Topics Logic

Question Consider the argument: I am more than six feet tall. Therefore, I am over five feet tall. Is this a sound argument? Is it circular? Tautologous?

Accepted:
September 14, 2008

Comments

There is no possible way that you can be over six foot tall yet not over five foot tall. So the argument "I am more than six feet tall. Therefore I am over five feet tall." is certainly a valid one [on a classical account of validity: see below].
Standardly these days, an argument is said to be sound if it (a) is valid, and (b) has a true premiss/true premisses. So whether the given argument is sound depends on whether the premiss is true or not (which we are not told!). If you are over six foot tall, the argument is a sound one (since it is valid): if not, then not.
Let's say that an argument is formally circular if its conclusion is identical to one of its premisses. The given argument is plainly not formally circular in this sense. As to whether it counts as "circular" in some other, looser sense, that will depend on how this looser sense is explicated.
Standardly these days, we use talk of tautologies/tautologousness in a narrow sense -- not for logical truth/logical validity generally but for those logical truths/logically valid arguments whose status depends only on the distribution of truth-functional propositional connectives in the proposition/argument. In this narrow sense, the given argument is not tautologically valid. For its validity doesn't depend on truth-functional structure.

Let me muddy the waters in hopes that Peter will say more.

According to at least some philosophers, it is simply impossible that something should contain water without containing H2O. If they are right, then given the notion of validity presupposed by Peter's (1), this is a valid argument:

The plastic jug in my refrigerator contains water.
Therefore, the plastic jug in my refrigerator contains H2O.

But this doesn't strike most of us as a valid argument, and it doesn't help to invoke standard notions of meaning, since "water" and "H2O" aren't connected by meaning.

One reply would be to invoke a notion of validity of the following sort: an argument is valid if there is no argument with the same logical form whose premises are true and whose conclusion is false. On that account, the little argument about water isn't valid. Needless to say, this raises tricky questions about the notion of "logical form," but it lets us honor the intuition that the water/H2O argument isn't valid, even though it's impossible for the premise to be true and the conclusion false. However, this raises the question that kept me from wading in here in the first place: granted that it's impossible to be over 6' tall without being over 5' tall, is the impossibility of the right sort to make for a valid argument? Or is it more like the water/H2O case?

Since Peter's views on matters logical carry more weight than mine, I'm hoping he'll add to his answer.

Let's call the argument about Allen's plastic jug "argument (A)". Then consider the following claims:

(1) Kripke's doctrine: It is impossible that something should contain water without containing H2O.

(2) The classical account of validity: An inference is valid if and only if it is impossible that the premiss(es) be true and conclusion false.

(3) Non-ambiguity: The sense of "impossible" as it occurs in Kripke's doctrine is the same as the sense of "impossible" as it occurs in the classical account of validity.

(4) The intuition: Argument (A) is not a valid argument.

Then, as Allen points out, we can't hold all these together. If we accept Kripke's doctrine, then it is impossible for the premiss of (A) to be true and the conclusion false. If we accept non-ambiguity and the classical account of validity, then it immediately follows that argument (A) is valid. And that clashes with the intuition. What to do?

Let's set aside the option of disputing (1). As it happens I think that the particular claim about water can be questioned: but I think that some other Kripkean examples of a posteriori metaphysical necessities/impossibilities are correct, and we could recruit those other examples to make parallel arguments leading to the same sort of problem.

We might, though, dispute (3). Maybe, to further explicate the notion of impossibility that occurs in (2) -- logical impossibility! -- we need to talk about some notion of a priori impossibility (impossibilities of a kind that we can come to recognize by arm-chair reflection). And that's a notion distinct from the notion of impossibility involved in (1).

Alternatively, we might bite the bullet and reject (4). After all, we have a whole bunch of initial hunches about the notion of validity -- including, say, hunches to the effect that in a valid argument, premisses should be relevant to their conclusions. And we already know that we can't satisfy all those hunches together. In regimenting and refining the notion of validity, something has to give. Maybe the simplest and smoothest way of explicating our general notion of validity is the classical way, and the simplest and smoothest way of explicating the notion of possibility that it calls on sustains the non-ambiguity thesis. So we just have to learn to live with rejecting (4).

Or maybe -- to continue stirring up the muddy waters that Allen has disturbed -- we should be "logical pluralists" (see Ch.2 of JC Beall and Greg Restall's Logical Pluralism). That is to say, maybe we should hold that there isn't one best way of explicating the notion of validity, but there are various alternative ways of fitting some subset of our initial hunches (and fulfilling other desiderata). Then the classical account comes in just as one explication among many. And on some such explications argument (A) counts as valid, on others as invalid, and there is no right answer to the question whether it is "really" valid. (Similarly for the original questioner's case, 'if you are over six foot tall, then you are over five foot tall'.)

Though these murky issues are, I still suspect, taking us some distance away from the original questioner's concerns (hence my original response applied the fine philosophical maxim, "where it doesn't itch, don't scratch!").

Question Consider the argument: I am more than six feet tall. Therefore, I am over five feet tall. Is this a sound argument? Is it circular? Tautologous? Accepted:September 14, 2008

Question Consider the argument: I am more than six feet tall. Therefore, I am over five feet tall. Is this a sound argument? Is it circular? Tautologous?

Accepted:
September 14, 2008