Logical validity
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Logical validity can at least roughly be defined as the property an argument (a set of sentences among which one is designated as the conclusion and the others as premises) has if it satisfies the following condition: if the sentences are true, then the conclusion has to be true as well. In other words, an argument is logically valid if it is in principle impossible for the premises to be true and the conclusion false at the same time.
This is only a rough definition because, though useful for grasping the general notion of validity, it is technically incorrect. Specifically, it runs into trouble when dealing with necessary or impossible sentences. For example, say you had the following argument:
- P1: Madrid is in Spain.
- P2: Red apples are gross.
- C: Therefore, 2+2=4.
Even though the premises clearly are unrelated to the conclusion, and certainly don't show that it's true, the conclusion is necessarily true all on its own. It's easy to see that this isn't a logical argument, and so not something we'd want to class as valid. The rough definition of validity given above would classify it as valid; but the more technical definition of validity used by logicians would not because the form of the argument,
- P1: X
- P2: Y
- C: Therefore, Z
is not valid. But, as discussed below, validity is a property of an argument's logical structure.^{[1]}
Explanation[edit]
Validity has absolutely nothing to do with the truth or falsity of the premises or the conclusion, apart from the fact that a valid argument with true premises cannot have a false conclusion and the corollary that no argument with true premises and a false conclusion can be valid. A logically valid argument can have false premises and a false conclusion though, or false premises and a true conclusion (and an argument with true premises and a true conclusion can perfectly well be invalid). A logically valid argument with true premises and (hence) a true conclusion is usually called sound.
In first-order logic validity coincides with provability (in accordance with Gödel's completeness theorem^{}), though predicate logic with polynary predicates is not effectively decidable. The coincidence of validity with provability is considered to be a very nice property of first-order logic. It is not the case for, say, arithmetics (in accordance with Gödel's incompleteness theorem^{}).
Validity is closely related to logical truth. A sentence is a logical truth if it is a tautology. A tautology is a statement which is true solely in virtue of its logical structure. Any logically valid argument can be recast as a logically true sentence.
An Example[edit]
Consider the following argument:
- P1: If it is raining, the road will be wet.
- P2: It is raining.
- C: Therefore, the road is wet.
In symbolic logic this can be expressed in the following way:
- P1: If X, then Y
- P2: X
- C: Therefore, Y
Where 'X' is "it is raining" and 'Y' is "the road is wet", with the first two statements being the premises and the final statement the conclusion.
Logical Validity considers only the structure of the argument, not whether the argument is actually true. In the above argument, it is valid, because the structure of the argument is true. Another argument which has the same structure is the following:
- P1: If there is creation there must be a creator.
- P2: There is creation.
- C: Therefore, there is a creator.
The above argument is logically valid, again, meaning that the argument is true based only on its structure. However, the argument is not sound, meaning that its premise ("If there is creation there must be a creator," and if you're a solipsist you can argue with "There is creation.") is not true, or is debatable.
Inversely, dismissing any otherwise-truthful proposition as false simply based on the invalidity of how it is argued is committing the fallacy fallacy. This is because there are cases where true conclusions can come out from false premises.
If an argument is valid it can be the following: False premises with true conclusion, false premises with false conclusion, and true.
Soundness[edit]
Soundness is related to validity and has the following requirements:
- The argument is valid.
- The argument has true premises.
Now, let's return to the original example argument:
- P1: If it is raining, the road will be wet.
- P2: It is raining.
- C: Therefore, the road is wet.
This argument is structurally valid. Now, for it to be sound it must be true that if it is raining the road will be wet and that it is raining. It is possible that it is not actually raining, so while the argument is valid, it is not sound. Also, we could imagine, for example, the road is covered with some kind of canopy which prevents the rain from hitting the road. Again making it unsound.
The above example is based on a simple argument structure, but arguments can have many many premises, which can make debating the soundness of an argument extremely difficult in some cases.