logicandlanguage.net

it is a semantic phenomenon in that it arises because of a problem with the meaning of the pathological expressions, whether those expressions are themselves semantic ones or not.

After reading the whole post, this does sound more plausible to me than it did at first. There certainly seems to be "something wrong" with the putative meanings given to these words, if they can give rise to such paradoxes. Can we just get around this by stipulating that the only horses are the well-founded horses, the way we do in set theory?

Would the upshot of that be that, say, Acer and Bandit aren't horses? I don't think I'm comfortable with that.

However, still in the spirit of your suggestion, we might try saying that the only American Paints are the well-founded American Paints, which would make neither Acer nor Bandit an American Paint.

I think this is best understood as a proposal to reform the language (by rejecting and replacing the definition of "American Paint" that I gave. (We could think of it as the basis of naive American Paint Theory.)

And then perhaps an additional reform proposal could include some of the horses as hyper--American Paints, as in Aczel's hypersets!?

This probably won't come as a surprise, but these horse concepts are the sort of circular concept that's dealt with by Gupta and Belnap's revision theory. I suppose they'd argue that the semantic paradoxes are just a subspecies of the set of paradoxes generated by concepts with circular definitions.

Would you like it to be known as Gillian's Paradox or as The Other Russell's Paradox?

How about "the lesser Russell's paradox"? (inspired by Cian Dorr referring to C.I. Lewis as "the lesser Lewis")

My friend Michael in Toronto thought of a speech-act paradox:

"I am being, like, SO sarcastic."

Obviously, it's basically a semantic paradox, but certainly a unique extension of the genre to non-literal content.

Nice example. Is your friend "Michael in Toronto" Michael Glanzberg perchance?

In 1976, I invented McBurney's Paradox: How on earth did he get anything named after himself?

-- Peter McBurney.

I think this doesn't quite capture the trouble with the liar sentence. Certainly we have cases where some predicate seems not to apply but I don't see why this is in and of itself problematic. For instance we can reformulate this paradox mathematically as follows.

In any group say x is a blah if the order of x divides the order of the group. Now let our group be the complex numbers and let our number be a square root of unity. Is this a blah? Well to figure that out we need to know if 2 divides infinity which doesn't make sense.

I don't see how your examples differ from this case. They are simply instances of cases where some part of a complex predicate is undefined. Just as above where we did not define what happens when the order of the group is infinite you simply failed to list the conditions for being a painted horse in those situations where parentage is not well founded.

The reason that the liar paradox is particularly troubling is that we have apparent reasons for believing that the sentence is the sort of thing that true and false apply to. When true and false clearly aren't defined (for a string of nonsense) we have no problem it is only because the sentence appears to be the sort of thing which is true or false do we have issues.

In particular the liar paradox provides deep problems for a propositional/referential account of truth. Since such a theory all but requires us to say the liar is the sort of thing which is true or false. I think the solution is to toss propositions in favor of mental representations but that is another discussion.

Initially, I would object that this can be considered semantic in nature due to the fact that the excercise becomes absolutely semantic if time travel is demonstrated to be impossible, and a minor number of possible cosmological behaviors surrounding time travel allow this specific scenario. Thus, it's still a semantic artifice to assmeble abstract symbols into a conflicting symbological construct.

The time travel element isn't essential. All you need is something that persists through change, because anything like that can cause some of its own changes. Then we'll have some group the Causeds who are defined in terms of being caused to be X by other Causeds. What if I cause myself to be X? Am I a Caused? That parallels the truth-teller.

I think you can do something similar with the liar. Design of chain of causes parallel exactly to the example above and then have the same thing be the causer and the final result, and define a group with characteristics parallel to how the example in the post goes. This doesn't rely on time travel.

Steve and Jeremy - the time-travel is really just an acidental feature of the example, but it also shouldn't surprise me that it sticks in someone's craw. (I have to confess that I kind of love that aspect of the example though.) I'll see if I can come up with a version which exploits an ordering that people don't mind seeing subverted so much.

Logicnazi - ok, so we have two cases - the Acer one is the one that is like the Truth-Teller, the Ain't Misbehavin' one is the one that is like the Liar. Your "blah" example is more like the Acer case, in that the answer to our question (is Acer an American Paint?/is "this sentence is true" true?/is this complex number blah?) is underdetermined. Your example differs (in a way you could change) from the Acer and Truth-Teller cases in that your definition is a conditional and not a biconditional (your definition of "blah" is a bit like Soames' definition of "smidget" from Understanding truth. Soames says something which seems compatible with what you are saying about the square root of unity (though I must confess I haven't a clue what that really is, - I understand the role it plays in your example though.) He says that the rules of the language just fall silent on those cases. Nothing about our definition decides it for us. It just isn't defined for those cases.) It's harder to take the "just undefined" route with the biconditional, I think. If we have a biconditional, haven't we defined it for everything? Numbers, ships, ceiling-wax etc...(I'm not really sure.)

But this is a bit beside the point, because the Ain't Misbehavin' case (the one which I claim is Liar like) does differ from your case in a way that matters: it isn't a case of underdetermination, but one of overdetermination. It follows from the definition of an even winner (given certain empirical facts) that Ain't Misbehavin' is BOTH an even winner and not an even winner. (Just as it follows from the T-schema, given certain empirical facts, that sentence L is both true and not true.) We might respond to the paradox by deciding to say that in such cases the object is not an even winner, but that would be to alter the definition - which is the kind of solution to the paradox that I favour. But if you don't think the paradox is a semantic problem, changing the definition would just be a way to alter the language to make it impossible to express a paradox that remains - it wouldn't be a way of solving it.

So I take it your first response rest on the idea that if we give a biconditional we completely determine the extension of the predicate. Yet consider the example I gave before written the following way we get a biconditional:

x divides order of G the least n s.t. x^n=1 divides the # of elements in G.

Sure this example is really about properties of pairs but it could easily be reformulated without and we do not have a paradox because 1 divides order of N isn't defined. So really our expectation of definition by biconditional is that one side is defined in all the cases the other is defined. Yet the property 'x is an american paint' is undefined for *everyone* who is descended from the time loop you gave so the left side is undefined (acer is an american paint) as well as the right side (acer's father is an american paint) and we have no problem.

A similar response is availible to the overdetermination case. As before we don't always expect things defined by biconditionals to be always defined. Suppose I offer the following rule, "an integer n (positive or negative) is a moneyshot n-1 is a moneyshot"(I had too). If I don't tell you whether any base case was a moneyshot it wouldn't make sense to insist that whether 5 was a moneyshot had a definite answer. Just because we can redefine the property in terms of a differnt property doesn't guarantee the property is defined.

Once again we expect the left side of a bi-conditional definition to be defined iff the right side is defined. Whether or not Ain't Misbehavin is an even winner isn't defined because whether or not his faither is an even winner isn't defined. Just as before it's fine to have both sides undefined.

I think this would be a little more clear if we spelled out the definition of an even winner. Let S^n denote the n'th sire. Now X is an even winnder iff S^2(X) is an even winner or a foundation stalion. Yet I can construct chains of sentences like this even for obviously undefined sentences.

define x is true "x is true" is true or Ey x="y is true" and y is a foundation truth. Where foundation truths are just the collection of true sentences in the language without the truth operator.. Yet surely when X=asdfjkhasd we don't think that X must be true or false because we can push back the question to another one involving X.

Ultimately I think the paradoxical nature of these examples just comes from the assumptions we come to about sire's and damns when we read the statements which makes us think it is well defined in all cases.

Jeremy, Gillian -

It seems to me that Jeremy's objection still requires a Caused to predate its Cause, unless one considers his causes and Caused (and Gillians', perhaps, if I understand Gillian's answer properly, ) to be similar to the Credit Paradox, i.e., the real world causal cycle that protests: You may not *have* credit, because you *have no* credit previously. (We all know that it doesn't really work this way, but it's certainly an explanation we recieve often as young'ns just starting out) I suppose I can only wait to see a re-statement that doesn't posit time travel to create the re-ordering of Causes and Caused, since I cannot discern one myself at the moment.