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"Sound," "Valid" and "True"

In logic, "sound", "valid", and "true" are not synonymous. 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 individual premise and the conclusion. It's exactly what it sounds like, but it does not address the validity of the logic.

 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 / 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.

"Validity" refers to the chain of reasoning, the logical part of the argument. An argument is valid only if it is impossible for all of the premises to be true and for the conclusion to be false. It does not rely on the truth of the premises or of the conclusion.

 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 / Valid logic: If all A are B, and all B are C, then all A are C.)

Technically speaking, even this breeds further distinctions, as "cogent" is the word you're properly supposed to use for inductive reasoning; only deductive arguments are "valid".

The argument as a whole is unsound. "Soundness" refers to the argument as a whole. The premises must be "true" and the logic must be "valid". (Using a fallacy results in an unsound argument, as does using false premises.) If these conditions are met, the 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.

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

Of course, the above only refers to deductive logic. When it comes to induction, things get a bit more dicey. (Literally.)

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: we only know that Brahim is Moroccan[1]. Brahim could 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 Brahim is in fact Muslim, then the argument becomes cogent: not only strong, but true.

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:

 Alex is human.

50.25% of humans are male.

Therefore, Rory is male.

Notes

  1. And male -- from his name -- but let's ignore that
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