Topics Logic

Question I'm struggling wit the following: I am reading an essay that states (repeatedly) that the following "p, p implies q, therefore q" is valid but that the following: "I judge that p, I judge that p implies q, therefore I judge that q" is "obviously" invalid. There is no explanation; apparently this is supposed to be transparent but I fail to see why this is obviously invalid. Why adding "I judge that" makes it invalid?

Accepted:
October 20, 2011

Comments

One sure way to prove invalidity is to describe a possible case where the premises of an argument are true and the conclusion false. To make things a bit more plausible, let's change the example slightly. The following is valid:

"q, not-p implies not-q, therefore p"

I pick this example because this argument (closely related to modus ponens) is one that people have a little more trouble seeing, or so my experience teaching logic suggests. So there could be and likely are cases where a person judges that q, and judges that not-p implies not-q, but has trouble with the logical leap and therefore fails to judge that p. That's a counterexample to the argument you're interested in. We have someone who judges the premises of a valid argument to be true but doesn't judge the conclusion to be true.

This isn't surprising. To judge something is (putting it a bit crudely) to be in a certain state of mind toward it. Being in the "judges that" state of mind toward the premises of an argument doesn't guarantee that someone will be in the same state of mind toward the conclusion, even if the conclusion happens to follow. Consider some argument with 10 premises P(1), P(2)...P(10) and a conclusion X. And suppose that X really does follow from P(1)... P(10). Your suggestion seems to tell us that anyone who judges the premises to be true will actually judge the conclusion to be true, even though, we'll suppose, the reasoning required to get from premises to conclusion is subtle and complex.

If you think about it, it's not at all unusual for people to miss seeing what their beliefs imply. Math is a particularly obvious case; getting someone to accept the principles of Euclidean geometry doesn't guarantee that they'll simply judge all the theorems to be true. But math is by no means the only case.

Another way to to put it: the problematic argument you judge to be valid needs another premise. A general version of the premise would be something like this:

Whenever an argument is valid and I judge that its premises are true, I always judge that its conclusion is true as well.

That's false for any human "I."