I. Background
A valid deductive argument is one where the truth of the conclusion follows necessarily from the truth of its premises. An argument is valid when it is impossible for the premises to be true and the conclusion false and invalid when it is possible for the conclusion to be false and the premises true.
When we speak of the truth and falsity of an argument's premises and conclusion, we are being somewhat deceptive because validity has more to do with an argument's form than with its content. The premises and conclusion do not actually have to be true; rather, when we analyse an argument for whether it is valid or invalid, we just have to suppose the premises are true. We ask ourselves: "If the premises were true, then is there any possible way for the conclusion to be false?" If so, then the argument is invalid. If not, then the argument is valid.
A few examples will help us along here, so let me begin with one we have all heard, and probably used in the classroom, before.
ARGUMENT A
A1. All humans are mortal.
A2. Socrates is a human.
A3. Therefore, Socrates is mortal.
The argument is valid because A3 necessarily follows from A1 and A2. Since all entities that are human fall within the category of entities that are mortal, and Socrates is one member of the category human, Socrates must also be a member of the category mortal.
So much for validity. A valid deductive argument with all true premises is sound. Argument A is valid and sound because, in fact, all humans are mortal and Socrates is (was) a human.
Validity and soundness seem to come apart. A deductive argument need not be sound to be valid because it is possible for a valid deductive argument to have false premises. For example:
ARGUMENT B
B1. If the moon is made of green cheese, then the present King of France is blonde.
B2. The moon is made of green cheese.
B3. Thus, the present King of France is blonde.
B2 is clearly false because the moon is not made of green cheese. In spite of B2's falsity, the conclusion necessarily follows from the premises resulting in a valid argument. Supposing it were true that the moon is made of green cheese and supposing that the present King of France is blonde if the moon is made of green cheese, then it necessarily follows that the present King of France is blonde. The argument is valid but unsound, because at least one of its premises is false.
Suppose we replace the antecedent of B1, and subsequently premise B2, with a true proposition. For example:
ARGUMENT B*
B1*. If Barack Obama is President of the United States, then the present King of France is blonde.
B2*. Barack Obama is President of the United States.
B3*. Thus, the present King of France is blonde.
B2* is clearly a true proposition because, in fact, Barack Obama is President of the United States. B1*, however, is false. Whenever the antecedent of a conditional is true and the consequent false, the entire conditional statement is false. Because the argument's form is the same as B and we have already accepted that such a form is valid, it is not possible for the premises to be true and the conclusion false. Like B, B* is valid but unsound.
A, B, and B* are valid deductive arguments, even though B and B* are unsound. Validity and soundness are challenging concepts to disentangle since even in our most perspicuous moments we have an urge to say that unsound premises yield an invalid argument or that a valid argument must also be sound. When we turn to thinking about validity and soundness embedded in an argument, a curious case seems to arise.
II. The Problem
Think about the following argument.
ARGUMENT C
C1. If C is valid, then C is unsound.
C2. C is valid.
C3. Hence, C is unsound.
Argument C has the same form of B and B*. C is undoubtedly valid given what we know of the argument form modus ponens ponens, but is it sound or unsound?
The conclusion says of C that it is unsound. So, if C3 is true, and most assuredly it is, at least one of the argument's premises is false. Yet, C2 cannot possibly be false because the argument's form is valid, and the conclusion necessarily follows from the argument's premises. Saying of itself that C is valid must necessarily be true.
If C2 cannot possibly be false, then C1 must be the culprit, the one false premise that yields an unsound argument. The falsity of C1 depends upon the conditional statement having a true antecedent and false consequent. If the consequent is false as it should be for the argument to be unsound, then C is sound and it has all true premises. The argument cannot possibly contain all true premises because C1 is false given that the conclusion is false and the argument is valid. But if C has all true premises and C3 is false, then C3 does not necessarily follow from the premises.
III. A Way Out?
To my mind, there seem to be three relatively unpalatable resolutions of the aporia, and I will address them ordinally from the most lamentable to the least lamentable. First, argument C might be a new counterexample of modus ponens. In his landmark paper, Vann McGee (1985) provided a counterexample to modus ponens showing that the conjunction of 'if φ, then ψ' and 'φ' need not necessitate the truth of ψ. Accordingly, with regards to C, if we were to believe C1 and C2, we would not need to accept that C3 follows because in accepting that conclusion at least one of the premises has to be false. The only likely candidate premise surely would have to be C1 and that means C3 is false and the argument is sound. C could not possibly be sound because all of its premises would then all have to be true, something that the conclusion denies.
A second possible resolution has it that C is valid and unsound. For C to be valid, it cannot be possible for C1 and C2 to be true and C3 to be false. If C3 is true, then the consequent of C1 is true and the conditional is trivially true. Any conditional with a true consequent is true, even if the antecedent is false. But, C3's saying 'C is unsound' means at least one of C's premises has to be false, so C2 would have to be false. If C2 is false, then C is invalid if and only if C is valid. That cannot be a correct accounting of matters, so there must be an alternative rendering of C's validity. If C3 is false, then at least one of its premises has to be false for the argument to be valid. The likeliest candidate of a premise's being false is C1, especially since we have determined already that it cannot possibly be C2. Indeed, C1 is false if C3 is false. But if C3 is false, then all of C's premises are true since C would be sound. Of course, C would not have all true premises in the case of C3's being false because C1, too, would be false. Thus, the second resolution seems to fail.
A third potential resolution of the problem is that C is invalid. If C is invalid, then it is unsound, which is exactly what the conclusion says of C. For C to be invalid, it is possible for C1 and C2 to be true and C3 to be false. C1 is trivially true when the antecedent is false. If the antecedent of C1 is false, then C2, the second premise of C, is also false, seeing as how it is the antecedent of C1, and the argument turns out valid. Therefore, the argument is invalid if and only if it is valid. Perhaps, then, C1's antecedent and consequent is true, thereby making the conditional and C2 true. If C1 and C2 are true, then C3 is true, too, and C is valid, which is exactly what we have been trying to avoid. It is worse than that, though, because since C3 says of C that it has at least one false premise and, as we have just manufactured it, C1 and C2 are true, it cannot possibly be that C1 and C2 are both true.
REFERENCES
McGee, V. (1985) 'A Counterexample to Modus Ponens' Journal of Philosophy 82.9: 462-471.