidents of the United States are white.
Therefore, Barack Obama is white.

Using a Syllogism to Test Premises

Besides the use of syllogisms to come up with conclusions from known true premises, a syllogism can also be used to test the truth of premises. If a syllogism is valid, but comes to a conclusion that is known to be untrue, then one or both of the premises must be untrue. For example:

No good thing has ever come from military research.
The Internet is a good thing.
Therefore, the Internet did not come from military research.

A valid deductive argument. However, the conclusion is known to be untrue; it is well documented that the Internet originated with United States military research in communications systems. Therefore, one or both of the premises must be false. Important: this technique does not tell you which premise is false, or whether both of them are, merely that at least one must be. In this example, it may be that good things can indeed come from military research; or that the Internet is not a good thing; or both (good things have come from military research, but the Internet is not one of them.)

By combining chains of logic using sets of syllogisms known to be sound, valid, and true, one can also prove the falsity of a hypothesis or logical fallacy with certainty, and establish larger absolute sound, valid, true statements:

All fish are aquatic creatures.
Hypothesis: All aquatic creatures are fish. (Converse Error fallacy)
All dolphins are aquatic creatures.
Therefore, all dolphins are fish. (Association Fallacy)

All dolphins are mammals.
Mammals are not fish.
Therefore, dolphins are not fish.

All dolphins are aquatic creatures.
Dolphins are not fish.
Therefore, not all aquatic creatures are fish.

Thus we know for certain that the statement "All fish are aquatic creatures, but not all aquatic creatures are fish." is sound, valid, and true.