There is more than one way to understand Russell’s famous logical puzzle about the present king of France, and moreover it’s one of those puzzles that is used to show more than one thing (even things that Russell never intended), but suppose you understand the puzzle like this:
“The present king of France is bald” must be either true or false. But it is not true, for there is no PKoF. But it is not false either, for then the contradictory statement would be true: “The PKoF is not bald”, which is not the case, since he doesn’t exist. In other words, both sides of the contradiction are false for the same reason: there is no King of France. Both sides of a contradiction are false! The principle of contradiction has no value!
The simpler response is that the affirmation “…is bald” is false, and the contradictory is true, precisely as it contradicts. It is true to say that the PKoF is “not bald” where “not bald” is taken as indeterminate to whether the PKoF exists or not. The PKoF is not bald because he is not anything at all. Our current president is not bald, but neither is last Tuesday or happiness, though for different reasons: the first because he has hair, the second because they are not the sort of things that can be bald or not. When we say “not bald”, both must be taken as possibilities, and the second possibility verifies the truth of the statement “the PKoF is not bald”.
This is generally true of truths that are generated out of the principle of contradiction. The falsity of the affirmation guarantees the truth of the denial only so far as the denial prescinds from whether something exists or not. To say “either p or not p” is deceptive, since “not p” must be taken as open to the existence or non-existence of the subject.
Just Thomism 
PatrickH said,
December 1, 2009 at 8:03 am
From Joseph Kenny on philosophy of nature: “Negation does not determine a subject. Non-seeing can be said even of non-beings, for example we say that the dragon does not see and we say the same of beings which are not apt to have sight, as stones.” He distinguishes negation from privation, which does determine a subject (in a certain way, and not all by itself, of course), and which is involved in all true generation.
I am attempting to learn Thomistic and Aristotelian philosophy, with a current focus on Phil of Nature. Have I understood your argument correctly? Is the comparison to Kenny apposite?
And perhaps I’m hallucinating, but in the post did you not mention something about “infinite names”? Does the above interpretation render their invocation unnecessary?
James Chastek said,
December 1, 2009 at 9:49 am
The original post mentioned infinite names, but it was an unreadable disaster. The non-determination of a subject, as such, is able to be said both of what exists and does not, and such non-determination is called an “infinite name”. This is important in negative theology: God is “not a body” or “immobile” in the sense that he is not the sort of thing that could either have or lack a body or be at motion or rest. Saying “God is a body” is an intrinsically ridiculous statement, like saying “last wednesday is orange”. God is immobile in a way similar to how last Tuesday is not orange.
The comparision to Kenny is very apt. The relevant differnce is between privation (which is necessarily of an existent subject) and negation (which is not necessarily of an existent subject.)
Bob the Chef said,
March 12, 2010 at 7:39 pm
This is an example of an existential fallacy, a syllogistic fallacy that can occur in a categorical syllogism. However, in Aristotelian logic, this is all fine and dandy. The property of being bald here is after all contingent on the existence of the king. But, the evaluation of the proposition as false has two different meanings when treated in the modern and the Aristotelian sense.
When I say false in the modern sense, I am saying false to the property qua property of the king. It assumes the existence of the king. If the king doesn’t exist, the proposition is meaningless, and therefore cannot be evaluated and does not have a truth value. This is not strange because after all. It’s a bit artificial to require a proposition to have a truth value. It’s as if you aren’t distinguishing between the real and concepts in the mind.
Now when I say false in the Aristotelian sense, I’m saying false to the entire enthymeme, which includes the unstated premise regarding the existence of the king. This is certainly valid, although how the syllogism is false is less clear with respect to the truth value alone.
James Chastek said,
March 13, 2010 at 7:49 pm
This brings up some new topics that would require new posts. As a summary, I’m uncomfortable with the term “meaningless” as some modern logicians use it. What signifies has meaning, and words and propositions, since they are signs, cannot fail to signify The ridiculous is not the same as the meaningless- in fact, to be ridiculous requires understanding significations. But that’s another topic that I’m not getting into.
For the moment, I’d limit myself to the example given in the post, and I see no reason to bring up the additional concept meaningless when true and false can do the work. One side of a contradiction is indeterminate to existence. If all you tell me is that SP is false, I know that ~SP is true, but not whether S exists.
On a different note, this still works with supposed meaningless statements like “donuts twinkle willingly” (which I think is ridiculous, but not meaningless). This sentence is false, and the contradictory true, but the truth of the contradictory does not require that the proposition not be ridiculous (though the contradictory- donuts do not twinkle willingly- is ridiculous too, but true when “not” is taken as expressing sheer negation). One can put T and F on a truth table here, so I don’t see the problem with assigning truth values. We would know that there is there was a true and false sense, but not the sense of the word “not” that makes it true.