Reprise on Validity - University of St Andrews



Reprise on Validity

1i. A valid argument can have false premises and a false conclusion, for example:

All fish are chips

All chips are Spanish

∴All fish are Spanish

In this argument the premises are false and the conclusion is false, but the argument has a valid form, (All A’s are B’s, All B’s are C’s, therefore all A’s are C’s) as there is no possible world in which the premises are all true and the conclusion is false.

ii. A valid argument can have false premises and a true conclusion, for example:

All cats are reptiles

All reptiles are mammals

∴All cats are mammals

In this argument the premises are false and the conclusion is true, but the argument has a valid form (All A’s are B’s, All B’s are C’s, therefore all A’s are C’s) as there is no possible world in which the premises are all true and the conclusion is false.

iii. A valid argument cannot have premises which are all true and a false conclusion, as the definition of a valid argument is an argument for which there is no possible world in which the premises are all true and the conclusion is false.

iv. All arguments with inconsistent premises are valid, as there is no possible world in which the premises are all true.

v. An argument can be valid if the conclusion is inconsistent, for example:

Sherlock is a detective

Sherlock is not a detective

∴1 + 8 = 96

Is valid because it has inconsistent premises, meaning there is no possible world in which the premises are all true.

2i. It is not possible to tell whether an argument is valid given only the information that the conclusion is inconsistent. The following examples show why:

Sherlock is a detective Sherlock is a detective

Sherlock is not a detective All cats are mammals

∴1 + 8 = 96 ∴1 + 8 = 96

The first argument is valid, as it has inconsistent premises and any argument with inconsistent premises is valid. However, the second argument is not valid, as there is a possible world in which the premises are true and the conclusion is false. Both these arguments have inconsistent conclusions, but one argument is valid and the other is invalid, so it is not possible to tell whether an argument is valid given only the information that the conclusion is inconsistent.

ii. It is not possible to tell whether an argument is valid given only the information that the set consisting of the premises and the conclusion of an argument is consistent. The following examples show why:

All cats are mammals All cats are mammals

All dogs are mammals All dogs are mammals

∴All cats and dogs are mammals All pigs are mammals

The first argument is of a valid form (All A’s are C’s, All B’s are C’s, therefore All A’s and B’s are C’s) but the second argument is of an invalid form (All A’s are D’s, All B’s are D’s, therefore All C’s are D’s).

iii. It is not possible to tell whether an argument is valid given only the information that the set consisting of the premises and the conclusion of an argument is inconsistent. The following examples show why:

All cats are mammals All cats are mammals

All dogs are mammals All cats are not mammals o

∴2 + 3 = 91 ∴Penguins are flightless birds

The first argument is invalid, as both the premises are true and the conclusion is false, while the second is valid as there is no possible world in which the premises are true.

iv. If the set consisting of the premises and the negation of the conclusion is consistent, then the argument is invalid. If the set containing the premises and the conclusion is consistent, and the argument is valid, then the set containing the premises and the negation conclusion will be inconsistent. So for the set containing the premises and the negation conclusion to be consistent, the set containing the premises and the conclusion must be inconsistent. If this inconsistency comes from the premises, then the set containing the premises and the negation of the conclusion will also be inconsistent. This means that if the argument is to be valid, and the set containing the premises and the negation of the conclusion is to be consistent, the conclusion must be inconsistent. However, if the conclusion is inconsistent it is false in all possible worlds, so the only way to make an argument with an inconsistent conclusion valid is to have inconsistent premises. However, if the premises are inconsistent then the set containing the premises and the negation of the conclusion will be inconsistent. Thus, if the set consisting of the premises and the negation of the conclusion is consistent, the argument must be invalid.

3.

Cretin-Mangler’s first claim, that for an argument to be invalid it must be the case that the premises are true and the conclusion false is incorrect. For an argument to be invalid there must be a possible world in which the premises are true and the conclusion is false, but an argument with an invalid form may have true premises and a true conclusion. For example,

All cats are mammalsjjjjjjjjjjjjjjjjjjjjjjjj

∴ Germany lost the Second World War

has true premises and a true conclusion, but is still an invalid argument, as there is a possible world in which the premises are true and the conclusion is false.

Cretin-Mangler’s second claim, that the argument “David is a werewolf, and all werewolves are philosophers, so David is a philosopher” must be valid, because it has false premises and a true conclusion is also incorrect. The argument is valid, but not because it has false premises and a true conclusion. It is valid because it is of the form: A is a B, All B’s are C’s, so A is a C, and for arguments of this form there is no possible world in which the premises are true and the conclusion is false.

If Cretin-Mangler’s test for validity were applied (the argument must have false premises and a true conclusion) arguments such as,

All cats are not mammalsjjjjjjjjjjjjjjjjjjj

∴ Germany won the Second World War

would be incorrectly identified as valid, when in fact this argument is of the same form as the argument that was just identified as invalid.

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