"Sound," "Valid" and "True"

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

The simplest version of an argument in formal logic is the syllogism: A set of at least two premises, and then a conclusion that follows from the premises according to a set of standard relationships (which are too complex to go into for this article).

A classic example syllogism:
Premise: All men are mortal.
Premise: Socrates is a man.
Conclusion: Therefore, Socrates is mortal.

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

Truth refers to the factual accuracy of each individual premise and conclusion. It has nothing to do with whether the argument as a whole is correct or not; a statement can be true even if it is used in an incoherent argument.

Premise: All dogs are animals.
Premise: All cats are animals.
Conclusion: Therefore, all turtles are animals.
Both of the premises are true, and the conclusion is true; all turtles are indeed animals. The truth of all three statements is not affected by the clearly invalid logic connecting them.

Validity refers to whether the chain of reasoning that connects the premises and conclusion is logical or not. An argument is valid if it is impossible for its conclusion to be false while the premises are true. It has nothing to do with whether the premises and conclusion are in fact true.

All animals are dogs. (This is a false premise.)
All dogs are terriers. (This is also a false premise.)
Although all three parts of this argument are obviously wrong, together they form a valid argument. IF all animals were dogs AND all dogs were terriers, then all animals would indeed be terriers.

Soundness refers to the argument as a whole. An argument is sound if the logic is valid and all the premises are true. If the argument is sound, then the conclusion must be accepted as true, by the definition of "valid". An unsound argument is one that contains a Logical Fallacy, making it invalid, or contains a false premise.

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

Strength and Cogency: Inductive logic

The above only refers to deductive logic. When it comes to induction, things get more complicated.

Inductive logic is drawing likely conclusions from true premises. All inductive arguments are, by their nature, invalid; induction relies on probability as a central element rather than certainty. Validity requires that true premises NEVER lead to a false conclusion, but a probabilistic premise specifically breaks that rule. This doesn't make them any less useful, and in fact most of the logic people make use of on a day to day basis is inductive, based on previous experience and 'rules of thumb' rather than strict, unbreakable truths.

As an example:
Brahim is Moroccan.
98.7% of Moroccans are Muslims.
Therefore, Brahim is Muslim.

This argument is invalid: even if both premises are true, the only thing we know for certain is that Brahim is Moroccan.note  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, given no other data about Brahim than that he is Moroccan, it is highly likely that Brahim is in fact Muslim.

So for inductive arguments, we don't worry about validity; instead we say the argument is strong if it is unlikely to have a false conclusion, provided the premises are true. And instead of saying the argument is sound, we say it is cogent if the logic is strong and the premises are all true.

If it is true that Brahim is Moroccan, and it is true that 98.7% of Moroccans are Muslim, then this argument cogent and we can safely assume Brahim is Muslim unless and until we find specific information to the contrary.

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 a "strong" argument, and if the premises are correct then it is cogent. But as it's only a tiny fraction past 50%, one wouldn't want to rely on that conclusion.

In the same way that an invalid deduction might still have a true conclusion (as with the turtles above), a weak inductive argument could still have a true conclusion. This should be fairly intuitive:

Barack Obama was the President of the United States.
2.22% 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, a strong argument can easily be cogent while still producing a false conclusion:

Barack Obama was the President of the United States.
97.7% of Presidents of the United States are white.
Therefore, Barack Obama is white.

That's why inductive reasoning is less reliable than deduction.

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.

Induction: A Part of Science

Induction allows us to make probabilistic conclusions about the world. If that makes you think of science, that's because it's part of the scientific method. The scientific method begins with a series of observations to learn about the world then uses inductive logic to reach a tentative conclusion about how that part of the world works. This tentative conclusion is your hypothesis.

The next step is to turn around and use deductive logic to see what would undermine the hypothesis, then run the experiment to test whether that is true. Ideally, you would test something necessary for your hypothesis. Or it may be something that should be true if your hypothesis is true. By testing this next idea, you have a new data point, and you return to inductive reasoning to shade your confidence in whether or not your hypothesis is true.

Thus, science never gets the certainty of deductive logic because it always has inductive logic mixed in. However, we can have overwhelming confidence in a conclusion which we have built up enough evidence in. This is reflected in an argument by philosopher David Hume which proved that it is impossible to ever derive certain knowledge by observing the world. Hume argued that we observe the world and expect future events to play out similarly. You throw an apple in the air and it falls. You throw an orange in the air and it falls. You throw an elephant in the air and it falls. This idea, that things you throw into the air will fall, is a prediction based on an assumption. The future will resemble the past.

How do you know the future will resemble the past? Because in the past, the future has resembled the past. This is circular reasoning, which makes it spurious. Thus, inductive logic and science can never give you certainty. No observation can ever give you certainty. The strength of science is that it has a means of continually reinforcing the confidence in its conclusions, not that it can ever remove all doubt.

Remember how in science, you try to disprove your hypothesis? Karl Popper, the philosopher of science, noted this is the key to science. Falsifiability is the core of a scientific statement. If you can't think of a way you might prove something wrong, it isn't science.