Friday, January 16, 2015

Do All Valid Arguments Have an Invalid Form?

The caption reveals a poor argument.  Not knowing something is not a good reason to believe that there are aliens.  Among other things the argument is invalid.
It is of the clearly invalid form
therefore, q

With that in mind here are some thoughts in light of a recent series of emails between a few friends on invalidity and validity.

A valid argument is a deductive argument such that if the premises were true then the conclusion must be true.  An invalid argument is a deductive argument such that if the premises were true it's still possible that the conclusion is false.

Take the following instance of the invalid argument form known as Affirming the Consequent:

If Tully is a woman, then Tully is a human.
Tully is a human.
Thus, Tully is a woman.

The form of the argument is clearly invalid:
If p, then q
Thus, p

If the premises were true, the conclusion could still be false.

Now note that an argument can instantiate more than one form and that no invalid argument instantiates a valid form.

For instance, the above argument also apparently instantiates the following form which is also invalid:
p (premise 1)
q (premise 2)
r (conclusion)

Moreover, it appears to instantiate the following form as well:

p (premises)
q (conclusion)

Again, this is also an invalid form.

My question is: do all valid arguments also instantiate an invalid form?  On a first glance it would seem that they do since every argument appears to take the form:

p (premise/premises)
q (conclusion)

Maybe that's right.  We then say that an invalid argument instantiates no valid forms but a valid argument instantiates at least one valid form.

Nonetheless, I raise a question about this.

Distinguish between the concrete argument that I give, say, at a philosophy conference that happens at a specific time and place with a specific utterance or writing.  The concrete argument that I give is embodied in token sentences which I express verbally or perhaps in writing.  We can abstract from that argument to consider the propositions (the extra-linguistic content of the sentences) and we can express the same argument in Spanish, Latin, etc.--that is, with different sentence types and tokens.  That set of propositions will consist of a certain form (say, Modus Ponens).

(Call this form #1)
If p, then q
thus, q

As I noted earlier, we can abstract further from the content of the variables to consider just the form of the premises themselves:

(Call this form #2)

And we can abstract further to the most general premise-conclusion form:

(Call this form #3)

But now consider the concrete argument that I give.  When I express the argument, does that argument instantiate the form of Modus Ponens (#1) as well as #2 and #3?  Or does it only instantiate #1?

It might be thought that if it instantiates #1 that it thereby instantiates #2 and #3, after all, the argument has premises and a conclusion.  But is that right?  Why should we think that?  Why not think that the concrete argument only instantiates the form of Modus Ponens because that is the only form that I intend for the argument to take?  Sure, the abstracted propositional argument also instantiates the invalid forms, but why think that the concrete argument does as well?  After all, I was trying to give an argument which takes the form of Modus Ponens not an argument with an invalid form of


Here is, then, a suggestion: A concrete argument only instantiates one form, the form at the infima species level of abstraction.

[A further aside: Does an argument only instantiate a certain form in virtue of abstracting?  If so, then we can say that the argument only has whatever form it instantiates at whatever level of abstraction has taken place in the mind of whoever is considering/presenting/etc. the argument.  This would make room for thinking that concrete arguments (always?) instantiate only one form.]

Does anything of serious consequence hang on the issue?  I don't know.

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